Download comparative study of forecasting models based on weather

Thesis on
COMPARATIVE STUDY OF FORECASTING MODELS
BASED ON WEATHER PARAMETERS
Submitted for the award of
DOCTOR OF PHILOSOPHY
Degree in
STATISTICS
SUBMITTED BY
Mohita Anand Sharma
UNDER THE SUPERVISION OF
Dr. J.B. Singh
Senior Professor Statistics
SHOBHIT INSTITUTE OF ENGINEERING & TECHNOLOGY
A DEEMED-TO-BE UNIVERSITY
MODIPURAM, MEERUT– 250110 (INDIA)
2012
Shobhit
University Campus :
NH-58, Modipuram,
Meerut 250110, INDIA
T. : + 91-121-2575091/92
F. : + 91-121-2575724
E. : [email protected]
U. : www.shobhituniversity.ac.in
University
Certificate
This is to certify that the thesis, entitled “Comparative Study Of Forecasting
Models Based On Weather Parameters” which is being submitted by Ms. Mohita
Anand Sharma for the award of Degree of Doctor of Philosophy in Statistics to the
Faculty of Humanities, Physical and Mathematical Sciences of Shobhit University,
Meerut, a deemed-to-be University, established by GOI u/s 3 of UGC Act 1956, is a record
of bonafide investigations and extensions of the problems carried out by her under my
supervision and guidance.
To the best of my knowledge, the matter embodied in this thesis is the original work
of the candidate herself and has not been submitted for the award of any other degree or
diploma of any University or Institution.
It is further certified that she has worked with me for the required period in the
Faculty of Humanities, Physical and Mathematical Sciences, Shobhit University, Meerut,
(U.P.), India.
Prof. J.B. Singh
(Supervisor)
Senior Professor Statistics
DECLARATION
I, hereby, declare that the work presented in this thesis, entitled “Comparative
Study Of Forecasting Models Based On Weather Parameters” in fulfillment of the
requirements for the award of Degree of Doctor of Philosophy, submitted in the Faculty
of Humanities, Physical and Mathematical Sciences at Shobhit University, Meerut, a
deemed-to-be University, established by GOI u/s 3 of UGC Act 1956, is an authentic
record of my own research work carried out under the supervision of Prof. J.B. Singh.
I also declare that the work embodied in the present thesis
(i)
Is my original work and has not been copied from any Journal/Thesis/Book, and
(ii) Has not been submitted by me for any other Degree or Diploma of any
University/ Institution.
[Mohita Anand Sharma]
ACKNOWLEDGEMENT
Research is well versed with booms and hiccups. But despite these, one
relishes at the faint end where one comes out of this entrenched mundane. During
this mercurial period one gets along with innumerable individuals to whom you owe
something or more. This is my endeavor here to figure at least few people who lent
their support for smooth accomplishment of my doctoral work.
Foremost, I would like to express my heartiest gratitude to Prof. J. B. Singh, my
guide, for providing his openhanded driving force behind my research activities. It is
the great opportunity to complete my doctoral program under his scholarly and
innovative guidance. I owe him for his efficient supervision, constant inspiration,
encouragement and stimulating discussions throughout the research work.
I am thankful to the Chancellor Dr. Shobhit Kumar, Pro-vice chancellor Kuwar
Shekhar Vijendra, Vice-Chancellor Prof. R. P. Agarwal and Dean Prof. S. C. Agarwal for
providing amiable environment for concluding research in the University. I would
like to acknowledge the significant contribution of Prof. Irene Sarkar and Prof. Sanjay
Sharma for their valuable guidance, encouragement and providing necessary support
in the development of models in this research work.
Sincere gratified to my parents Late Mohan Swaroop Anand and Late Nisha
Anand, who instilled inspiration in my life, who brought me up, nurtured and
imparted me the real virtues of humanity, empathy and kindness. They are my real
motivators who sacrificed in order to bring me to the present position and blessed
me with their grace and affection. I wish they were alive to see me achieve this goal
but I am sure they must be blessing me from heaven.
Especial express thanks to my loving and caring younger brother Surya Anand,
for his gigantic over all support from day one. I am profoundly thankful for the love
and extreme affection of my elder sister Sanchita, collogue Anshul Ujaliyan and best
friends Mahima & Nida to encourage me constantly. Special thanks to
Ms. Rajni
Nayyar and her family to devote the imperative time to facilitate me direct/indirect.
I am immensely thankful to Indian Methodology Department, Dehradun for
providing the valuable data for this research work. This murky world is a difficult
place to walk without blessings and teachings of some people. I am grateful to my
teachers who bestowed upon me the real lessons of life. I would like to thank all my
family and friends who have directly or indirectly contributed in my research
endeavor.
Honest recognition to my In-laws Shree Madan Pal Sharma and Smt. Brijbala
Sharma along with complete family to encourage me and taking care of my
children.
With an endless word of thanks to my backbone - my lovable husband Mr.
Prashant Kumar Sharma, as he toughs each and every aspect of my life. The credible
love of my kids had boosted me with their charming smiles and activities throughout
the day since birth.
Finally, but most importantly, I pay my reverence to the GOD, preserver and
protector with whose grace I stand tall at present. He showed me the right path in all
ups and downs and moments of despair throughout the tenure of this work. I bow my
head in complete submission before Him.
CONTENTS
Proem
(i)
List of Tables
(ii)-(iii)
List of Figures
(iv)-(vii)
Chapter 1: Introduction
1-6
1.1
Scope
1
1.2
Motivation
1
1.3
Overview
2
1.4
Contribution
3
1.5
Objectives
4
1.6
Study area
5
Chapter 2: Review of literature
7-25
2.1
Probability Distribution
2.2
Multiple Regression (MR)
13
2.3
Autoregressive Integrated Moving Average (ARIMA)
16
2.4
Artificial Neural Network (ANN)
17
2.5
Comparison among MR, ARIMA and ANN
20
Chapter 3: Fitting of Probability Distribution
8
26-85
3.1
Introduction
26
3.2
Descriptive Statistics
26
3.3
Methodology
48
3.3.1 Fitting the probability distribution
48
3.3.2 Testing the goodness of fit
48
3.3.3 Identification of best fit probability distribution
52
Probability Distribution Pattern
54
3.4.1 Introduction
54
3.4.2 Rainfall
54
3.4
3.5
3.4.3 Maximum temperature
58
3.4.4 Minimum temperature
62
3.4.5 Relative humidity at 7 AM
66
3.4.6 Relative humidity at 2 PM
70
3.4.7 Pan evaporation
74
3.4.8 Bright sunshine
78
Conclusion
82
Chapter 4: Weather forecasting models
86-94
4.1
Introduction
86
4.2
Correlation Analysis
86
4.3
Methodology for forecasting models
87
4.3.1 Multiple Linear Regressions
87
4.3.2 Autoregressive Integrated Moving Average
88
4.3.3 Artificial Neural Network
90
4.3.4 Hybrid Approach
92
4.3.5 Performance Evaluation Criteria
93
Development of forecasting model for weather parameters
95
4.4.1 Introduction
95
4.4.2 Rainfall
95
4.4
4.5
4.4.3 Maximum temperature
100
4.4.4 Minimum temperature
103
4.4.5. Relative humidity at 7 A.M.
108
4.4.6 Relative humidity at 2 P.M.
111
4.4.7 Pan evaporation
114
4.4.8 Bright sunshine
119
Comparison of prediction ability of forecasting models
124
4.5.1 Introduction
124
4.5.2 Rainfall
124
4.6
4.5.3 Maximum temperature
128
4.5.4 Minimum temperature
128
4.5.5. Relative humidity at 7 A.M.
128
4.5.6 Relative humidity at 2 P.M.
138
4.5.7 Pan evaporation
138
4.5.8 Bright sunshine
138
Conclusion
148
Chapter 5: Identification of precise weather forecasting model 149-180
5.1
Introduction
149
5.2
Validation of weather forecasting model
149
5.2.1 Rainfall
149
5.2.2 Maximum temperature
154
5.2.3 Minimum temperature
158
5.2.4. Relative humidity at 7 A.M.
162
5.2.5 Relative humidity at 2 P.M.
167
5.2.6 Pan evaporation
171
5.2.7 Bright sunshine
175
Conclusion
180
5.3
Chapter 6: Summary and future scope
181-183
6.1
Summary
181
6.2
Future scope
183
Bibliography
184-200
APPENDICES
(A)
Procedure followed for Stepwise Regression Analysis
(B)
Programs for developing ANN models
225-238
(C)
Programs for developing Hybrid MLR_ANN models
239-246
List of Reprints (Attached) of Publications
201-224
PROEM
Copious business and economic time series are non-stationary and contains
trend and seasonal discrepancy. So an accurate forecasting of such time series will
always be an important chore for effective decisions in marketing, production,
weather forecasting and many other sectors. Since weather forecasting is the most
crucial and challenging operational errands accepted worldwide. There are many
methodologies that decompose a time series linear and non-linear forms which will
always require forecasting.
We have indicated the open problems and scope of the research work in the
first chapter, which delineate enthuse that drive us to identify the anticipated
methods. Further, summing a concise contribution of the toil, intimating the objective
implicated for the study area. Literature assess is briefly illustrated with recent
progress in prediction of weather parameters in the second chapter.
Third chapter presents the descriptive statistics of the weather data seasonally
and weekly considered for the monsoon months for the study, moreover presenting
the methodology for fitting the probability distribution for each weather parameter
using goodness of fit tests.
The fourth chapter dwells with the methodology of traditional and proposed
hybrid forecasting models developed for each weather parameter, ensuring their
predictive ability graphically.
Chapter fifth corroborates the precise weather forecasting model for each
parameter by comparing the models through performance evaluation criteria. Finally,
the last chapter windups along with the future span of work.
LIST OF TABLES
Table
No.
Title
Page
No.
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9(a)
3.9(b)
3.9(c)
3.9(d)
3.10(a)
3.10(b)
Summary of statistics for Rainfall.
Summary of statistics for Maximum Temperature.
Summary of statistics for Minimum Temperature.
Summary of statistics for Relative Humidity at 7 AM.
Summary of statistics for Relative Humidity at 2 PM.
Summary of statistics for Pan Evaporation.
Summary of statistics for Bright Sunshine.
Description of various probability distribution functions.
Distributions fitted for Rainfall data sets.
Distributions with highest score for Rainfall data sets.
Parameters of the distributions fitted for Rainfall data sets.
Best fit probability distribution for Rainfall.
Distributions fitted for Maximum Temperature data sets.
Distributions with highest score for Maximum Temperature data
sets.
Parameters of the distributions fitted for Maximum Temperature
data sets.
Best fit probability distribution for Maximum Temperature data
sets.
Distributions fitted by the tests for Minimum Temperature data
sets.
Distributions with highest score for Minimum Temperature data
sets.
Parameters of the distributions fitted for Minimum Temperature
data sets.
Best fit probability distribution for Minimum Temperature.
Distributions fitted for Relative Humidity at 7 AM data sets.
Distributions with highest score for Relative Humidity at 7 AM
data sets.
Parameters of the distributions fitted for Relative Humidity at
7 AM data sets.
Best fit probability distribution for Relative Humidity at 7 AM.
Distributions fitted for Relative Humidity at 2 PM data sets.
Distributions with highest score for Relative Humidity at 2 PM
data sets.
27
31
34
37
40
43
46
49
56
56
57
58
59
60
3.10(c)
3.10(d)
3.11(a)
3.11(b)
3.11(c)
3.11(d)
3.12(a)
3.12(b)
3.12(c)
3.12(d)
3.13(a)
3.13(b)
60
62
63
63
64
66
67
67
68
69
71
72
LIST OF TABLES (CONTINUED)
Table
Title
No.
3.13(c) Parameters of the distributions fitted for Relative Humidity at 2 PM
data sets.
3.13(d) Best fit probability distribution for Relative Humidity at 2 PM.
3.14(a) Distributions fitted for Pan Evaporation data sets.
3.14(b) Distributions with highest score for Pan Evaporation data sets.
3.14(c) Parameters of the distributions fitted for Pan Evaporation data sets.
3.14(d) Best fit probability distribution for Pan Evaporation.
3.15(a) Distributions fitted for Bright Sunshine data sets.
3.15(b) Distributions with highest score for Bright Sunshine data sets.
3.15(c) Parameters of the distributions fitted for Bright Sunshine data sets.
3.15(d) Best fit probability distribution for Bright Sunshine.
4.1
Inter correlation coefficient between weather parameters for total data
set.
5.1
Comparison of the performance of forecasting models for Rainfall.
5.2
Comparison of the performance of forecasting models for Maximum
Temperature.
5.3
Comparison of the performance of forecasting models for Minimum
Temperature.
5.4
Comparison of the performance of forecasting models for Relative
Humidity at 7 AM.
5.5
Comparison of the performance of forecasting models for Relative
Humidity at 2 PM.
5.6
Comparison of the performance of forecasting models for Pan
Evaporation.
5.7
Comparison of the performance of forecasting models for Bright
Sunshine.
Page
No.
72
74
75
75
76
77
79
80
80
81
87
153
158
162
163
167
171
176
LIST OF FIGURES
Figure
Title
No.
3.1
Mean, standard deviation and range of weekly Rainfall.
3.2
50 years weekly Rainfall for monsoon period.
3.3
Mean, standard deviation and range of weekly Maximum
Temperature.
3.4
50 years weekly Maximum Temperature for monsoon period
3.5
Mean, standard deviation and range of weekly Minimum
Temperature.
3.6
50 years weekly Minimum Temperature for monsoon period
3.7
Mean, standard deviation and range of weekly Relative Humidity
at 7 AM.
3.8
50 years weekly Relative Humidity at 7 AM for monsoon period
3.9
Mean, standard deviation and range of weekly Relative Humidity
at 2 PM.
3.10 50 years weekly Relative Humidity at 2 PM for monsoon period
3.11 Mean, standard deviation and range of weekly Pan Evaporation.
3.12 50 years weekly Pan Evaporation for monsoon period
3.13 Mean, standard deviation and range of weekly Bright Sunshine.
3.14 50 years weekly Bright Sunshine for monsoon period
4.1
An (m x n x o) artificial neural network structure, showing a
multilayer perceptron.
4.2
Plots of autocorrelation and partial autocorrelation coefficients and
time lags of weakly average Rainfall parameter.
4.3
Artificial neural network structure for weekly average Rainfall
prediction parameter
4.4
Mapping of the number of epochs obtained for desired goal for
ANN model for Rainfall parameter
4.5
Hybrid MLR_ANN structure for weekly average Rainfall
prediction parameter.
4.6
Mapping of the number of epochs obtained for desired goal for
hybrid MLR_ANN model for Rainfall parameter.
4.7
Plots of autocorrelation and partial autocorrelation coefficients and
time lags of weakly average Maximum Temperature parameter.
Page
No.
28
29
31
32
34
35
37
38
40
41
43
44
46
47
91
96
97
97
99
99
101
LIST OF FIGURES (CONTINUED)
Figure
Title
No.
4.8
Artificial neural network structure for weekly average Maximum
Temperature prediction parameter
4.9
Mapping of the number of epochs obtained for desired goal for
ANN model for Maximum Temperature
4.10 Plots of autocorrelation and partial autocorrelation coefficients and
time lags of weakly average Minimum Temperature parameter.
4.11 Artificial neural network structure for weekly average Minimum
Temperature prediction parameter.
4.12 Mapping of the number of epochs obtained for desired goal for
ANN model for Minimum Temperature parameter.
4.13 Hybrid MLR_ANN structure for weekly average Minimum
Temperature prediction parameter
4.14 Mapping of the number of epochs obtained for desired goal for
Hybrid MLR_ANN model for Minimum Temperature parameter.
4.15 Plots of autocorrelation and partial autocorrelation coefficients and
time lags of weakly average Relative Humidity at 7 AM
parameter.
4.16 Artificial neural network structure for weekly average Relative
Humidity at 7 AM prediction parameter.
4.17 Mapping of the number of epochs obtained for desired goal for
ANN model for Relative Humidity at 7 AM parameter.
4.18 Plots of autocorrelation and partial autocorrelation coefficients and
time lags of weakly average Relative Humidity at 2 PM parameter.
4.19 Artificial neural network structure for weekly average Relative
Humidity at 2 PM prediction parameter.
4.20 Mapping of the number of epochs obtained for desired goal for
ANN model for Relative Humidity at 2 PM parameter.
4.21 Plots of autocorrelation and partial autocorrelation coefficients and
time lags of weakly average Pan Evaporation parameter.
4.22 Artificial neural network structure for weekly average Pan
Evaporation prediction parameter.
4.23 Mapping of the number of epochs obtained for desired goal for
ANN model for Pan Evaporation parameter.
4.24 Hybrid MLR_ANN structure for weekly average Pan Evaporation
prediction parameter.
4.25 Mapping of the number of epochs obtained for desired goal for
hybrid MLR_ANN model for Pan Evaporation parameter.
Page
No.
102
102
104
105
105
106
107
109
110
110
112
113
113
115
116
117
118
118
LIST OF FIGURES (CONTINUED)
Figure
Title
No.
4.26 Plots of autocorrelation and partial autocorrelation coefficients and
time lags of weakly average Bright Sunshine parameter.
4.27 Artificial neural network structure for weekly average Bright
Sunshine prediction parameter.
4.28 Mapping of the number of epochs obtained for desired goal for
ANN model for Bright Sunshine parameter.
4.29 Hybrid MLR_ANN structure for weekly average Bright Sunshine
prediction parameter.
4.30 Mapping of the number of epochs obtained for desired goal for
hybrid MLR_ANN Bright Sunshine parameter.
4.31 Plots of the Actual and Predicted weekly average Rainfall for
training data set using Multiple Linear Regression, ARIMA,
Artificial Neural Network, Hybrid MLR_ARIMA and Hybrid
MLR_ANN models.
4.32 Plots of the Actual and Predicted weekly average Maximum
Temperature for training data set using Multiple Linear Regression,
ARIMA and Artificial Neural Network models.
4.33 Plots of the Actual and Predicted weekly average Minimum
Temperature for training data set using Multiple Regression,
ARIMA, Artificial Neural Network, Hybrid MLR_ARIMA and
Hybrid MLR_ANN models.
4.34 Plots of the Actual and Predicted weekly average Relative Humidity
7AM for training data set using Multiple Linear Regression,
ARIMA and Artificial Neural Network models.
4.35 Plots of the Actual and Predicted weekly average Relative Humidity
2PM for training data set using Multiple Linear Regression,
ARIMA and Artificial Neural Network models.
4.36 Plots of the Actual and Predicted weekly average Pan Evaporation
for training data set using Multiple Linear Regression, ARIMA,
Artificial Neural Network, Hybrid MLR_ARIMA and Hybrid
MLR_ANN models.
4.37 Plots of the Actual and Predicted weekly average Bright Sunshine
for training data set using Multiple Linear Regression, ARIMA,
Artificial Neural Network, Hybrid MLR_ARIMA and Hybrid
MLR_ANN models.
Page
No.
120
121
121
122
123
125
129
132
135
139
142
145
LIST OF FIGURES (CONTINUED)
Figure
Title
No.
5.1
Plots of the Actual and Predicted weekly average Rainfall for
testing data set using Multiple Linear Regression, ARIMA,
Artificial Neural Network, Hybrid MLR_ARIMA and Hybrid
MLR_ANN models.
5.2
Plots of the Actual and Predicted weekly average Maximum
Temperature (OC) for testing data set using Multiple Linear
Regression, ARIMA and Artificial Neural Network models.
5.3
Plots of the Actual and Predicted weekly average Minimum
Temperature (OC) for testing data set using Multiple Linear
Regression, ARIMA, and Artificial Neural Network, Hybrid
MLR_ARIMA and Hybrid MLR_ANN models.
5.4
Plots of the Actual and Predicted weekly average Relative
Humidity 7AM for testing data set using Multiple Linear
Regression, ARIMA and Artificial Neural Network models.
5.5
Plots of the Actual and Predicted weekly average Relative
Humidity 2PM for testing data set using Multiple Linear
Regression, ARIMA, Artificial Neural Network models.
5.6
Plots of the Actual and Predicted weekly average Pan Evaporation
for testing data set using Multiple Linear Regression, ARIMA,
Artificial Neural Network, Hybrid MLR_ARIMA and Hybrid
MLR_ANN models.
5.7
Plots of the Actual and Predicted weekly average Bright Sunshine
for testing data set using Multiple Linear Regression, ARIMA,
Artificial Neural Network, Hybrid MLR_ARIMA and Hybrid
MLR_ANN models.
Page
No.
150
155
159
164
168
172
177
INTRODUCTION
1.1 Scope
1.2 Motivation
1.3 Overview
1.4 Contribution
1.5 Objectives
1.6 Study area
CHAPTER 1
INTRODUCTION
1.1
Scope
Weather forecasting is an important issue in the field of meteorology all over the world.
Several factors contribute significantly to increase the forecasting accuracy; one among them is
the development of statistical methods for enhancing the scope and accuracy of model predictions.
Numerous efforts have been devoted to develop and improve the existing time series weather
forecasting models by using different techniques. The role of statistical methodology for
predicting the weather parameters is considered to be most important for their precise estimates.
Although, high-speed computers, meteorological satellites, and weather radars are tools that had
played major roles in improving weather forecasts. But the improvement in initial conditions is the
result of an increased number of observations and better use of the observations in computational
techniques. Since, many efforts have been made by researchers to identify the best precise weather
forecasting models. The combinations of linear and non-linear models are one of the most popular
and widely used hybrid models for improving the forecasting accuracy. The present study is
planned to investigate the potential for using the existing Multiple Linear Regression,
Autoregressive Integrated Moving Average and Artificial Neural Network models to forecast
weather parameters. A comparative study of the existing and proposed weather forecasting models
is performed to identify the precise and reliable weather forecasting models.
1.2
Motivation
The prediction of weather conditions can have significant impacts on various sectors of
society in different parts of the country. Forecasts are used by government and industry to protect
life and property and to improve the efficiency of operations, and by individuals to plan a wide
range of daily activities. The notable improvement in forecast accuracy has been achieved since
the 1950s, that is, a direct outgrowth of technological developments, basic and applied research,
and the application of new knowledge and methods by weather forecasters. The advance
knowledge of weather parameters in a particular region is advantageous in effective planning.
Several studies on forecasting weather variables based on time series data in reference to a
particular region have been carried out at national and international level both in the farm and nonfarm sectors. It was observed that the combination of two or more computational models/ hybrid
models decomposes a time series into linear and non-linear form and prove to be better approach
in comparison to single models for the reason that hybrid model produces small forecasting error
in terms of accuracy.
In contrary some of the studies also mentioned that hybrid approaches are not always
better. Such uncertainty in weather forecasting models open up new opportunities for the selection
of precise forecasting model. These aspects motivate this thesis, to explore the existing
opportunities to identify the precise weather forecasting model. Predictions of weather parameters
provide by such identify models based on time series data will be of particular interest to the
weather forecasters.
1.3
Overview
The prime contribution of this thesis is to compare the existing weather forecasting model
and to select the precise model based on their predictive ability. The methodology consists of four
stages for each study period data of weather parameters which are
(i)
Computation of descriptive statistics.
(ii)
Statistical analysis to identify the best fit probability distribution.
(iii)
Development of weather forecasting models and comparison of their predictive
ability.
(iv)
Identification of precise and reliable weather forecasting model.
These four components correspond to reduce forecasting errors by relaxing certain
assumptions of traditional forecasting techniques. These components are interlinked to each other.
The first component of the methodology explains the details of different measures of
general statistics of time series data to explore the real situation of the different weather
parameters. The objective of this phase is to understand the distribution pattern of weather data.
The second component of the methodology is concerned with the fitting of the suitable probability
distribution to each weather parameter independently by using different goodness of fit test. This
methodology further establishes the analytical devised and testing procedure for future application.
Weather forecasting models were developed using time series weather data and their predictive
ability was compared using graphical and numerical performance evaluation criteria in the third
stage of the methodology. Finally hybrid models were developed and appropriate forecasting
model was identifying for future application of researchers in the related field.
1.4
Contribution
This thesis is divided into six chapters starting with acknowledgement, table of contents
and appendix in the end. Chapter first consists of a brief introduction which outlines the scope,
motivation, describes overview of the proposed methodology with objectives and study area of
the thesis as well as summarizing the major contributions in brief. Chapter second describes the
recent advances in predicting weather parameters which provide the brief review of literature in
the related field. Chapter third explain the descriptive statistics of seasonal and weekly weather
data and presents the methodology for fitting the probability distribution to weather parameters
using testing of goodness of fit. The procedure of identifying the best fit probability distribution
was explained for each weather parameter. Chapter fourth describes the methodology of Multiple
Linear Regression (MLR), Autoregressive Integrated Moving Average (ARIMA), Artificial
Neural Network (ANN) and hybrid forecasting models in brief. Forecasting models developed for
each weather variables are presented and hybrid model of Multiple Linear Regression with
ARIMA and ANN is proposed. Finally a comparison of prediction ability of forecasting model is
presented graphically for each weather parameters.
Chapter fifth describes a comparison of models which are designed to identify the precise
weather forecasting model. Finally the precise weather forecasting model is identified based on
minimum value of mean error, mean absolute error, root mean square error and prediction error
and maximum value of correlation coefficients. The last chapter presents the main summary of
the thesis and discusses direction for future work. Bibliography is presented just after the last
chapter. Graphs for the comparison of the actual and predicted weekly average weather
parameters for training and testing data set using all the statistical techniques are inclusive in the
respective chapters. The details of the computer programme used through standard software’s
SAS and Matlab in the thesis are presented in the form of appendix at the end along with the
research papers already published in related to our research work on weather forecasting models.
1.5
Objective
The advance knowledge of weather parameters in a particular region is very helpful in
sound planning. A reliable prediction of Indian monsoon in a region on seasonal and interseasonal time series is not only scientifically challenging but also important for the future
planning. The role of statistical techniques for predicting the weather parameters at a particular
place and time depends on an understanding of the past time series data. The transient behavior of
weather parameters over a particular period of time makes difficult to predict correctly and
consistently. Indian economy in general and especially in the field of Agriculture and Industry
depends upon weather conditions. The frequent fluctuation in weather parameters in different part
of India is being faced by the government/ non-government planning agencies.
In recent time, the concept of combined models/ hybrid weather forecasting model is
introduced to increase the accuracy in prediction. The problem to identify the precise weather
forecasting model seems to be interesting. Thus providing reliable prediction and forecasting of
weather parameters in the Himalaya in particular and of India in general is an important challenge
for planners and scientists. Keeping in view a comparative study of weather forecasting models
and to propose hybrid model for seasonal and inter-seasonal time series data is planned with the
following objectives:
1.6
(i)
To study the distribution pattern of weather parameters.
(ii)
Development of weather forecasting models.
(iii)
To compare the predictive ability of the developed model.
(iv)
To identify the precise and reliable weather forecasting model.
Study Area
The present study is based on a time series weather data of 50 years observed at the
Pantnagar station and collected from the IMD approved meteorology observatory Dehradun, India.
India is situated in the east direction of earth and lies between latitude 220°00’N and longitude
770°00’W. Pantnagar station is located at 29°N latitude and 79°3’ E longitudes approximately
243.89 meters above mean sea level, in the Tarai region of Uttarakhand and lies within the
Shivalik Ranges of the Himalayan foothills. On an average the region has a humid subtropical
climate having hot summers (40-42OC) and cold winters (2-4OC) with monsoon rains occurring
from June to September. July is the rainiest month followed by August. In September, due to the
depression from the Bay of Bengal, the local weather is affected, causing heavy rains. With the
withdrawal of monsoon in September, the intensity of rainfall rapidly decreases till in November it
becomes practically rainless. Rain Gauge Station suggest that the annual average rainfall in and
around Pantnagar is of the order of 1400 mm. More than 80% of the rain is received from southwest monsoon during these four month period from June to September, and the rainfall of rainy
season is significantly different from that of dry season. Winter precipitation in the region,
associated with the passage of Western disturbances, is in the form of snowfall in the Higher
Central Himalaya.
The average monsoon season in and around Pantnagar region, ranges between
15 to 20
weeks. 17 weeks data set from 4th June to 30th September of each year is considered as inter
season monsoon periods for our study. The data comprises of seven parameters viz. rainfall,
maximum and minimum temperature, relative humidity at 7:00 AM and 2:00 PM, pan evaporation
and bright sunshine hours, collected during the monsoon months on a time series weather data of
850 weeks from 1961 to 2010.
REVIEW OF LITERATURE

Probability Distribution

Multiple Regression

Autoregressive Integrated Moving Average

Artificial Neural Network

Comparison among MR, ARIMA and ANN
CHAPTER 2
REVIEW OF LITERATURE
Weather is a continuous, data-intensive, multi-dimensional, dynamic and chaotic process,
and these properties make weather forecasting a formidable challenge. It is one of the most
imperative and demanding operational responsibilities carried out by meteorological services all
over the world. At present, the assessment of the nature and causes of seasonal climate variability
is still conception. Since, it is a complicated procedure that includes numerous specialized fields of
know-how (Guhathakurata, 2006); therefore, in the field of meteorology all decisions are to be
taken in the visage of uncertainty associated with local of and global climatic variables. Several
authors have discussed the vagueness associated with the weather systems. Chaotic features
associated with the atmospheric phenomena also have attracted the attention of the modern
scientists (Sivakumar 2001; Sivakumar et al. 1999; Men et al. 2004). Different scientists over the
globe have developed stochastic weather models. It is often used to predict and warn about natural
disasters that are caused by abrupt change in climate conditions. The variables defining weather
conditions vary continuously with time, forming time series of each parameter and can be used to
develop a forecasting model either statistically or using some other means that uses this time series
data (Chatfield 1994; Montgomery and Lynwood 1996).
Weather prediction modeling involves a combination of computer models, observation and
knowledge of trends and patterns. Generally, two methods are used to forecast weather: (a) the
empirical approach and (b) the dynamical approach (Lorenz, 1969). The first approach is based
upon the occurrence of analogues and is often referred to by meteorologists as analogue
forecasting. This approach is useful for predicting local-scale weather if recorded cases are
plentiful. The second approach is based upon equations and forward simulations of the
atmosphere, and is often referred to as computer modeling. Because of the grid coarseness, the
dynamical approach is only useful for modeling large-scale weather phenomena and may not
predict short-term weather efficiently. Many weather prediction systems use a combination of
empirical and dynamical techniques. At macro level, weather forecasting is usually done using the
data gathered by remote sensing satellites. Weather parameters like maximum temperature,
minimum temperature, extent of rainfall, cloud conditions, wind streams and their directions, are
projected using images taken by these meteorological satellites to asses future trends. The
satellites-based systems are inherently costlier and require complete support system. Moreover,
such systems are capable of providing only such information, which is usually generalized over a
larger geographical area.
The successful weather predictions are performed since early 1920’s. The practical use of
numerical weather prediction starts in the middle of nineteenth century. A number of forecast
models, both global and regional are being used to create forecasts. This chapter is intended to
provide the brief review of literature in the field of weather forecasting models in general and
specially for comparative study of weather forecasts models. Thus, the chapter is divided into five
parts and the research work done in each field is being reviewed in different sections.
2.1
Probability distribution
Analysis of weather data strongly depends on its probability distribution pattern.
Establishing a probability distribution that provides a good fit to the weather parameter has long
been a topic of interest in the fields of hydrology, meteorology and other fields. Several studies
have been conducted in India and abroad on weather analysis and best fit probability distribution
function such as normal, log-normal, gumbel, weibull and Pearson type distribution were
identified.
Fisher (1924) studied the influence of rainfall on the yield of wheat in Rothamasted. He
showed that it is the distribution of rainfall during a season rather than its total amount which
influence the crop yield. Tippet (1929) subsequently applied the technique on sunshine
distribution and found that sunshine has beneficial effect throughout the year on wheat crop.
Another useful line of work relating to the study of rainfall distribution was introduced by
Manning (1956). He transformed the skew frequency distribution of rainfall to approximate
closely to the theoretical normal distribution showing that fifteen observations were enough to get
a reasonable good estimate of the distribution and confidence limit.
Further, Rao et al. (1963) have used the extreme value distribution (EVD) on rainfall
(Chow, 1964) to predict the flood and drought situations in the parts of India. Abraham (1965) has
applied the fisher’s method to see the joint relationship of crop yield and weather variable (rainfall
and temperature). Rai and Jay (1966) studied humidity and upper winds temperature over Madras
in relation to precipitation occurrence and found the vertical distribution of temperature and
humidity associated with dry or wet days over the same area.
Benson (1968) adopted a large scale planning for improved flood plain management and
expending water resources development and he suggested adopting a procedure where records are
available for all government agencies. Along with Pearson type I, Gumble’s and log normal
distribution, the log Pearson type III distribution has been selected as the based method with
provision for departure from the base method were justified continuing study leading towards
improvements or revision of method is recommended. Kulkarni and Pant (1969) studied the
cumulative frequency distribution of rainfall of different intensities during south-west monsoon
for 20 stations in India. The distribution was found to be exponential and curves were fitted to
observed date by the method of least square.
Mooley et al. (1970) have studied statistical distribution of rainfall during south-west and
north-east monsoon season at representative stations in India. Gamma distribution has been fitted
to rainfall data and has been tested by Chi-square test. Bhargava et al. (1974) showed that for a
number of crops the distribution of rainfall over the season has a great influence on the yield.
Krishnan and Kushwaha (1972) studied the mathematical distribution of accumulated rainfall for
2 pentads, 4 pentads,…., 20 pentads commencing from the onset of monsoon in respect of a
typical arid zone stations. In case of Jaipur the distribution beyond a month is normal while for
Jodhpur, distribution is not normal at all. Raman Rao et al. (1975) analyzed the daily rainfall data
collected at Bijapur for the year from 1921 to 1970.
Parthsarthy and Dhar (1976) have studied the trends and periodicities in the annual
rainfall of the metrological subdivisions of Madhya Pradesh for the 60 years period. It was seen
that the frequency distribution of annual rainfall of east and west M.P. is normal. Significant
increase of 15% of the mean annual rainfall per 30 years was observed in west M.P. Cunnane et
al. (1978) and Gringorten (1963) have used plotting rule for extreme probability paper to study
the extreme value analysis. Mukherjee et al. (1979) made the studies to improve the weather
bulletin and a beginning in this direction is made with a detailed study of rainfall even within same
district. They observed that there is a wide variation in the intensity and distribution of rainfall.
Mukherjee et al. (1980) have conducted the study of monthly seasonal and annual rainfall
distribution for 16 stations in Pune and 11 stations in Ahmed Nagar for 50 years period. Combined
study of two district shows that the rainfall distribution in western part of Pune is same as western
part of Ahmed Nagar while rainfall in eastern part of Pune is same as that in eastern part of
Ahmed Nagar. Kulandaivelu (1984) analyzed the daily precipitation data of Coimbatore for a
period of 70 years for weekly totals by fitting incomplete Gamma distribution model. The data
indicate the likely commencement of rains, period of drought length of growing season and end of
growing season. Based on the assured rainfall at (50%) probability level, suitable cropping system
was suggested for Coimbatore.
Phien and Ajirajah (1984) showed that for the annual flood, annual maximum rainfall,
annual stream flow and annual rainfall, the log-Pearson type III distribution was highly suitable
after evaluating by Chi-square and Kolmogorov- Smirnov tests. Biswas and Khambete (1989)
computed the lowest amount of rainfall at different probability level by fitting gamma distribution
probability model to week by week total rainfall of 82 stations in dry farming tract of Maharashtra.
Rao and Singh (1990) studied the distributions of weather variables and developed methodology
for forecasting extreme values of weather variables at Pantnagar. They observed that the square
root model ( y  a  b x  cx) is approximate to predict wheat yield based on metrological
observations.
Gumble distribution was applied by Mukherjee et al. (1991) to estimate return period of
reoccurrence of highest one day rainfall. Lin et al. (1993) stated that in accordance with the
probability distribution all stations in same area can be classified in different clusters and special
characteristic among a clusters can have spatial relationship to a certain extent in that cluster.
Chapman (1994) evaluated five daily rainfall generating models with several methods and
analyzed that Srikanthan-McMahon model performed well when calibrated with long rainfall
records. Nese and Jon (1994) estimated the potential effect of the biases on the mean and standard
deviation of a temperature distribution; biasing simulations were performed on various normal
distributions. In addition, it is shown that these biases can affect other relevant climatic statistics.
Duan et al. (1995) suggested that for modeling daily rainfall amounts, the Weibull and to a lesser
extent the exponential distribution is suitable.
Extreme value analysis was done for seven stations of Krishna Godavari agro-climatic
zone in Andhra Pradesh of India, Kulshrestha et al. (1995). Similar analysis was made for 14
stations of Gujarat state of India and ascertained the most suitable type of distribution,
Kulshrestha et al. (1999). Statistical distribution has been used to define extremes with given
return periods (Aggarwal et al. 1988, Bhatt et al. 1996). Upadhaya and Singh (1998) stated that it
is possible to predict rainfall fairly accurate using various probability distributions for certain
returns periods although the rainfall varies with space, time and have erratic nature. Sen and
Eljadid (1999) reported that for monthly rainfall in arid regions, gamma probability distribution is
best fit hence which enables one to construct regional maps for the area of two gamma parameters,
shape and scale.
Rai and Chandrahas (1996) studied the effect of intensity and distribution pattern of
weather parameters at different stages of crop growth and rice yield. They found that temperature
and sunshine hours are effective at the growing phase, whereas, sunshine hours were found
ineffective during early growth phase. Ogunlela (2001) studied the stochastic analysis of rainfall
event in Illorin using probability distribution functions. He concluded that the log-Peasson type III
distribution best described the peak daily rainfall data for Ilorin. Kar (2002) has predicted the
extreme rainfall for mid central table zone of Orissa using Extreme value Type-I distribution and
was concluded that extreme value type-I distribution was a good fit for predicting the one day
maximum rainfall. Tao et al. (2002) recommended generalized extreme value model as the most
suitable distribution after a systematic assessment procedure for representing extreme-value
process and its relatively simple parameter estimation.
Topaloglu (2002) reported that gumbel probability model estimated by the method of
moments and evaluated by chi-square tests was found to be the best model in the Seyhan river
basin. Fowler et al. (2003) have used two methods to assess rainfall extremes and their
probabilities. One of the method comprised percentile approach (Kar, 2002) and the other used the
statistical distributions of rainfall (Hennersy et al., 1997). Salami (2004) studied the
meteorological data for Texas and found that Gumbel distribution fits adequately for both
evaporation and temperature data, while for precipitation data log-Pearson type III distribution
conforms more accurate. Lee (2005) indicated that log-Pearson type III distribution fits for 50% of
total station number for the rainfall distribution characteristics of Chia-Nan plain area. Bhakar et
al. (2006) observed the frequency analysis of consecutive days peaked rainfall at Banswara,
Rajasthan, India, and found gamma distribution as the best fit as compared by other distribution
and tested by Chi-square value. Deidda and Puliga (2006) found for left-censored records of
Sardinia, that some weak are evident for the generalized Pareto distribution.
Kwaku et al. (2007) revealed that the log-normal distribution was the best fit probability
distribution for one to five consecutive days’ maximum rainfall for Accra, Ghana. Hanson et al.,
(2008) analysis indicated that Pearson type III distribution fits the full record of daily precipitation
data and Kappa distribution best describes the observed distribution of wet-day daily rainfall.
Olofintoye et al. (2009) examined that 50% of the total station number in Nigeria follows logPearson type III distribution for peak daily rainfall, while 40% and 10% of the total station follows
Pearson type III and log-Gumbel distribution respectively. Sharma and Singh (2010) studied the
distribution pattern for extreme value rainfall for Pantnagar data. Generalized extreme value
distribution was observed in most of the weekly period as best fit probability distribution.
There are a wide variety of previous studies which have explored the probability
distribution of daily rainfall for the purpose of rainfall frequency analysis. However, we are
unaware of any studies that have used recent developments. This research seeks to reexamine the
question of which continuous distribution best fits the weekly average monsoon weather variable.
Our primary objective is to determine a suitable distribution of monsoon season for each weekly
average weather variables using different probability distributions.
2.2
Multiple Regression (MR)
Forecasting models based on time series data are being developed for prediction of the
different variables. Regression is a statistical empirical technique and is widely used in business,
the social and behavioral sciences, the biological sciences, climate prediction, and many other
areas. Linear and non-linear multiple regression models of different orders are also being used for
predicting purpose based on the time series data. These models can consider more than one
predictor for rainfall prediction. There are some limitations of multiple regression approach such
as multiple collinearly, inter relation, extreme observation and non-linear relationship between
dependent and independent variables.
Goulden (1962) found the relationship between monthly average of weather parameters
and crop yield using multiple regression technique. Ramchandran (1967) made an analysis of the
normal rainfall of 167 observatory station distributed over India and the neighborhood country,
using regression equation representing monthly and annually rainfall as a linear function of
latitude, longitude and elevation above sea level. Bali (1970) found more precise results with the
help of regression method for calculating average yields and explained the inadequacy of currently
employed methods for forecasting crop yield in India. Huda et al. (1975) reported that a second
degree multiple regression can be employed for studying the relationship between rice yield and
weather variables.
Huda et al. (1976) applied second degree multiple regression equation to quantify the
relationship between maize yield and meteorological data and it was found that maize yield was
affected differently by different weather variables during different stages of growth. Singh et al.
(1979) gave hints for forecasting the yield rate by traditional and objective methods, on the basis
of biometrical character as well as weather parameters. Regression studies on the relationship of
crop yield with weather factor have been made. Agrawal et al. (1980) studied regression models
for forecasting the yield of rice in Raipur district on weekly data using weather variables. Khatri et
al. (1983) used regression analysis for crop estimation surveys on historic and rainfall data for
developing the forecasting model with the help of stepwise regression analysis. Hastenrath (1988)
developed statistical model using regression method to predict Indian summer monsoon rainfall
anomaly.
Singh (1988) developed a suitable pre-harvesting forecasting model with the help of
multiple regression techniques for sugarcane yield. Singh and Bapat (1988) developed a preharvest forecast model using stepwise regression for selection of yield attribute to entering finally
in forecast model. Pal (1995) studied the relationship between weather parameters and yields
using linear multiple regression and second degree multiple regression equation based on time
series weather data. Sparks (1997) developed a multiple regression model for time series data to
predict a production in arid climate at high evaluations. Shashi Kumar et al. (1998) showed that
the principal components regression gives better precision for the estimates than ordinary least
square regression analysis.
Vaccari et al. (1999) modeled plant motion time-series and nutrient recovery data for
advanced life support using multi variable polynomial regression. Hassani et al. (2003) proposed
human height prediction model based on multiple polynomial regression that was used
successfully to forecast the growth potentials of height with precision and was helpful in children
growth study. Sen (2003) has presented long-range summer monsoon rainfall forecast model
based on power regression technique with the use of Ei Nino, Eurasian snow cover, North West
Europe temperature, Europe pressure gradient, 50 hpa Wind pattern, Arabian sea SST, east Asia
pressure and south Indian ocean temperature in previous year. The experimental results showed
that the model error was 4%. Nkrintra et al. (2005) described the development of a statistical
forecasting method for SMR over Thailand using multiple linear regression and local polynomial
based non-parametric approaches. SST, sea level pressure (SLP), wind speed, EiNino Southern
Oscillation (ENSO), IOD was chosen as predictors. The experiments indicated that the correlation
between observed and forecast rainfall was 0.6.
Sohn et al. (2005) has developed a prediction model for the occurrence of heavy rain in
South Korea using multiple linear regression, decision tree and artificial neural network. They
used 45 synoptic factors generated by the numerical model as potential predictors. Anderson et al.
(2006) examines the possibility of forecasting traffic volumes by using a multiple linear regression
model to perform what is termed direct demand forecasting and obtained consistent results from
the traditional four-step methodology. Zaw and Naing (2008) performed the modeling of monthly
rainfall prediction over Myanmar by applying the polynomial regression equation and compared
with multiple linear regression model. Experiments indicated that the prediction model based on
MPR has higher accuracy than MLR.
Radhika and Shashi (2009) used time series data of daily maximum temperature and
found non-linear regression method suitable to train support vector machines (SVMs) for weather
prediction. Kannan et al. (2010) computed values for rainfall fall in the ground level using five
years input data by Karl Pearson correlation coefficient and predicted for future years rainfall fall
in ground level by multiple linear regression. Ghani and Ahmad (2010) applied six types of linear
regression including stepwise multiple regression to select the suitable controlled variables in
forecast fish landing.
2.3 Autoregressive Integrated Moving Average (ARIMA)
Two popular models for seasonal time series are multiplicative seasonal ARIMA
(autoregressive-integrated-moving average) models (Box and Jenkins 1976) and ARIMA
component (structural) models. Despite the rising popularity of ARIMA component models in the
time series literature of recent years, empirical studies comparing these models with seasonal
ARIMA models have been relatively rare.
Cottrell et al. (1995) proposed a systematic methodology to determine which weights are
nonsignificant and to eliminate them to simplify the architecture. They tried to combine the
statistical techniques of linear and nonlinear time series with the connectionist approach. Zhang
and Qi (2003) investigated as how to effectively model time series with both seasonal and trend
patterns. They found that combined detrending and deseasonalization is the most effective data
preprocessing approach. Campbell and Diebold (2005) used simple time-series approach to
modeling and forecasting daily average temperature in U.S. cities and found it useful for the
vantage point of participants in the weather derivatives market.
Iqbal et al. (2005) made the study on ARIMA to forecast the area and production of wheat
in Pakistan. Further, suggesting that the scope of higher area and production lies in adequate
availability of inputs, educating and training the farming community, soil conservation and
reclamation, and especially the supportive government policies regarding wheat cultivation in the
country. Zhou and Hu (2008) proposed a hybrid modeling and forecasting approach based on the
grey and the Box–Jenkins autoregressive moving average (ARMA) models to forecast the
gyrodrift concluding that the hybrid method has a higher forecasting precision to the complex
problems than the single method. Kal et al. (2010) developed a framework to determine the
optimal inventory policy under the environment that the leadtime demand is generated by the
ARIMA process.
Alnaa and Ahiakpor (2011) considered ARIMA model to predict inflation in Ghana.
Inflation was predicted highest for the months of March, April and May. Further, suggesting that
inflation has a long memory and that once the inflation spiral is set in motion, it will take at least
12 periods (months) to bring it to a stable state. Badmus and Ariyo (2011) utilized ARIMA for
forecasting the cultivated area and production of maize in Nigeria. They concluded that the total
cropped area can be increased in future, if land reclamation and conservation measures are
adopted.
Saima et al. (2011) explained a hybrid fuzzy time series model is proposed that will
develop an Interval type 2 fuzzy model based on ARIMA. IT2-FLS is utilized here for handling
the uncertainty in the time series data to obtain accurate forecasting result. Ghosh et al. (2012)
developed a model based on ARIMA to depict the future prospects of coal based thermal power
sector of India. The evidence showed that India needs to identify alternative sources of power
generation to grow without damaging world and maintaining sustainability.
2.4
Artificial Neural Network (ANN)
An Artificial Neural Network is a powerful data modeling tool that provides a
methodology for solving many types of non-linear problems that are difficult to solve by
traditional techniques. Neural Network makes very few assumptions as opposed to normality
assumptions commonly found in statistical methods. From a statistician’s point of view neural
networks are analogous to nonparametric, nonlinear and regression models. The ANN approach
has several advantages over conventional phenomenological or semi-empirical models, since they
require known input data set without any assumptions (Gardner and Dorling, 1998; Nagendra
and khare, 2006). It exhibits rapid information processing and is able to develop a mapping of the
input and output variables. Such a mapping can subsequently be used to predict desired outputs as
a function of suitable inputs (Nagendra and Khare, 2006).
The ANNs use many simplifications over actual biological neurons that help us to use the
computational principles employed in the massively parallel machine (Haykin 1999). The neural
networks adaptively change their synaptic weights through the process of learning. Feed Forward
Neural Networks with Back-propagation (BKP) of error have been used in past for modeling and
forecasting various parameters of interest using time series data, Cottrell el al. (1995) used the
time series modeling to provide a method for weight elimination in ANNs. Since the last few
decades, ANN a voluminous development in the application field of ANN has opened up new
avenues to the forecasting task involving atmosphere related phenomena (Gardner and Dorling,
1998; Hsieh and Tang, 1998). The prediction in an artificial neural network method (ANN)
always takes place according to any data situation (without limitation) based on initial training as
indicated by Adielsson (2005).
Thus, for forecasting, certain statistical techniques can be combined with the connectionist
approach of ANN exploiting the information contained in linear or nonlinear time series. The
knowledge that an ANN gains about a problem domain is encoded in the weights assigned to the
connections of the ANN. The ANN can then be thought as a black box, taking in and giving out
information (Roadknight et al. 1997). ANN non-linear models have been widely used for
resolving forecast problem as identified by Hill et al. (1996), Faraway and Chatfield (1998),
Kaashoek and Van Dijk (2001), Tseng et al. (2002), Altun et al. (2007), Fallah-Ghalhary
(2009), Wu et al. (2010) and El-Shafie et al.(2011).
Hu (1964) initiated the implementation of ANN, an important Soft Computing
methodology in weather forecasting. Forecasting the behavior of complex system has been a broad
application domain for neural networks. In particular such as electric load forecasting (Park et al.
1991), economic forecasting (Refenes et al. 1994), forecasting natural physical phenomena
(Weigend et al., 1994), river flow forecasting (Atiya et al. 1996) and forecasting student
admission in colleges (Puri et al. 2007) have been widely studied.
A successful application of ANN to rainfall forecasting has been done by French et al.
(1992) who applied a neural network to forecast one-hour-ahead, two-dimensional rainfall fields
on a regular grid. Moro et al. (1994) applied a neural network approach for weather forecasting for
local data. Kalogirou et al. (1997) implemented ANN to reconstruct the rainfall time series over
Cyprus. Kuligowski and Barros (1998) analyzed a precipitation forecasts model using neural
network approach. Lee et al. (1998) applied Artificial Neural Network in rainfall prediction by
splitting the available data into homogeneous subpopulations. Wong et al. (1999) constructed
fuzzy rule bases with the aid of SOM and back propagation neural networks and then with the help
of the rule base developed predictive model for rainfall over Switzerland using spatial
interpolation.
Atiya et al. (1997) studied the generalization performance in large network, which means
producing appropriate outputs for those input samples not encountered during training process of
the network is best described by training data size and number of synaptic weights. Share prices
are also taken as time series to forecast stock prices of the future (Mathur et al. 1998). Maqsood
et al. (2002a, 2002b) used neurocomputing based weather monitoring and analysis models.
Anmala et al. (2000) reported that recurrent networks may perform better than standard feed
forward networks in predicting monthly runoff. Sahai et al. (2000) applied the ANN technique to
five time series of June, July, August, September monthly and seasonal rainfall. The previous five
years values from all the five time series were used to train the ANN to predict for the next year.
They found good performance in predicting rainfall.
Toth et al. (2000) investigated the capability of ANN in short-term rainfall forecasting
using historical rainfall data as the only input information. Kishtawal et al. (2003) assessed the
feasibility of a nonlinear technique based on genetic algorithm, an Artificial Intelligence technique
for the prediction of summer rainfall over India. Guhathakurta (2006) was the first ever to
implement ANN technique to predict summer monsoon rainfall over a state of India. Miao et al.
(2006) developed almost seven different methods for ANN and each one can be used in a different
analysis rather than classical statistical methods in identifying factors influencing corn yield and
grain quality variability. Chattopadhyay (2007) analyzed that neural network with three nodes in
the hidden layer is found to be the best predictive model for possibility of predicting average
summer-monsoon rainfall over India. Paras el al. (2007) concluded that neural networks are
capable of modeling a weather forecast system. Statistical indicators chosen are capable of
extracting the trends, which can be considered as features for developing the models.
Hayati et al. (2007) showed that multi-layer perceptron (MLP) network has minimum
forecasting error for each season and can be considered as a good method for temperature
forecasting. Kumar et al. (2007) presented reasonably good Artificial Intelligence approaches for
regional rainfall forecasting for Orissa state, India on monthly and seasonal time series scale. The
study emphasizes the value of using large scale climate teleconnections for regional rainfall
forecasting and the significance of Artificial intelligence approaches in predicting the uncertain
rainfall. Hung et al. (2009) developed the ANN model and applied for real time rainfall
forecasting and flood management is Bangkok, Thailand, Resulting that ANN forecasts have
superiorly over the ones obtained by the persistent model. Rainfall forecast for Bangkok from 1 to
3 h ahead were found highly satisfactory. Sharma et al. (2011) proposed hybrid MR with ANN
for the Himalayan monsoon data, suggesting that hybrid techniques can be used as a reliable
rainfall forecasting tool in the Himalaya.
2.5
Comparison among of MR, ARIMA and ANN
The comparative evaluation of the performance of all the three models has been conducted
using forecasting weather variable and many other time series data. Many studies have been
conducted by Lek (1996), Starett (1998), Manel (1999), Salt (1999), Ozesmi (1999), Gail (2005),
Diane (2007) with their co-authors and Pastor (2005) to compare two methods to show that in
predicting the dependent variable, the ANN method results are more accurate than Multiple Linear
Regression. The performance of ANN and traditional statistical methods were also compared and
discussed by Kumar (2005), Pao (2006), Wang and Elhag (2007), Zhang (2001) and Wang et al.
(2010).
Dutta and Shekhar (1988) compared neural networks to a multiple regression model in an
application to predict bond ratings based on ten financial variables. The neural network model
consistently outperformed the regression model yielding a success rate of 88.3 percent versus 64.7
percent for regression. In addition, the neural network was never off by more than one rating while
the regression model was often off by several ratings.
Marquez et al. (1991) compared the performance of neural network models to various
regression models. They tested the data with the correct regression model, two other regression
models that were one-step in either direction away from the correct model on a ladder of
expression, and two neural network models. The neural network was generally within two percent
of the mean absolute percentage error of the current model for the linear and inverse cases. The
neural network performance on the log data was poorer and in general the neural networks did not
perform quite as well as the other regression models but the authors concluded that neural
networks have considerable potential as an alternative to regression.
Specht and Donald (1991) examined the use of neural network to perform the function of
multiple linear regressions. They compared the use of neural networks with standard linear
regression in four cases: the regression model was correctly specified with all assumptions valid;
the regression model was correctly specified, but the data contained an outlier; the regression
variables exhibited multicollinearity; and the regression model was incorrectly specified by
omitting an interaction. The authors concluded that neural networks are robust and relatively
insensitive to problems with bad data, model assumptions, and faulty model construction.
Chang et al. (1991) used neural networks to forecast rainfall based on time series data. The
data showed both seasonal and cyclical components that were incorporated into the input data set.
The forecasts were for one month into its future. Based on the mean square error, neural network
outperformed the unnamed statistical approach. Dulibar (1991) performed a study to predict the
performance of carriers in a particular segment of the transportation sector. In the study, she
compared a neural network model to several regression models. The measure of performance used
was percent capacity utilization. The neural network performed better than some of the regression
models and not as well as others. According to the author, a possible reason for the poorer
performance of the neural network model is that he firms involved in the study were explicitly
included in the regression models but not in the Neural Network model.
Raghupathi et al. (1991) found that a neural network provided 86 % correct classifications
and would therefore likely provide a good model for the bankruptcy prediction process.
Salchenberger et al. (1992) used neural networks to predict thrift failures. They compared a
neural network to the more traditional logit model. They used five financial ratios in attempting to
classify an institution as one that would fail or not fail. For each data set created, the neural
network performed at least as well as the logit model. Also, as the classification cutoff was
lowered, the neural network committed less type I errors than the logit model.
Tam and Kiang (1992), neural networks were compared to several popular discriminant
analysis methods in a bank failure classification application. The sample consisted of 59 matched
pairs of Texas banks. The neural network model showed better predictive accuracy on the test set
than the other methods. Wu and Yen (1992) proposed a neural network structure and the
associated design-oriented procedure for neural network development for regression applications.
The proposed methodology is illustrated by two practical applications. One is a linear regression
case concerning the relation between marginal cost and cumulative production; the other is a
nonlinear regression case concerning the yield of wheat corresponding to the application rate of
fertilizer. They compared the results of the regression techniques with those of neural networks.
Fletcher and Goss (1993) used financial ratios to compare neural networks to a logit
model. The output of the models represented the probability that a particular firm would fail. The
neural network was 82.4 % accurate with a 0.5 cutoff versus 77 % for the logit model. Similar
results were found at other cutoff values. Also, the neural network had less error variance and
lower prediction risk than the logit model.
Yi and Prybutok (1996) compared neural networks to multiple regression and ARIMA
models in an application to predict the maximum ozone concentration in a large metropolitan area.
The independent variables consisted of nine meteorological and auto emission measures. The
neural network model was statistically superior to both the regression and ARIMA models.
Empirical results have shown that Neural Networks outperform linear regression as data quality
varies was described by Bansal et al. (1993) and Marquez et al. (1991). Michaelides et al. (1995)
compared the performance of ANN with multiple linear regression in estimating missing rainfall
data over Cyprus.
Goh (1996, 1998) studied the comparison between ANN and multiple regressions in
construction, management, and engineering. Comrie (1997) studied multiple regression models
and neural networks are examined for a range of cities under different climate and ozone regimes,
enabling a comparative study of the two approaches, resulting neural network techniques are better
than regression models for daily ozone prediction. Ostensibly, Neural Network is simply an
extension of regression modeling which can be referred to a flexible non-linear regression models,
Menamin and Stuart (1997).
Venkatesan et al. (1997) have used neural network technique to predict monsoon rainfall
of India using few predictions and compared the results with linear regression techniques, showing
that the model based on neural network technique performed better. Baker (1998) compared linear
regression and neural network methods for forecasting educational spending and found Neural
Network provide comparable prediction accuracy. Man-Chung et al. (1998) proposed conjugate
gradient with multiple linear regression (MLR) weight initialization requires a lower computation
cost and learns better than steepest decent with random initialization for financial time series data
collected from Shanghai Stock Exchange.
Ranasinghe et al. (1999) compared ANN and multiple regression analysis in estimating
willingness to pay for urban water supply found that forecasting error of the best ANN model was
about half of the best multiple regression model. Hippert et al. (2000) proposed a hybrid
forecasting system that combines linear models and multilayer neural networks to forecast hourly
temperatures based on the past observed temperatures and the maximum and minimum forecast
temperatures supplied by the weather service.
Mathur et al. (2001) made a comparative study of neural network and regression models
for predicting stock prices and results verified the suitability and superiority of neural network
model over regression model. Victor (2001) described a neural network forecasting model as an
alternative to regression model for messy data problems and limitations in variable structure
specification. Zhang (2003) applied a hybrid methodology that combines both ARIMA and ANN
models to take advantage of the unique strength of ARIMA and ANN models in linear and
nonlinear modeling.
Maqsood et al. (2004) developed neural network based ensemble models and applied for
hourly weather forecasting of solution Saskatchewan. The experimental results show that the
ensemble network can be trained effectively without excessively compromising the performance.
Further, compared to the regression models, the ensemble networks forecast the weather parameter
with higher accuracy. Suhartono et al. (2005) made a comparative study of forecasting models for
trend and seasonal time series, concluding that the more complex model does not always yield
better forecast than the simpler one, especially on the testing samples. Also showing FFNN model
always yields better forecast in training data and it indicates an over fitting problem.
Taskaya-Temizel et al. (2005) suggested that the use of a nonlinear component may
degenerate the performance of hybrid methods and that a simpler hybrid comprising linear AR
model with a TDNN outperforms the more complex hybrid in tests on benchmark economic and
financial time series. Somvanshi et al. (2006) examined two fundamental different approaches
ARIMA and ANN for designing a model and predict the behavioral pattern in rainfall phenomena
based on past behavior. The study revealed that ANN model can be used as an approximate
forecasting tool to predict the rainfall, that out performs the ARIMA model.
Pandey et al. (2008) made a comparative study of neural network & fuzzy time series
forecasting techniques for crop yield and observed that neural network produces more accurate
results in comparison of fuzzy time series methods. Pao (2008) studied multiple linear regressions
and neural networks models with seven explanatory variables of corporation’s feature and three
external macro-economic control variables to analyze the important determinants of capital
structures of high-tech and traditional industries in Taiwan, respectively. The ANN models
achieve a better fit and forecast than the regression models for debt ratio. Aladag et al. (2009)
studied a new hybrid approach combining ERNN and ARIMA modes to time series data resulting
with best forecasting accuracy. Kulshrestha et al. (2009) examined that ANN gives more accurate
results to predict the probability of extreme rainfall than the probability by Fisher-Tippet Type II
distribution.
Khashei and Bijari (2011) proposed hybrid methodologies c
ombining linear models such as ARIMA and nonlinear models such as ANNs together and
found them more effective than traditional hybrid methodologies. Sharma and Singh (2011)
studied the forecasting models to make comparison among the model to identify the appropriate
model for the prediction of rainfall, concluding that ANN approach is better than other models.
Zaefizadeh et al. (2011) showed that in the ANN technique the mean deviation index of
estimation significantly was one-third of its rate in the MLR, because there was a significant
interaction between genotype and environment and its impact on estimation by MLR method.
Therefore, they recommended ANN approach as better predictor of yield in Barley than in
multiple linear regression. Recently, Ghodsi and Zakerinia (2012) used ARIMA, ANN and fuzzy
regression to analyse price forecasting. Fuzzy regression was found to be best method in
forecasting. Teri and Onal (2012) conducted MLR and ANN to forecast monthly river flow in
Turkey. The performance of the models suggested that the flow could be forecasting easily from
available flow data using ANN.
After a thorough review about the forecasting model, it becomes necessary to compare the
efficiency of different weather forecasting model and to identify the more precise weather
forecasting model. In the present work MLR, ARIMA and ANN models were used and hybrid
model of MLR with ARIMA and ANN, and a comparative analysis was done using different
analytical methods for weekly average time series data so as to identify the appropriate model.
FITTING OF
PROBABILITY DISTRIBUTION

Introduction

Descriptive Statistics

Methodology

Probability Distribution Pattern

Conclusion
CHAPTER 3
FITTING OF PROBABILITY DISTRIBUTION
3.1
Introduction
Establishing a best fit probability distribution for different parameter has long been a topic
of interest in the field of meterology. The investigation of weather parameter distribution strongly
depends upon their distribution pattern. The present study is planned to identify the best fit
probability distribution based on distribution pattern for different data set. The 16 probability
distribution are identified out of large number of commonly used probability distribution for such
type of study. The descriptive statistics are computed first for each weather parameters for
different study periods. The parameters are discussed and explained through tables and graphs.
The test statistics D, A2 and  2 are computed for all 16 probability distribution. The best fit
probability distribution is identified based on highest ranks computed through all the three tests
independently. The best fit probability distribution so obtained is presented with their test statistic
value in each study period. It was further weighted using highest scores of selected probability
distribution for each study period. The combination of total test score of all the three test statistics
was computed for all the 16 probability distributions. The distribution having the maximum score
obtained from 18 set of data is identified. The parameters of the best fit probability distribution for
different data set of each weather parameters are presented. Fitted distribution was used to
generate random number for each data set. Finally, the best fit probability distribution for each
weather parameters was identified using the least square method.
3.2
Descriptive Statistics
The weekly data of seven parameters viz. rainfall, maximum and maximum temperature,
relative humidity at 7.00 am and 2.00 pm. bright sunshine hours and pan evaporation for the four
monsoon months were recorded. The monsoon season in this region lies between 15 to 20 weeks.
Keeping this point in view 17 weeks weather data from 4th June to 30th September of each year
was considered for the present study.
The descriptive statistics of the seasonal and weekly weather data set was computed
resulting the mean, standard deviation, skewness coefficient and coefficient of variation for all the
seven parameters. Minimum and maximum weekly value is also presented for each weather
parameter. The standard deviation indicate about the fluctuation of the parameter. The coefficient
of skewness are computed for all parameter which explain about the shape of the curve. The
coefficient of variation was computed for each parameters which explain the variability in the
data. The details of each parameter was also presented in the form of graphs. The period-wise
details of each weather parameters is presented and discussed in the subsequent sections.
Rainfall (mm)
The study period-wise summary of rainfall is presented in table 3.1 along with mean,
standard deviation, skewness, coefficient of variation, maximum and minimum values.
Table 3.1. Summary of statistics for Rainfall.
Parameters
Study
Period
Data
(From – To)
Mean
Standard
Deviation
Skewness
Coefficient
of
variation
Maximum
(weekly
total)
Minimum
(weekly
total)
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
4June-30Sep
4 june-10 june
11 june-17 june
18 june-24 june
25 june-1 july
2 july-8 july
9 july-15 july
16 july-22 july
23 july-29 july
30 july-5 aug
6 aug-12 aug
13 aug-19 aug
20 aug-26 aug
27 aug-2 sep
3 sep-9 sep
10 sep-16 sep
17sep-23 sep
24 sep-30 sep
73.20
29.07
41.21
46.32
57.17
75.44
102.50
96.77
109.31
74.83
93.99
104.07
99.55
75.55
79.26
60.43
62.25
36.59
80.40
53.92
57.11
51.61
52.47
86.77
88.37
78.75
98.87
59.20
87.17
94.04
87.83
82.09
81.90
68.96
92.92
62.98
0.4402
3.1482
2.3822
1.941
1.4613
1.8573
1.1507
1.6998
1.2496
0.8854
1.8554
1.4451
1.4541
1.8776
1.5354
1.5986
1.7897
1.9873
0.4022
1.8546
1.3859
1.1142
0.9177
1.1501
0.8621
0.8139
0.9045
0.7911
0.9275
0.9036
0.8823
1.0865
1.0333
1.1411
1.4926
1.7211
443.20
263.60
291.80
217.40
245.20
361.60
355.10
396.00
434.20
223.30
443.20
422.80
395.20
413.20
353.20
296.20
347.60
230.40
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.80
0.00
0.90
0.00
3.10
0.00
0.00
0.00
0.00
0.00
0.00
Where, the mean of seasonal rainfall of 50 years was 73.20 mm and mean of weekly
rainfall was varying from 29.07 mm in first week of June to 109.31 mm in last week of July. The
maximum value of weekly rainfall lies between 217.4 mm in third week of June in year 1975 to
443.20 mm in fourth week of August in year 2000.
The weekly minimum value of the rainfall in most of the weeks in 50 years was zero
except for the year 2002 which was the highest in third week of August. The standard deviation
for seasonal rainfall for 50 years was 80.40 mm while the weekly variation of standard deviation
ranging from 51.61 mm in third week of June to 98.87 mm in fourth week of July. The graphical
representation of the weekly rainfall is shown in figure 3.1 and weekly rainfall statistics for
seasonal 850 weeks of total 50 years is also presented in figure 3.2.
Figure 3.1 Mean, standard deviation and range of weekly Rainfall.
The maximum value of coefficient of variation for weekly data was observed 1.8546 in the
first week of June which indicates maximum fluctuation in the rainfall data set, that is, a large
variation in the occurrence of rainfall during the 50 years was observed. The measure of skewness
in seasonal data was 0.4402 and ranging from 0.8854 in first week of August to 3.1482 in first
week of June, which further shows a large degree of asymmetry of a distribution around its mean.
Maximum Temperature (OC)
The summary of statistics for maximum temperature were presented in table 3.2 along with
mean, standard deviation, skewness, coefficient of variation, maximum and minimum values,
where, the mean of weekly maximum temperature of 50 years seasonally was 33.08 OC and mean
of weekly maximum temperature was varying from 31.78 OC in second week of September to
37.75 OC in first week of June. The maximum value of seasonal maximum temperature was 43.20
O
C in two years 1966 and 1967 and that of weekly maximum temperature was lying between
34.00 OC in third week of September in year 2007 to 43.20 OC in first week of June in two years
1966 and 1967.
It was moreover observed that the minimum value of the seasonal maximum temperature
was 23.60 OC in the year 2005. The weekly minimum value of the maximum temperature was
between 23.60 OC in fourth week of June in year 2005 to 30.20 OC in first week of June in year
2002. The standard deviation for seasonal maximum temperature for 50 years was 2.67 OC while
the weekly variation of standard deviation ranging from 1.19 OC in fourth week of September to
3.33 OC in first week of June.
The maximum value of coefficient of variation for weekly data was observed as 0.1018 in
the second week of June which indicates a large fluctuation in the maximum temperature data set.
The measure of skewness in seasonal data was -0.2994 and ranging from -1.3196 in first week of
September to 0.9789 in second week of July indicating the degree of asymmetry of a distribution
around its mean. The graphical representation of the weekly maximum temperature is shown in
figure 3.3 and weekly maximum temperature statistics for seasonal 850 weeks of total 50 years is
also presented in figure 3.4.
Table 3.2. Summary of statistics for Maximum Temperature.
Parameters
Study
Period
Data
(From – To)
Mean
Standard
Deviation
Skewness
Coefficient
of
variation
Maximum
Minimum
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
4June-30Sep
4 june-10 June
11 june-17 june
18 june-24 june
25 june-1 july
2 july-8 july
9 july-15 july
16 july-22 july
23 july-29 july
30 july-5 aug
6 aug-12 aug
13 aug-19 aug
20 aug-26 aug
27 aug-2 sep
3 sep-9 sep
10 sep-16 sep
17sep-23 sep
24 sep-30 sep
33.08
37.75
36.44
35.53
33.83
33.80
32.84
32.25
32.10
32.23
32.35
31.88
31.89
32.11
31.79
31.78
31.85
31.91
2.67
3.33
3.71
3.01
2.55
1.89
1.95
1.48
1.43
1.39
1.26
1.35
1.19
1.32
1.79
1.39
1.26
1.46
-0.2994
-0.3093
-0.5374
0.4169
-0.7018
0.5194
0.9789
-0.2906
-0.1761
0.1455
0.5677
-0.3130
0.0737
-0.5801
-1.3196
-0.3879
-0.9759
-0.7634
0.0239
0.0881
0.1018
0.0847
0.0754
0.0558
0.0591
0.0459
0.0444
0.0433
0.0390
0.0425
0.0373
0.0411
0.0565
0.0436
0.0397
0.0458
43.20
43.20
42.70
42.90
41.90
39.30
39.70
34.90
34.90
36.00
36.00
34.10
35.40
35.10
35.30
34.80
34.00
34.20
23.60
30.20
24.40
29.30
23.60
29.80
29.60
28.60
28.20
29.20
29.90
28.50
29.10
28.60
24.80
28.20
28.20
28.10
Figure 3.3 Mean, standard deviation and range of weekly Maximum Temperature.
Minimum Temperature (OC)
The summary of statistics for minimum temperature in different period is presented in
table 3.3 along with mean, standard deviation, skewness, coefficient of variation, maximum and
minimum values, where, the mean of minimum temperature of 50 years seasonally was 24.43 OC
and mean of weekly minimum temperature was varying from 21.57 OC in last week of September
to 25.28 OC in fourth week of June. The maximum value of seasonal minimum temperature was
29.20 OC in year 1995 and for weekly minimum temperature lies between 24.50 OC in third week
of September in year 1998 to 29.20 OC in second week of June in year 1995.
It was further observed that the minimum value of the seasonal minimum temperature was
17.20 OC in the year 1984. The weekly minimum value of the minimum temperature was between
17.20OC in last week of September in year 1984 to 23.20 OC in second, third and fourth week of
July in years 1976, 1979 and 1981, respectively and also in first week of August in year 1975.
The standard deviation for seasonal minimum temperature for 50 years was 1.51OC while
the weekly variation of standard deviation ranging from 0.68 OC in third week of July to 1.81 OC
in third week of June. The maximum value of coefficient of variation for weekly data was
observed to be 0.0714 in the last week of September which indicates fluctuation in the minimum
temperature data set.
The measure of skewness in seasonal data was -0.6409 and ranging from -2.3776 in third
week of August to 0.0132 in third week of June indicating the degree of asymmetry of a
distribution around its mean. The graphical representation of the weekly minimum temperature is
shown in figure 3.5 and weekly minimum temperature statistics for seasonal 850 weeks of total 50
years is also presented in figure 3.6.
Table 3.3. Summary of statistics for Minimum Temperature.
Parameters
Study
Period
Data
(From – To)
Mean
Standard
Deviation
Skewness
Coefficient
of
variation
Maximum
Minimum
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
4June-30Sep
4 june-10 June
11 june-17 june
18 june-24 june
25 june-1 july
2 july-8 july
9 july-15 july
16 july-22 july
23 july-29 july
30 july-5 aug
6 aug-12 aug
13 aug-19 aug
20 aug-26 aug
27 aug-2 sep
3 sep-9 sep
10 sep-16 sep
17sep-23 sep
24 sep-30 sep
24.43
24.07
24.70
25.01
25.28
25.22
25.25
25.04
25.17
25.11
25.01
24.80
24.65
24.45
24.01
23.44
22.54
21.57
1.51
1.75
1.54
1.81
1.70
1.17
0.89
0.68
0.93
0.80
0.77
0.90
0.70
0.81
0.70
0.80
1.34
1.54
-0.6409
-0.2130
0.0132
2.0372
2.1990
-1.4444
-0.0492
-0.4531
0.2800
0.1502
-0.3468
-2.3776
-0.5365
-2.1114
-0.4238
-1.4321
-1.4456
-0.7552
0.0192
0.0727
0.0625
0.0723
0.0673
0.0463
0.0350
0.0272
0.0367
0.0318
0.0308
0.0361
0.0284
0.0331
0.0292
0.0341
0.0596
0.0714
29.20
28.30
29.20
33.50
34.00
27.90
27.30
26.20
27.20
27.10
26.60
26.40
26.50
25.80
25.30
24.80
24.50
24.90
17.20
18.80
21.50
21.50
20.00
20.10
23.20
23.20
23.20
23.20
22.70
20.30
22.60
20.70
22.20
20.10
17.60
17.20
Figure 3.5 Mean, standard deviation and range of weekly Minimum Temperature.
Relative Humidity at 7AM (%)
The study period wise summary of Relative Humidity at 7 AM is presented in table 3.4
along with mean, standard deviation, skewness, coefficient of variation, maximum and minimum
values, where, the mean of Relative Humidity at 7 AM of 50 years seasonally was 86.71 % and
mean of weekly average Relative Humidity at 7 AM was varying from 66.80 % in first week of
June to 91.65 % in fourth week of August.
The maximum value of seasonal average Relative Humidity at 7 AM was 98 % in four
years 1988, 1995, 1996 and 1998 and for weekly average Relative Humidity at 7 AM lies between
87 % in first week of June in years 1971 and 1984 to 98 % in second and third week of August in
year 1988 and 1995 respectively and also in first week of September in the two years 1996 and
1998.
It was moreover pragmatic that the least value of the seasonal average Relative Humidity
at 7 AM was 38 % in the year 1966. The weekly minimum value of the Relative Humidity at 7
AM was between 38 % in first week of June in year 1966 to 85 % in second week of September in
two years 1981 and 2008. The standard deviation for seasonal average Relative Humidity at 7 AM
for 50 years was 9.35 % while the weekly variation of standard deviation ranging from 2.97 % in
second week of September to 12.16 % in first week of June.
The maximum value of coefficient of variation for weekly data was observed as 0.1820 in
the first week of June which indicates fluctuation in the Relative Humidity at 7 AM data set. The
measure of skewness in seasonal data was -0.6237 and ranging from -1.7081 in fourth week of
July to 0.0113 in first week of June indicating the degree of asymmetry of a distribution around its
mean. The graphical representation of the weekly Relative Humidity at 7 AM is shown in figure
3.7 and weekly Relative Humidity at 7 AM statistics for seasonal 850 weeks of total 50 years is
also presented in figure 3.8.
Table 3.4. Summary of statistics for Relative Humidity at 7 AM.
Parameters
Study
Period
Data
(From – To)
Mean
Standard
Deviation
Skewness
Coefficient
of
variation
Maximum
Minimum
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
4June-30Sep
4 june-10 June
11 june-17 june
18 june-24 june
25 june-1 july
2 july-8 july
9 july-15 july
16 july-22 july
23 july-29 july
30 july-5 aug
6 aug-12 aug
13 aug-19 aug
20 aug-26 aug
27 aug-2 sep
3 sep-9 sep
10 sep-16 sep
17sep-23 sep
24 sep-30 sep
86.71
66.80
72.75
78.93
84.49
86.64
87.71
89.65
90.20
90.67
90.64
90.96
91.65
90.89
91.60
91.61
90.44
88.43
9.35
12.16
11.87
10.79
6.26
4.85
6.23
4.00
5.05
3.20
3.59
3.39
3.10
3.65
3.29
2.97
3.25
5.23
-0.6237
0.0113
-0.1560
-0.9558
-1.0339
-0.4663
-1.3950
-0.4663
-1.7081
-0.1685
-0.48305
-0.2252
-0.6682
-0.6405
-0.7568
-0.2493
-0.6645
-1.4181
0.0286
0.1820
0.1632
0.1367
0.0740
0.0560
0.0710
0.0446
0.0560
0.0353
0.0396
0.0373
0.0338
0.0402
0.0359
0.0325
0.0359
0.0591
98
87
94
95
95
96
96
95
97
96
98
98
97
97
98
96
96
95
38
38
46
50
63
75
63
80
69
84
81
83
83
79
80
85
80
69
Figure 3.7 Mean, standard deviation and range of weekly average Relative Humidity at 7 AM.
Relative Humidity at 2 PM (%)
The summary of statistics for Relative Humidity at 2 PM for different study periods is
presented in table 3.5 along with mean, standard deviation, skewness, coefficient of variation,
maximum and minimum values, where, the mean of Relative Humidity at 2 PM of 50 years
seasonally was 65.70 % and mean of weekly average Relative Humidity at 2 PM was varying
from 39.98 % in first week of June to 74.45 % in eleventh week, that is, third week of August.
The maximum value of coefficient of variation for seasonal average Relative Humidity at 2
PM was 0.0537. The maximum value of coefficient of variation for weekly data was observed to
be 0.3903 in the first week of June which indicates a relatively high fluctuation in the Relative
Humidity at 2 PM data set.
The measure of skewness in seasonal data was -0.4778 and ranging from -1.0936 in fourth
week of June to 0.3368 in fourth week of August indicating the degree of asymmetry of a
distribution around its mean. The maximum value of seasonal average Relative Humidity at 2 PM
was 92 % in year 1988 and for weekly average Relative Humidity at 2 PM lies between 72 % in
first week of June in year 1962 to 92 % in second week of August in year 1988.
It was further observed that the minimum value of the seasonal average Relative Humidity
at 2 PM was 16 % in the two years 1965 and 2005. The weekly minimum value of the Relative
Humidity at 2 PM was between 16 % in second week of June in two years 1965 and 2005 to 62 %
in fourth week of August in year 1973. The standard deviation for seasonal average Relative
Humidity at 2 PM for 50 years was 13.60 % whereas the weekly variation of standard deviation
ranging from 5.01 % in twelfth week, that is, fourth week of August to 16.47 % in second week of
June.
The graphical representation of the weekly average Relative Humidity at 2 PM is shown in
figure 3.9 and weekly Relative Humidity at 2 PM statistics for seasonal 850 weeks of total 50
years is also presented in figure 3.10.
Table 3.5. Summary of statistics for Relative Humidity at 2 PM.
Parameters
Study
Period
Data
(From – To)
Mean
Standard
Deviation
Skewness
Coefficient
of
variation
Maximum
Minimum
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
4June-30Sep
4 june-10 June
11 june-17 june
18 june-24 june
25 june-1 july
2 july-8 july
9 july-15 july
16 july-22 july
23 july-29 july
30 july-5 aug
6 aug-12 aug
13 aug-19 aug
20 aug-26 aug
27 aug-2 sep
3 sep-9 sep
10 sep-16 sep
17sep-23 sep
24 sep-30 sep
65.70
39.98
47.15
55.70
63.29
65.18
70.70
72.09
73.34
73.13
72.63
74.45
72.97
70.54
70.26
69.85
65.63
60.01
13.60
15.60
16.47
13.92
10.33
10.44
8.91
6.77
7.36
6.43
7.10
6.21
5.01
7.22
7.86
8.03
8.02
9.66
-0.4778
0.3116
0.1156
-0.7909
-1.0936
-0.3999
-0.4001
0.2866
-0.1840
-1.0567
0.1527
0.1023
0.3368
-0.3006
-0.2773
0.1521
-0.5267
-0.4532
0.0537
0.3903
0.3494
0.2499
0.1632
0.1602
0.1260
0.0938
0.1004
0.0879
0.0978
0.0834
0.0686
0.1024
0.1118
0.1150
0.1221
0.1610
92
72
80
79
82
88
85
85
88
85
92
89
87
83
85.6
89
81
78
16
17
16
20
24
35
50
61
56.6
50
59
61
62
53
50
54
42
36
Figure 3.9 Mean, standard deviation and range of weekly average Relative Humidity at 2 PM.
Pan Evaporation (mm)
The summary of statistics for Pan Evaporation is presented in table 3.6 along with mean,
standard deviation, skewness, coefficient of variation, maximum and minimum values, where, the
mean of Pan Evaporation of 50 years seasonally was 5.31 mm and mean of weekly average Pan
Evaporation was varying from 3.83 mm in third week of September to 10.59 mm in first week of
June.
The maximum value of seasonal average Pan Evaporation was 18.50 mm in year 1967 and
for weekly average Pan Evaporation lies between 5.20 mm in third week of September in year
1996 to 18.50 mm in first week of June in year 1967. It was also observed that the minimum value
of the seasonal average Pan Evaporation was zero. The weekly minimum value of the Pan
Evaporation was between zero mm in second and fourth week of July in the same year 1976 and
second week of August in year 1972 to 4.20 mm in first week of June in year 1971.
The standard deviation for seasonal average Pan Evaporation for 50 years was 2.67mm
with the weekly variation of standard deviation ranging from 0.76 mm in last week of September
to 2.94 mm in second week of June. The maximum value of coefficient of variation for weekly
data was observed as 0.4402 in the fourth week of July indicating fluctuation in the Pan
Evaporation data set.
The measure of skewness in seasonal data was -0.4380 and weekly average is ranging from
-0.0483 in last week of September to 0.9927 in third week of August indicating the degree of
asymmetry of a distribution around its mean. The graphical representation of the weekly average
Pan Evaporation is shown in figure 3.11 and weekly Pan Evaporation statistics for seasonal 850
weeks of total 50 years is also presented in figure 3.12.
Table 3.6. Summary of statistics for Pan Evaporation.
Parameters
Study
Period
Data
(From – To)
Mean
Standard
Deviation
Skewness
Coefficient
of
variation
Maximum
Minimum
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
4June-30Sep
4 june-10 June
11 june-17 june
18 june-24 june
25 june-1 july
2 july-8 july
9 july-15 july
16 july-22 july
23 july-29 july
30 july-5 aug
6 aug-12 aug
13 aug-19 aug
20 aug-26 aug
27 aug-2 sep
3 sep-9 sep
10 sep-16 sep
17sep-23 sep
24 sep-30 sep
5.31
10.59
9.00
8.19
6.26
5.89
5.29
4.88
4.12
4.15
4.04
4.14
4.04
4.07
3.97
3.88
3.83
3.85
2.67
2.78
2.94
2.67
1.83
2.17
1.92
1.75
1.82
1.23
1.45
1.49
1.20
1.26
1.04
0.91
0.83
0.76
-0.4380
0.1739
0.5974
0.7330
0.7244
0.4327
-0.2450
0.7634
0.3147
-0.2453
-0.6027
0.9927
-0.0942
0.2181
-0.2653
-0.9353
-0.3608
-0.0483
0.1646
0.2623
0.3264
0.3259
0.2921
0.3676
0.3633
0.3585
0.4402
0.2957
0.3573
0.3608
0.2958
0.3087
0.2608
0.2332
0.2163
0.1978
18.50
18.50
17.20
15.90
12.10
11.40
9.50
10.20
9.80
6.50
6.70
9.50
6.40
7.60
6.30
5.50
5.20
5.60
0.00
4.20
2.90
3.60
2.10
1.40
0.00
1.50
0.00
1.60
0.00
1.60
1.50
1.80
1.60
1.20
1.90
2.30
Figure 3.11 Mean, standard deviation and range of weekly average Pan Evaporation.
Bright Sunshine (hours)
The summary of Bright Sunshine for different study period is presented in table 3.6 along
with mean, standard deviation, skewness, coefficient of variation, maximum and minimum values,
where, the mean of Bright Sunshine of 50 years seasonally was 6.38 hours and mean of weekly
average Bright Sunshine was varying from 5.30 hours in fourth week of July to 8.60 hours in first
week of June.
The maximum value of seasonal average Bright Sunshine was 11.60 hours in the two years
1986 and 2009 and for weekly average Bright Sunshine was between 8.80 hours in third week of
August in year 1965 to 11.60 hours in first and third week of June in years 1986 and 2009
respectively. It was also observed that the minimum among the seasonal average Bright Sunshine
was 0.70 hours in the year 2009. The weekly minimum value of the Bright Sunshine was between
0.70 hours in third week of August in the year 2009 to 4.20 hours in first week of June in year
2000.
The standard deviation for seasonal average Bright Sunshine for 50 years was 2.16 hours
with the weekly variation of standard deviation ranging from 1.72 hours in fourth week of July to
2.21 hours in third week of June. The maximum value of coefficient of variation for weekly data
was observed in the second week of August which indicates a reasonable fluctuation in the Bright
Sunshine data set.
The measure of skewness in seasonal data was 0.1935 and weekly average is ranging from
-1.2914 in second week of June to 0.4982 in second week of august indicating the measure of
asymmetry of a distribution around its mean. The graphical representation of the weekly average
Bright Sunshine is shown in figure 3.13 and weekly Bright Sunshine statistics for seasonal 850
weeks of total 50 years is also presented in figure 3.14.
Table 3.7. Summary of statistics for Bright Sunshine.
Parameters
Study
Period
Data
(From – To)
Mean
Standard
Deviation
Skewness
Coefficient
of
variation
Maximum
Minimum
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
4June-30Sep
4 june-10 June
11 june-17 june
18 june-24 june
25 june-1 july
2 july-8 july
9 july-15 july
16 july-22 july
23 july-29 july
30 july-5 aug
6 aug-12 aug
13 aug-19 aug
20 aug-26 aug
27 aug-2 sep
3 sep-9 sep
10 sep-16 sep
17sep-23 sep
24 sep-30 sep
6.38
8.60
7.79
7.12
6.19
6.15
5.66
5.54
5.30
5.66
5.56
5.41
5.52
6.02
6.03
6.49
7.31
8.02
2.16
1.80
1.97
2.21
1.74
1.92
2.10
1.77
1.72
1.80
2.07
1.73
1.89
2.02
2.20
2.06
1.90
1.87
0.1935
-0.5602
-1.2914
-0.4005
0.0237
-0.4944
-0.2721
-0.3552
0.2553
-0.1008
0.4982
-0.6706
0.1980
0.0019
-0.1090
-0.1149
-0.4497
-0.9273
0.1060
0.2092
0.2528
0.3107
0.2812
0.3126
0.3705
0.3197
0.3255
0.3186
0.3711
0.3192
0.3429
0.3360
0.3644
0.3166
0.2601
0.2328
11.60
11.60
11.30
11.60
10.00
9.30
9.70
8.90
9.10
9.30
11.30
8.80
9.30
9.90
9.50
10.70
10.60
10.30
0.70
4.20
1.30
1.70
2.40
1.70
0.90
1.40
1.20
2.00
0.90
0.70
1.60
1.60
2.40
2.60
2.30
3.20
Figure 3.13 Mean, standard deviation and range of weekly average Bright Sunshine.
3.3
Methodology
Weather parameter data was analyzed to identify the best fit probability distribution for
each period of study. Three statistical goodness of fit test were carried out in order to select the
best fit probability distribution on the basis of highest rank with minimum value of test statistic.
The appropriate probability distributions are identified for the different dataset using maximum
overall score based on sum of individual point score obtained from three selected goodness of fit
test. Random numbers were generated for actual and estimated weekly weather parameters for
each period of study using the parameters of selected distributions.
3.3.1 Fitting the probability distribution
The probability distributions viz. normal, lognormal, gamma, weibull, pearson, generalized
extreme value were fitted to the data for evaluating the best fit probability distribution for weather
parameters. In addition, the different forms of these distributions were also tried and thus total 16
probability distributions viz. normal, lognormal (2P, 3P), gamma (2P, 3P), generalized gamma
(3P, 4P), log-gamma, weibull (2P, 3P), pearson 5 (2P, 3P), pearson 6 (3P, 4P), log-pearson 3,
generalized extreme value were applied to find out the best fit probability distribution The
description of various probability distribution functions viz. density function, range and the
parameter involved are presented in table 3.8.
3.3.2 Testing the goodness of fit
The goodness of fit test measures the compatibility of random sample with the theoretical
probability distribution. The goodness of fit tests is applied for testing the following null
hypothesis:
HO: the weather parameter data follow the specified distribution
HA: the weather parameter data does not follow the specified distribution.
Table 3.8. Description of various probability distribution functions.
Distribution
Gamma
(3P)
Gamma
(2P)
Probability density function
 1
x  

x   
f ( x) 
exp 


  ( )

 1
f ( x) 
 x
   ( )
Range



  x  
   x 
 


exp 
Generalized
Extreme
Value
1 
11 k
1 
k
k0
 exp  1  kz   1  kz 
f ( x)  
 1 exp   z  exp   z 
k 0

Generalized
Gamma
(4P)
k 1
 x   
k x   
f ( x) 
exp   
k

 
 ( )
Generalized
Gamma
(3P)
LogGamma

k 1

kx
f ( x) 
exp  
k
 ( )

 
x



1 k
x    0

  x  
for k  0
for k  0
k



  x  


 1  In( x   )   2 

 

 
 2 
exp 
f ( x) 
 x   
Lognormal
(2P)
0  x  
  x  
2
 1  In( x )   2 

 

 
 2 
 x
( k  0)
  shape parameter
(  0)
  scale parameter
(   0)
  shape parameter
  scale parameter
(  0)
(  0)
  scale parameter
(  0)
  shapeparameter
(  0)
  location parameter (  0
yields thetwo parameter
lognormaldistribution)
2
 1
 In( x)   
f ( x) 


x  ( )   
1
k  shape parameter
Generalized gamma distribution )
exp 
f ( x) 
  scale parameter (  0)
k  shape parameter
  location parameter
x
where z 

yields the three parameter
 1
 In  x  
exp  

 
x   ( )

  shape parameter
(  0)
  scale parameter
(   0)
  location parameter (  0
yieldsthetwo parameter
gamma distribution )
  Gamma function
  location parameter (  0
k
 In  x 
f ( x) 
Lognormal
(3P)
LogPearson 3
Parameters
0  x  e
  0   shape
 In( x)    
exp  
 e  x     0
 

parameter (  0)
  scale parameter (   0)
  location parameter
Table 3.8. Continued
Distribution
Probability density function
Normal
 1  x   2 
f ( x) 
exp  
 
 2
 2    
Pearson 5
(3P)
Pearson 5
(2P)
Pearson 6
(4P)
Pearson 6
(3P)
Range
  mean
 x     standard Deviation
(  0)
1
f ( x) 

 x   
 1
  ( )  ( x   )  
exp  
f ( x) 
exp   

x
 1
  ( )  x  
  shape parameter (  0)
  scale parameter (   0)
 x  
  location parameter (  0
yields thetwo parameter
pearson 5 distribution)
 1
f ( x) 
 x      1
 
 B(1,  2 ) 1  ( x   )   1 2
  x  
 1
x   1
f ( x) 
 
 B (1,  2 ) 1  x   1 2
Weibull (3P)
P( x) 
 1
  x  


  
Weibull (2P)
P( x) 
 1
x
 
  
  x 
exp  
  




  x  
exp    
    
Parameters


  x  
1  shape parameter (1  0)
 2  shape parameter ( 2  0)
  scale parameter
(   0)
  location parameter (  0
yields thethree parameter
pearson 6 distribution)
  shape parameter (  0)
  scale parameter (   0)
  location parameter (  0
yieldsthetwo parameter
weibull distribution)
The following goodness-of-fit tests viz. Kolmogorov-Smirnov test and Anderson-Darling
test were used along with the chi-square test at  (0.01) level of significance for the selection of
the best fit probability distribution. The distribution function of these tests is explained in brief in
the next section.
Kolmogorov-Smirnov Test
The Kolmogorov-Smirnov statistic (D) is defined as the largest vertical difference between
the theoretical and the empirical cumulative distribution function (ECDF):

D  max  F (x i ) 
1in 
i 1
n
,
i
n


 F (x i ) 
Where,
Xi = random sample, i =1, 2,….., n.
(1)
CDF= Fn (x)=
1

. Number of observations  x
n

(2)
This test is used to decide if a sample comes from a hypothesized continuous distribution.
Anderson-Darling Test
The Anderson-Darling statistic (A2) is defined as
1 n
2
A =-n (2i - 1).[In F(X i ) + In(1 - F(X n-i+1 ))]
n i=1
(3)
It is a test to compare the fit of an observed cumulative distribution function to an expected
cumulative distribution function. This test gives more weight to the tails then the KolmogorovSmirnov test.
Chi-Squared Test
The Chi-Squared statistic is defined as
χ
2
Oi - Ei 
=
2
k
i=1
(4)
Ei
Where,
Oi = observed frequency,
Ei = expected frequency,
‘i’= number of observations (1, 2, …….k)
Ei is calculated by the following computation
Ei =F(x 2 ) - F(x1 )
(5)
F is the CDF of the probability distribution being tested.
The observed number of observation (k) in interval ‘i’ is computed from equation given below
k  1  log 2 n
(6)
Where, n is the sample size.
This test is for continuous sample data only and is used to determine if a sample comes from a
population with a specific distribution.
3.3.3 Identification of best fit probability distribution
The three goodness of fit test mentioned above were fitted to the weather parameters data
treating different data set. The test statistic of each test were computed and tested at (  =0.01)
level of significance. Accordingly the ranking of different probability distributions were marked
from 1 to 16 based on minimum test statistic value. The distribution holding the first rank was
selected for all the three tests independently. The assessments of all the probability distribution
were made on the bases of total test score obtained by combining the entire three tests. Maximum
score 16 was awarded to rank first probability distribution based on the test statistic and further
less score were awarded to the distribution having rank more than 1, that is 2 to 16. Thus the total
score of the entire three tests were summarized to identify the fourth distribution on the bases of
highest score obtained.
The probability distribution having the maximum score was included as a fourth
probability distribution in addition to three probability distributions which were previously
identified. Thus on the bases of the four identified probability distribution the procedure for
obtaining the best fitted probability distribution is explained below:
Generating random numbers
The four probability distributions identified for each data set were used to select the best
probability distribution. The parameters of these four probability distributions were used to
generate the random numbers.
Least square method
The least square method was used to identify the best fit probability. The random numbers
were generated for the distributions and residuals (R) were computed for each observation of the
data set.



R    Yi  Yi 
i=1 

n
(7)
Where, Yi = the actual observation

Yi = the estimated observation
( i = 1, 2,….., n )
The distribution having minimum sum of residuals was considered to be the best fit
probability distribution for that particular data set. Finally the best fit probability distributions for
weather parameters on different sets of data were obtained and the best fit distribution for each set
of data was identified. The convergence and performance of the best fit probability distribution
had been evaluated on the basis of the Easy fit 5.5 version of software.
3.4
Probability Distribution
3.4.1 Introduction
The methodology presented above is applied to the 50 years weather data as classified into 18
data sets. These 18 data sets are classified as 1 seasonal and 17 weekly to study the distribution pattern
at different levels. The test statistic D, A2 and  2 for each data set are computed and the
combination of total test score are obtained for each data set for all probability distributions.
The distribution is identified using maximum overall score based on sum of individual point
score obtained from three selected goodness of fit test. The distributions identified which are having
highest score and the best fit are listed in counter form where the parameter of these identified
distribution for each data set of weather parameters are also mentioned.
These values of the parameter are used to generate random numbers for each data set and
the least square method is used for the weather parameters analysis. The residuals are computed
for each data set of weather parameters. Sum of the deviation are obtained for all identified
distribution. The probability distribution having minimum deviation is treated as the best selected
probability distribution for the individual data set. The discussions on the results for the individual
parameters are mentioned below.
3.4.2 Rainfall
The test statistic D, A2 and  2 for each data set, of rainfall is computed for 16 probability
distribution. The probability distribution having the first rank along with their test statistic is
presented in table 3.9(a). It is observed that for seasonal rainfall, Pearson 5 (3P) distribution is
fitted using Kolmogorov Smirnov and Anderson Darling tests based on first rank. Similarly,
Normal distribution is fitted using Chi-square test for seasonal rainfall. Thus these probability
distributions are identified as the best fit based on these three tests independently.
The fourth probability distribution identified, which is having highest score, is presented in
table 3.9(b) with their scores. Those distributions which are having same highest score are also
included in the selected probability distribution. The probability distributions fitted for the fourth
week data set, that is, last week of June are Generalized Extreme Value distribution and
Lognormal (3P) distribution based on highest score. While for ninth week data set, that is, in first
week of August, Weibull (2P, 3P) distributions are fitted having 36 as highest score and for the
thirteenth week, that is, in last week of August, Gamma (2P) distribution and Generalized Extreme
Value distribution, having 38 as the highest score are selected.
It is observed that Generalized Extreme Value probability distribution is fitted in more than
50% weeks. The distributions identified are thus listed in table 3.9(c) where the parameter of these
identified distribution for each data set is mentioned.
These values of the parameter are used to generate random numbers for each data set and
the least square method is used for selecting the best fit probability distribution. The probability
distribution having minimum deviation is treated as the best selected probability distribution for
the individual data set for the rainfall as presented in \table 3.9(d).
Normal distribution represents the best fitted distribution for seasonal rainfall and is also
observed in the sixth week data set, that is, second week of July. Generalized Extreme Value is
observed six times in the weekly data sets, means, first, second, tenth, fifteenth, twelfth and thirteenth
weeks, that is, first week of June, second week of June, August and September, and last two weeks of
august, respectively, indicating the highest contribution of the distribution.
Further, we observe that Gamma (3P) distribution, Log-Pearson 3 distribution, Pearson 6
(3P) distribution and Lognormal (3P) distribution are found as the best fitted probability
distributions for the weekly rainfall data sets.
Table 3.9(a). Distributions fitted for Rainfall data sets.
Study
period
Kolmogorov Smirnov
Test ranking first position
Anderson Darling
Chi-square
Distribution
Statistic
Distribution
Statistic
Distribution
Statistic
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Pearson 5 (3P)
0.0681
Pearson 5 (3P)
0.2976
Normal
1.3956
Gen. Extreme
0.1762
Gen. Extreme
2.1837
Gen. Extreme
8.3819
Gen. Extreme
Gen. Extreme
0.1371
0.0717
Gen. Extreme
Gen. Extreme
1.1632
0.3888
Gen. Extreme
Pearson 6 (3P)
2.9090
2.5659
Gen. Extreme
Pearson 6 (3P)
0.0735
0.0912
Gen. Extreme
Pearson 6 (3P)
0.3163
0.5049
Lognormal (3P)
Gen. Gamma (4P)
0.8179
0.4317
Gen. Gamma (4P)
Gen. Extreme
0.0777
0.0931
Gen. Extreme
Gen. Extreme
0.4326
0.4959
Normal
Gen. Gamma (3P)
2.5651
0.7131
Lognormal (3P)
Weibull (2P)
0.0702
0.0719
Lognormal (3P)
Log-Pearson 3
0.3222
0.2836
Lognormal (3P)
Lognormal (3P)
1.0616
0.9351
Lognormal (3P)
Gamma (2P)
0.0584
0.0689
Gen. Extreme
Log- Pearson 3
0.1731
0.2110
Gen. Extreme
Log-Pearson 3
0.9746
0.0377
Gen. Extreme
Gamma (2P)
0.0792
0.1048
Gen. Extreme
Gen. Extreme
0.4724
0.8361
Gen. Extreme
Gamma (2P)
1.9015
2.1996
Gen. Extreme
0.0966
Gen. Extreme
0.6122
Gen. Extreme
1.7454
Weibull (3P)
Gamma (3P)
0.0852
0.1337
Pearson 6 (3P)
Pearson 6 (3P)
0.5338
1.1343
Lognormal (3P)
Pearson 6 (3P)
1.0211
2.4038
Gamma (3P)
0.1596
Pearson 6 (3P)
2.4205
Gen. Extreme
3.9689
Table 3.9(b). Distributions with highest score for Rainfall data sets.
Study Period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Distributions with highest Score
Distribution
Score
Pearson 5 (3P)
42
Gen. Extreme
42
Gen. Extreme
43
Gen. Extreme
39
Gen. Extreme and Lognormal (3P)
39
Gamma (2P)
34
Gen. Extreme
38
Gen. Extreme
41
Lognormal (3P)
42
Weibull (2P) and Weibull (3P)
36
Gen. Extreme
41
Gamma (2P)
43
Gen. Extreme
39
Gamma (2P) and Gen. Extreme
38
Gen. Extreme
40
Pearson 6 (3P)
38
Pearson 6 (3P)
39
Gen. Extreme
35
Table 3.9(c). Parameters of the distributions fitted for Rainfall data sets.
Study Period
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
Distributions
Normal
Pearson 5(3P)
Gen. Extreme Value
Gen. Extreme Value
Gen. Extreme Value
Pearson 6 (3P)
Gen. Extreme Value
Lognormal (3P)
Gamma (2P)
Gen. Gamma (4P)
Pearson 6 (3P)
Gen. Extreme Value
Gen. Gamma (4P)
Normal
Gen. Extreme Value
Gen. Gamma (3P)
Lognormal (3P)
Log-Pearson 3
Lognormal (3P)
Weibull (2P)
Weibull (3P)
Gen. Extreme Value
Lognormal (3P)
Gamma (2P)
Log-Pearson 3
Gen. Extreme Value
Gamma (2P)
Gen. Extreme Value
Gen. Extreme Value
Lognormal (3P)
Pearson 6 (3P)
Weibull (3P)
Gamma (3P)
Pearson 6 (3P)
Gamma (3P)
Gen. Extreme Value
Pearson 6 (3P)
Parameters
=29.44 =73.195
=112.06 =33906.0 =--232.11
k=0.57245 =12.389 =5.8622
k=0.40315 =22.496 =13.506
k=0.31177 =24.99 =20.906
1=0.57131 2=2.8403E+8 =2.2429E+10
k=0.16122 =33.769 =31.333
=0.80884 =3.9156 =-10.454
=0.75596 =99.791
k=1.5013 =0.31028 =197.88 =0.01
1=0.63702 2=4.6454E+7 =5.5026E+9
k=0.10111 =62.632 =59.431
k=3.1445 =0.19166 =270.22 =0.01
=88.365 =102.5
k=0.18365 =47.572 =58.857
k=0.98854 =1.4834 =64.094
=0.79731 =4.5807 =-21.17
=2.7764 =-0.64072 =5.6879
=0.73597 =4.2048 =-11.153
=1.0891 =79.975
=1.2235 =79.444 =0.33401
k=0.21148 =50.642 =51.512
=0.73608 =4.4708 =-19.338
=1.2248 =84.969
=12.356 =-0.30187 =7.9246
k=0.1263 =58.585 =57.435
=0.84718 =89.184
k=0.24491 =45.313 =35.09
k=0.2002 =49.471 =38.627
=1.4308 =3.3643 =-1.496
1=0.53587 2=2.4525E+5 =2.7661E+7
=0.64552 =61.049 =0.01
=0.23816 =209.68 =0.01
1=0.27517 2=7.8369E+7 =1.7728E+10
=0.18131 =139.08 =0.01
k=0.56455 =16.239 =6.8411
1=0.21422 2=3.4782E+7 =5.9395E+9
Table 3.9(d). Best fit probability distribution for Rainfall.
STUDY PERIOD
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
BEST-FIT
Normal
Gen. Extreme Value
Gen. Extreme Value
Pearson 6 (3P)
Lognormal (3P)
Pearson 6 (3P)
Normal
Gen. Gamma (3P)
Lognormal (3P)
Log-Pearson 3
Gen. Extreme Value
Log-Pearson 3
Gen. Extreme Value
Gen. Extreme Value
Gen. Extreme Value
Pearson 6 (3P)
Pearson 6 (3P)
Gamma (3P)
3.4.3 Maximum Temperature
The test statistic D, A2 and  2 for each data set, of maximum temperature is computed for
16 probability distribution. The probability distribution having the first rank along with their test
statistic is presented in table 3.10(a). It is observed that for seasonal maximum temperature,
Weibull (2P) distribution is fitted using Kolmogorov Smirnov and Chi-square tests based on first
rank. Similarly, Log Pearson 3 is fitted using Anderson Darling test for seasonal maximum
temperature. Thus these probability distributions are identified as the best fit based on these three
tests independently.
The fourth probability distribution identified, which is having highest score, is presented in
table 3.10(b) with their scores. The Log-Pearson 3 distribution is fitted having the highest score as
43 for the seasonal maximum temperature data set and also for third and seventh week with the
score of 44 and 47 respectively. Pearson 5 (3P) is observed consecutively in the fifth and sixth
week with the score 40 and 44, respectively, and is also in the eleventh week with a score of 37.
Similarly, Weibull (3P) is observed successively in the last two weeks and also in the fourteenth
week, that is, first week of September. The distributions identified are thus listed in table 3.10(c),
where, the parameter of these identified distribution for each data set are mentioned.
To generate random numbers for each data set these values of the parameter are used and
the least square method is used for selecting the best fit probability distribution. The probability
distribution having minimum deviation is treated as the best selected probability distribution for
the individual data set for the maximum temperature as presented in table 3.10(d).
Table 3.10(a). Distributions fitted for Maximum Temperature data sets.
Study
period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Kolmogorov Smirnov
Test ranking first position
Anderson Darling
Chi-square
Distribution
Statistic
Distribution
Statistic
Distribution
Statistic
Weibull (2P)
0.0746
Log Pearson 3
0.4322
Weibull (2P)
2.2681
Gen. Extreme
0.0661
Gen. Extreme
0.1641
Pearson 6 (4P)
0.3980
Pearson 6 (4P)
Pearson 5 (3P)
0.0647
0.0785
Pearson 6 (4P)
Log Pearson 3
0.3325
0.2532
Pearson 5 (2P)
Log Pearson 3
0.2498
1.4033
Gen. Extreme
Pearson 5 ( 3P)
0.1139
0.0710
Log Normal (3P)
Pearson 5 (3P)
1.1606
0.1673
Log Normal (3P)
Weibull (3P)
4.2667
0.3935
Gen. Extreme
Weibull (3P)
0.0500
0.0611
Pearson 5 (3P)
Log Pearson 3
0.1330
0.1767
Log Pearson 3
Log Pearson 3
0.7313
1.0778
Pearson 5 (3P)
Pearson 6 (3P)
0.0667
0.0871
Gen. Extreme
Gen. Gamma (3P)
0.2656
0.2481
Gen. Extreme
Weibull (2P)
0.3337
1.7466
Weibull (3P)
Pearson 5 (3P)
0.0775
0.0754
Gen. Extreme
Log Normal (3P)
0.3098
0.4681
Gen. Extreme
Log Pearson 3
1.6984
0.3302
Weibull (2P)
Gen. Extreme
0.0631
0.0736
Pearson 6 (4P)
Weibull (2P)
0.3450
0.2783
Pearson 5 (3P)
Weibull (2P)
2.3160
2.7992
Gen. Extreme
0.0785
Weibull (3P)
0.3401
Weibull (3P)
3.8631
Weibull (3P)
Weibull (3P)
0.0557
0.0691
Log Pearson 3
Weibull (3P)
0.1649
0.2918
Pearson 5 (2P)
Weibull (3P)
1.3690
1.9102
Weibull (3P)
0.0642
Weibull (3P)
0.2485
Weibull (3P)
1.4818
Weibull (2P) distribution represents the best fitted probability distribution for seasonal
maximum temperature and is also observed in the ninth and twelfth week data set, that is, first and
fourth week of August, respectively. Further, Log-Pearson 3 is observed consecutively in the sixth
and seventh week data set, that is, second and third week of July, respectively.
Similarly, Weibull (3P) is observed successively in the sixteenth and seventeenth week,
that is, in last two weeks of September and also in the fifth and fourteenth week, that is, in first
week of July and September, respectively. Further, we observe that Generalized Gamma (3P)
distribution, Generalized Extreme Value, Pearson 5 (2P, 3P) distribution and Lognormal (3P)
distribution are found as the best fitted probability distributions for the weekly maximum
temperature data sets.
Table 3.10(b). Distributions with highest score for Maximum Temperature data sets.
Study Period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Distributions with highest Score
Distribution
Score
Log Pearson 3
43
Gen. Extreme Value
44
Pearson 6 (4P)
45
Log Pearson 3
44
Log Normal (3P)
44
Pearson 5 (3P)
40
Pearson 5 (3P)
44
Log Pearson 3
47
Log Gamma
45
Pearson 6 (4P)
41
Gen. Extreme Value
39
Pearson 5 (3P)
37
Pearson 6 (4P)
36
Weibull (2P)
44
Weibull (3P)
45
Gen. Gamma (3P)
36
Weibull (3P)
47
Weibull (3P)
48
Table 3.10(c). Parameters of the distributions fitted for Maximum Temperature data sets.
Study Period Distributions
Parameters
Seasonal
Log-Pearson 3
=25.842 =-0.00473 =3.6207
Weibull (2P)
=50.1 =33.395
1 week
Gen. Extreme Value
k=-0.41111 =3.5854 =36.758
Pearson 6 (4P)
1=1530.1 2=2181.0 =142.91 =-62.564
2 week
Pearson 5 (2P)
=87.078 =3137.7
1=1.5633E+6 2=2.6044E+5 =289.5 =Pearson 6 (4P)
1701.3
3 week
Log-Pearson 3
=100.04 =0.00839 =2.7275
Pearson 5 (3P)
=68.215 =1634.5 =11.209
4 week
Gen. Extreme Value
k=-0.35246 =2.3597 =33.096
Lognormal (3P)
=0.02174 =4.7508 =-81.87
Table 3.10(c). Continued
Study Period
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
Distributions
Pearson 5 (3P)
Weibull (3P)
Gen. Extreme Value
Log-Pearson 3
Pearson 5 (3P)
Log-Pearson 3
Weibull (3P)
Gen. Extreme Value
Log-Gamma
Pearson 5 (3P)
Gen. Gamma (3P)
Pearson 6 (3P)
Pearson 6 (4P)
Weibull (2P)
Parameters
=69.116 =1040.8 =18.517
=2.597 =5.1432 =29.223
k=-0.05566 =1.6168 =31.989
=7.0789 =0.02172 =3.3361
=21.315 =171.78 =24.383
=23.506 =-0.00954 =3.6966
=5.0304 =6.9459 =25.872
k=-0.28636 =1.4404 =31.595
=6023.6 =5.7572E-4
=331.76 =8638.6 =5.9613
k=1.0026 =543.3 =0.0603
1=2984.7 2=668.66 =7.2093
1=3429.0 2=950.5 =10.414 =-5.3796
=28.524 =32.749
Gen. Extreme Value
k=-0.10418 =1.113 =31.808
Weibull (3P)
=2.4245 =3.2317 =29.477
Log-Pearson 3
=22.25 =-0.00908 =3.6631
Lognormal (3P)
Pearson 5 (3P)
=0.03054 =3.792 =-12.46
=319.73 =7827.0 =7.3207
Pearson 5 (3P)
=308.23 =6381.1 =11.114
Pearson 6 (4P)
Weibull (2P)
1=4004.5 2=10022.0 =157.47 =-31.041
=33.748 =32.317
Gen. Extreme Value
k=-0.53325 =1.4321 =31.809
Weibull (2P)
=29.152 =32.632
Gen. Extreme Value
k=-0.60001 =1.873 =31.454
Weibull (3P)
=20.417 =29.794 =2.7487
Gen. Gamma (3P)
k=1.0009 =528.38 =0.0605
Log-Pearson 3
Pearson 5 (2P)
Weibull (3P)
=14.722 =-0.01149 =3.6271
=521.2 =16533.0
=5.682 =7.2696 =25.058
Weibull (3P)
=1.5522E+7 =1.5144E+7 =-1.5144E+7
Weibull (3P)
=41.383 =48.366 =-15.802
Table 3.10(d). Best fit probability distribution for Maximum Temperature data sets.
STUDY PERIOD
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
BEST-FIT
Weibull (2P)
Pearson 6 (4P)
Pearson 5 (2P)
Pearson 5 (3P)
Log Normal (3P)
Weibull (3P)
Log Pearson 3
Log Pearson 3
Pearson 5 (3P)
Weibull (2P)
Gen Extreme value
Pearson 5 (3P)
Weibull (2P)
Gen. Extreme
Weibull (3P)
Gen. Gamma (3P)
Weibull (3P)
Weibull (3P)
3.4.4 Minimum Temperature
The test statistic D, A2 and  2 for each data set, of minimum temperature is computed for
16 probability distribution. The probability distribution having the first rank along with their test
statistic is presented in table 3.11(a). It is observed that for seasonal minimum temperature,
Lognormal (3P) distribution is fitted using Kolmogorov Smirnov, Weibull (3P) distribution is
fitted using Anderson Darling tests and Gamma (3P) distribution is fitted using Chi-square test
based on first rank. Thus these probability distributions are identified as the best fit based on these
three tests independently.
The fourth probability distribution identified which is having highest score, is presented in
table 3.11(b) with their scores. Those distributions which are having same highest score are also
included in the selected probability distribution.
Table 3.11(a). Distributions fitted by the tests for Minimum Temperature data sets.
Test ranking first position
Anderson Darling
Study
period
Kolmogorov Smirnov
Distribution
Statistic
Distribution
Statistic
Distribution
Statistic
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Lognormal (3P)
0.0606
Weibull (3P)
0.1507
Gamma (3P)
0.2870
Lognormal (3P)
0.0667
Lognormal (3P)
0.2740
Weibull (3P)
3.4178
Weibull (2P)
Gamma (3P)
0.0883
0.0785
Log- Pearson 3
Gen. Extreme
0.6013
0.3980
Weibull (2P)
Weibull (2P)
2.0818
1.6087
Gen. Extreme
Gen. Extreme
0.1264
0.1562
Gen. Extreme
Weibull (3P)
1.7918
1.4011
Weibull (2P)
Gamma (2P)
2.5028
4.8657
Gamma (3P)
Gen. Gamma (4P)
0.0886
0.0712
Gen. Extreme
Gen. Extreme
0.3089
0.4068
Gen. Extreme
Pearson 6 (4P)
3.6625
1.5668
Gen. Extreme
Normal
0.0576
0.0710
Gen. Extreme
Lognormal (2P)
0.2106
0.3093
Gen. Extreme
Normal
2.1897
0.7627
Gen. Extreme
Gen. Extreme
0.0751
0.1462
Normal
Weibull (3P)
0.2347
1.5743
Pearson 5 (3P)
Weibull (3P)
0.4314
5.8003
Weibull (2P)
Weibull (2P)
0.1372
0.1203
Weibull (2P)
Weibull (3P)
1.1944
0.3749
Weibull (2P)
Pearson 5 (3P)
5.3154
3.3573
Weibull (2P)
0.0887
Log-Pearson 3
0.3438
Gen. Extreme
3.2287
Weibull (3P)
Gen. Extreme
0.0896
0.0723
Weibull (3P)
Weibull (3P)
0.6710
0.3678
Gamma (2P)
Pearson 5 (2P)
1.2229
1.5721
Weibull (3P)
0.0859
Weibull (3P)
0.3871
Weibull (2P)
2.6457
Chi-square
Table 3.11(b). Distributions with highest score for Minimum Temperature data sets.
Study Period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Distributions with highest Score
Distribution
Score
Lognormal (3P)
36
Lognormal (3P) and Pearson 6 (4P)
42
Log-Pearson 3
42
Gen. Extreme and Gen. Gamma (4P)
39
Gen. Extreme
43
Weibull (3P)
34
Log-Gamma
39
Normal
38
Gen. Extreme
46
Normal
39
Gen. Extreme
35
Pearson 6 (4P)
40
Weibull (2P)
45
Weibull (3P)
42
Gen. Extreme
43
Gamma (2P)
39
Pearson 6 (4P)
33
Weibull (3P)
46
The probability distributions fitted for the first week data set, that is, first week of June are
Lognormal (3P) and Pearson 6 (4P) distributions based on the highest score as 42. While for the
third week of June, Generalized Extreme value and Generalized Gamma (4P) distribution having 39
as the highest score are selected. The distributions identified are thus listed in table 3.11(c) where
the parameter of these identified distribution for each data set are mentioned.
These values of the parameter are further utilized to generate random numbers for each
data set and the least square method is worn for selecting the best fit probability distribution. The
probability distribution having minimum deviation is treated as the best selected probability
distribution for the individual data set for the minimum temperature as presented in table 3.11(d).
Weibull (3P) distribution represents the best fitted distribution for seasonal minimum temperature
and is also observed in the fifteenth week data set, that is, second week of September.
Further, we observe that Generalized Extreme Value is observed repetitively in sixth,
seventh and eighth week, that is, in second, third and fourth week of July, respectively, also in the
fourth week, that is, last week of June. Further, it was observed, Weibull (2P) appeared four times
among the 17 weeks, that is, in second, twelfth, fourteenth and seventeenth week. Moreover,
Gamma (2P, 3P) distributions, Normal distribution, Pearson 5 (2P, 3P) distributions and Pearson 6
(4P) distribution are obtained as the best fitted probability distributions for the weekly minimum
temperature data sets.
Table 3.11(c). Parameters of the distributions fitted for Minimum Temperature data sets.
Study Period
Seasonal
1 week
2 week
3 week
Distributions
Gamma (3P)
Lognormal (3P)
Weibull (3P)
Lognormal (3P)
Pearson 6 (4P)
Weibull (3P)
Log-Pearson 3
Weibull (2P)
Gamma (3P)
Parameters
=417.41 =0.02319 =14.749
=0.02324 =2.9883 =4.5826
=9.1205 =3.7283 =20.897
=0.02953 =4.0735 =-34.706
1=2.1298E+6 2=4.1770E+5 =201.05 =-1001.1
=4.9884 =8.4329 =16.313
=91.984 =-0.00655 =3.8076
=19.808 =25.259
=8.8541 =0.56396 =20.017
Table 3.11(c). Continued
Study Period
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
Distributions
Parameters
Gen. Extreme Value
Gen. Gamma (4P)
Weibull (2P)
Gen. Extreme Value
Weibull (2P)
Gamma (2P)
Gen. Extreme Value
Weibull (3P)
Gamma (3P)
Gen. Extreme Value
Log-Gamma
Gen. Extreme Value
Gen. Gamma (4P)
Normal
Pearson 6 (4P)
Gen. Extreme Value
k=-0.04248 =1.3598 =24.28
k=0.66312 =23.481 =0.04531 =19.635
=21.248 =25.449
k=-0.09791 =1.1745 =24.703
=22.761 =25.681
=466.36 =0.05407
k=-0.43734 =1.1081 =24.927
=10.717 =11.146 =14.543
=189.35 =0.06414 =13.095
k=-0.27557 =0.88526 =24.93
=8468.0 =3.8121E-4
k=-0.39862 =0.71759 =24.841
k=1.2543 =139.59 =0.19991 =14.797
=0.68218 =25.044
1=7.5120E+5 2=1.0096E+6 =596.77 =-418.98
k=-0.16051 =0.85926 =24.791
Lognormal (2P)
Normal
Gen. Extreme Value
Normal
Pearson 5 (3P)
Gen. Extreme Value
Pearson 6 (4P)
Weibull (3P)
Weibull (2P)
=0.03148 =3.2226
=0.79908 =25.106
k=-0.34377 =0.79003 =24.759
=0.77006 =25.008
=402.16 =6301.9 =9.2897
k=-0.47302 =0.81554 =24.607
1=6.8551E+7 2=1.2744E+7 =541.25 =-2886.7
=29.428 =20.996 =4.1713
=42.758 =24.921
Pearson 5 (3P)
Weibull (2P)
Weibull (3P)
Gen. Extreme Value
Log-Pearson 3
Weibull (2P)
Gamma (2P)
Weibull (3P)
Gen. Extreme Value
Pearson 5 (2P)
Pearson 6 (4P)
Weibull (3P)
Weibull (2P)
=1117.8 =31616.0 =-3.9018
=32.235 =24.83
=8.6285E+7 =5.0049E+7 =-5.0049E+7
k=-0.43381 =0.74913 =23.815
=15.75 =-0.00742 =3.295
=40.329 =24.304
=859.04 =0.02729
=24.716 =16.039 =7.7422
k=-0.71079 =1.43 =22.358
=251.28 =5641.6
1=5.4524E+6 2=2.5203E+6 =806.8 =-1722.9
=2.2691E+8 =2.1828E+8 =-2.1828E+8
=16.24 =22.186
Weibull (3P)
=10.426 =13.7 =8.5124
Table 3.11(d). Best fit probability distribution for Minimum Temperature.
STUDY PERIOD
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
BEST-FIT
Weibull (3P)
Pearson 6 (4P)
Weibull (2P)
Gamma (3P)
Gen. Extreme Value
Gamma (2P)
Gen. Extreme Value
Gen. Extreme Value
Gen. Extreme Value
Normal
Pearson 5 (3P)
Pearson 6 (4P)
Weibull (2P)
Pearson 5 (3P)
Weibull (2P)
Weibull (3P)
Pearson 5 (2P)
Weibull (2P)
3.4.5 Relative Humidity at 7AM
The test statistic D, A2 and  2 for each data set, of average relative humidity at 7 AM is
computed for 16 probability distribution. The probability distribution having the first rank along
with their test statistic is presented in table 3.12(a). It is observed that for seasonal average relative
humidity at 7 AM, Weibull (3P) distribution is fitted using Kolmogorov Smirnov and Anderson
Darling tests and Log-Pearson 3 distribution is fitted using Chi-square test based on first rank.
Thus these probability distributions are identified as the best fit based on these three tests
independently.
The fourth probability distribution identified which is having the highest score, is presented
in table 3.12(b) with their scores. Those distributions which are having same highest score are also
included in the selected probability distribution. The probability distributions fitted for the second
week of June are Generalized Extreme value and Generalized Gamma distribution based on
highest score as 46. While for fifteenth week data set, that is, second week of September, LogPearson 3 and Normal distributions are fitted having 37 as the highest score are selected.
Table 3.12(a). Distributions fitted for Relative Humidity at 7 AM data sets.
Study
period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Test ranking first position
Anderson Darling
Kolmogorov Smirnov
Chi-square
Distribution
Weibull (3P)
Statistic
0.0644
Distribution
Weibull (3P)
Statistic
0.2282
Distribution
Log-Pearson 3
Statistic
2.4455
Gen. Gamma (2P)
0.0809
Gen. Extreme
0.4724
Pearson 5 (2P)
3.1401
Gen. Gamma (4P)
Gen. Extreme
0.0817
0.0645
Gen. Gamma (4P)
Weibull (3P)
0.3753
0.3266
Gen. Extreme
Weibull (3P)
2.5458
0.9433
Weibull (3P)
Gen. Extreme
0.1067
0.0754
Weibull (3P)
Weibull (3P)
0.4077
0.2839
Weibull (3P)
Pearson 5 (3P)
0.6927
0.5172
Weibull (3P)
Gen. Extreme
0.1003
0.1175
Gen. Extreme
Gen. Extreme
0.5463
0.5201
Log-gamma
Pearson 5 (3P)
0.9289
0.7461
Weibull (2P)
Weibull (2P)
0.1140
0.0953
Weibull (3P)
Gen. Extreme
0.3695
0.5176
Weibull (3P)
Weibull (2P)
3.3202
1.2428
Gen. Extreme
Gen. Extreme
0.0917
0.0819
Log-Pearson 3
Gen. Extreme
0.3199
0.2937
Gen. Extreme
Pearson 6 (4P)
0.7057
0.7896
Weibull (3P)
Gen. Extreme
0.0889
0.0999
Weibull (3P)
Weibull (3P)
0.3774
0.3997
Weibull (2P)
Pearson 5 (3P)
0.9981
3.9081
Weibull (3P)
0.1165
Weibull (3P)
0.6708
Pearson 5 (3P)
1.7364
Gen. Extreme
Gen. Extreme
0.0931
0.1110
Gen. Extreme
Gen. Extreme
0.5232
0.5244
Weibull (2P)
Normal
0.9164
1.6683
Log-Pearson 3
0.1056
Log-Pearson 3
0.5137
Log-Pearson 3
2.3387
Table 3.12(b). Distributions with highest score for Relative Humidity at 7 AM data sets.
Study Period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Distributions with highest Score
Distribution
Score
Weibull (3P)
42
Log-Gamma
42
Gen. Extreme and Gen. Gamma (4P)
46
Weibull (3P)
46
Weibull (3P)
47
Log-Pearson 3
35
Gen. Extreme Value
42
Gamma (3P)
36
Weibull (3P)
45
Gen. Extreme
43
Gen. Extreme
46
Normal
41
Weibull (3P)
44
Gen. Extreme
34
Weibull (3P)
42
Log-Pearson 3 and Normal
37
Normal
38
Log-Pearson 3
47
Table 3.12(c). Parameters of the distributions fitted for Relative Humidity at 7 AM data sets.
Study Period
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
Distributions
Log-Pearson 3
Weibull (3P)
Gen. Extreme
Gen. Gamma (3P)
Log-Gamma
Pearson 5 (2P)
Gen. Extreme Value
Gen. Gamma (4P)
Gen. Extreme Value
Weibull (3P)
Weibull (3P)
Gen. Extreme Value
Log-Pearson 3
Pearson5 (3P)
Weibull (3P)
Gen. Extreme Value
Log-gamma
Weibull (3P)
Gamma (3P)
Gen. Extreme Value
Pearson 5 (3P)
Weibull (2P)
Weibull (3P)
Gen. Extreme Value
Weibull (2P)
Gen. Extreme Value
Log-Pearson 3
Gen. Extreme Value
Normal
Pearson 6 (4P)
Weibull (2P)
Weibull (3P)
Gen. Extreme Value
Pearson 5 (3P)
Weibull (3P)
Pearson 5 (3P)
Weibull (3P)
Gen. Extreme Value
Log-Pearson 3
Normal
Weibull (2P)
Gen. Extreme Value
Normal
Log-Pearson 3
Parameters
=8.5299 =-0.00988 =4.5465
=16.171 =33.249 =54.544
k=-0.23796 =12.069 =62.183
k=0.99828 =29.997 =2.2137
=496.56 =0.00843
=28.444 =1835.6
k=-0.35205 =12.544 =68.866
k=6.8227 =0.27514 =46.462 =42.88
k=-0.73561 =11.967 =77.559
=6.1010E+7 =4.8501E+8 =-4.8501E+8
=25.424 =128.95 =-41.738
k=-0.49866 =5.3036 =85.431
=11.507 =-0.01676 =4.653
=300.04 =25916.0 =-0.02967
=7.4322 =31.766 =56.865
k=-0.58918 =6.5341 =86.52
=3503.9 =0.00128
=4.1714E+7 =1.9459E+8 =-1.9459E+8
=209.79 =0.27939 =31.041
k=-0.48214 =4.3697 =88.611
=376.91 =29802.0 =10.246
=18.978 =92.549
=6.1271E+7 =2.2208E+8 =-2.2208E+8
k=-0.37372 =3.3895 =89.667
=32.887 =92.033
k=-0.42925 =3.8021 =89.633
=10.99 =-0.01207 =4.6388
k=-0.37052 =3.5769 =89.887
=3.3925 =90.956
1=5.3567E+5 2=58139.0 =83.891 =-682.0
=34.646 =92.952
=17.665 =45.223 =47.786
k=-0.43112 =3.8265 =89.876
=498.21 =41655.0 =7.1435
=8.8403 =28.468 =63.942
=618.57 =51280.0 =8.5325
=9.0316 =26.325 =66.64
k=-0.37424 =3.1488 =90.672
=41.964 =-0.00503 =4.7282
=2.9724 =91.606
=35.621 =92.888
k=-0.43177 =3.3948 =89.546
=3.2459 =90.442
=1.4453 =-0.05156 =4.5549
Table 3.12(d). Best fit probability distribution for Relative Humidity at 7 AM.
STUDY PERIOD
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
BEST-FIT
Log-Pearson 3
Gen. Gamma (3P)
Gen. Gamma (4P)
Weibull (3P)
Weibull (3P)
Log-Pearson 3
Weibull (3P)
Gamma (3P)
Weibull (2P)
Gen. Extreme Value
Gen. Extreme Value
Gen. Extreme Value
Weibull (2P)
Gen. Extreme Value
Pearson 5 (3P)
Log-Pearson 3
Normal
Log-Pearson 3
The distributions identified are thus listed in table 3.12(c), where the parameter of these
identified distribution for each data set are mentioned. Random numbers are generated using the
parametric values for each data set and the least square method is worn for selecting the best fit
probability distribution. The probability distribution having minimum deviation is treated as the
best selected probability distribution for the individual data set for the average relative humidity at
7 AM as presented in table 3.12(d). Log-Pearson 3 distribution represents the best fitted
distribution for seasonal average relative humidity at 7 AM and is also observed in the fifth,
fifteenth and seventeenth week data set, that is, first week of July, second and last week of
September, respectively.
Further, we observe that Generalized Extreme Value is obtained in recurrence form in the
ninth, tenth and eleventh week, that is, in first three weeks of August and also in the thirteenth week,
that is, in last week of August. Moreover, Gamma (3P) distribution, Generalized Gamma (3P, 4P)
distributions, Normal distribution, Pearson 5 (3P) distribution, Weibull (2P, 3P) distributions are
found as the best fitted probability distributions for the weekly average relative humidity at 7 AM
data sets.
3.4.6 Relative Humidity at 2 PM
The test statistic D, A2 and  2 for each data set, of average relative humidity at 2 PM is
computed for 16 probability distribution. The probability distribution having the first rank along
with their test statistic is presented in table 3.13(a). It is observed that for seasonal average relative
humidity at 2 PM, Generalized Extreme Value is fitted using Kolmogorov Smirnov test, Weibull
(2P) distribution is fitted using Anderson Darling test and Weibull (3P) distribution is fitted using
Chi-square test based on first rank. Thus these probability distributions are identified as the best fit
based on these three tests independently.
The fourth probability distribution identified which is having highest score is presented in
table 3.13(b) with their scores. Those distributions which are having same highest score are also
included in the selected probability distribution. The Probability distributions with the highest
score as 45 for seasonal average relative humidity at 2 PM are three, that is, Generalized Extreme
Value and Weibull (2P, 3P) distributions. Moreover, the probability distributions fitted, based on
the highest score as 41 for the third week data set, that is, third week of June are Pearson 6 (4P)
and Weibull (3P) distributions.
While for fourth week data set, that is, fourth week of June, Generalized Gamma (4P) and
Lognormal (3P) distributions are fitted having 37 as the highest score. Similarly, for the sixth
week, that is, second week of July, Generalized Extreme Value and Log-Pearson 3 distributions
are having 38 as the highest score are selected. Further, for ninth week data set, that is, first week
of August, Normal and Pearson 6 (4P) distributions are selected having 40 as the highest score fit.
Also, with the highest score of 37 in the twelfth week, that is, fourth week of August, Lognormal
(2P) and Pearson 5 (2P) distributions are selected. The distributions identified are thus listed in
table 3.13(c) where the parameter of these identified distribution for each data set are mentioned.
These values of the parameter are used to generate random numbers for each data set and
the least square method is considered for selecting the best fit probability distribution. The
probability distribution having minimum deviation is treated as the best selected probability
distribution for the individual data set for the average relative humidity at 2 PM as presented in
table 3.13(d). Weibull (2P) distribution represents the best fitted distribution for seasonal average
relative humidity at 2 PM and is also observed in the tenth and fifteenth week data set, that is,
second week of August and September, respectively.
Further, we observe Weibull (3P) distribution as recurrence in the sixteenth and seventeenth
week, which are the last two weeks of September, and is also observed in the fifth and ninth week,
that is, first week of July and August, respectively. Besides, Generalized Extreme Value,
Generalized Gamma (4P) distribution, Log-Pearson 3 distribution, Normal distribution, Pearson 5
(2P) distribution, Pearson 6 (4P) distributions are found as the best fitted probability distributions for
the weekly average relative humidity at 2 PM data sets.
Table 3.13(a). Distributions fitted for Relative Humidity at 2 PM data sets.
Test ranking first position
Anderson Darling
Study
period
Kolmogorov Smirnov
Distribution
Statistic
Distribution
Statistic
Distribution
Statistic
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Gen. Extreme
0.0712
Weibull (2P)
0.3734
Weibull (3P)
2.0229
Gen. Extreme
0.0875
Gen. Extreme
0.3977
Weibull (2P)
0.5576
Log-Pearson 3
Weibull (3P)
0.0758
0.0570
Log-Pearson 3
Weibull (3P)
0.3080
0.1785
Lognormal (2P)
Gamma (3P)
1.2683
0.8789
Gen. Extreme
Log-Pearson 3
0.0995
0.0779
Weibull (3P)
Weibull (3P)
0.3831
0.2791
Gamma (2P)
Log-Pearson 3
1.2667
2.4067
Gen. Extreme
Gen. Extreme
0.0796
0.0881
Gen. Extreme
Gen. Extreme
0.2436
0.3657
Pearson 6 (4P)
Gen. Extreme
0.6941
0.5427
Normal
Weibull (3P)
0.0749
0.1113
Gen. Extreme
Weibull (3P)
0.2524
0.3488
Weibull (2P)
Pearson 6 (4P)
1.0323
3.6623
Weibull (2P)
Log-Pearson 3
0.1026
0.0672
Normal
Weibull (3P)
0.4765
0.2489
Weibull (2P)
Gen. Extreme
2.2518
2.5681
Pearson 5 (2P)
Gamma (3P)
0.0826
0.0915
Lognormal (2P)
Gen. Extreme
0.3529
0.3627
Normal
Gen. Extreme
1.5444
4.0812
Gen. Gamma (4P)
0.0692
Normal
0.2625
Weibull (3P)
0.6171
Weibull (2P)
Weibull (3P)
0.0842
0.0465
Gen. Extreme
Weibull (3P)
0.2904
0.1661
Gen. Gamma (4P)
Gamma (3P)
0.7798
1.0619
Weibull (3P)
0.0632
Weibull (3P)
0.1706
Weibull (3P)
1.2512
Chi-square
Table 3.13(b). Distributions with highest score for Relative Humidity at 2 PM data sets.
Study Period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Distributions with highest Score
Distribution
Score
Gen. Extreme and Weibull (2P, 3P)
45
Gen. Extreme
46
Gen. Extreme
41
Pearson 6 (4P) and Weibull (3P)
41
Gen. Gamma (4P) and Lognormal (3P)
37
Log-Pearson 3
46
Gen. Extreme Value and Log-Pearson 3
38
Gen. Extreme
48
Normal
42
Normal and Pearson 6 (4P)
40
Normal
44
Weibull (3P)
44
Lognormal (2P) and Pearson 5 (2P)
37
Gen. Extreme
41
Gen. Gamma (4P)
45
Gen. Extreme
44
Weibull (3P)
40
Weibull (3P)
48
Table 3.13(c). Parameters of the distributions fitted for Relative Humidity at 2 PM data sets.
Study Period
Seasonal
1 week
2 week
3 week
4 week
5 week
Distributions
Gen. Extreme Value
Weibull (2P)
Weibull (3P)
Gen. Extreme Value
Weibull (2P)
Gen. Extreme Value
Log-Pearson 3
Lognormal (2P)
Gamma (3P)
Pearson 6 (4P)
Weibull (3P)
Gamma (2P)
Gen. Extreme Value
Gen. Gamma (4P)
Lognormal (3P)
Weibull (3P)
Log-Pearson 3
Weibull (3P)
Parameters
k=-0.52018 =3.8843 =64.855
=21.659 =67.161
=8.9724 =27.338 =39.844
k=-0.14233 =14.596 =33.374
=2.8071 =44.234
k=-0.21603 =16.209 =40.701
=9.6018 =-0.12479 =4.984
=0.3828 =3.7858
=240.94 =0.92958 =-168.36
1=10915.0 2=5529.0 =426.83 =-786.39
=26.877 =302.82 =-240.97
=37.54 =1.6858
k=-0.51157 =10.857 =60.881
k=3.107 =191.32 =80.612 =-373.76
=0.02348 =6.0912 =-378.52
=24.603 =206.72 =-138.95
=3.9062 =-0.08697 =4.5031
=6.1205 =58.567 =10.777
Table 3.13(c). Continued
Study Period
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
Distributions
Gen. Extreme Value
Log-Pearson 3
Pearson 6 (4P)
Gen. Extreme Value
Gen. Extreme Value
Normal
Weibull (2P)
Parameters
k=-0.43316 =9.59 =68.177
=8.1847 =-0.04591 =4.626
1=22.745 2=4.5724E+7 =9.2934E+7 =24.56
k=-0.15802 =6.3781 =69.278
k=-0.32505 =7.5835 =70.865
=7.3628 =73.338
=11.591 =76.209
Normal
=6.4266 =73.128
Pearson 6 (4P)
Weibull (3P)
1=2.4128E+6 2=3.6815E+5 =550.43 =-3534.3
Normal
=7.1024 =72.632
Weibull (2P)
=12.408 =75.186
Gen. Extreme Value
k=-0.27237 =6.2423 =72.21
Log-Pearson 3
Weibull (3P)
=443.55 =-0.00397 =6.069
=3.0131 =18.909 =57.572
Lognormal (2P)
=0.06761 =4.2878
Normal
Pearson 5 (2P)
=5.0058 =72.972
=219.35 =15934.0
Gamma (3P)
=154.71 =0.58977 =-20.676
Gen. Extreme
k=-0.36324 =7.5515 =68.249
Gen. Gamma (4P)
k=18.772 =8.3733 =368.55 =-341.2
Normal
Weibull (3P)
=7.855 =70.258
=4.9722 =36.763 =36.513
Gen. Extreme Value
k=-0.26115 =8.0205 =66.905
Gen. Gamma (4P)
Weibull (2P)
k=2.7474 =0.73193 =22.975 =52.467
=10.378 =72.787
Gamma (3P)
=173.41 =0.61761 =-41.471
Weibull (3P)
=7.3775 =52.812 =16.088
Weibull (3P)
=6.7096 =58.176 =5.7427
=22.354 =116.68 =-40.776
Table 3.13(d). Best fit probability distribution for Relative Humidity at 2 PM.
STUDY PERIOD
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
BEST-FIT
Weibull (2P)
Gen. Extreme Value
Log-Pearson 3
Pearson 6 (4P)
Gen. Gamma (4P)
Weibull (3P)
Pearson 6 (4P)
Gen. Extreme Value
Normal
Weibull (3P)
Weibull (2P)
Log-Pearson 3
Pearson 5 (2P)
Gen. Extreme Value
Normal
Weibull (2P)
Weibull (3P)
Weibull (3P)
3.4.7 Pan Evaporation
The test statistic D, A2 and  2 for each data set, of average Pan Evaporation is computed
for 16 probability distribution. The probability distribution having the first rank along with their
test statistic is presented in table 3.14(a). It is observed that for seasonal average pan evaporation,
Log-Pearson 3 is fitted using Kolmogorov Smirnov test, Generalized Extreme Value is fitted using
Anderson Darling test and Gamma (3P) distribution is fitted using Chi-square test based on first
rank. Thus these probability distributions are identified as the best fit based on these three tests
independently.
The fourth probability distribution identified, which is having the highest score, is presented
in table 3.14(b) with their scores. Those distributions which are having same highest score are also
included in the selected probability distribution. The probability distributions selected, based on
the highest fit score as 35 for the sixth week data set, that is, second week of July are Pearson 6
(3P) and Generalized Gamma (3P) distributions.
Table 3.14(a). Distributions fitted for Pan Evaporation data sets.
Test ranking first position
Anderson Darling
Study
period
Kolmogorov Smirnov
Distribution
Statistic
Distribution
Statistic
Distribution
Statistic
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Log-Pearson 3
0.0537
Gen. Extreme
0.1404
Gamma (3P)
1.0902
Log-Pearson 3
0.0596
Normal
0.1440
Log-Pearson 3
0.6899
Pearson 6 (3P)
Weibull (3P)
0.0626
0.1092
Pearson 5 (3P)
Gamma (3P)
0.1640
0.3565
Lognormal (2P)
Gen. Extreme
1.0880
0.6749
Pearson 5 (3P)
Lognormal (3P)
0.0744
0.0982
Gen. Extreme
Pearson 5 (3P)
0.3528
0.3740
Log-Gamma
Gen. Gamma (3P)
1.2723
0.5740
Pearson 6 (3P)
Weibull (2P)
0.0845
0.0985
Normal
Pearson 5 (3P)
0.4287
0.5113
Pearson 6 (3P)
Gamma (2P)
1.3534
2.3513
Normal
Gen. Extreme
0.0659
0.1014
Normal
Gen. Extreme
0.2283
0.3822
Normal
Gen. Extreme
1.4441
1.8532
Gen. Gamma (4P)
Normal
0.0842
0.0984
Gen. Extreme
Gen. Extreme
0.2938
0.5842
Weibull (3P)
Normal
0.7581
0.3715
Gen. Extreme
Normal
0.1113
0.0955
Normal
Normal
0.7725
0.4676
Gen. Gamma (4P)
Normal
2.7838
1.2692
Gen. Extreme
0.0557
Weibull (3P)
0.2112
Pearson 6 (4P)
0.5223
Weibull (3P)
Gen. Gamma (4P)
0.0870
0.0708
Weibull (3P)
Gen. Gamma (4P)
0.3080
0.1977
Gamma(2P)
Lognormal (2P)
1.7154
0.6335
Weibull (2P)
0.0911
Gen. Extreme
0.3561
Log Gamma
0.4024
Chi-square
Table 3.14(b). Distributions with highest score for Pan Evaporation data sets.
Study Period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Distributions with highest Score
Distribution
Score
Weibull (3P)
45
Normal
43
Pearson 6 (3P)
40
Gen. Extreme
41
Gen. Extreme
45
Lognormal (3P)
34
Pearson 6 (3P) and Gen. Gamma (3P)
35
Gen. Extreme
40
Normal
42
Gen. Extreme
48
Gen. Extreme
40
Gen. Extreme
45
Weibull (3P)
44
Normal
48
Weibull (3P)
45
Weibull (3P)
45
Gen. Gamma (4P)
41
Gen. Extreme and Weibull (2P)
43
Table 3.14(c). Parameters of the distributions fitted for Pan Evaporation data sets.
Study Period
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
Distributions
Gamma (3P)
Gen. Extreme Value
Log-Pearson 3
Weibull (3P)
Log-Pearson 3
Normal
Lognormal (2P)
Pearson 5 (3P)
Pearson 6 (3P)
Gamma (3P)
Gen. Extreme Value
Weibull (3P)
Gen. Extreme Value
Log-Gamma
Pearson 5 (3P)
Gen. Gamma (3P)
Lognormal (3P)
Pearson 5 (3P)
Gen. Gamma (3P)
Normal
Pearson 6 (3P)
Gamma (2P)
Gen. Extreme Value
Pearson 5 (3P)
Weibull (2P)
Normal
Gen. Extreme value
Gen. Gamma (4P)
Gen. Extreme Value
Weibull (3P)
Gen. Extreme Value
Normal
Gen. Extreme Value
Gen. Gamma (4P)
Normal
Weibull (3P)
Normal
Gen. Extreme Value
Pearson 6 (4P)
Weibull (3P)
Gamma (2P)
Weibull (3P)
Gen. Gamma (4P)
Lognormal (2P)
Gen. Extreme Value
Log-Gamma
Weibull (2P)
Parameters
=176.56 =0.06712 =-6.5449
k=-0.46176 =0.94758 =5.071
=5.3329 =-0.07625 =2.0609
=6.9709 =5.419 =0.24249
=7.448 =-0.10341 =3.0937
=2.7786 =10.592
=0.33983 =2.1431
=53.101 =1082.3 =-11.77
1=9.4905 2=3.9194E+5 =3.7171E+5
=4.3022 =1.2891 =2.6401
k=-0.01007 =2.1805 =6.9489
=1.9242 =5.4847 =3.3242
k=-0.06906 =1.5321 =5.4741
=34.648 =0.0517
=66.621 =946.37 =-8.1624
k=0.9814 =7.1148 =0.79576
=0.13189 =2.7742 =-10.276
=100.88 =2128.3 =-15.42
k=1.0123 =7.9218 =0.69854
=1.9227 =5.292
1=8.5077 2=1.7018E+8 =1.0754E+8
=7.7831 =0.62674
k=-0.14136 =1.5403 =4.1801
=49.793 =579.97 =-7.0088
=3.1558 =5.3419
=1.8146 =4.122
k=-0.41167 =1.3065 =3.7864
k=7.3788 =0.35509 =5.7372
k=-0.52272 =1.5798 =3.703
=3.5914 =4.5836
k=-0.16384 =1.3216 =3.562
=1.493 =4.138
k=-0.4002 =1.2408 =3.6924
k=4.1478 =0.48649 =4.1257 =1.1742
=1.1954 =4.042
=3.5458 =4.1785 =0.27855
=1.2559 =4.068
k=-0.40987 =1.1073 =3.6661
1=4.3526E+5 2=2.8077E+5 =273.81 =-420.49
=4.6889 =4.5552 =-0.1909
=18.396 =0.21092
=20.584 =15.172 =-10.901
k=16.334 =0.11141 =3.3346 =1.7597
=0.23367 =1.3172
k=-0.34383 =0.79331 =3.6023
=41.233 =0.03222
=5.7794 =4.117
While for the seventeenth week, that is, last week of September, Generalized Extreme
Value and Weibull (2P) distributions are having 43 as the highest fit score are selected. The
distributions identified are thus listed in table 3.14(c) where the parameter of these identified
distribution for each data set are mentioned. The least square method is utilized for selecting the
best fit probability distribution after generating random number for each data set with the help of
the parametric values obtained. The probability distribution having minimum deviation is treated
as the best selected probability distribution for the individual data set for the average pan
evaporation as presented in table 3.14(d).
Table 3.14(d). Best fit probability distribution for Pan Evaporation.
STUDY PERIOD BEST-FIT
Gamma (3P)
Seasonal
1 Week
Log-Pearson 3
2 Week
Pearson 6 (3P)
Weibull (3P)
3 Week
4 Week
Gen. Extreme Value
5 Week
Gen. Gamma (3P)
6 Week
Normal
7 Week
Pearson 5 (3P)
8 Week
Normal
9 Week
Gen. Extreme
10 Week
Gen. Gamma (4P)
11 Week
Normal
12 Week
Gen. Gamma (4P)
13 Week
Normal
14 Week
Weibull (3P)
15 Week
Gamma (2P)
16 Week
Gen. Gamma (4P)
Weibull (2P)
17 Week
Gamma (3P) distribution represents the best fitted distribution for seasonal average pan
evaporation. Further, we observe Normal distribution plays a vital role by appearing four times as
the best fit in the weekly data set in sixth, eighth, eleventh and thirteenth week, that is, in second and
last week of July and also in third and last week of August, respectively. Further, Generalized
Gamma (4P) distribution is observed thrice in the weekly data set, means, in the tenth, twelfth and
sixteenth week, that is, second and fourth week of August and third week of September,
respectively.
Besides, Gamma (2P), Generalized Extreme Value, Generalized Gamma (3P) distribution,
Log-Pearson 3 distribution, Normal distribution, Pearson 5 (3P) distribution, Pearson 6 (3P)
distribution and Weibull (2P, 3P) are obtained as the best fitted probability distributions for the
weekly average pan evaporation data sets.
3.4.8 Bright Sunshine
The test statistic D, A2 and  2 for each data set, of average bright sunshine is computed for
16 probability distribution. The probability distribution having the first rank along with their test
statistic is presented in table 3.15(a). It is observed that for seasonal average Bright Sunshine,
Generalized Extreme Value is fitted using Kolmogorov Smirnov and Anderson Darling tests and
Log-Gamma distribution is fitted using Chi-square test based on first rank. Thus these probability
distributions are identified as the best fit based on these three tests independently.
The fourth probability distribution identified which is having highest score is presented in
table 3.15(b) with their scores. Those distributions which are having same highest score are also
included in the selected probability distribution. The Probability distributions with the highest
score as 45 for seasonal average bright sunshine are Generalized Extreme Value and Log-Gamma
distributions. Moreover, the probability distributions selected, based on the highest fit score as 36
for the eleventh week data set, that is, third week of August are Weibull (3P) and Generalized
Gamma (4P) distributions. While for last week of August, Generalized Extreme Value and
Pearson 6 (4P) distributions having 36 as the highest score are selected. The distributions
identified are thus listed in table 3.15(c) where the parameter of these identified distribution for
each data set are mentioned.
Random numbers are generated using the parameter values for each data set and the least
square method is inculcated for selecting the best fit probability distribution. The probability
distribution having minimum deviation is treated as the best selected probability distribution for
the individual data set for the average bright sunshine and is presented in table 3.15(d).
Log-Gamma distribution represents the best fitted distribution for seasonal average bright
sunshine. Further, we observe Generalized Extreme Value plays an essential role by appearing six
times in the weekly data set, means, first, fifth, fourteenth, eleventh, eighth and seventeenth week,
that is, in the first week of June, July and September, third week of August, last week of July and
September, respectively.
Further, Log-Pearson 3 distribution is observed thrice in the weekly data set, in the sixth,
thirteenth and sixteenth week, that is, second week of July, last week of August and third week of
September, respectively. In addition, Gamma (2P, 3P) distributions, Generalized Gamma (3P)
distribution, Pearson 5 (2P) distribution, Pearson 6 (4P) distribution and Weibull (2P, 3P)
distributions are found as the best fitted probability distributions for the weekly average bright
sunshine data sets.
Table 3.15(a). Distributions fitted for Bright Sunshine data sets.
Test ranking first position
Study
period
Kolmogorov Smirnov
Distribution
Statistic
Distribution
Statistic
Distribution
Statistic
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Gen. Extreme
0.0709
Gen. Extreme
0.1333
Log Gamma
0.0521
Gen. Extreme
0.0619
Gen. Extreme
0.1545
Gen. Extreme
0.8832
Weibull (3P)
Weibull (3P)
0.0852
0.0769
Weibull (3P)
Weibull (3P)
0.4103
0.3180
Normal
Gamma (2P)
3.8528
1.2523
Gamma (3P)
Gen. Extreme
0.0536
0.0605
Gen. Extreme
Gen. Extreme
0.1784
0.2168
Pearson 5 (3P)
Gen. Extreme
0.9833
0.5146
Log-Pearson 3
Weibull (3P)
0.0546
0.0772
Log-Pearson 3
Weibull (3P)
0.1712
0.2748
Log-Pearson 3
Pearson 6 (4P)
0.4699
1.1762
Gen. Gamma (3P)
Gen. Gamma (4P)
0.1039
0.0755
Gen. Extreme
Gen. Extreme
0.7188
0.1798
Weibull (2P)
Pearson 5 (2P)
2.6207
0.9543
Pearson 6 (4P)
Gen. Extreme
0.0674
0.0752
Gen. Extreme
Gen. Gamma (4P)
0.3030
0.4851
Pearson 5 (2P)
Pearson 6 (3P)
0.7440
0.8082
Lognormal (2P)
0.0719
Log-Pearson 3
0.3558
Gamma (2P)
0.9435
Gen. Gamma (4P)
Gen. Extreme
Gen. Extreme
Log-Pearson 3
Gen. Extreme
0.0725
0.0628
0.0900
0.0647
0.0704
Log-Pearson 3
Gen. Extreme
Gen. Extreme
Log-Pearson 3
Gen. Extreme
0.2354
0.4214
0.3967
0.1547
0.2137
Pearson 6 (4P)
Weibull (2P)
Weibull (2P)
Gen. Extreme
Log-Pearson 3
2.1774
1.2300
0.2969
0.2675
0.1466
Anderson Darling
Chi-square
Table 3.15(b). Distributions with highest score for Bright Sunshine data sets.
Study Period
Seasonal
1 Week
2 Week
3 Week
4 Week
5 Week
6 Week
7 Week
8 Week
9 Week
10 Week
11 Week
12 Week
13 Week
14 Week
15 Week
16 Week
17 Week
Distributions with highest Score
Distribution
Score
Gen. Extreme and Log-Gamma
45
Gen. Extreme
48
Gen. Gamma (4P)
42
Weibull (3P)
42
Gen. Extreme
45
Gen. Extreme
48
Log-Pearson 3
45
Weibull (3P)
46
Gen. Gamma (3P)
44
Gen. Gamma (4P)
46
Gen. Extreme
37
Weibull (3P) and Gen. Gamma (4P)
36
Gen. Gamma (3P)
43
Gen. Extreme and Pearson 6 (4P)
36
Gen. Extreme
46
Gen. Extreme
46
Log-Pearson 3
47
Gen. Extreme
45
Table 3.15(c). Parameters of the distributions fitted for Bright Sunshine data sets.
Study Period
Seasonal
1 week
2 week
3 week
4 week
5 week
6 week
7 week
8 week
9 week
10 week
Distributions
Gen. Extreme Value
Log-Gamma
Gen. Extreme Value
Gen. Gamma (4P)
Normal
Weibull (3P)
Gamma (2P)
Weibull (3P)
Gamma (3P)
Gen. Extreme Value
Pearson 5 (3P)
Gen. Extreme Value
Log-Pearson 3
Pearson 6 (4P)
Weibull (3P)
Gen. Extreme Value
Gen. Gamma (3P)
Weibull (2P)
Gen. Extreme Value
Gen. Gamma (4P)
Pearson 5 (2P)
Gen. Extreme Value
Parameters
k=-0.19099 =0.65064 =6.104
=302.87 =0.0061
k=-0.5255 =1.9833 =8.1687
k=4.5392E+7 =3.6791 =1.4983E+8 =-1.4983E+8
=1.9683 =7.786
=42.341 =65.371 =-56.739
=10.361 =0.6874
=5.5538 =11.374 =-3.388
=99.275 =0.17517 =-11.197
k=-0.24962 =1.7275 =5.5468
=186.61 =4420.6 =-17.66
k=-0.48463 =2.103 =5.6586
=1.9754 =-0.33814 =2.3106
1=34811.0 2=84746.0 =671.58 =-270.33
=5.5963 =9.1116 =-2.8707
k=-0.10922 =1.5379 =4.5656
k=0.98179 =9.0462 =0.56163
=3.1144 =5.8867
k=-0.33192 =1.8814 =5.0503
k=7.6955 =0.18288 =6.6278 =1.865
=7.2398 =35.996
k=-0.12606 =1.8538 =4.7016
Table 3.15(c). Parameters of the distributions fitted for Bright Sunshine data sets.
Study Period
10 week
11 week
12 week
13 week
14 week
15 week
16 week
17 week
Distributions
Pearson 5 (2P)
Pearson 6 (4P)
Gen. Extreme Value
Gen. Gamma (4P)
Pearson 6 (3P)
Weibull (3P)
Gamma (2P)
Gen. Gamma (3P)
Lognormal (2P)
Log-Pearson 3
Gen. Extreme Value
Gen. Gamma (4P)
Log-Pearson 3
Pearson 6 (4P)
Gen. Extreme Value
Weibull (2P)
Gen. Extreme Value
Weibull (2P)
Gen. Extreme Value
Log-Pearson 3
Gen. Extreme Value
Log-Pearson 3
Parameters
=4.645 =21.397
1=71.073 2=88.787 =15.705 =-7.1531
k=-0.51525 =1.8675 =4.9998
k=9.3109 =0.41883 =9.1555 =-1.8893
1=7.1213 2=1.1589E+8 =8.7428E+7
=12.653 =18.439 =-12.289
=8.5054 =0.64853
k=0.98578 =8.2386 =0.64853
=0.37365 =1.6434
=8.0178 =-0.1333 =2.7121
k=-0.27324 =2.0501 =5.2841
k=6.1523 =0.28286 =7.5382 =1.3255
=4.8308 =-0.17463 =2.5739
1=9730.1 2=13924.0 =216.92 =-145.57
k=-0.34374 =2.3213 =5.2994
=2.7832 =6.7243
k=-0.35206 =2.1715 =5.8174
=3.3259 =7.1501
k=-0.45339 =2.063 =6.7856
=2.1593 =-0.20805 =2.3975
k=-0.74188 =2.0831 =7.7856
=1.7031 =-0.21526 =2.4145
Table 3.15(d). Best fit probability distribution for Bright Sunshine.
STUDY PERIOD BEST-FIT
Seasonal
Log-Gamma
1 Week
Gen. Extreme Value
2 Week
Weibull (3P)
Gamma (2P)
3 Week
4 Week
Gamma (3P)
5 Week
Gen. Extreme Value
6 Week
Log-Pearson 3
7 Week
Weibull (3P)
8 Week
Gen. Extreme
9 Week
Pearson 5 (2P)
10 Week
Pearson 6 (4P)
11 Week
Gen. Extreme
12 Week
Gen. Gamma (3P)
13 Week
Log-Pearson 3
14 Week
Gen. Extreme
15 Week
Weibull (2P)
16 Week
Log-Pearson 3
Gen. Extreme Value
17 Week
3.5
Conclusion
In this chapter before identifying the best fit probability distribution, the descriptive
statistics are computed for each weather parameters for different study period. The result of
weather parameters analysis for identifying the best fit probability distribution revealed that the
distribution pattern for different data set can be identified out of a large number of commonly used
probability distributions by using different goodness of fit tests. After observing the weekly
weather parameters independently we can conclude them as follows:
Rainfall
The data represented that the maximum value of weekly rainfall is 443.20 mm in fourth
week of August in year 2000. Normal distribution represents the best fitted probability distribution
for seasonal rainfall and is also observed in the second week of July. Moreover, Generalized
Extreme Value is observed six times in the weekly data, in the first week of June, second week of
June, August and September, and also, in the last two weeks of August, indicating the highest
contribution of the distribution. In addition, Gamma (3P) distribution, Log-Pearson 3 distribution,
Pearson 6 (3P) distribution and Lognormal (3P) distribution are pragmatic as the best fitted
probability distributions for the weekly rainfall data sets.
Maximum Temperature
The data offered that the seasonal maximum temperature ranged between 23.6 OC in the
year 2005 to 43.2 OC in the years 1966 and 1967. Weibull (2P) distribution represents the best
fitted distribution for seasonal maximum temperature and is also observed in the ninth and twelfth
week, that is, first and fourth week of August. Further, Log-Pearson 3 is observed consecutively in
the second and third week of July. Similarly, Weibull (3P) is observed successively in the last two
weeks of September and also in the first week of July and September. Moreover, we observe that
Generalized Gamma (3P) distribution, Generalized Extreme Value, Pearson 5 (2P, 3P) distributions
and Lognormal (3P) distribution as the best fitted probability distributions for the weekly maximum
temperature data sets.
Minimum Temperature
The seasonal minimum temperature ranged between 17.2 OC in the year 1984 to 29.2 OC in
the year 1995. Weibull (3P) distribution represents the best fitted distribution for seasonal
minimum temperature and is also observed in the second week of September. Further, we observe
that Generalized Extreme Value is obtained repetitively in three weeks, that is, second, third and
fourth week of July, also in the last week of June. Also, Weibull (2P) appeared four times among
the 17 weeks, that is, second week of June, first week of September and fourth week of August
and September. Besides, Gamma (2P, 3P) distributions, Normal distribution, Pearson 5 (2P, 3P)
distributions and Pearson 6 (4P) distribution are obtained as the best fitted probability distributions
for the weekly minimum temperature data sets.
Relative Humidity at 7AM
The data explains the seasonal average relative humidity at 7 AM ranged between 38%
(minimum) to 98% (maximum). Log-Pearson 3 distribution represents the best fitted distribution
for seasonal average relative humidity at 7 AM and is also observed in the first week of July,
second and last week of September. Likewise, we observe that Generalized Extreme Value is
obtained in recurrence form in the first three weeks of August and also in the last week of August.
Moreover, Gamma (3P) distribution, Generalized Gamma (3P, 4P) distributions, Normal
distribution, Pearson 5 (3P) distribution, Weibull (2P, 3P) distributions are found as the best fitted
probability distributions for the weekly average relative humidity at 7 AM data sets.
Relative Humidity at 2 PM
The seasonal average relative humidity at 2 PM ranged between 16 % in the years 1965 and
2005 to 92 % in the year 1988. Weibull (2P) distribution represents the best fitted distribution for
seasonal average relative humidity at 2 PM and is also observed in the tenth and fifteenth week data
set, that is, second week of August and September respectively. As well as, we observe Weibull
(3P) distribution as recurrence in last two weeks of September, and are also observed in first week
of July and August. Besides, Generalized Extreme Value, Generalized Gamma (4P) distribution,
Log-Pearson 3 distribution, Normal distribution, Pearson 5 (2P) distribution, Pearson 6 (4P)
distributions are obtained as best fitted probability distributions for the weekly average relative
humidity at 2 PM data sets.
Pan Evaporation
The data shows the seasonal average Pan Evaporation ranged between zero mm to 18.5
mm in the year 1967. Gamma (3P) distribution represents the best fitted distribution for seasonal
average pan evaporation. Additionally, we observe Normal distribution plays a vital role by
appearing four times as best fit in the weekly data set, that is, in second and last week of July and
also in third and last week of August. Moreover, Generalized Gamma (4P) distribution is observed
thrice in the weekly data set, that is, second and fourth week of August and third week of
September respectively. Besides, Gamma (2P), Generalized Extreme Value, Generalized Gamma
(3P) distribution, Log-Pearson 3 distribution, Normal distribution, Pearson 5 (3P) distribution,
Pearson 6 (3P) distribution and Weibull (2P, 3P) are obtained as the best fitted probability
distributions for the weekly average pan evaporation data sets.
Bright Sunshine
The seasonal average bright sunshine ranged between 0.70 hours in the year 2009 to 11.6
hours in the years 1986 and 2009. Log-Gamma distribution represents the best fitted distribution
for seasonal average bright sunshine. Further, we observe Generalized Extreme Value plays an
essential role by appearing six times in the weekly data set, in the first week of June, July and
September and also in third week of August as well as the last week of July and September.
Moreover, Log-Pearson 3 distribution is observed thrice in the weekly data set, that is, second
week of July, last week of August and third week of September, respectively. In addition, Gamma
(2P, 3P) distributions, Generalized Gamma (3P) distribution, Pearson 5 (2P) distribution, Pearson
6 (4P) distribution and Weibull (2P, 3P) distributions are found as the best fitted probability
distributions for the weekly average bright sunshine data sets.
The best fit probability distributions for seasonal and weekly weather parameters for
different study period are different. In general, Generalized Extreme Value,
Weibull (2P, 3P)
distributions are most commonly the best fit probability distribution for most of the weeks among
the different weather parameters.
WEATHER FORECASTING MODELS

Introduction

Correlation Analysis

Methodology for forecasting models

Development of forecasting model for weather
parameters

Comparison of prediction ability of forecasting models

Conclusion
CHAPTER 4
WEATHER FORECASTING MODELS
4.1
Introduction
The early civilizations used reoccurring astronomical and meteorological events to monitor
seasonal changes in the weather. In arid and non-arid regions, the most dominant meteorological
parameter is rainfall which reflects wet and dry period characteristics and is measured at point
locations but assumed to represent the surrounding areas. The occurrence of rainfall depends on
several other weather parameters.
This chapter describes all the seven weather parameters under study for which the weekly
data of seven parameter viz. rainfall, maximum and maximum temperature, relative humidity at
7.00 am and 2.00 pm, bright sunshine hours and pan evaporation for the four monsoon months
were recorded. The monsoon season in this region ranges between 15 to 20 weeks, thus 17 weeks
weather data from 4th June to 30th September of 50 years (1961-2010) was considered for the
present study.
4.2
Correlation Analysis
The Inter correlation coefficient between different parameters based on 50 years data set is
computed and presented in table 4.1. From table 4.1 it was observed that maximum temperature is
positively correlated with minimum temperature but highly positively correlated with pan
evaporation and bright sunshine. Relative humidity at 7 am is also highly positively correlated
with relative humidity at 2 pm.
There is negative correlation between rainfall and maximum temperature, minimum
temperature, pan evaporation and bright sunshine hours while rainfall is positively correlated with
relative humidity at 7 am and 2 pm. It can be observed that there is highest correlation between
relative humidity at 7 am and relative humidity at
and minimum temperature.
2 pm and lowest correlation among rainfall
Table 4.1. Inter correlation coefficient between weather parameters for total data set.
Maximum
Minimum
Relative Relative Pan
Temperature Temperature humidity humidity Evaporation
(7 am)
(2 pm)
Maximum temperature
1.00000
0.19218
-0.82554 -0.83234 0.77801
Minimum Temperature
0.19218
1.00000
-0.11481
0.07373 0.14586
Relative Humidity (7 am)
-0.82554
-0.11481
1.00000
0.83736 -0.77588
Relative Humidity (2 pm)
-0.83234
0.07373
0.83736
1.00000 -0.73340
Pan Evaporation
0.77801
0.14586
-0.77588
-0.73340 1.00000
Bright Sunshine
0.57776
-0.14112
-0.48374
-0.66752 0.43054
Rainfall
-0.47340
-0.00666
0.36459
0.52490 -0.28520
Parameters
4.3
Bright
Sunshine
Rainfall
0.57776 -0.47340
-0.14112 -0.00666
-0.48374 0.36459
-0.66752 0.52490
0.43054 -0.28520
1.00000 -0.53330
-0.53330 1.00000
Methodology for forecasting Models
The methodology for forecasting models viz. Multiple Linear Regression (MLR),
Autoregressive Integrated Moving Average (ARIMA) and Artificial Neural Network (ANN)
models are given in brief in the next subsections. Hybrid approach for developing weather
forecasting model and their performance evaluation criteria is also discussed in this section.
4.3.1 Multiple Linear regression model
Multiple regression analysis is to include a number of independent parameters at the same
time for predicting the significance of a dependent parameter, (Snedecor and Cochran, 1967). In
the study, the multiple linear regression equation fitted to the weekly weather parameters treating
one as independent parameter and six other as independent parameters are given below in
generalized form.
Y = β 0 + β1X1 + β 2 X 2 + ............... + β 6 X 6 + ε
(4.1)
Where:
0 = Intercept,
i = regression coefficient of ith independent parameters, ( i = 1,2,…, 6),
 = error term,
Xi = ith weather parameter.
To identify the significant parameters for predicting the dependent parameter based on the
six independent parameters, stepwise regression analysis was used. Stepwise process starts with a
simple regression model in which most extremely correlated one independent parameter was only
incorporated at first in the company of a dependent parameter. Correlation coefficient is further
examined in the practice to find an additional independent parameter that explains the major
portion of the error remaining from the initial regression model. Until the model includes all the
significant contributing parameters, the procedure keeps on repeating. The possible bias in the
stepwise regression procedure fallout from the consideration of only one parameter at a time.
4.3.2 Autoregressive Integrated Moving Average Model
The equally spaced univariate time series data, transfer function data, and intervention data
are analyzed and forecast using the Autoregressive Integrated Moving-Average (ARIMA) or
autoregressive moving-average (ARMA) model. An ARIMA model predicts a value in a response
time series as a linear combination of its own past values, past errors (also called shocks or
innovations), and current and past values of other time series.
The ARIMA approach was first popularized by Box and Jenkins (1976), and ARIMA
models are often referred to as Box-Jenkins models. ARIMA (p, d, q) models are the extension of
AR model that use three components for modeling the serial correlation in the time series data.
The first component is the autoregressive (AR) term, where there is a memory of past events and it
uses the ‘p’ lags of the time series. The second component is the integration (I) term which
accounts for stabilizing or making the data stationary, making it easier to forecast. Each
integration order corresponds to differencing the time series. I (d) means differencing the data‘d’
times. The third component is the moving average (MA) term of the forecast errors, such that the
longer the historical data, the more accurate the forecasts will be, as it learns overtime. The
MA
(q) model uses the ‘q’ lags of the forecast errors to improve the forecast.
A dependent weekly parameter time series data, Yt : 1  t  n , mathematically the pure
ARIMA model is written as:
Wt   
 B
a
 B t
(4.2)
where,
t = indexes time of weekly parameter.
Wt = the response series Y t = 1  B Yt .
d
 = the mean time of weekly parameter.
B = the backshift operator, that is, BX t  X t 1
 B 1  1B1  ............  pBp = the autoregressive operator.
 B 1  1B1  ............  qBq = the moving average operator.
a t = the independent disturbance (random error).
d = the degree of differencing.
In this ARIMA (p, d, q) modeling, the foremost step is to decide whether the time series is
stationary or non-stationary. If it is non-stationary, it is transformed into a stationary time series by
applying appropriate degree of differencing by selecting suitable value of ‘d’. The appropriate
values of p and q are chosen by examining the autocorrelation function (ACF) and partial
autocorrelation function (PACF) of the time series data set.
4.3.3 Artificial Neural Network Model
Artificial Neural Networks are massively parallel adaptive networks of simple non-linear
computing elements called neurons which are intended to abstract and model some of the
functionality of the human nervous system in an attempt to partially capture some of its
computational strengths. Artificial Neural Network (ANN) is loosely based on biological neural
systems, in that; they are made up of an interconnected system of neurons. Also, a neural network
can identify patterns adaptively between input and output data set in a somewhat analogous
fashion to the learning process. Neural networks are highly robust with respect to underlying data
distributions and no assumptions are made about relationships between parameters.
Artificial Neural Networks (ANNs) provide a methodology for solving many types of nonlinear problems that are difficult to solve by traditional techniques. In Artificial Neural Network
Software all inputs and outputs are normalized between 0 and 1. Appropriate process of
normalization and denormalization of data is needed before and after the program execution. The
best and the simplest way is to divide it by the maximum for normalization and after the program
execution the result is to be multiplied by the same amount.
There are many neural network models, but the basic structure involves a system of
layered, interconnected nodes and neurons are presented in figure 4.1. The nodes are arranged to
form an input layer, with neurons in each hidden layer connected to all neurons in neighboring
layers. The input layer supplies data to the hidden layer and does not contain activation or transfer
functions. A typical feed-forward network might use a dot-product activation function that, for
each neuron Bj ( j = 1, 2, …..,n) in the hidden layer, is computed as:
m
B j   w ijAi  w ojAo
(4.3)
i 1
with input nodes Ai ( i = 1, 2,…., m) and weights Wij between nodes Ai and neurons Bj. The bias
node (Ao) typically has a constant input of 1, with a matching weight Woj.
A similar
calculation is made for each neuron Ck ( k = 1, 2, …., o) in the output layer
(o = 1 for the
example in figure 4.1), using weights Wjk between neurons Bj and Ck ( with Wok and Bo for the
bias). Each neuron value is subsequently passed through a transfer function, which may be linear
or nonlinear (Zurada, 1992). A common choice of nonlinear transfer function is a sigmoid, of the
general form:
  u   1 eu 
1
(4.4)
Where, u = Bj (or Ck ).
Ai (i  1, 2,...., m)
Figure 4.1.
B j ( j  1, 2,....., n)
C k (k  1, 2,....., O)
An (m x n x o) artificial neural network structure, showing a multilayer perceptron.
Nonlinearities are incorporated into the network via the activation and transfer functions in
each neuron. Complexities in the data are captured through the number of neurons in the hidden
layer. In adoption of Neural Network for practical purpose, it is desired to restrict the connections
between neurons. This is done by fixing some of the weights to zero, so that, they can dropped out
from the calculations, the working principal for subsequent adjustment of weight is in accordance
with the error propagation in the network. If increasing a given weight leads to more error, we
adjust the weight downwards and if increasing a weight leads to less error, we adjust the weight
upwards. Adjustment of all the up or down continues throughout this process until the weights and
error settled down.
To avoid over fitting to the data, a neural network is usually trained on a subset of inputs
and outputs to determine weights, and subsequently validated on the remaining (quasiindependent) data to measure the accuracy of prediction.
4.3.4 Hybrid Approach
In recent times, the concept of combined model instead of single time series model is being
prepared for prediction purpose. Several researchers have used a hybrid principal component and
ANN approach to improve the accuracy of the prediction results of their long range forecasting
investigations. In the present investigation, the hybrid approach for weather forecasting is tried to
improve the accuracy of prediction. Several studies show that the techniques of combinations of
ANN with ARIMA offer a competitive edge over each of the individual model. Taskaya et al.
(2007) doughty the degrade performance of ARIMA neural network hybrids, if the relationship
between the linear and non-linear components is different from additive assumption. The
combination of MLR with ARIMA and MLR with ANN is proposed in the present study. The
hybrid of multiple linear regression with ARIMA and ANN techniques to analyze the weekly
weather parameters of all the seven parameter studied and included in the comparative study to
identify the best precise weather forecasting model.
Hybrid Model of Multiple Linear Regression and Autoregressive Integrated Moving Average
(MLR_ARIMA)
The composition of a multiple linear regression with autoregressive integrated moving
average model is proposed to develop a new hybrid model in this section. It is assumed that the
predictive performance improves by integrating two single models. For this purpose the selected
significantly contributed parameters obtained through stepwise regression analysis are used to
develop the MLR_ARIMA model and their performance is compared with all other models.
Hybrid Model of Multiple Linear Regression and Artificial Neural Network (MLR_ANN)
It has been observed in the current researches that a single model may not be sufficient to
identify all the characteristics of the time series data. The hybrid models decompose a time series
into linear and non-linear form and prove to be better approach in comparison to single model. In
this section the hybrid model of multiple linear regression with neural network approach is
proposed to yield more accurate results. Similar to previous model, the significantly contributed
parameters selected through stepwise regression analysis in multiple linear regression model are
used to develop the hybrid MLR_ANN model.
4.3.5 Performance evaluation criteria
Many analytical methods have been proposed for the evaluation and inter-comparison of
different models, which can be evaluated in terms of graphical representation and numerical
computations.
The graphical performance criteria involves: A linear scale plot of the predicted and observed
weather parameters for training and testing data sets for all the models.
The numerical performance criterion involves:
1 N 

Mean error (BIAS):
 Yi  Yi 

N i 1 

(4.5)
1 N 
Mean absolute error (MAE):  Yi  Yi
N i 1
(4.6)
Root mean square error (RMSE):
1 N

 Y i  Yi 

N i 1 

2
(4.7)

Y i - Yi
Prediction error (PE):
Yi
(4.8)
Correlation Coefficient (r): This is obtained by performing a regression between the predicted
values and the actual values and is computed by




Y

Y
Y

Y
i
i 
 i

i 
i 1 


N
r
Y  Y 
N
i 1
where,
i
i
2



Y

Y
i
i


i 1 

N
2
(4.9)
implies the average over the whole test set, N is the total number of forecast


outputs. Yi and Y i are the actual and predicted values respectively for i = 1, 2, …..,N, Yi and Y i are
the mean values of the actual and predicted values respectively.
For the best prediction, the BIAS, MAE and RMSE values should be small and PE should
be sufficiently small i.e., close to 0. But ‘r’ should be found closer to 1 (between 0 - 1) for
indicating better agreement between observed and predicted values. The recital of weather
forecasting models had been evaluated on the basis of Mat lab 7.0.1 version, students’ academic
SAS version and Microsoft Excel.
4.4
Development of forecasting model for weather parameters
4.4.1 Introduction
The most common problems, that a contemporary data analyst encounters, is pulling out of
significant conclusions about a intricate system using data from a solitary measured parameter.
The methodology presented above was applied to the 50 years monsoon weather data for the
months of June, July, August and September. The weather data was further classified into weeks
for further analysis. A total of 850 data sets (17 weeks x 50 years) of weekly parameters were
used. The most significantly contributed parameters were selected using stepwise regression
analysis based on training data set of 35 years i.e., 595 data sets and the remaining data sets (15
years) are used in testing of the developed models which are used for comparing the real and
predicted values.
4.4.2 Rainfall
The multiple linear regression model is fitted to predict the weekly rainfall as dependent
parameter taking the other weekly independent parameters as maximum temperature, minimum
temperature, relative humidity at 7 am and 2 pm, pan evaporation and bright sunshine. The most
significantly contributed parameters are selected using stepwise regression analysis based on 595
data set of 35 years and the best fit multiple regression model is given below:
Y = 391.90674 - 8.65592X1 -1.89389X 3 + 2.46673X 4 + 4.70662X5 - 8.50348X 6
(4.10)
where, maximum temperature (X1), relative humidity at 7 am (X3), relative humidity at 2
pm (X4), pan evaporation (X5), bright sunshine (X6) are observed with coefficient of determination
as 37.66% as
most contributing parameters for predicting the rainfall (Y), as dependent
parameter. The parameter minimum temperature (X2) seems to have least control over rainfall and
hence does not appear in the proposed multiple regression model. The stepwise regression
procedure for selecting the significant parameters for the rainfall parameter is mentioned in
Appendix A (a).
We next performed an ARIMA modeling of the data using all the weather parameters. The
rainfall parameter time series data set is stationary so we do not require any transformation in the
data set. Then we used the autocorrelation function (ACF) and partial autocorrelation function
(PACF) of the weekly rainfall time series (see figure 4.2) to estimate the values of ‘p’ and ‘q’ of
the ARIMA model. Note that while both the ACF and PACF have significant terms at lags 1and
17, they have maximum correlation coefficient (0.152) at lag 1. This possibly suggests that an
ARIMA of order 1 is the best fit for the rainfall weather data set. Using an iterative model building
process of identification, estimation, and diagnostic checking using all the weather parameters, we
finally selected an ARIMA (1, 0, 1) model as the most appropriate fit for the observed data.
Figure 4.2. Plots of autocorrelation and partial autocorrelation coefficients and time lags of
weakly Rainfall parameter.
Then an ANN model building process was performed using all the weather parameters. We
selected the best suited architecture of Feed Forward Neural Network Model for our weekly
rainfall data by comparing methods and changing the layer and number of neurons in each
network. This proposed model had an input environment with all the weather parameters, three
hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and 12
neurons in the third hidden layer) and one neuron in the output layer (see figure 4.3).
Figure 4.3.
Artificial neural network structure for weekly Rainfall prediction parameter
Figure 4.4. Mapping of the number of epochs obtained for desired goal for ANN model for
Rainfall parameter
We used a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a
tan sigmoid activation function in the first, second and third hidden layer and a log sigmoid
activation function in the output layer. A set of random values distributed uniformly between -1 to
+1 are used to initialize the weight of the neural network model. 515 epochs are used to train the
neural network model with 0.003 goal (see figure 4.4). The program used for training the 595 data
sets using all the weather parameters for ANN model for predicting the rainfall parameter is
mentioned in Appendix B (a).
We next performed a hybrid MLR_ARIMA modeling of the data using the five weather
parameters, significantly selected through stepwise regression earlier, for predicting the rainfall
weather parameter. We used the same ACF and PACF of the rainfall weather parameter, having
maximum correlation coefficient (0.152) at lag 1 (see figure 4.2). Suggesting that an
MLR_ARIMA (1, 0, 1) model as the most appropriate fit using the significantly selected weather
parameters data set.
Finally we performed a hybrid MLR_ANN model building process using the same, more
significant parameters as suggested by the multiple regression model. We selected the same best
suited architecture of Feed Forward Neural Network Model as obtained while developing the
ANN model for weekly rainfall data set. This proposed hybrid MLR_ANN model had different
input environment, that is, only the significant parameters selected through stepwise regression
analysis are used, but the same three hidden layers (8 neurons in the first hidden layer, 10 neurons
in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in the output
layer as considered earlier while developing the ANN model (see figure 4.5). We used the same
Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a tan sigmoid
activation function in the first, second and third hidden layer and a log sigmoid activation function
in the output layer. A new different set of random values distributed uniformly between -1 to +1
are used to initialize the weight of the MLR_ANN model. 415 epochs are used to train the neural
network model with the same 0.003 goal (see figure 4.6). The program used for training the 595
data sets using the significantly selected parameter for the hybrid MLR_ANN model of dependent
rainfall weather parameter is mentioned in Appendix C (a).
Figure 4.5.
Hybrid MLR_ANN structure for weekly Rainfall prediction parameter.
Figure 4.6. Mapping of the number of epochs obtained for desired goal for hybrid MLR_ANN
model for Rainfall parameter.
4.4.3 Maximum temperature
The multiple regression model is fitted to predict the weekly maximum temperature as
dependent parameter taking the other weekly independent parameters as rainfall, minimum
temperature, relative humidity at 7 am and 2 pm, pan evaporation and bright sunshine. The most
significantly contributed parameters are selected using stepwise regression analysis based on 595
data set of 35 years and the best fit multiple regression model is given below:
Y = 34.9866 - 0.0025X1 + 0.3343X 2 - 0.0778X3 - 0.0703X 4 + 0.2465X5 + 0.0683X6
(4.11)
where, rainfall (X1), minimum temperature (X2), relative humidity at 7 am (X3) and 2 pm
(X4), pan evaporation (X5), bright sunshine (X6) are observed as 84.32% contributing parameters
for predicting the maximum temperature (Y), as dependent parameter. The stepwise regression
procedure for selecting the significant parameters for the maximum temperature is mentioned in
Appendix A (b).
We next performed an ARIMA modeling of the data using all the weather parameters. The
maximum temperature time series data set is found to be stationary data set. Then, we use the
autocorrelation function (ACF) and partial autocorrelation function (PACF) of the weekly
maximum temperature time series (see figure 4.7) to estimate the parameters (p and q) of the
ARIMA model. Note that while both the ACF and PACF have significant terms at lags 1, 16 and
17, they have maximum correlation coefficient (0.463) at lag 1. This possibly suggests that an
ARIMA of order 1 is best fit for the maximum temperature data. Using an iterative model building
process of identification, estimation, and diagnostic checking using all the weather parameters, we
finally selected an ARIMA (1, 0, 1) model as the most appropriate fit for the observed data.
Further, we performed an ANN model building process using all the same weather
parameters. We selected the best suited architecture of Feed Forward Neural Network Model for
our weekly maximum temperature data by comparing methods and changing the layer and number
of neurons in each network. This proposed model had an input environment with all the weather
parameters, three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second
hidden layer and 12 neurons in the third hidden layer) and one neuron in the output layer (see
figure 4.8). We used a Scaled Conjugate Gradient Algorithm for training this multilayer
perceptron, a tan sigmoid activation function in all the four layers, that is, the three hidden layers
and an output layer. A set of random values distributed uniformly between -1 to +1 are used to
initialize the weight of the neural network model. 386 epochs are used to train the neural network
model with 0.00000117083 goal (see figure 4.9). The program used for training the 595 data sets
using all the weather parameter for ANN model for predicting the maximum temperature
parameter is mentioned in Appendix B (b).
Since in the stepwise regression analysis, all the parameters are selected as significant
parameters, therefore, there is no possibility of developing the hybrid models for maximum
temperature weather parameter, that is, the hybrid model will be same as ARIMA and ANN
model.
Figure 4.7. Plots of autocorrelation and partial autocorrelation coefficients and time lags of
weakly Maximum Temperature parameter.
Figure 4.8. Artificial neural network structure for weekly Maximum Temperature prediction
parameter.
Figure 4.9. Mapping of the number of epochs obtained for desired goal for ANN model for
Maximum Temperature.
4.4.4 Minimum temperature
The multiple regression model is fitted to predict the weekly minimum temperature as
dependent parameter taking the other weekly independent parameters as maximum temperature,
rainfall, relative humidity at 7 am and 2 pm, pan evaporation and bright sunshine. The most
significantly contributed parameters are selected using stepwise regression analysis based on 595
data set of 35 years and the best fit multiple regression model is given below:
Y = 2.64644 + 0.52100X1 - 0.01740X 3 + 0.09727X 4  0.09121X 6
(4.12)
where, minimum temperature (Y) as the dependent parameter having 30.13% contribution
of the significant parameters maximum temperature (X1), relative humidity at 7 am (X3), relative
humidity at 2 pm (X4) and bright sunshine (X6). The parameters rainfall (X2) and pan evaporation
(X5) seems to have least control over minimum temperature and hence does not appear in the
proposed multiple regression model.
The stepwise regression procedure for selecting the
significant parameters for the minimum temperature parameter is mentioned in Appendix A (c).
We next performed an ARIMA modeling of the data using all the weather parameters. The
minimum temperature time series data set is stationary, so no transformation of the data set is
required. We then used the autocorrelation function (ACF) and partial autocorrelation function
(PACF) of the weekly minimum temperature time series (see figure 4.10) to estimate the values of
‘p’ and ‘q’ of the ARIMA model. Note that while both the ACF and PACF have significant terms
commonly at lags 1, 15, 16 and 17, they have maximum correlation coefficient (0.435) at lag 1.
This possibly suggests that an ARIMA of order 1 is the best fit for the minimum temperature data
set. Using an iterative model building process of identification, estimation, and diagnostic
checking using all the weather parameters, we finally selected an ARIMA (1, 0, 1) model as the
most appropriate fit for the observed data. But after applying ARIMA (1, 0, 1) it is observed that
there is lag 18 in the autocorrelation plots, so we applied ARIMA (1, 0, 18) model and obtained
the appropriate fit for the observed data.
Then an ANN model building process was performed using all the weather parameters. We
selected the best suited architecture of Feed Forward Neural Network Model for our weekly
minimum temperature data by comparing methods and changing the layer and number of neurons
in each network. This proposed model had an input environment with all the weather parameters,
three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and
12 neurons in the third hidden layer) and one neuron in the output layer (see figure 4.11). We used
a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a tan sigmoid
activation function for all the hidden and output layers. A set of random values distributed
uniformly between -1 to +1 are used to initialize the weight of the neural network model. 594
epochs are used to train the neural network model with 0.000001717 goal (see figure 4.12). The
program used for training the 595 data sets using all the weather parameters for ANN model for
predicting the minimum temperature parameter is mentioned in Appendix B (c).
Figure 4.10. Plots of autocorrelation and partial autocorrelation coefficients and time lags of
weakly Minimum Temperature parameter.
Figure 4.11. Artificial neural network structure for weekly Minimum Temperature prediction
parameter.
Figure 4.12.
Mapping of the number of epochs obtained for desired goal for ANN model for
Minimum Temperature parameter.
We next performed a hybrid MLR_ARIMA modeling of the data using the four weather
parameters, significantly selected through stepwise regression earlier, for predicting the minimum
temperature weather parameter. We used the same ACF and PACF of the minimum temperature
weather parameter (see figure 4.10), we applied ARIMA (1, 0, 18) model and obtained the
appropriate fit using the significantly selected weather parameters data set as earlier applied for
ANN model.
Figure4.13.
Hybrid MLR_ANN structure for weekly Minimum Temperature prediction
parameter.
Finally we performed a hybrid MLR_ANN model building process using the same, more
significant parameters as suggested by the multiple regression model. We selected the same best
suited architecture of Feed Forward Neural Network Model as obtained while developing the
ANN model for weekly minimum temperature data set. This proposed hybrid MLR_ANN model
had different input environment, that is, only the significant parameters selected through stepwise
regression analysis are used, but the same three hidden layers (8 neurons in the first hidden layer,
10 neurons in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in
the output layer (see figure 4.13) are considered.
We used the same Scaled Conjugate Gradient Algorithm for training this multilayer
perceptron, a tan sigmoid activation function for all the hidden and output layers. A new different
set of random values distributed uniformly between -1 to +1 are used to initialize the weight of the
MLR_ANN model. 1028 epochs are used to train the neural network model with the same
0.000001717 goal (see figure 4.14). The program used for training the 595 data sets using the
significantly selected parameter for the hybrid MLR_ANN model of dependent Minimum
Temperature weather parameter is mentioned in Appendix C (b).
Figure 4.14. Mapping of the number of epochs obtained for desired goal for Hybrid MLR_ANN
model for Minimum Temperature parameter.
4.4.5. Relative Humidity at 7 AM
The multiple regression model is fitted to predict the weekly average relative humidity at 7
am as dependent parameter taking the other weekly independent parameters as maximum
temperature, minimum temperature, rainfall, relative humidity at 2 pm, pan evaporation and bright
sunshine. The most significantly contributed parameters are selected using stepwise regression
analysis based on 595 data set of 35 years and the best fit multiple regression model is given
below:
Y = 116.06296 -1.31618X1 - 0.21386X2 - 0.01111X3 + 0.33298X 4 - 0.60294X5 + 0.29476X6
(4.13)
where, relative humidity at 7 am (Y) as the dependent parameter having 79.04%
contribution of the significant parameters maximum temperature (X1), minimum temperature (X2),
rainfall (X3), relative humidity at 2 pm (X4) pan evaporation (X5) and bright sunshine (X6). The
stepwise regression procedure for selecting the significant parameter for the relative humidity at 7
am parameter is mentioned in Appendix A (d).
We next performed an ARIMA modeling of the data using all the weather parameters.
Further, seeing the relative humidity at 7 AM time series data set it is clear that the data is
stationary and, therefore, does not require any transformation. We used the autocorrelation
function (ACF) and partial autocorrelation function (PACF) of the weekly average relative
humidity at 7 AM time series (see figure 4.15) to estimate the parameters (p and q) of the ARIMA
model. Here, while both the ACF and PACF have common significant terms at lags 1, 15, 16 and
17, but autocorrelation function (ACF) has maximum correlation coefficient (0.531) at lag 17 and
partial autocorrelation function (PACF) have maximum correlation coefficient (0.513) at lag 1.
Using an iterative model building process of identification, estimation, and diagnostic checking
using all the weather parameters, we finally selected an ARIMA (1, 0, 1) model as the most
appropriate fit for the observed data. But after applying ARIMA (1, 0, 1) it is observed that there
is lag 10 in the autocorrelation plots, so we applied ARIMA (1, 0, 10) model and obtained the
appropriate fit for the observed data without lags.
Next, we performed an ANN model building process using all the same weather
parameters. We selected the best suited architecture of Feed Forward Neural Network Model for
our weekly relative humidity at 7 am by comparing methods and changing the layer and number of
neurons in each network. This proposed model had an input environment with all the parameters,
three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and
12 neurons in the third hidden layer) and one neuron in the output layer (see figure 4.16). We used
a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a tan sigmoid
activation function in the first, second and third hidden layer and a log sigmoid activation function
in the output layer.
A set of random values distributed uniformly between -1 to +1 are used to initialize the
weight of the neural network model. 747 epochs are used to train the neural network model with
0.0000166 goals (see figure 4.17). The program used for training the 595 data sets using all the
weather parameters for ANN model for predicting relative humidity at 7 AM parameter is
mentioned in Appendix B (d). Since in the stepwise regression analysis all the parameters are
selected as significant parameters, therefore, there is no possibility of developing the hybrid
models for relative humidity at 7 AM weather parameter, that is, the hybrid model will be same as
ARIMA and ANN model.
Figure 4.15.
Plots of autocorrelation and partial autocorrelation coefficients and time lags of
weakly average Relative Humidity at 7 AM parameter.
Figure 4.16.
Artificial neural network structure for weekly average Relative Humidity at 7 AM
prediction parameter.
Figure 4.17. Mapping of the number of epochs obtained for desired goal for ANN model for
Relative Humidity at 7 AM parameter.
4.4.6. Relative Humidity at 2 PM
The multiple regression model is fitted to predict the weekly average relative humidity at 2
pm as dependent parameter taking the other weekly independent parameters as maximum
temperature, minimum temperature, relative humidity at 7 am, rainfall, pan evaporation and bright
sunshine. The most significantly contributed parameters are selected using stepwise regression
analysis based on 595 data set of 35 years and the best fit multiple regression model is given
below:
Y = 46.85286 -1.85133X1 +1.74419X 2 + 0.51835X 3 + 0.02415X 4 - 0.40761X5 -1.02706X 6
(4.14)
where, relative humidity at 2 pm (Y) as the dependent parameter having 84.45%
contribution of the significant parameters maximum temperature (X1), minimum temperature (X2),
relative humidity at 7 am (X3), rainfall (X4), pan evaporation (X5) and bright sunshine (X6). The
stepwise regression procedure for selecting the significant parameters for the relative humidity at 2
pm parameter is mentioned in Appendix A (e).
We next performed an ARIMA modeling of the data using all the weather parameters. The
relative humidity at 2 pm time series data set required no transformation in the data since it is a
stationary data set. We then used the autocorrelation function (ACF) and partial autocorrelation
function (PACF) of the weekly average relative humidity at 2 pm time series (see figure 4.18) to
estimate the values of ‘p’ and ‘q’ of the ARIMA model. Note that while both the ACF and PACF
have common significant terms at lags 1, 15, 16 and 17, they have maximum correlation
coefficient (0.622) at lag 1. This possibly suggests that an ARIMA of order 1 is best fit for the
data. Using an iterative model building process of identification, estimation, and diagnostic
checking using all the weather parameters, we finally selected an ARIMA (1, 0, 0) model as the
most appropriate fit for the observed data.
Further we performed an ANN model building process using all the same weather
parameters. We selected the best suited architecture of Feed Forward Neural Network Model for
our weekly relative humidity at 2 PM by comparing methods and changing the layer and number
of neurons in each network. This proposed model had an input environment with all the weather
parameters, three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second
hidden layer and 12 neurons in the third hidden layer) and one neuron in the output layer (see
figure 4.19). We used a Scaled Conjugate Gradient Algorithm for training this multilayer
perceptron, a tan sigmoid activation function in the three hidden layer and the output layer. A set
of random values distributed uniformly between -1 to +1 are used to initialize the weight of the
neural network model. 544 epochs are used to train the neural network model with 0.00003155
goals (see figure 4.20). The program used for training the 595 data sets using all the weather
parameter for ANN model for predicting relative humidity at 2 pm parameter is mentioned in
Appendix B (e).
Figure 4.18. Plots of autocorrelation and partial autocorrelation coefficients and time lags of
weakly average Relative Humidity at 2 PM parameter.
Since in the stepwise regression analysis all the parameters are selected as significant
parameters, therefore, there is no possibility of developing the hybrid models for relative humidity
at 2 pm weather parameter, that is, the hybrid model will be same as ARIMA and ANN model.
Figure 4.19.
Artificial neural network structure for weekly average Relative Humidity at 2 PM
prediction parameter.
Figure 4.20. Mapping of the number of epochs obtained for desired goal for ANN model for
Relative Humidity at 2 PM parameter.
4.4.7. Pan Evaporation
The multiple regression model is fitted to predict the weekly average pan evaporation as
dependent parameter taking the other weekly independent parameters as maximum temperature,
minimum temperature, relative humidity at 7 am and 2 pm, rainfall, and bright sunshine. The most
significantly contributed parameters are selected using stepwise regression analysis based on 595
data set of 35 years and the best fit multiple regression model is given below:
Y = -4.10879 + 0.53040X1 - 0.07685X3 - 0.02902X4 + 0.00351X5
(4.15)
where, maximum temperature (X1), relative humidity at 7 am (X3), relative humidity at 2
pm (X4) and rainfall (X5) are observed with coefficient of determination as 69.22% as most
contributing parameters for predicting the pan evaporation (Y), as dependent parameter. The
parameter minimum temperature (X2) and bright sunshine (X6) seems to have least control over
pan evaporation and hence does not appear in the proposed multiple regression model. The
stepwise regression procedure for selecting the significant parameters for the pan evaporation
parameter is mentioned in Appendix A (f).
We next performed an ARIMA modeling of the data using all the weather parameters. The
pan evaporation time series data set is found to be stationary, so we did not require any
transformation of the data set. We then used the autocorrelation function (ACF) and partial
autocorrelation function (PACF) of the weekly average pan evaporation time series (see figure
4.21) to estimate the parameters (p and q) of the ARIMA model. Note that, both the ACF and
PACF have common significant terms at lags 1, 15, 16 and 17, but autocorrelation function (ACF)
has maximum correlation coefficient (0.605) at lag 17 and partial autocorrelation function (PACF)
have maximum correlation coefficient (0.548) at lag 1. Using an iterative model building process
of identification, estimation, and diagnostic checking, we finally selected an ARIMA (1, 0, 17)
model as the most appropriate fit for the observed data.
Further, we performed an ANN model building process using all the weather parameters.
We selected the best suited architecture of Feed Forward Neural Network Model for our weekly
pan evaporation by comparing methods and changing the layer and number of neurons in each
network. This proposed model had an input environment with all the weather parameters, three
hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and 12
neurons in the third hidden layer) and one neuron in the output layer (see figure 4.22).
Figure 4.21. Plots of autocorrelation and partial autocorrelation coefficients and time lags of
weakly average Pan Evaporation parameter.
Figure 4.22. Artificial neural network structure for weekly average Pan Evaporation prediction
parameter.
We used a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a
tan sigmoid activation function in the first, second and third hidden layer and a log sigmoid
activation function in the output layer. A set of random values distributed uniformly between -1 to
+1 are used to initialize the weight of the neural network model. 117 epochs are used to train the
neural network model with 0.00000285 goals (see figure 4.23). The program used for training the
595 data sets using all the weather parameters for ANN model for predicting pan evaporation
parameter is mentioned in Appendix B (f).
We next performed a hybrid MLR_ARIMA modeling of the data using the four weather
parameters, significantly selected through stepwise regression earlier, for predicting the pan
evaporation weather parameter. We used the same ACF and PACF having maximum correlation
coefficient (0.605) at lag 17 and (0.548) at lag 1, respectively (see figure 4.21). Suggesting that an
MLR_ARIMA (1, 0, 17) model as the most appropriate fit using the significantly selected weather
parameters data set.
Figure 4.23. Mapping of the number of epochs obtained for desired goal for ANN model for Pan
Evaporation parameter.
Finally we performed a hybrid MLR_ANN model building process using the same, more
significant parameters as suggested by the multiple regression model. We selected the same best
suited architecture of Feed Forward Neural Network Model as obtained while developing the
ANN model for weekly average pan evaporation data set. This proposed hybrid MLR_ANN
model had different input environment, that is, only the significant parameters selected through
stepwise regression analysis, but the same three hidden layers (8 neurons in the first hidden layer,
10 neurons in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in
the output layer (see figure 4.24). We used the same Scaled Conjugate Gradient Algorithm for
training this multilayer perceptron, a tan sigmoid activation function in the first, second and third
hidden layer and a log sigmoid activation function in the output layer. A new different set of
random values distributed uniformly between -1 to +1 are used to initialize the weight of the
MLR_ANN model. 76 epochs are used to train the neural network model with 0.00000285 goals
(see figure 4.25). The program used for training the 595 data sets using the significantly selected
parameter for the hybrid MLR_ANN model of dependent pan evaporation weather parameter is
mentioned in Appendix C (c).
Figure 4.24.
Hybrid MLR_ANN structure for weekly average Pan Evaporation prediction
parameter.
Figure 4.25. Mapping of the number of epochs obtained for desired goal for hybrid MLR_ANN
model for Pan Evaporation parameter.
4.4.8. Bright Sunshine
The multiple regression model is fitted to predict the weekly average bright sunshine as
dependent parameter taking the other weekly independent parameters as maximum temperature,
minimum temperature, relative humidity at 7 am and 2 pm, pan evaporation and rainfall. The most
significantly contributed parameters are selected using stepwise regression analysis based on 595
data set of 35 years and the best fit multiple regression model is given below:
Y = 7.86562 + 0.12606X1 - 0.13358X2 + 0.03621X3 - 0.07713X 4 - 0.00597X6
(4.16)
where, bright sunshine (Y) as the dependent parameter having 46.68% contribution of the
significant parameters maximum temperature (X1), minimum temperature (X2) , relative humidity
at 7 am (X3), relative humidity at 2 pm (X4) and rainfall (X6). The parameter pan evaporation (X5)
seems to have least control over bright sunshine and hence does not appear in the proposed
multiple regression model. The stepwise regression procedure for selecting the significant
parameters for the bright sunshine parameter is mentioned in Appendix A (g).
We next performed an ARIMA modeling of the data using all the weather parameters. The
bright sunshine time series data set is found to be stationary so we did not require any
transformation of the data set. We used the autocorrelation function (ACF) and partial
autocorrelation function (PACF) of the weekly average bright sunshine time series (see figure
4.26) to estimate the parameters (p and q) of the ARIMA model. Note that while both the ACF
and PACF have common significant terms at lags 1 and 17, they have maximum correlation
coefficient (0.356) at lag 1. This possibly suggests that an ARIMA of order 1 is best fit for the
bright sunshine data set. Using an iterative model building process of identification, estimation,
and diagnostic checking using all the weather parameters, we finally selected an ARIMA (1, 0, 0)
model as the most appropriate fit for the observed data.
Then we performed an ANN model building process using all the same weather
parameters. We selected the best suited architecture of Feed Forward Neural Network Model for
our weekly bright sunshine by comparing methods and changing the layer and number of neurons
in each network. This proposed model had an input environment with all the weather parameters,
three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and
12 neurons in the third hidden layer) and one neuron in the output layer (see figure 4.27).
We used a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a
tan sigmoid activation function in the first, second and third hidden layer and a purelin activation
function in the output layer. A set of random values distributed uniformly between -1 to +1 are
used to initialize the weight of the neural network model. 570 epochs are used to train the neural
network model with 0.00000219 goals (see figure 4.28). The program used for training the 595 data
sets using all the weather parameters for ANN model for predicting bright sunshine parameter is
mentioned in Appendix B (g).
Figure 4.26.
Plots of autocorrelation and partial autocorrelation coefficients and time lags of
weakly average Bright Sunshine parameter.
Figure 4.27. Artificial neural network structure for weekly average Bright Sunshine prediction
parameter.
Figure 4.28. Mapping of the number of epochs obtained for desired goal for ANN model for
Bright Sunshine parameter.
We next performed a hybrid MLR_ARIMA modeling of the data using the five weather
parameters, significantly selected through stepwise regression earlier, for predicting the bright
sunshine weather parameter. We used the same ACF and PACF of the bright sunshine weather
parameter, having maximum correlation coefficient (0.356) at lag 1 (see figure 4.26). Suggesting
that an ARIMA (1, 0, 0) model as the most appropriate fit using the significantly selected weather
parameters data set.
Figure 4.29.
Hybrid MLR_ANN structure for weekly average Bright Sunshine prediction
parameter.
Finally, we performed a hybrid MLR_ANN model building process using the same, more
significant parameters as suggested by the multiple regression model. We selected the same best
suited architecture of Feed Forward Neural Network Model as obtained while developing the
ANN model for weekly average bright sunshine data set. This proposed hybrid MLR_ANN model
had different input environment, that is, only the significant parameters selected through stepwise
regression analysis, but the same three hidden layers (8 neurons in the first hidden layer, 10
neurons in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in the
output layer (see figure 4.29) are considered.
We used the same Scaled Conjugate Gradient Algorithm for training this multilayer
perceptron, a tan sigmoid activation function in the first, second and third hidden layer and a
purelin activation function in the output layer as considered while developing the ANN model. A
new different set of random values distributed uniformly between -1 to +1 are used to initialize the
weight of the MLR_ANN model. 1257 epochs are used to train the neural network model with the
same 0.00000219 goal (see figure 4.30). The program used for training the 595 data sets using the
significantly selected parameter for the hybrid MLR_ANN model of dependent bright sunshine
weather parameter is mentioned in Appendix C (d).
Figure 4.30. Mapping of the number of epochs obtained for desired goal for hybrid MLR_ANN
Bright Sunshine parameter.
4.5
Comparison of prediction ability of forecasting models
4.5.1. Introduction
Recognition of pertinent model is an important errand, obtained through comparing the
predictive ability of forecasting models. The methodology presented above was applied to training
data set of 35 years, that is, 595 data sets, weekly weather data for the monsoon period. For the
training data sets, the comparison among the developed models is made for each weather
parameters by comparing the real and predicted values.
The comparison is made on the basis of the analytical methods, which can be evaluated in
terms of graphical representation in the form of linear scale plot and numerical computations
through mean error (BIAS), mean absolute error (MAE), root mean square error (RMSE),
prediction error (PE), and correlation coefficient (r). Here the comparison among the predicted and
observed values for all the seven parameters for training date set are discussed graphically in the
next sub sections.
4.5.2. Rainfall
The process developed through MLR, ARIMA, ANN, hybrid MLR_ARIMA and hybrid
MLR_ANN in section 4.4.2 are compared using the graphical method. The comparison among the
developed models is made to identify the finest model for the rainfall weather parameter by
comparing the real and predicted values. In figure 4.31, we provide a comparison of the actual
rainfall with the predicted rainfall using all the five developed models, for 595 data sets used for
training purposes, observing that predicted values by ANN and Hybrid MLR_ANN models are
tending more towards the actual values of weekly rainfall.
4.5.3. Maximum Temperature
The procedure developed through multiple linear regression, ARIMA and neural network in
section 4.4.3 are compared using the graphical method. The comparison among the developed models
is presented to identify the finest model for the maximum temperature weather parameter by
comparing the real and predicted values. In figure 4.32, we provide a comparison of the weekly actual
maximum temperature with the predicted maximum temperature using the three developed models, for
595 data sets used for training purposes. The values obtained through ANN model are tending more to
overlap the actual weekly maximum temperature values. A few points among them can be observed
viz. the last week of 1961, second week of 1965, 1973, first week of 1980 and seventh week of 1982,
1986.
4.5.4. Minimum Temperature
The course of action developed through MLR, ARIMA, ANN, hybrid MLR_ARIMA and
hybrid MLR_ANN in section 4.4.4 are compared using the graphical method. The comparison
among the developed models is discussed to identify the finest model for the minimum
temperature weather parameter by comparing the real and predicted values. In figure 4.33, we
provide a comparison of the weekly actual minimum temperature with the predicted minimum
temperature using all the developed models, for 595 data sets used for training purposes. For one
or two weeks of few years the predicted values obtained through the Hybrid MLR_ANN model
tended more towards the actual values of minimum temperature.
4.5.5. Relative Humidity at 7 AM
The guiding principle developed through multiple regression, ARIMA and neural network
in section 4.4.5 are compared using the graphical method. The comparison among the developed
models is fundamental, so as to identify the finest model for the relative humidity at 7 am, weather
parameter by comparing the real and predicted values. In figure 4.34, we provide a comparison of
the weekly actual relative humidity at 7 am with the predicted relative humidity at 7 am using the
three developed models, for 595 data sets used for training purposes. The predicted values
obtained through ANN model gives more precise results as can be seen graphically at few point’s
viz., first week of 1964, 1965, 1966, 1975, 1992; seventh week of 1969; second week of 1979,
1987 and twelfth week of 1982.
4.5.6. Relative Humidity at 2 PM
The procedure developed through multiple regression, ARIMA and neural network in
section 4.4.6 are compared using the graphical method. The comparison among the developed
models is presented, so as to identify the finest model for the relative humidity at 2 pm, weather
parameter by comparing the real and predicted values. In figure 4.35, we provide a comparison of
the weekly actual relative humidity at 2 pm with the predicted relative humidity at 2 pm using the
three developed models, for 595 data sets used for training purposes, indicating ANN model as
preferred model.
4.5.7. Pan Evaporation
The path developed through multiple regression, ARIMA, ANN, hybrid MLR_ARIMA and
hybrid MLR_ANN in section 4.4.7 are compared using the graphical method. The comparison
among the developed models are primary, so as to identify the premium model for the pan
evaporation weather parameter by comparing the real and predicted values. Figure 4.36, presents a
comparison of the weekly actual pan evaporation with the predicted pan evaporation using all the
developed models, for 595 data sets used for training purposes. The values obtained through ANN
and hybrid MLR_ANN models are tending to partly cover the actual weekly pan evaporation but we
can examine that hybrid MLR_ANN is leaning more towards the actual values of pan evaporation,
as observed in few point’s viz., first week of 1961, 1962, 1964, 1965, 1967, 1980, 1984; third week
of 1973, 1992, 1994; sixth week of 1972 and tenth week of 1972, 1975.
4.5.8. Bright Sunshine
The path developed through MLR, ARIMA, ANN, hybrid MLR_ARIMA and hybrid
MLR_ANN in section 4.4.8 are compared using the graphical method. The comparison among the
developed models are primary, so as to identify the premium model for the bright sunshine
weather parameter by comparing the real and predicted values. Figure 4.37, is presented with a
comparison of the weekly actual bright sunshine with the predicted bright sunshine using all the
developed models, for 595 data sets used for training purposes indicating hybrid MLR_ANN
model as a preferred model.
4.6
Conclusion
Complexity of the nature of weekly weather parameters record has been studied using the
Multiple Linear Regression, Autoregressive Integrated Moving Average, Artificial Neural
Network, Hybrid MLR_ARIMA and Hybrid MLR_ANN techniques. The weekly weather
parameters data for the months of June, July, August and September over a period of 35 years of
pantnagar region was used to develop and train the models. Since, the parameter selection (input
pattern) in the models is always a challenging task, so to reduce this complexicity we proposed the
hybrid model by introducing those parameters only which are found significant using stepwise
regression analysis to obtain valid non-bias results. The result showed that the variation during the
four months was among all the parameters and correlation between relative humidity at 7 am and
relative humidity at 2 pm was maximum (0.83736) and it was minimum between rainfall and
minimum temperature (-0.00666).
The above mentioned five models were developed for each weather parameter. In
developing the ANN and hybrid MLR_ANN models, the three hidden layers and scale conjugate
gradient algorithm for training were same for all the weather parameters, expect the input
parameters and the weight and biases considered. Observing the graphical presentation of each
weather parameter, it was concluded that ANN model is a preferred model in comparison to the
MLR and ARIMA models for all the weekly weather parameter. Finally, the study reveals that
hybrid MLR_ANN model can be used as an appropriate forecasting tool to estimate the weather
parameters, in contrast to the MLR, ARIMA, ANN and hybrid MLR_ARIMA models.
IDENTIFICATION OF PRECISE
WEATHER FORECASTING MODEL

Introduction

Validation of weather forecasting model

Conclusion
CHAPTER 5
IDENTIFICATION OF PRECISE
WEATHER FORECASTING MODEL
5.1
Introduction
The performance of the entire existing weather forecasting model was evaluated in the
previous chapter and the comparison of prediction ability of forecasting models indicates the
better performance of Artificial Neural Network model. All the seven weather parameters were
used to develop the MLR, ARIMA and ANN models. The most significantly contributed variables
selected for each weather parameter using stepwise regression analysis were used to develop the
hybrid MLR_ARIMA and hybrid MLR_ANN models. The comparison among the real and
predicted values, to identify the precise weather forecasting is made in this section and applied to
testing
quasi-independent 255 data sets of 15 years weekly weather data for the
monsoon period which was not used while developing the model.
The comparison is made on the basis of the analytical methods, which can be evaluated in
terms of graphical representation in the form of linear scale plot and numerical computations
through mean error (BIAS), mean absolute error (MAE), root mean square error (RMSE),
prediction error (PE), and correlation coefficient (r). The details of the numerical computational
methodology are used as explained in section 4.3.5. The appropriateness effectiveness of these
models is demonstrated by comparing the actual value of all the seven weather parameters with
their predicted value and results are presented in next section.
5.2.
Validation of weather forecasting model
5.2.1 Rainfall
A comparison of the performance of the actual weekly rainfall with its predicted value
using the proposed models are presented with those of other traditional forecasting models,
graphically in figure 5.1.
It is clearly perceptible that the predicted values of the 17 weeks of weekly monsoon
rainfall for the first three years 1996 to 1998 varied with slight fluctuation from the actual values.
Further, it identify that there is a large variation among few monsoon weeks of the two years, 1999
and 2000, since the predicted values are in a trend form but the actual value are having high
variation. Next, in the years 2001 and 2002 the actual values are slightly smaller than the predicted
values, but a large variation again due to the actual values different from the trend pragmatic
through previous years are observed for two to three monsoon weeks of the years 2003, 2004 and
2005.
Presently, it is also observed that in the last monsoon weeks of 2003 and beginning
monsoon weeks of 2004, very less variation is observed between the actual value and predicted
value through hybrid MLR_ANN model. Further, similar observation is identified for the few
monsoon weeks in the years 2007 and 2009. Thus indicating the hybrid MLR_ANN model is a
proficient predictor in comparison to the other considered models.
The estimates viz. mean error, mean absolute error, root mean square error, prediction
error and correlation coefficient are presented in table 5.1 for the testing data set. The bias for
testing data set is the least for hybrid MLR_ANN model than that obtained from hybrid
MLR_ARIMA, ANN, ARIMA and MLR models.
Table 5.1. Comparison of the performance of forecasting models for Rainfall parameter.
Techniques
BIAS
MAE
RMSE
PE
CC
MLR
3.39615
53.91620
75.58739
0.66798
0.60251
ARIMA
1.57248
52.87119
74.90918
0.65503
0.61154
ANN
-6.23647
49.74627
74.74657
0.61632
0.61441
MLR_ARIMA
3.59415
53.90974
75.50548
0.66790
0.60344
MLR_ANN
-6.28196
49.33765
74.34915
0.61125
0.61894
The MAE further explains that the hybrid MLR_ANN model is more precise than ANN
model. The hybrid MLR_ANN model shows a smaller value for RMSE as compared to those of
the other models. The PE obtained for testing data from MLR, hybrid MLR_ARIMA, ARIMA and
ANN models is 0.66798, 0.66790, 0.65503 and 0.61632, respectively, while for hybrid
MLR_ANN is 0.61125 which is the least and indicating it as precise prediction model. Further, the
correlation coefficient is found to be highest for hybrid MLR_ANN model in comparison to other
models. Thus, the graphical representation as well as the numerical estimates both favored, the
hybrid multiple linear regression with artificial neural network (MLR_ANN) model as a preferred
performance in comparison to the other models, concluding that this hybrid technique can be used
as an effective rainfall forecasting tool in the Himalaya.
5.2.2
Maximum Temperature
Comparison of the performance of the weekly maximum temperature parameter is
presented graphically in figure 5.2. The weekly monsoon maximum temperature for the years
1996, 1999, 2000, 2001, 2002, 2005, 2006 2007 and 2008 clearly shows some variation among the
predicted value and actual values for most of the weeks of each year. But for the years 1997, 1998,
2003, 2004, 2009 and few starting weeks of 2010 very less variation is observed among the actual
values and predicted value through ANN model, indicating that ANN predicts comparatively
better than the other traditional models.
The estimates viz. mean error, mean absolute error, root mean square error, prediction
error and correlation coefficient are also presented in table 5.2 for the same data set. Since in the
stepwise regression analysis all the parameters are selected as significant parameters, therefore,
there is no possibility of developing the hybrid models for maximum temperature in the study
area. The bias obtained for testing data set is smaller for artificial neural network than the values
that are obtained from autoregressive integrated moving average and multiple linear regression.
Table 5.2. Comparison of the performance of forecasting models for Maximum Temperature.
Techniques
BIAS
MAE
RMSE
PE
CC
MLR
0.87990
1.05235
1.45396
0.03228
0.88964
ARIMA
0.87050
1.04583
1.45192
0.03208
0.88881
ANN
0.79330
1.00275
1.40719
0.03076
0.88995
The MAE measure for testing data set for multiple linear regression model is 1.05235 and
for ARIMA model is 1.04583, while the same error measure is considerably lower at 1.00275 for
the artificial neural network model. The ANN model shows a lesser value for RMSE compared to
those of MLR and ARIMA models. The PE obtained for testing data from multiple linear
regression model is 0.03228 and ARIMA is 0.03208 and through artificial neural network model it
is 0.03076 which is smaller representing it as preferred prediction model. Moreover, the neural
network model had the highest correlation coefficient among all the models. These numerical
estimates hold up that the ANN model has a superior performance in comparison to ARIMA and
MLR model which coincides with the previous results.
5.2.3
Minimum Temperature
Comparison of the performance of the proposed model is presented with those of other
forecasting model of the weekly minimum temperature parameter graphically in figure 5.3. The
weekly minimum temperature for the monsoon season shows fluctuations among the actual and
predicted values. The predicted values developed through hybrid MLR_ANN models following a
trend based on previous years, showed least variation to the actual values for few weeks of the
years 1997, 1998, 1999, 2000, 2004 and 2009. Thus the hybrid MLR_ANN model proved to be
better performer and can be further analyzed more precisely by evaluating through numerical
performance of each model. The estimates viz. mean error, mean absolute error, root mean square
error, prediction error and correlation coefficient are presented in table 5.3 for the testing data set.
Table 5.3. Comparison of the performance of forecasting models for Minimum Temperature.
BIAS
MAE
RMSE
PE
CC
MLR
-0.79700
1.09592
1.38227
0.04416
0.44361
ARIMA
-0.71713
1.08877
1.29726
0.04387
0.49334
ANN
-0.77922
1.08745
1.33547
0.04382
0.49699
MLR_ARIMA
-0.72705
1.08911
1.31870
0.04388
0.46572
MLR_ANN
-0.72980
1.03726
1.29656
0.04179
0.50767
Techniques
It can be seen from table 5.3 that the bias for testing data set is -0.72980 for hybrid
MLR_ANN model which is neither the lowest nor the highest among all the other compared
models. Since, the bias does not prove to be a perfect criterion for comparison among the
developed models, so we move forward to the other considered performance evaluation criteria’s.
Next, the MAE for testing data set is 1.09592, 1.08911, 1.08877 and 1.08745 for MLR, hybrid
MLR_ARIMA, ARIMA and ANN respectively; while the same error measure is lowest as
1.03726 for the hybrid MLR_ANN model. The RMSE is also found to be least as 1.29656 for the
hybrid MLR_ANN model in comparison to the other forecasting models. The PE obtained for
testing data from MLR model is 0.04416, hybrid MLR_ARIMA is 0.04388, ARIMA model is
0.04387 and ANN is 0.04382 and through hybrid MLR_ANN model is 0.04179 which is lesser
indicating it as preferred prediction model. Further, the correlation coefficient is observed highest
for hybrid MLR_ANN model. As the graphical representation as well as the numerical estimates
sustains that the hybrid MLR_ANN model has a preferred performance. It can be concluded that
this hybrid technique can be used as a reliable minimum temperature forecasting contrivance in
the Himalaya.
5.2.4
Relative Humidity at 7 AM
A comparison of the performance of the actual values with the predicted values of relative
humidity at 7 am is presented graphically in figure 5.4. The predicted values of the 17 weeks of
weekly average relative humidity at 7 am monsoon season, showed least variation of predicted
ANN values to the actual values for most of the weeks of all the 15 years, except for September
weeks of 1998, 2003 and 2007, and few weeks of 1999, 2004, 2008 and 2010. Thus, it can be
competently said that ANN model is an efficient performer than the other developed models.
In table 5.4, the estimates viz. mean error, mean absolute error, root mean square error,
prediction error and correlation coefficient for the testing data set are also given. Since in the
stepwise regression analysis, all the parameters are selected as significant parameters, therefore,
there is no possibility of developing the hybrid models for Relative Humidity at 7 AM weather
parameter in the present study area.
Table 5.4. Comparison of the performance of forecasting models for Relative Humidity at 7 AM .
Techniques
BIAS
MAE
RMSE
PE
CC
MLR
1.13423
3.16400
4.11119
0.03663
0.89181
ARIMA
1.13182
3.08161
4.01749
0.03568
0.89820
ANN
1.06157
2.92431
3.87023
0.03385
0.90479
It can be observed from table 5.4 that the bias for testing data set is lesser for artificial
neural network than the values that are obtained from ARIMA and MLR. The MAE measure for
testing data set for multiple linear regression model is 3.16400 and for ARIMA model is 3.08161,
while the same error measure is considerably lower at 2.92431 for the artificial neural network
model. The artificial neural network model also shows a smaller value for RMSE judge against to
those of MLR and ARIMA models. The PE obtained for testing data from MLR, ARIMA and
ANN models is 0.03663, 0.03568 and 0.03385, indicating ANN model as the lowest among the
entire three prediction models. Further, the correlation coefficient is observed to be highest for
artificial neural network model as 0.90479 in comparison to 0.89181 and 0.09820 for multiple
linear regression and ARIMA models respectively. These numerical estimates maintain that the
ANN model has a preferred performance and can be used as a forecasting tool for relative
humidity at 7 am in the Himalayas range, which validate the previous findings.
5.2.5
Relative humidity at 2 PM
A comparison of the performance of the actual relative humidity at 2 pm with the predicted
relative humidity at 2 pm using the three forecasting models is presented graphically in figure 5.5.
The weekly average relative humidity at 2 pm of the monsoon season showed less variation
between the actual values and predicted values through ANN for most of the weeks of all years. A
reasonable variation of predicted values to the actual value can be observed in a week of 1996,
2005, 2006 and 2007. Thus, it can be said that ANN model is a better performer in comparison to
the other developed models.
The table 5.5, represents the estimates viz. mean error, mean absolute error, root mean
square error, prediction error and correlation coefficient for the testing data set. Since in the
stepwise regression analysis all the parameters are selected as significant parameters, therefore,
there is no possibility of developing the hybrid models for Relative Humidity at 2 PM weather
parameter in the present study area.
Table 5.5. Comparison of the performance of forecasting models for Relative Humidity at 2 PM.
Techniques
BIAS
MAE
RMSE
PE
CC
MLR
1.71266
3.47877
4.93385
0.05245
0.93181
ARIMA
1.58552
3.38308
4.83330
0.05100
0.93333
ANN
0.45569
3.28471
4.55504
0.04952
0.93479
It can be seen from the above table that the bias for testing data set is the least for artificial
neural network model than the other considered models. The MAE measure for testing data set for
multiple linear regression model is 3.47877 and for ARIMA model is 3.38308, while the same
error measure is considerably lower at 3.28471 for the artificial neural network model. The
artificial neural network model shows a smaller value for RMSE compared to those of multiple
linear regression and ARIMA models. The PE obtained for testing data is 0.05245 and 0.05100
from multiple linear regression model and ARIMA, respectively, and through artificial neural
network model is 0.4952 which is smaller and can be preferred as predictive model.
Moreover, the correlation coefficient is observed to be highest for artificial neural network
model. Thus, these numerical estimates support the graphical presentation indicating that the
artificial neural network model has a preferred performance in comparison to multiple linear
regression and autoregressive integrated moving average models.
5.2.6 Pan Evaporation
Comparison on the performance of the proposed model is presented with those of other
forecasting model for the weekly average pan evaporation parameter graphically in figure 5.6. The
monsoon weekly average pan evaporation showed the least variation among the actual and
predicted values obtained through hybrid MLR_ANN for most of the weeks of almost all the
years, except for the two years 1999 and 2000, since these two years also followed a prediction
pattern developed from the previous years, are reasonably different, from the actually observed
values of weekly average pan evaporation. Thus, indicating that the hybrid MLR_ANN is a better
performer in comparison to the other models.
The estimates viz. mean error, mean absolute error, root mean square error, prediction
error and correlation coefficient are presented in table 5.6 for the testing data set. The bias for
testing data set is the highest for hybrid MLR_ANN model than the values that are obtained from
hybrid MLR_ARIMA, ANN, ARIMA and MLR models.
Table 5.6. Comparison of the performance of forecasting models for Pan Evaporation.
BIAS
MAE
RMSE
PE
CC
MLR
-0.61433
1.12179
1.50834
0.20349
0.80829
ARIMA
-0.55696
1.09292
1.46588
0.19826
0.80699
ANN
-0.42118
1.07216
1.43977
0.19449
0.80355
MLR_ARIMA
-0.58701
1.09695
1.48079
0.19899
0.80593
MLR_ANN
-0.40118
1.07019
1.40800
0.19413
0.81262
Techniques
The MAE for testing data set for multiple linear regression model is 1.12179 and for
hybrid MLR_ARIMA model is 1.09695, ARIMA model is 1.09292 and ANN is 1.07216 while the
same error measure is noticeably lower as 1.07019 for the hybrid MLR_ANN model. Moreover,
the hybrid MLR_ANN model shows a smaller value for RMSE as 1.40800 in comparison to the
other models. The PE obtained for testing data from multiple linear regression model is 0.20349,
hybrid MLR_ARIMA is 0.19899, ARIMA model is 0.19826 and ANN is 0.19449 and through
hybrid MLR_ANN model is 0.19413 which is the smallest, indicating it as preferred prediction
model. Further, the correlation coefficient is found to be the highest for neural network model.
These numerical estimates also support the graphical presentation, indicating that, the hybrid
multiple linear regression with artificial neural network (MLR_ANN) model has a favored
performance and can be used as a reliable pan evaporation forecasting tool in the Himalaya.
5.2.7
Bright Sunshine
Comparison of the performance of the proposed model is presented with those of other
forecasting model of the weekly average bright sunshine parameter graphically in figure 5.7. It can
be identified that the weekly average bright sunshine of the monsoon season reflects variation
between the actual and predicted values through all the five considered models, except for few
weeks among the years, a less variation is observed between the actual and predicted value
through hybrid MLR_ANN model. Indicating hybrid MLR_ANN model as a preferred performer
than the other methods considered and developed.
The estimates viz. mean error, mean absolute error, root mean square error, prediction
error and correlation coefficient are presented in table 5.7 for the testing data set. The bias for
testing data set is the highest for hybrid MLR_ANN model than the values that are obtained from
hybrid MLR_ARIMA, ANN, ARIMA and MLR models.
Table 5.7.
Comparison of the performance of forecasting models for Bright Sunshine.
Techniques
BIAS
MAE
RMSE
PE
CC
MLR
0.11512
1.21750
1.48545
0.20045
0.79686
ARIMA
0.07957
1.19586
1.46565
0.19689
0.79746
ANN
0.24863
1.22118
1.46282
0.20106
0.81908
MLR_ARIMA
0.08059
1.19610
1.46593
0.19693
0.79742
MLR_ANN
0.30745
1.17726
1.43552
0.19383
0.81924
The MAE for testing data set for MLR model is 1.21750 and for hybrid MLR_ARIMA
model is 1.19610, ARIMA model is 1.19586 and ANN model is 1.22118 while the same error
measure is noticeably lower as 1.17726 for the hybrid MLR_ANN model. The hybrid MLR_ANN
model also shows a smaller value for RMSE as 1.43552 compared to those of MLR, ARIMA,
ANN and hybrid MLR_ARIMA models as 1.48545, 1.46565, 1.46282 and 1.46593 respectively,
for testing data set.
In general, a good forecast may have a relatively high BIAS value but a relatively low
MAE and RMSE values (if the predicted variable is well correlated with independent variables) or
low BIAS value and high MAE and RMSE (if the predicted variable is poorly correlated with
independent variables).
The PE obtained for testing data from MLR model is 0.20045, hybrid MLR_ARIMA
model is 0.19693, ARIMA model is 0.19689 and ANN model is 0.20106 and through hybrid
MLR_ANN model is 0.19383 which is the smallest indicating it as preferred prediction model.
Further, the correlation coefficient is found to be the highest for hybrid MLR_ANN model as
0.81924 than the other models. These numerical estimates, thus, support that the hybrid multiple
linear regression with artificial neural network (MLR_ANN) model has a better performance and
can be preferred as a reliable bright sunshine forecasting tool in the Himalaya.
5.3
CONCLUSION
Hybrid MLR_ANN, hybrid MLR_ARIMA, artificial neural network, autoregressive
moving average and multiple linear regression models are used to study the impediment of the
environment of weekly weather parameters. The weekly weather parameters data for the months
of June, July, August and September over a period of 15 years of pantnagar region was used to
identify the precise weather forecasting models. Since, the parameter assortment in the models is
always an exigent task, so we introduced those parameters for hybrid model building only which
are found significant using stepwise regression analysis during the training period.
The comparison among the five models shows the trend, based on previous years but the
actual values were fluctuating. The proposed hybrid MLR_ANN model was observed as precise
model in comparison to MLR, ARIMA, ANN and hybrid MLR_ARIMA models. In view of the
fact that, all the prediction models are consistent but the finest model is the lone having least mean
absolute error and root mean square error, prediction error and high correlation coefficient, as
observed in hybrid MLR_ANN model. It was observed that the ANN model is also precise
weather forecasting model as compared to MLR and ARIMA models which coincides with the
previous findings. At last, the study reveal that hybrid MLR_ANN model maintained can be used
as an appropriate forecasting interest to estimate the weather parameters, in contradiction to the
multiple linear regression, ARIMA, ANN and hybrid MLR_ARIMA models.
SUMMARY AND FUTURE SCOPE

Summary

Future scope
CHAPTER 6
SUMMARY AND FUTURE SCOPE
6.1.
Summary
The role of statistical techniques for providing reliable predictions of weather parameters is
considered to be most important in the field of metrology all over the world. These predictions
influence the agricultural as well as the industrial strategies. In India, the month of June, July,
August and September are identified as the summer monsoon month. The summer monsoon in
early June marks the beginning of the principal raining season for the Himalaya. The monsoon
season in and around Indian Metrological Department (IMD) Pantnagar observatory which is
situated in the foot hills of Himalayas, ranges between 15 to 20 weeks. Accordingly our study is
based on a time series weather data set, collected at the IMD observatory at Pantnagar, India, over
a period of 50 years. Pantnagar, located at 29 N, 79.45 E approximately 293.89 meter above mean
sea level, in the tarai region of Uttarakhand. Assuming that the monsoon season in and around
Pantnagar ranges between 15 to 20 weeks: we consider a 17 weeks data set for our study during
1961-2010. The weekly data comprises of seven weather parameters, viz., Rainfall, Maximum and
Minimum Temperature, Relative Humidity at
7 AM and 2 PM, Bright Sunshine and Pan
Evaporation, collected during monsoon months June to September.
Thus providing reliable prediction and forecasting of weather parameters in the Himalayas
in particular and of India in general is an important challenges for planners and scientists. The
present study is planned with the following objectives:
1.
To study, the distribution pattern of weather parameters.
2.
Prediction of weather parameters using different forecasting model.
3.
To compare the prediction ability of these model.
4.
To identify the precise weather forecasting model.
5.
To study the reliability of the developed model by comparing the forecast value with its
observed value.
In this research work we took up a study to identify the precise and reliable weather
forecasting model through comparison of several existing and proposed models. The thesis is
divided into seven chapters and the salient results obtained and the main significance of study is
summarized in the following paragraphs.
Probability Distribution: The descriptive statistics are computed for each weather parameters for
different study period. The best fit probability distribution was identified out of a large number of
commonly used probability distribution by using different goodness of fit tests. It was observed
that the best fit probability distribution obtained for the weather parameters data sets are different.
For seasonal weather parameters Normal distribution, Weibull (3P) distribution, LogPearson 3 distribution, Gamma (3P) and Log-Gamma distribution represents the best fit
distribution for Rainfall, Maximum temperature, Relative humidity at 7AM, Pan Evaporation and
Bright Sunshine respectively, while Weibull (2P) was fitted for both Maximum Temperature and
Relative Humidity at PM. The best fit probability distributions for weekly weather parameters for
different study periods are different. In general, Generalized Extreme Value distribution, Weibull
(2P, 3P) distributions are most commonly best fit probability distribution for most of the weeks
among the different weather parameters.
Forecasting Models: The weekly weather parameters data for the months of June, July, August
and September over a period of 50 years of pantnagar region was worn in which 35 years data was
employed to develop and train the models and 15 years was used to test and validate the developed
models. The variation during the four months was among all the parameters and correlation
between relative humidity at 7 am and relative humidity at 2 pm was maximum (0.83736) and it
was minimum between rainfall and minimum temperature (-0.00666). Since, the parameter
selection in the models is always a difficult task, so we introduced those parameters for hybrid
model building only which are found significant using stepwise regression analysis during the
training period. Hybrid MLR_ANN, hybrid MLR_ARIMA, Artificial Neural Network, ARIMA
and Multiple Linear Regression models are used to study the environment of weekly weather
parameters. The models were developed and compared individually for each weather parameter
graphically and numerically through Mean Error, Mean Absolute Error, Root Mean Square Error,
Prediction Error and Correlation Coefficient. Finally, the study divulges that Hybrid MLR_ANN
model is an appropriate forecasting interest to estimate the weather parameters, in disparity to the
multiple linear regression, ARIMA, ANN and hybrid MLR_ARIMA models.
6.2.
Future Scope
Weather forecasting is an art using astronomical and meteorological events to monitor
changes in the weather. Although with modern technology, particularly computers and weather
satellites and the availability of data provided by coordinated meteorological observing networks,
has resulted in enormous improvements in the accuracy of weather forecasting. But still with the
ever growing demand for more accurate and reliable weather forecasts, the field opens-up to
additional investigation. However there are some observations in this regard and issues related to
our research investigation which can be addressed in future.
In contrast to the traditional hybrid linear and non-linear methodologies, we can generally
say that the performance of the hybrid MLR_ANN model will not be worse than either of the
components used in isolation, so that it can be applied as an appropriate methodology for
combination linear and non-linear models for time series forecasting. Further, we can say that
more hybrid models can be developed using different ANN ensemble and by using more weather
parameters. Moreover, the inclusion of other seasonal factors may also improve the forecasting
accuracy. It is expected that instead of developing hybrid models based on two techniques, the
integrated hybrid model may be developed by combining more than two techniques together. This
integrated hybrid model may produce more precise model for future and may be an upcoming
topic for further studies.
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www.google.com
APPENDIX A(a)
Procedure followed for Stepwise Regression Analysis for
Dependent Variable: Rainfall
Stepwise Selection: Step 1
Variable Relative Humidity 2PM Entered: R-Square = 0.2996 and C (p) = 71.6676
Source
Analysis of Variance
Sum of
Mean
DF
Squares
Square
Model
Error
Corrected Total
1
593
594
Variable
Intercept
Relative Humidity 2PM
963290
2252448
3215738
963290
3798.39389
F Value
Pr > F
253.60
<.0001
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
-119.44984
2.89509
12.16003
0.18180
366524
963290
96.49
253.60
<.0001
<.0001
Bounds on condition number: 1, 1
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 2
Variable Bright Sunshine Entered: R-Square = 0.3444 and C (p) = 31.2555
Analysis of Variance
Sum of
Mean
Source
DF
Squares
Square
F Value
Pr > F
Model
Error
Corrected Total
2
592
594
1107452
2108286
3215738
553726
3561.29453
155.48
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Relative Humidity 2PM
Bright Sunshine
6.22513
1.95314
-9.84784
22.99585
0.23001
1.54782
260.97892
256785
144161
0.07
72.10
40.48
0.7867
<.0001
<.0001
Bounds on condition number: 1.7074, 6.8295
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Relative Humidity 7AM Entered: R-Square = 0.3545 and C (p) = 23.7044
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
3
591
594
1139917
2075821
3215738
379972
3512.38808
Variable
Intercept
Relative Humidity 7AM
Relative Humidity 2PM
Bright Sunshine
F Value
Pr > F
108.18
<.0001
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
66.26725
-1.38703
2.81011
-9.17876
30.19235
0.45622
0.36281
1.55283
16920
32465
210708
122722
4.82
9.24
59.99
34.94
0.0286
0.0025
<.0001
<.0001
Bounds on condition number: 4.3072, 27.888
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 4
Variable Maximum temperature Entered: R-Square = 0.3666 and C (p) = 14.2837
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
4
590
594
1178736
2037002
3215738
294684
3452.54544
Variable
Parameter
Estimate
Standard
Error
F Value
Pr > F
85.35
<.0001
Type II SS
F Value
Intercept
379.40528
98.06595
51678
14.97
Maximum Temperature
-6.25287
1.86477
38820
11.24
Relative Humidity 7AM
-2.28785
0.52608
65295
18.91
Relative Humidity 2PM
2.35796
0.38415
130078
37.68
Bright Sunshine
-8.74845
1.54488
110716
32.07
Bounds on condition number: 4.9125, 61.686
--------------------------------------------------------------------------------------------------
Pr > F
0.0001
0.0009
<.0001
<.0001
<.0001
Stepwise Selection: Step 5
Variable Pan Evaporation Entered: R-Square = 0.3766 and C (p) = 6.7459
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
5
589
594
1211156
2004582
3215738
242231
3403.36528
F Value
71.17
Pr > F
<.0001
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
391.90674
Maximum Temperature
-8.65592
Relative Humidity 7AM
-1.89389
Relative Humidity 2PM
2.46673
Pan Evaporation
4.70662
Bright Sunshine
-8.50348
97.44921
2.00849
0.53770
0.38303
1.52496
1.53589
55045
63212
42223
141151
32420
104323
16.17
18.57
12.41
41.47
9.53
30.65
<.0001
<.0001
0.0005
<.0001
0.0021
<.0001
Variable
Bounds on condition number: 5.1343, 98.501
-------------------------------------------------------------------------------------------------All variables left in the model are significant at the 0.1500 level.
No other variable met the 0.1500 significance level for entry into the model.
Summary of Stepwise Selection
Step
1
2
3
4
5
Variable
Entered
Relative Humidity 2PM
Bright Sunshine
Relative Humidity 7AM
Maximum Temperature
Pan Evaporation
Number
Partial
Model
Variables In R-Square R-Square C (p)
1
2
3
4
5
0.2996 0.2996
0.0448 0.3444
0.0101 0.3545
0.0121 0.3666
0.0101 0.3766
71.6676
31.2555
23.7044
14.2837
6.7459
F Value
Pr > F
253.60
40.48
9.24
11.24
9.53
<.0001
<.0001
0.0025
0.0009
0.0021
APPENDIX A(b)
Procedure followed for Stepwise Regression Analysis for
Dependent Variable: Maximum Temperature
Stepwise Selection: Step 1
Variable Relative humidity 7AM Entered: R-Square = 0.7151 and C (p) = 477.2735
Analysis of Variance
Source
DF
Model
Error
Corrected Total
1
593
594
Sum of
Squares
3097.61259
1234.05268
4331.66528
Mean
Square
3097.61259
2.08103
F Value
Pr > F
1488.50
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Intercept
Relative humidity 7AM
53.93488
-0.23778
0.53853
0.00616
20873
3097.61259
10030.3 <.0001
1488.50 <.0001
Pr > F
Bounds on condition number: 1, 1
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 2
Variable Pan Evaporation Entered: R-Square = 0.7720 and C (p) = 266.0520
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
2
592
594
3343.92268
987.74260
4331.66528
Variable
Intercept
Relative humidity 7AM
Pan Evaporation
Mean
Square
1671.96134
1.66848
F Value
Pr > F
1002.08
<.0001
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
44.80651
-0.15463
0.36545
0.89273
0.00879
0.03008
4203.00312
516.16007
246.31008
2519.05
309.36
147.63
<.0001
<.0001
<.0001
Bounds on condition number: 2.5379, 10.152
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Relative Humidity 2PM Entered: R-Square = 0.8034 and C (p) = 150.1939
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
3
591
594
3480.07053
851.59475
4331.66528
Mean
Square
1160.02351
1.44094
F Value
Pr > F
805.05
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
Intercept
Relative humidity 7AM
Relative Humidity 2PM
Pan Evaporation
44.14902
-0.09487
-0.06352
0.29327
0.83238
0.01022
0.00653
0.02892
4053.61208 2813.17 <.0001
124.06086
86.10 <.0001
136.14785
94.49 <.0001
148.17111
102.83 <.0001
F Value
Pr > F
Bounds on condition number: 3.9748, 30.293
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 4
Variable Minimum Temperature Entered: R-Square = 0.8374 and C (p) = 24.8486
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
4
590
594
3627.17774
704.48753
4331.66528
Mean
Square
906.79444
1.19405
F Value
Pr > F
759.43
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS F Value
Pr > F
Intercept
Minimum Temperature
Relative humidity 7AM
Relative Humidity 2PM
Pan Evaporation
35.78241
0.34132
-0.07114
-0.08928
0.23873
1.06880 1338.35354 1120.86
0.03075 147.10722 123.20
0.00955
66.26731
55.50
0.00639 233.44846 195.51
0.02678
94.87433
79.46
<.0001
<.0001
<.0001
<.0001
<.0001
Bounds on condition number: 4.1845, 48.364
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 5
Variable Rainfall Entered: R-Square = 0.8417 and C (p) = 10.5253
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
5
589
594
3646.03416
685.63112
4331.66528
729.20683
1.16406
F Value
Pr > F
626.43
<.0001
Variable
Parameter
Estimate
Standard
Error
F Value
Pr > F
Intercept
Rainfall
Minimum Temperature
Relative humidity 7AM
Relative Humidity 2PM
Pan Evaporation
35.85264
-0.00295
0.32834
-0.07620
-0.07638
0.24756
1.05543
1343.24493 1153.93
0.00073325
18.85641
16.20
0.03053
134.61214
115.64
0.00951
74.69547
64.17
0.00707
135.76660
116.63
0.02653
101.32602
87.05
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Type II SS
Bounds on condition number: 4.9388, 73.488
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 6
Variable Bright Sunshine Entered: R-Square = 0.8432 and C (p) = 7.0000
Analysis of Variance
Sum of
Mean
Source
DF
Squares
Square
F Value
Pr > F
Model
Error
Corrected Total
6
588
594
Variable
Intercept
Rainfall
Minimum Temperature
Relative humidity 7AM
Relative Humidity 2PM
Pan Evaporation
Bright Sunshine
3652.41694
679.24833
4331.66528
Parameter
Estimate
34.98664
-0.00252
0.33439
-0.07787
-0.07036
0.24658
0.06837
608.73616
1.15518
526.96
<.0001
Standard
Error
Type II SS
1.11408
0.00075311
0.03053
0.00950
0.00750
0.02644
0.02909
1139.25818 986.21
12.93483 11.20
138.62525 120.00
77.56213 67.14
101.72363 88.06
100.50376 87.00
6.38279
5.53
F Value
Bounds on condition number: 5.5926, 104.02
---------------------------------------------------------------------------------------------
Pr > F
<.0001
0.0009
<.0001
<.0001
<.0001
<.0001
0.0191
All variables left in the model are significant at the 0.1500 level.
All variables have been entered into the model.
Summary of Stepwise Selection
Step
1
2
3
4
5
6
Variable
Entered
Relative humidity 7AM
Pan Evaporation
Relative Humidity 2PM
Minimum Temperature
Rainfall
Bright Sunshine
Number
Partial
Model
Variables In R-Square R-Square
1
2
3
4
5
6
0.7151
0.0569
0.0314
0.0340
0.0044
0.0015
0.7151
0.7720
0.8034
0.8374
0.8417
0.8432
C (p)
F Value Pr > F
477.273 1488.50 <.0001
266.052 147.63 <.0001
150.194
94.49 <.0001
24.8486 123.20 <.0001
10.5253
16.20 <.0001
7.0000
5.53 0.0191
APPENDIX A(c)
Procedure followed for Stepwise Regression Analysis for
Dependent Variable: Minimum Temperature
Stepwise Selection: Step 1
Variable Maximum Temperature Entered: R-Square = 0.0421 and C (p) = 217.5398
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
1
593
594
62.20914
1415.79260
1478.00175
Variable
Intercept
Maximum Temperature
Mean
Square
62.20914
2.38751
F Value
26.06
Pr > F
<.0001
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
20.25823
0.11984
0.78396
0.02348
1594.25897
62.20914
667.75
26.06
<.0001
<.0001
Bounds on condition number: 1, 1
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 2
Variable Relative Humidity 2PM Entered: R-Square = 0.2887 and C (p) = 11.3595
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
592
594
426.74307
1051.25867
1478.00175
213.37154
1.77577
Variable
Parameter
Estimate
Standard
Error
Intercept
Maximum Temperature
Relative Humidity 2PM
-0.76196
0.55257
0.10114
1.61540
0.03636
0.00706
F Value
Pr > F
120.16
<.0001
Type II SS
0.39509
410.09498
364.53393
F Value
Pr > F
0.22
230.94
205.28
0.6373
<.0001
<.0001
Bounds on condition number: 3.2251, 12.9
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Bright Sunshine Entered: R-Square = 0.2987 and C (p) = 4.9410
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
3
591
594
441.48427
1036.51748
1478.00175
147.16142
1.75384
Variable
Parameter
Estimate
Standard
Error
Intercept
Maximum Temperature
Relative Humidity 2PM
Bright Sunshine
0.51214
0.55250
0.09160
-0.09958
1.66446
0.03614
0.00775
0.03435
F Value Pr > F
83.91
Type II SS
0.16605
409.98422
245.12691
14.74120
<.0001
F Value
Pr > F
0.09
233.76
139.77
8.41
0.7584
<.0001
<.0001
0.0039
Bounds on condition number: 3.9343, 26.6
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 4
Variable Relative Humidity 7AM Entered: R-Square = 0.3013 and C (p) = 4.7836
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
4
590
594
445.26193
1032.73982
1478.00175
111.31548
1.75041
F Value
Pr > F
63.59
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value Pr > F
Intercept
Maximum Temperature
Relative humidity 7AM
Relative Humidity 2PM
Bright Sunshine
2.64644
0.52100
-0.01740
0.09727
-0.09121
2.20810
0.04199
0.01185
0.00865
0.03479
2.51435
269.50114
3.77766
221.37195
12.03446
1.44 0.2312
153.96 <.0001
2.16 0.1423
126.47 <.0001
6.88 0.0090
Bounds on condition number: 4.9125, 61.686
--------------------------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1500 level.
No other variable met the 0.1500 significance level for entry into the model.
Summary of Stepwise Selection
Step
1
2
3
4
Variable
Entered
Maximum Temperature
Relative Humidity 2PM
Bright Sunshine
Relative humidity 7AM
Number
Partial
Model
Variables In R-Square R-Square
1
2
3
4
0.0421
0.2466
0.0100
0.0026
0.0421
0.2887
0.2987
0.3013
C (p)
F Value
Pr > F
217.540
26.06 <.0001
11.3595 205.28 <.0001
4.9410
8.41 0.0039
4.7836
2.16 0.1423
APPENDIX A(d)
Procedure followed for Stepwise Regression Analysis for
Dependent Variable: Relative Humidity at 7 AM
Stepwise Selection: Step 1
Variable Maximum Temperature Entered: R-Square = 0.7151 and C (p) = 208.3268
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
1
593
594
39178
15608
54786
Mean
Square
39178
26.32032
F Value
Pr > F
1488.50
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
Intercept
Maximum Temperature
186.94704
-3.00741
2.60296
0.07795
135767
39178
F Value
Pr > F
5158.24 <.0001
1488.50 <.0001
Bounds on condition number: 1, 1
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 2
Variable Relative Humidity 2PM Entered: R-Square = 0.7660 and C (p) = 67.4853
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
2
592
594
41967
12819
54786
Variable
Intercept
Maximum Temperature
Relative Humidity 2PM
Mean
Square
20983
21.65334
F Value
Pr > F
969.06
<.0001
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
128.80293
-1.81043
0.27977
5.64091
0.12697
0.02465
11290
4402.23745
2789.17590
521.38
203.31
128.81
<.0001
<.0001
<.0001
Bounds on condition number: 3.2251, 12.9
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Pan Evaporation Entered: R-Square = 0.7804 and C (p) = 29.1147
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
3
591
594
42755
12030
54786
Mean
Square
F Value
14252 700.12
20.35615
Pr > F
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Maximum Temperature
Relative Humidity 2PM
Pan Evaporation
118.78898
-1.34028
0.25027
-0.71005
5.70115
0.14444
0.02437
0.11410
8837.36012
1752.60870
2147.56886
788.29066
434.14
86.10
105.50
38.72
<.0001
<.0001
<.0001
<.0001
Bounds on condition number: 4.4397, 32.356
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 4
Variable Rainfall Entered: R-Square = 0.7870 and C (p) = 12.5982
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
4
590
594
43117
11669
54786
10779
19.77784
F Value
Pr > F
545.01
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
Intercept
Maximum Temperature
Rainfall
Relative Humidity 2PM
Pan Evaporation
119.67441
-1.42546
-0.01290
0.28687
-0.62239
5.62340
0.14376
0.00302
0.02550
0.11432
F Value
Pr > F
8957.43103 452.90
1944.38902
98.31
361.56017 18.28
2503.59168 126.59
586.18727 29.64
<.0001
<.0001
<.0001
<.0001
<.0001
Bounds on condition number: 4.5267, 51.506
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 5
Variable Bright Sunshine Entered: R-Square = 0.7896 and C (p) = 7.4216
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square F Value
Model
5
43257 8651.37901
Error
589
11529
19.57350
Corrected Total
594
54786
Variable
Intercept
Maximum Temperature
Rainfall
Relative Humidity 2PM
Pan Evaporation
Bright Sunshine
441.99
Pr > F
<.0001
Parameter
Estimate
Standard
Error
Type II SS
115.86488
-1.43045
-0.01089
0.31277
-0.60811
0.31774
5.77261
0.14303
0.00309
0.02715
0.11386
0.11875
7885.47968 402.87 <.0001
1957.70893 100.02 <.0001
242.83120
12.41 0.0005
2597.72731 132.72 <.0001
558.38604 28.53 <.0001
140.13310
7.16 0.0077
F Value
Pr > F
Bounds on condition number: 4.5274, 76.775
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 6
Variable Minimum Temperature Entered: R-Square = 0.7904 and C (p) = 7.0000
Analysis of Variance
Sum of
Mean
Source
DF
Squares
Square
F Value Pr > F
Model
Error
Corrected Total
6
588
594
43304
11482
54786
7217.36338
19.52637
369.62
Variable
Parameter
Estimate
Standard
Error
Intercept
Maximum Temperature
Minimum Temperature
Rainfall
Relative Humidity 2PM
Pan Evaporation
Bright Sunshine
116.06296
-1.31618
-0.21386
-0.01111
0.33298
-0.60294
0.29476
5.76706 7908.60919
0.16063 1311.05245
0.13743
47.28526
0.00309 252.18817
0.03007 2395.09251
0.11377
548.45500
0.11952
118.75498
Type II SS
<.0001
F Value
Pr > F
405.02
67.14
2.42
12.92
122.66
28.09
6.08
<.0001
<.0001
0.1202
0.0004
<.0001
<.0001
0.0139
Bounds on condition number: 5.7236, 114.04
--------------------------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1500 level.
All variables have been entered into the model.
Summary of Stepwise Selection
Step
1
2
3
4
5
6
Variable
Entered
Maximum Temperature
Relative Humidity 2PM
Pan Evaporation
Rainfall
Bright Sunshine
Minimum Temperature
Number
Partial
Model
Variables In R-Square R-Square C (p)
1
2
3
4
5
6
0.7151
0.0509
0.0144
0.0066
0.0026
0.0009
0.7151
0.7660
0.7804
0.7870
0.7896
0.7904
F Value
208.327 1488.50
67.4853 128.81
29.1147
38.72
12.5982
18.28
7.4216
7.16
7.0000
2.42
Pr > F
<.0001
<.0001
<.0001
<.0001
0.0077
0.1202
APPENDIX A(e)
Procedure followed for Stepwise Regression Analysis for
Dependent Variable: Relative Humidity at 2 PM
Stepwise Selection: Step 1
Variable Maximum Temperature Entered: R-Square = 0.6899 and C (p) = 581.3400
Analysis of Variance
Source
Model
Error
Corrected Total
DF
Sum of
Squares
1
593
594
79294
35636
114930
Variable
Parameter
Estimate
Intercept
Maximum Temperature
207.83164
-4.27851
Mean
Square
F Value
Pr > F
79294
1319.49
60.09429
<.0001
Standard
Error
3.93313
0.11778
Type II SS
167795
79294
F Value
Pr > F
2792.20
1319.49
<.0001
<.0001
Bounds on condition number: 1, 1
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 2
Variable Minimum Temperature Entered: R-Square = 0.7698 and C (p) = 281.4895
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
592
594
88469
26460
114930
44235
44.69677
F Value
Pr > F
989.66
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Maximum Temperature
Minimum Temperature
156.25954
-4.58359
2.54574
4.94592
0.10379
0.17768
44614
87175
9175.42530
998.16
1950.36
205.28
<.0001
<.0001
<.0001
Bounds on condition number: 1.0439, 4.1758
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Relative humidity 7AM Entered: R-Square = 0.8076 and C (p) = 140.5917
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
3
591
594
92813
22117
114930
30938
37.42265
F Value
Pr > F
826.71
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Maximum Temperature
Minimum Temperature
Relative humidity 7AM
61.95949
-2.94557
2.27424
0.53384
9.85360
0.17926
0.16452
0.04955
1479.65519
10104
7150.90949
4343.70133
39.54
270.00
191.09
116.07
<.0001
<.0001
<.0001
<.0001
Bounds on condition number: 3.7196, 25.149
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 4
Variable Bright Sunshine Entered: R-Square = 0.8324 and C (p) = 48.8003
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
4
590
594
95664
19266
114930
Mean
Square
23916
32.65388
F Value
Pr > F
732.41
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
Intercept
Maximum Temperature
Minimum Temperature
Relative humidity 7AM
Bright Sunshine
61.71801
-2.34854
1.81464
0.53482
-1.31787
9.20442
0.17923
0.16136
0.04629
0.14104
1468.13280 44.96
5606.88575 171.71
4129.69912 126.47
4359.54612 133.51
2850.99934 87.31
F Value
Pr > F
<.0001
<.0001
<.0001
<.0001
<.0001
Bounds on condition number: 4.2612, 42.321
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 5
Variable Rainfall Entered: R-Square = 0.8424 and C (p) = 12.9769
Analysis of Variance
Source
DF
Model
Error
Corrected Total
5
589
594
Sum of
Squares
96814
18116
114930
Mean
Square
19363
30.75732
F Value
Pr > F
629.53
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Maximum Temperature
Minimum Temperature
Relative humidity 7AM
Rainfall
Bright Sunshine
49.13831
-2.09240
1.76062
0.55662
0.02307
-1.03245
9.16702
0.17892
0.15685
0.04506
0.00377
0.14462
883.75694
4206.57361
3875.11862
4692.75771
1149.72761
1567.47160
28.73
136.77
125.99
152.57
37.38
50.96
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Bounds on condition number: 4.5083, 62.612
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 6
Variable Pan Evaporation Entered: R-Square = 0.8445 and C (p) = 7.0000
Analysis of Variance
Sum of
Mean
Source
DF
Squares
Square
F Value Pr > F
Model
Error
Corrected Total
Variable
6
588
594
97056
17874
114930
Parameter
Estimate
16176
30.39725
Standard
Error
532.15
<.0001
Type II SS
F Value
Pr > F
Intercept
46.85286
9.14906
797.17553
26.23
<.0001
Maximum Temperature
-1.85133
0.19729
2676.73218
88.06
<.0001
Minimum Temperature
1.74419
0.15604
3797.88959
124.94
<.0001
Relative humidity 7AM
0.51835
0.04680
3728.50755
122.66
<.0001
Rainfall
0.02415
0.00377
1246.78191
41.02
<.0001
Pan Evaporation
-0.40761
0.14432
242.47563
7.98
0.0049
Bright Sunshine
-1.02706
0.14379
1550.87910
51.02
<.0001
Bounds on condition number: 5.5465, 102.7
--------------------------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1500 level.
All variables have been entered into the model.
Summary of Stepwise Selection
Step
1
2
3
4
5
6
Variable
Entered
Maximum Temperature
Minimum Temperature
Relative humidity 7AM
Bright Sunshine
Rainfall
Pan Evaporation
Number
Partial
Model
Variables In R-Square R-Square
1
2
3
4
5
6
0.6899
0.0798
0.0378
0.0248
0.0100
0.0021
0.6899
0.7698
0.8076
0.8324
0.8424
0.8445
C (p)
F Value Pr > F
581.340 1319.49 <.0001
281.489
205.28 <.0001
140.592
116.07 <.0001
48.8003
87.31 <.0001
12.9769
37.38 <.0001
7.0000
7.98 0.0049
APPENDIX A(f)
Procedure followed for Stepwise Regression Analysis for
Dependent Variable: Pan Evaporation
Stepwise Selection: Step 1
Variable Maximum Temperature Entered: R-Square = 0.6528 and C (p) = 72.6197
Analysis of Variance
Source
Model
Error
Corrected Total
DF
1
593
594
Sum of
Squares
Mean
Square
3055.56776
1625.05738
4680.62514
3055.56776
2.74040
F Value Pr > F
1115.01
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Maximum Temperature
-22.73697
0.83988
0.83990
0.02515
2008.26447
3055.56776
732.84
1115.01
<.0001
<.0001
Bounds on condition number: 1, 1
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 2
Variable Relative humidity 7AM Entered: R-Square = 0.6846 and C (p) = 13.8232
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
592
594
3204.44481
1476.18034
4680.62514
1602.22240
2.49355
Variable
Parameter
Estimate
Intercept
Maximum Temperature
Relative humidity 7AM
-4.47870
0.54616
-0.09767
Standard
Error
2.49508
0.04495
0.01264
F Value
Pr > F
642.55
<.0001
Type II SS
F Value
Pr > F
8.03440
368.11018
148.87705
3.22
147.63
59.70
0.0732
<.0001
<.0001
Bounds on condition number: 3.5101, 14.04
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Rainfall Entered: R-Square = 0.6876 and C (p) = 10.1554
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
3
591
594
3218.32405
1462.30110
4680.62514
Mean
Square
1072.77468
2.47428
F Value
Pr > F
433.57
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
Intercept
Maximum Temperature
Relative humidity 7AM
Rainfall
-5.95940
0.58167
-0.09613
0.00237
2.56285
0.04722
0.01261
0.00100
13.37857
375.44478
143.85708
13.87924
F Value
Pr > F
5.41
151.74
58.14
5.61
0.0204
<.0001
<.0001
0.0182
Bounds on condition number: 3.9035, 26.17
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 4
Variable Relative Humidity 2PM Entered: R-Square = 0.6922 and C (p) = 3.4119
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
4
590
594
3239.73488
1440.89027
4680.62514
809.93372
2.44219
Variable
Intercept
Maximum Temperature
Relative humidity 7AM
Relative Humidity 2PM
Rainfall
Parameter
Estimate
-4.10879
0.53040
-0.07685
-0.02902
0.00351
F Value Pr > F
331.64
<.0001
Standard
Error
Type II SS
F Value
Pr > F
2.62176
0.05001
0.01412
0.00980
0.00107
5.99821
274.75956
72.38297
21.41083
26.51822
2.46
112.51
29.64
8.77
10.86
0.1176
<.0001
<.0001
0.0032
0.0010
Bounds on condition number: 4.5211, 59.696
--------------------------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1500 level.
No other variable met the 0.1500 significance level for entry into the model.
Summary of Stepwise Selection
Step
Variable
Entered
1 Maximum Temperature
2 Relative humidity 7AM
3 Rainfall
4 Relative Humidity 2PM
Number
Partial
Variables In R-Square
1
2
3
4
0.6528
0.0318
0.0030
0.0046
Model
R-Square
0.6528
0.6846
0.6876
0.6922
C (p)
F Value Pr > F
72.6197 1115.01 <.0001
13.8232
59.70 <.0001
10.1554
5.61 0.0182
3.4119
8.77 0.0032
APPENDIX A(g)
Procedure followed for Stepwise Regression Analysis for
Dependent Variable: Bright Sunshine
Stepwise Selection: Step 1
Variable Relative Humidity 2PM Entered: R-Square = 0.4143 and C (p) = 55.0909
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
1
593
594
1051.50791
1486.50536
2538.01328
F Value Pr > F
1051.50791
2.50675
419.47
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Relative Humidity 2PM
12.76168
-0.09565
0.31239
0.00467
4183.56363
1051.50791
1668.92
419.47
<.0001
<.0001
Bounds on condition number: 1, 1
-------------------------------------------------------------------------------------------------Stepwise Selection: Step 2
Variable Rainfall Entered: R-Square = 0.4518 and C (p) = 15.7398
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
592
594
1146.64727
1391.36601
2538.01328
573.32364
2.35028
Variable
Parameter
Estimate
Intercept
Relative Humidity 2PM
Rainfall
11.98536
-0.07684
-0.00650
Standard
Error
0.32616
0.00540
0.00102
F Value
Pr > F
243.94
<.0001
Type II SS
F Value
Pr > F
3173.63613
475.26076
95.13936
1350.32
202.21
40.48
<.0001
<.0001
<.0001
Bounds on condition number: 1.4277, 5.7107
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 3
Variable Minimum Temperature Entered: R-Square = 0.4589 and C (p) = 9.9418
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
3
591
594
1164.58858
1373.42470
2538.01328
Mean
Square
388.19619
2.32390
F Value
Pr > F
167.05
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Minimum Temperature
Relative Humidity 2PM
Rainfall
14.57439
-0.11087
-0.07521
-0.00660
0.98662
0.03990
0.00540
0.00102
507.10468
17.94131
450.04584
97.97309
218.21
7.72
193.66
42.16
<.0001
0.0056
<.0001
<.0001
Bounds on condition number: 1.4446, 11.66
------------------------------------------------------------------------------------------------Stepwise Selection: Step 4
Variable Relative humidity 7AM Entered: R-Square = 0.4618 and C (p) = 8.6718
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
4
590
594
1172.11215
1365.90113
2538.01328
293.02804
2.31509
F Value
Pr > F
126.57
<.0001
Variable
Parameter
Estimate
Standard
Error
Type II SS
F Value
Pr > F
Intercept
Minimum Temperature
Relative humidity 7AM
Relative Humidity 2PM
Rainfall
12.90426
-0.08534
0.02245
-0.08930
-0.00627
1.35205
0.04227
0.01246
0.00950
0.00103
210.88565
9.43668
7.52357
204.69657
85.77070
91.09
4.08
3.25
88.42
37.05
<.0001
0.0439
0.0719
<.0001
<.0001
Bounds on condition number: 4.4776, 43.059
--------------------------------------------------------------------------------------------------
Stepwise Selection: Step 5
Variable Maximum Temperature Entered: R-Square = 0.4668 and C (p) = 5.2364
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
5
589
594
1184.61776
1353.39551
2538.01328
Parameter
Estimate
Variable
Intercept
Maximum Temperature
Minimum Temperature
Relative humidity 7AM
Relative Humidity 2PM
Rainfall
7.86562
0.12606
-0.13358
0.03621
-0.07713
-0.00597
Mean
Square
236.92355
2.29779
Standard
Error
2.54542
0.05404
0.04691
0.01374
0.01080
0.00103
F Value
Pr > F
103.11
<.0001
Type II SS
21.94101
12.50562
18.62797
15.96364
117.10102
76.53762
F Value
Pr > F
9.55
5.44
8.11
6.95
50.96
33.31
0.0021
0.0200
0.0046
0.0086
<.0001
<.0001
Bounds on condition number: 5.8389, 93.791
-------------------------------------------------------------------------------------------------All variables left in the model are significant at the 0.1500 level.
No other variable met the 0.1500 significance level for entry into the model.
Summary of Stepwise Selection
Step
1
2
3
4
5
Variable
Entered
Relative Humidity 2PM
Rainfall
Minimum Temperature
Relative humidity 7AM
Maximum Temperature
Number
Partial
Model
Variables In R-Square R-Square
1
2
3
4
5
0.4143
0.0375
0.0071
0.0030
0.0049
0.4143
0.4518
0.4589
0.4618
0.4668
C (p)
F Value
55.0909 419.47
15.7398
40.48
9.9418
7.72
8.6718
3.25
5.2364
5.44
Pr > F
<.0001
<.0001
0.0056
0.0719
0.0200
APPENDIX B(a)
Program for developing ANN model for Rainfall
% Training for Rainfall
% Creating the feed forward network
net=newff([0.0236 0.0432;0.0172 0.0335;0.038 0.098; 0.016 0.092;0.0000 0.0185;
0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','logsig'},'trainscg');
% Enter the pattern and target
p = [0.0381 0.0265 0.074 0.044 0.0122 0.0066
0.0347 0.0233 0.080 0.052 0.0065 0.0069
…………………………..…………………….
0.0328 0.024 0.093 0.065 0.0041 0.0088
0.032 0.0229 0.094 0.064 0.0034 0.0078];
p = p';
t = [0.0056
0.0738
………
0.0272
0.0112];
t = t';
% set initial weights&biases
net.IW{1,1} = [-0.8754 0.5691 -0.5960 -0.7580 -0.2941 -0.5808
0.6511 -0.2726 -0.0521 0.5720 -0.6031 0.5401
0.5375 0.3341 0.0824 0.2887 0.0589 -0.9999
0.3523 0.2958 0.9987 0.8578 0.1649 0.2789
-0.4838 -0.3115 0.6352 0.8444 0.6760 -0.9644
0.1908 -0.8439 0.2789 -0.5051 0.8061 0.1336
0.0303 0.3956 0.0722 -0.1205 0.0424 0.9686
-0.8503 0.9714 -0.9321 0.3208 -0.3348 0.1436];
net.LW{2,1} =[ 0.1669 0.6254 0.5631 0.0461 -0.0918 -0.2504 0.0376 -0.1060
0.3322 0.6177 -0.2275 -0.1279 0.0275 -0.0126 0.2725 0.0139
-0.2849 0.7109 -0.1916 -0.9661 0.9048 0.3261 0.6344 -0.6618
0.6657 -0.0168 0.7077 -0.1694 0.2635 0.2823 -0.1229 0.8654
0.5925 0.8494 -0.8233 0.2653 0.2465 0.1414 0.6385 0.1062
0.0052 -0.2024 -0.2001 0.9764 0.1687 -0.6857 0.8503 0.7989
0.9569 0.7751 0.4519 0.2634 0.2132 -0.5242 -0.2326 0.4896
-0.0316 0.0919 -0.7745 -0.4365 -0.0007 -0.4960 -0.0084 -0.9707
0.2303 0.5802 0.8999 -0.6458 0.7177 -0.5838 0.9549 -0.4680
0.2209 -0.2903 -0.6861 0.1895 -0.2585 0.2821 -0.6412 -0.0255];
net.LW{3,2} =
[-0.8486 -0.6747 0.3305 0.3452 -0.2362 0.7200 0.3338 0.3797 -0.2056 -0.3913
0.8301 -0.6520 0.0935 0.4716 -0.1065 -0.1097 0.3176 -0.6392 -0.8581 -0.0295
-0.0533 -0.0064 0.8642 0.6741 0.8208 -0.4858 -0.0578 0.1573 -0.2212 -0.5869
-0.2314 0.1757 0.7960 0.7165 0.8207 -0.0764 0.0396 0.3412 0.9951 -0.1821
-0.6669 -0.7202 -0.3719 -0.8201 -0.6210 0.4288 -0.6957 0.3734 -0.6647 -0.2793
0.4430 -0.1451 -0.1824 -0.2177 0.2350 -0.7160 -0.6020 -0.8999 -0.2222 0.6489
0.0563
0.9196
-0.1114
-0.5283
-0.4075
-0.5730
-0.5688
0.4672
-0.2678
0.1231
0.6867
-0.6748
0.6196
0.2423
0.0344
-0.3697
-0.3770
0.1384
-0.6303
-0.1273
0.9198
0.2298
0.0153
-0.6400
-0.5125
0.8340
0.5722
-0.9242
0.0430
0.8806
-0.5146
0.9841
-0.1578
0.0088
0.4637
0.6674
0.4519 0.8357 0.9454 0.8613
0.6740 0.6118 -0.6070 0.3347
0.4817 0.5853 0.4767 -0.6444
-0.7207 0.0067 -0.2896 0.1226
-0.8477 -0.5013 0.2867 0.4182
-0.4899 0.4453 0.5295 -0.4088];
net.LW{4,3} = [-0.2489 0.1706 -0.8142 0.9684 0.7488 -0.1751
0.5437 -0.0984 0.3068 -0.3575];
net.b{1} = [0.3660
-0.4437
-0.4755
-0.9644
0.9308
-0.6543
-0.2085
-0.3939];
net.b{2} = [ -0.0064
-0.4747
0.9436
-0.9568
-0.9078
0.6698
0.4679
0.2685
0.5845
-0.5981];
net.b{3} = [ 0.3816
-0.2013
0.2325
-0.6459
-0.3839
0.4836
-0.1686
0.1700
-0.3120
-0.0293
-0.6155
-0.0767];
net.b{4}=[0.0487];
% set the training parameters
net.trainParam.show=100;
0.8994
0.8689
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.003;
% train the network
net=train(net,p,t);
% simulate the trained network
y=sim(net,p)
APPENDIX B(b)
Program for developing ANN model for Maximum Temperature
% Training for Maximum temperature
% Creating the feed forward network
net = newff([0.0000 0.4432;0.0172 0.0340;0.0380 0.0980;0.0160 0.0920;0.0000
0.0185;0.0007 0.0116], [8,10,12,1],{'tansig','tansig','tansig','tansig'},'trainscg');
% Enter the pattern and target
p = [0.0056 0.0265 0.0740 0.0440 0.0122 0.0066
0.0738 0.0233 0.0800 0.0520 0.0065 0.0069
…….…………………………………………..….
0.0272 0.0240 0.0930 0.0650 0.0041 0.0088
0.0112 0.0229 0.0940 0.0640 0.0034 0.0078];
p = p';
t = [0.0381
0.0347
..…….
0.0328
0.032];
t = t';
% set initial weights&biases
net.IW{1,1} = [0.1830 0.9827 -0.8075 0.9702 -0.2300 -0.1041
0.5743 0.6219 -0.7227 0.3095 -0.1905 0.4341
0.9856 -0.7181 0.2101 0.5207 -0.6333 0.0682
-0.9022 -0.7562 0.0774 -0.2469 -0.1813 -0.7190
-0.6815 -0.3828 -0.3076 0.6001 -0.4974 -0.1432
0.4774 -0.1611 -0.9719 0.5146 0.7935 0.6651
-0.6774 0.9253 -0.9237 0.3601 0.5258 0.8976
0.4031 0.8722 0.6227 0.5242 0.7912 0.1363];
net.LW{2,1} = [0.7155 -0.4739 0.8768 -0.2460 -0.1839 -0.0928 0.8877 0.8881
-0.7649 -0.3336 -0.7183 -0.0781 -0.9393 0.4054 -0.8248 0.7563
-0.0680 0.3087 0.3761 0.3995 0.4740 -0.4405 -0.8136 -0.7103
0.3111 0.8226 -0.5903 -0.3393 -0.5643 0.8982 0.7213 0.7331
0.2030 -0.8793 -0.6909 0.0901 -0.1530 0.3457 0.8256 0.0985
0.5692 0.9367 0.4832 -0.5163 -0.0804 -0.3912 -0.7015 0.8838
0.1234 -0.7528 -0.5495 -0.5984 -0.5395 -0.9752 -0.0063 0.9674
-0.3328 -0.3471 0.7951 0.4012 0.8578 -0.0749 -0.0243 0.8066
-0.1534 -0.7072 -0.8250 0.7536 -0.6620 -0.4152 0.7060 0.6676
0.0302 -0.0263 -0.6002 -0.6233 0.6482 -0.9846 -0.2759 0.4045];
net.LW{3,2} =
[-0.7892 0.7948 -0.5644 0.1354 -0.3702 0.8356 -0.5480 0.1078 0.4876 -0.4043
0.5195 -0.4049 0.3418 0.7638 0.5408 0.1434 0.3398 -0.6380 0.5458 0.7920
0.4372 -0.9754 -0.0865 0.4747 -0.0022 0.0400 -0.7902 0.3215 0.9271 -0.5078
0.8674 0.9659 0.0349 0.5260 -0.0902 0.7690 -0.3179 0.4917 -0.2053 0.4285
-0.6505 0.9109 0.1923 0.2575 -0.7420 -0.2751 0.8238 -0.5874 0.4226 -0.0624
0.0962 0.0151 0.3048 0.2957 -0.6096 0.2236 0.0455 0.8854 -0.4382 0.6825
0.6181 0.0326 -0.7992 -0.2536 0.4944 -0.3513 0.3241 0.8275 -0.3580 0.8405
-0.2734 0.4844 0.2711 -0.7802 -0.6452 -0.1765 -0.3122 -0.7835 -0.0471 -0.7535
0.8305 0.2894 -0.1686 0.3789 0.2623 -0.4946 -0.7184 -0.3682 -0.7482 0.6708
0.3940 -0.7313 0.0449 0.0082 0.3848 -0.4528 -0.6386 0.4990 -0.8370 -0.6484
0.4221 -0.4647 -0.9156 -0.8721 -0.3680 -0.4029 -0.6183 0.3187 -0.5224 -0.3911
0.8366 -0.5711 0.9745 0.3079 -0.1101 0.1189 0.9379 -0.2884 -0.1105 0.2908];
net.LW{4,3} = [-0.0692 0.1473 0.1255 0.0563 0.2041
-0.0443 0.1328 -0.1395 0.1926];
net.b{1} = [-0.2063
-0.9503
0.0815
0.0754
0.5538
0.1737
-0.3515
-0.0912];
net.b{2} = [-0.9913
0.5159
0.1396
-0.5876
-0.1088
0.2714
0.7654
-0.9691
-0.4134
0.9016];
net.b{3} = [0.8224
-0.4396
-0.1421
-0.8167
0.5647
-0.0764
0.1147
-0.6510
0.8410
0.1498
0.4667
0.8328];
net.b{4} = [-0.0534];
% set the training parameters
net.trainParam.show=100;
0.0952
0.1070
0.1474
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.00000117083;
% train the network
net = train(net,p,t);
% simulate the trained network
y = sim(net,p)
APPENDIX B(c)
Program for developing ANN model for Minimum Temperature
% Training for Minimum temperature
% Creating the feed forward network
net=newff([0.0236 0.0432;0.0000 0.4432;0.038 0.098; 0.016 0.092;0.0000 0.0185;
0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','tansig'},'trainscg');
% Enter the pattern and target
p = [ 0.0381 0.0056 0.074 0.044 0.0122 0.0066
0.0347 0.0738 0.08 0.052 0.0065 0.0069
………………………………
0.0328 0.0272 0.093 0.065 0.0041 0.0088
0.0320 0.0112 0.094 0.064 0.0034 0.0078];
p = p';
t = [0.0265
0.0233
………
0.024
0.0229];
t= t';
% set initial weights & biases
net.IW{1,1} = [0.5536 0.0430 0.2723 -0.3851 0.0014 0.0443
0.9695 0.4651 0.5682 0.2038 0.1346 0.5991
0.5100 -0.7832 0.7564 0.2457 -0.4487 -0.1632
0.9535 -0.7101 -0.6487 0.1373 -0.7912 -0.1160
-0.5104 -0.5715 -0.0948 0.2581 0.5745 -0.6275
-0.2897 -0.4698 -0.2563 0.1590 0.3275 -0.2306
0.6673 0.1785 -0.8307 0.7457 0.8579 0.9837
-0.0878 -0.5797 0.6688 -0.0553 0.0985 0.9830];
net.LW{2,1} = [-0.6010 0.0488 0.4981 0.1298 -0.4060 0.4664 0.1657 -0.6048
-0.1352 -0.8005 -0.9650 0.5459 -0.3057 -0.8117 -0.4227 0.2894
0.8353 0.0013 0.7041 0.7583 -0.4720 -0.9661 -0.3515 0.7300
-0.6360 -0.2446 -0.9679 -0.3216 0.7613 -0.4419 -0.9072 0.0408
0.9010 0.4517 0.0144 -0.1730 0.2443 -0.7843 0.7669 0.1533
0.3172 0.3161 -0.6449 0.0063 -0.4995 0.0023 0.8137 0.6739
0.9079 0.9721 0.3042 -0.7417 0.6953 0.5513 -0.6486 -0.4126
0.0550 -0.6494 -0.6131 -0.7381 0.3837 -0.7441 0.0888 0.9754
0.8614 -0.6895 0.8226 0.0267 -0.4730 -0.8976 -0.4635 -0.6496
-0.3869 0.2148 -0.2552 0.7695 0.5063 0.6647 0.3153 0.4653];
net.LW{3,2}=
[0.3018 -0.9096 -0.1047 0.6253 0.4674 -0.5164 0.1890 0.8921 -0.1206 0.8019
0.6692 -0.2640 -0.2445 -0.9090 0.3149 0.7342 0.4179 0.5778 -0.8540 -0.9360
-0.3564 -0.2943 0.0032 -0.1677 0.5503 0.4887 0.8478 0.2316 0.2608 0.0614
-0.9227 -0.6927 -0.5703 -0.8114 -0.0176 -0.8090 -0.7305 0.3228 0.3272 -0.6274
0.0439 0.8080 -0.0190 -0.4571 0.3789 0.0237 -0.9561 -0.3868 0.4882 -0.8610
0.3212 -0.7708 0.2412 0.7825 -0.0516 -0.1919 0.0745 -0.4586 0.1078 -0.7197
-0.4023 0.0042 -0.9565
-0.9942 -0.3228 0.7342
0.5418 -0.5751 0.4136
0.1952 0.8292 0.2899
-0.8530 -0.6065 -0.0009
-0.7381 0.3235 -0.6587
0.5778
-0.2424
0.5352
0.0719
0.0653
0.1451
0.0245
0.8829
0.5687
-0.5043
0.3079
-0.5644
0.5371
-0.4090
-0.6623
-0.1085
0.0607
0.6793
-0.9470
-0.9720
-0.6044
0.5173
0.4455
-0.7501
net.LW{4,3} = [0.5012 -0.1792 -0.1174 -0.0282 0.3103
0.6469 0.0914 0.5573 -0.1817];
net.b{1} = [ -0.8705
-0.7203
-0.6073
0.1588
0.2553
0.9526
-0.4646
-0.0111];
net.b{2} = [ 0.9235
0.5283
-0.0836
0.6827
-0.3115
0.2728
0.4533
-0.1677
0.4202
-0.8918];
net.b{3} = [-0.8857
-0.5030
0.1551
0.8718
0.5743
-0.0888
-0.0141
-0.5285
0.8396
0.3056
-0.5666
-0.8621];
net.b{4} = [0.3720];
% set the training parameters
net.trainParam.show=100;
-0.9675
0.2503
0.4483
0.4314
0.0003
0.5707
0.5835
0.4133
0.5935
0.2180
-0.1015
0.1169
-0.8114
-0.6446
-0.0937
0.4337
-0.5388
-0.3810];
0.0855 -0.0692 -0.3121 -
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.000001717;
% train the network
net=train(net,p,t);
% simulate the trained network
y=sim(net,p)
APPENDIX B(d)
Program for developing ANN model for Relative Humidity at 7 AM
% Training for Relative Humidity at 7 AM
% Creating the feed forward network
net = newff([0.0236 0.0432; 0.0172 0.0335;0.0000 0.4432;0.016 0.092; 0.0000 0.0185;
0.0007 0.0116], [8,10,12,1],{'tansig','tansig','tansig','logsig'},'trainscg');
% Enter the pattern and target
p = [ 0.0381 0.0265 0.0056 0.044 0.0122 0.0066
0.0347 0.0233 0.0738 0.052 0.0065 0.0069
…………………………………………………..
0.0328 0.0240 0.0272 0.065 0.0041 0.0088
0.0320 0.0229 0.0112 0.064 0.0034 0.0078];
p = p';
t = [0.074
0.080
…..…
0.093
0.094];
t = t';
% set initial weights & biases
net.IW{1,1} = [0.3993 0.0559 -0.5671 -0.5619 0.1204 0.4636
0.6472 0.8795 -0.3328 0.0796 0.8470 -0.5849
0.1825 0.0479 0.6551 -0.0231 -0.9869 -0.0046
-0.0719 0.1821 -0.4123 -0.0032 0.1074 -0.3365
0.0785 -0.3795 0.4819 -0.1454 0.8002 0.3035
0.6397 0.5071 0.4646 0.1765 -0.2689 0.4831
0.0287 0.0059 0.0119 0.0260 -0.5120 0.8247
-0.2506 0.3626 0.9474 -0.0514 -0.4404 -0.4746];
net.LW{2,1} = [0.9260 0.0727 0.2154 0.8846 0.1803 0.1020 -0.6001 0.9212
-0.0342 -0.8027 0.5245 -0.2896 -0.0707 -0.0686 0.1523 0.4920
-0.3674 -0.6659 -0.5690 0.9120 -0.3458 -0.7657 0.7953 -0.6038
0.7335 -0.2760 0.2560 0.1025 -0.7507 -0.5497 0.4720 0.0999
0.1976 0.5876 0.1307 0.1320 0.0643 0.1058 0.7823 0.6258
-0.7387 -0.8299 -0.7319 0.7221 -0.9057 0.3150 0.5930 -0.7231
0.1501 -0.3159 -0.0717 0.5868 -0.2530 -0.6144 0.6100 -0.3937
0.7887 -0.7121 -0.5279 0.2268 -0.8491 -0.2602 -0.3849 0.7716
0.1960 -0.2286 -0.3955 0.8952 0.6934 0.4108 -0.0136 -0.2832
-0.0316 -0.1811 0.8871 -0.8321 -0.7273 -0.3724 -0.7060 -0.6157];
net.LW{3,2} =
[0.9303 -0.0139 0.0380 0.4166 0.7025 0.5978 -0.4936 -0.2215 -0.6515 0.8029
-0.1378 -0.6000 0.2000 -0.0268 0.6782 -0.2308 0.4191 0.0127 0.0514 0.5411
-0.1286 -0.7915 0.7275 -0.4092 -0.1055 0.6563 -0.1320 0.0308 -0.5607 0.4708
-0.0443 -0.2298 -0.4990 0.8830 0.8076 -0.3750 -0.1501 0.7050 0.8523 -0.6972
0.7379 0.3482 -0.5548 -0.5223 -0.1830 -0.6471 -0.4609 -0.8649 -0.5113 0.5506
-0.5910 0.3072 -0.2210 0.4513 -0.3957 0.8798 -0.2536 0.3912 -0.7545 0.2224
0.5711
0.3076
0.7320
0.5271
0.8774
-0.6874
-0.9355
0.7616
-0.4140
0.7713
-0.1706
-0.3568
0.1034
-0.3087
0.2274
-0.4005
0.0936
-0.0178
-0.2205
0.4447
-0.4210
-0.3363
0.2312
-0.2641
-0.8573
-0.7018
-0.6343
0.6316
-0.2708
0.6061
-0.5789
0.2237
0.7362
-0.1129
0.7386
-0.0261
0.3808
0.3231
0.7188
-0.6021
0.3247
-0.1514
net.LW{4,3}= [0.9768 -0.8569 -0.9742 0.4239 0.5066
0.2937 0.0338 -0.5995 -0.4561];
net.b{1} = [-0.0096
-0.1848
-0.9375
0.2745
-0.1984
-0.9723
0.2074
0.6717];
net.b{2} = [-0.1418
0.2738
0.3458
-0.6403
-0.2777
0.0127
0.8046
0.0245
0.5307
0.0546];
net.b{3} = [-0.8557
0.5106
0.2926
0.8038
-0.5511
-0.1513
0.1840
-0.2164
0.7711
0.1041
0.7744
0.8390];
net.b{4} = [ -0.7024];
% set the training parameters
0.5864
-0.0388
0.3095
0.7165
0.3825
0.5295
-0.4221
0.8255
-0.0006
0.7476
-0.5177
0.3511
0.6002 -0.3358
0.5794
0.7911
0.6939
-0.2067
0.5844
-0.1898];
0.1514
net.trainParam.show=100;
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.0000166;
% train the network
net =train(net,p,t);
% simulate the trained network
y = sim(net,p)
APPENDIX B(e)
Program for developing ANN model for Relative Humidity at 2 PM
% Training for Relative Humidity at 2 PM
% Creating the feed forward network
net = newff([0.0236 0.0432; 0.0172 0.0335;0.038 0.098;0.0000 0.4432;0.0000 0.0185;
0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','tansig'},'trainscg');
% Enter the pattern and target
p = [0.0381 0.0265 0.0740 0.0056 0.0122 0.0066
0.0347 0.0233 0.0800 0.0738 0.0065 0.0069
…………………………………………………..
0.0328 0.0240 0.0930 0.0272 0.0041 0.0088
0.0320 0.0229 0.0940 0.0112 0.0034 0.0078];
p = p';
t = [0.044
0.052
……..
0.065
0.064];
t = t';
% set initial weights&biases
net.IW{1,1} = [0.7359 0.1843 0.9574 0.0541 -0.5302 0.1915
-0.1483 -0.3240 0.5386 -0.6799 -0.2885 0.4064
-0.0541 0.8281 -0.4575 -0.1265 -0.6093 0.3627
0.2008 0.0897 0.7760 -0.4324 -0.3566 0.3755
0.7916 -0.1845 0.6725 -0.5910 0.9761 -0.5051
0.1901 0.5141 -0.5460 0.3930 -0.2015 -0.4294
-0.3855 0.9105 -0.0842 0.0241 0.6178 -0.1599
0.7207 0.5913 0.8029 -0.2480 0.5532 -0.3286];
net.LW{2,1} = [ -0.1515 0.0597 0.3223 0.0325 0.4205 0.0592 0.0392 0.9734
-0.5860 0.8262 0.1230 -0.0356 0.2368 0.3968 -0.8404 -0.6009
0.4300 -0.7895 -0.4803 0.2303 -0.6325 -0.0268 0.9145 0.2492
0.2832 0.7708 0.6287 0.4624 -0.9252 -0.2994 -0.8239 0.5658
-0.0173 -0.0276 -0.5713 -0.7164 0.0090 0.1134 0.2793 0.6552
0.3992 -0.2867 0.8809 -0.5434 0.7010 -0.6220 -0.1316 0.8412
-0.1605 -0.0369 0.3897 -0.4651 0.7298 -0.2298 0.5777 -0.4485
-0.0646 -0.6122 0.1675 0.5054 0.1979 -0.4009 -0.0385 -0.0661
-0.2886 0.5084 0.1057 0.4251 -0.4550 0.7736 0.0845 -0.9929
0.6689 0.2743 -0.4505 0.5538 -0.4464 0.3179 0.4715 -0.5543];
net.LW{3,2} =
[0.6800 -0.3204 0.5834 -0.5676 -0.1674 0.6121 0.6261 -0.4545 0.6760 -0.7316
-0.1814 0.5979 0.6621 0.6532 0.6253 -0.5038 0.7562 -0.2141 0.1145 -0.5323
-0.0929 0.7605 0.7157 -0.8474 -0.5593 -0.6711 -0.0027 -0.3229 -0.1120 0.5892
0.1920 -0.6137 -0.3324 0.4786 -0.7747 0.7508 0.3353 -0.8811 -0.5460 -0.7564
0.1314 0.5747 -0.0391 0.6691 -0.7323 -0.4133 0.8818 -0.3188 0.3723 -0.8967
0.1848 -0.2698 0.4607 -0.8323 -0.0265 -0.7108 -0.8618 -0.5132 -0.2076 0.2117
-0.7576
0.8972
-0.8588
0.0226
0.5123
0.4206
0.6499
0.4395
0.5257
0.2483
-0.5329
-0.5751
-0.8528
-0.5422
0.5042
-0.0519
-0.4689
-0.8644
-0.6629
-0.4070
0.0594
0.7665
-0.3846
-0.0663
0.0937
0.4045
-0.2184
0.3992
-0.1855
0.6990
0.6108
-0.8160
-0.4312
-0.0655
-0.8265
-0.1432
-0.0616
0.1586
-0.6882
0.4354
-0.4937
0.2059
net.LW{4,3} = [-0.0766 -0.4366 -0.3554 -0.8038 -0.3609
0.1600 -0.5794 -0.1973 0.0354];
net.b{1} = [-0.9832
0.0068
0.9386
-0.4354
-0.7773
0.4431
0.8719
-0.0728];
net.b{2}= [0.8723
0.5704
-0.0178
-0.3930
-0.0104
0.3372
-0.6547
-0.7064
-0.4553
0.5818];
net.b{3} = [-0.7931
0.4295
0.3078
-0.6710
-0.4580
-0.1164
0.0229
0.5187
-0.0030
0.2545
0.4883
0.7336];
net.b{4} = [-0.1122];
% set the training parameters
net.trainParam.show=100;
0.5420
-0.2606
-0.9036
0.8971
-0.4698
0.8181
-0.6451 -0.1679
0.6038 -0.1116
0.6119 0.0779
0.5170 0.1730
-0.1262 0.8056
0.6722 0.3270];
0.6728 -0.8659 -0.8034
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.00003155;
% train the network
net = train(net,p,t);
% simulate the trained network
y = sim(net,p)
APPENDIX B(f)
Program for developing ANN model for Pan Evaporation
% Training for Pan Evaporation
% Creating the feed forward network
net=newff([0.0236 0.0432;0.0172 0.0335;0.0380 0.0980; 0.016 0.092; 0.0000 0.4432;
0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','logsig'},'trainscg');
% Enter the pattern and target
p = [0.0381 0.0265 0.074 0.044 0.0056 0.0066
0.0347 0.0233 0.08 0.052 0.0738 0.0069
……………….………………………………..
0.0328 0.024 0.093 0.065 0.0272 0.0088
0.032
0.0229 0.094 0.064 0.0112 0.0078];
p = p';
t = [0.0122
0.0065
…….
0.0041
0.0034];
t = t';
% set initial weights&biases
net.IW{1,1} = [ -0.4092 -0.9839 -0.5397 0.2725 0.5089 -0.3987
-0.8912 -0.6087 0.7342 -0.2753 -0.9905 -0.8467
-0.4415 0.8863 0.6940 -0.7260 -0.2904 0.6507
0.7042 -0.4639 -0.2243 -0.8664 -0.7879 -0.1902
-0.4116 0.9310 -0.9683 -0.8323 0.0182 0.4231
0.0729 0.9679 -0.3609 -0.6115 0.0171 0.4061
-0.2921 -0.6636 0.8394 0.8251 -0.8558 0.0283
0.8973 -0.5499 0.6815 -0.1661 -0.9342 0.4265];
net.LW{2,1} = [-0.0233 -0.7616 0.3765 0.8237 -0.2273 -0.6645 -0.6830 0.0294
-0.2323 0.6478 -0.5732 0.0491 -0.8631 -0.6252 -0.1364 -0.0355
0.6712 0.7559 0.7828 -0.9838 -0.1100 -0.1012 0.7689 -0.2783
-0.2397 0.1351 0.9838 -0.7191 0.2285 -0.1063 -0.5203 0.0590
0.8100 -0.6106 -0.5614 -0.5072 0.4814 -0.7074 -0.7250 0.3408
0.5096 -0.5903 0.6014 -0.6320 -0.6054 0.1244 -0.8258 0.0195
-0.8097 -0.6649 0.6742 -0.9578 0.5613 0.7347 0.1129 -0.0748
-0.0109 -0.9295 -0.1454 -0.7372 -0.5355 -0.3679 0.8520 -0.9246
0.6237 -0.6560 0.0031 -0.5621 0.1688 -0.8662 -0.3380 -0.6366
-0.9506 0.0001 -0.6194 0.0373 0.0526 -0.7463 0.2886 -0.2908];
net.LW{3,2} =
[-0.0625 0.5352 0.3186 0.7122 -0.0415 0.5259 -0.7025 -0.0979 0.2204 -0.5865
0.7363 -0.2367 0.0692 0.8259 -0.2239 -0.3696 -0.6591 -0.2831 0.7458 -0.4855
0.1753 -0.6956 0.1714 -0.1048 0.4919 0.1439 -0.7313 0.8375 -0.6558 0.8587
0.1097 -0.2748 0.0113 0.0071 0.1185 -0.7124 0.4206 -0.5631 -0.9303 0.1635
0.3956 -0.8018 -0.5243 0.3055 -0.1133 0.6456 -0.3963 -0.9289 0.4450 0.4664
-0.0344 0.8625 0.4825 0.7005 -0.6820 0.4453 -0.5271 -0.4359 -0.3107 -0.2429
0.0025 -0.7645 0.2320 -0.4138 0.6134
-0.7840 0.0236 0.8105 0.5238 0.3795
-0.7037 0.2739 -0.0718 0.6901 0.8057
-0.8171 -0.1247 0.2110 -0.1482 0.1574
0.0832 -0.6319 -0.0688 -0.0815 0.7360
-0.2731 0.0732 0.6095 0.2183 -0.1434
-0.1513
0.0365
-0.2848
-0.1230
0.0706
0.8660
0.4661
0.1130
-0.6999
0.2671
-0.4168
0.2675
-0.1002
-0.0853
-0.2539
-0.3619
0.4522
-0.1427
0.1569
-0.8366
-0.1903
0.2990
-0.7653
-0.7780
-0.3581
0.9953
0.9215
-0.3592
0.0038
0.1498];
net.LW{4,3} = [0.4256 -0.0068 0.0028 0.0153 0.0247 -0.0619 -0.1291 -0.6879
0.0065 0.2692 0.0086 0.0468];
net.b{1}= [-0.9901
0.1540
-0.0596
0.9989
-0.8190
-0.4594
0.0302
0.3135];
net.b{2} = [-0.7062
0.5665
-0.2220
0.6866
-0.3450
0.4251
-0.7662
-0.0620
0.6925
-0.9648];
net.b{3} = [ 0.8081
-0.5466
-0.1421
-0.8709
-0.4226
0.2599
0.0083
-0.7717
-0.7710
-0.3026
0.4866
-0.8168];
net.b{4} = [ -0.8644];
% set the training parameters
net.trainParam.show=100;
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.00000285;
% train the network
net=train(net,p,t);
% simulate the trained network
y=sim(net,p)
APPENDIX B(g)
Program for developing ANN model for Bright Sunshine
% Training for Bright Sunshine
% Creating the feed forward network
net=newff([0.0236 0.0432; 0.0172 0.0335; 0.038 0.098;0.016 0.0920;0.0000 0.0185;
0.0000 0.4432],[8,10,12,1],{'tansig','tansig','tansig','purelin'},'trainscg');
% Enter the pattern and target
p = [0.0381 0.0265 0.074 0.044 0.0122 0.0056
0.0347 0.0233 0.08 0.052 0.0065 0.0738
…………………………………..
0.0328 0.024 0.093 0.065 0.0041 0.0272
0.032 0.0229 0.094 0.064 0.0034 0.0112];
p = p';
t = [0.0066
0.0069
……..
0.0088
0.0078];
t = t';
% set initial weights&biases
net.IW{1,1} = [-0.8535 0.5669 -0.1547 -0.5520 -0.7675 0.4131
0.1915 -0.6454 -0.0462 0.3010 0.5427 0.9058
-0.2946 -0.9411 -0.2110 -0.2481 0.9858 -0.8979
-0.3001 -0.2426 0.4265 0.8768 0.2576 0.6613
-0.5004 0.0477 0.9804 0.8947 -0.7140 0.5993
0.0643 -0.8281 0.9127 -0.5197 0.1732 -0.8671
-0.3453 -0.5308 0.6877 0.0677 -0.6298 -0.6339
0.5272 0.6893 -0.9391 -0.9730 0.9129 0.3312];
net.LW{2,1} = [-0.0165 -0.8781 0.1787 0.4896 0.6519 0.3350 0.0164 0.5407
0.8321 0.9109 -0.5751 0.0363 0.3604 -0.4380 -0.0497 0.1764
0.6101 -0.1766 -0.1908 0.8318 0.6970 -0.1683 0.0181 0.5219
0.5407 -0.1219 -0.5670 0.4181 0.3672 -0.4965 -0.9079 -0.9077
0.7888 -0.9364 0.6827 -0.2635 0.2550 -0.7381 0.7438 -0.3497
-0.8629 -0.9476 -0.8186 0.4906 -0.4659 0.3562 0.4617 -0.5893
-0.3644 -0.6823 0.2343 -0.2374 -0.7231 -0.8462 -0.0139 0.2752
0.0644 0.3444 -0.3105 -0.6914 0.4428 0.5727 0.6512 -0.7346
0.1995 0.2863 -0.8312 0.7299 0.7664 0.0098 -0.0303 0.1676
0.3587 -0.1578 0.7947 -0.7489 0.6266 0.9098 -0.5373 -0.7023];
net.LW{3,2} =
[-0.9401 -0.5537 -0.1126 -0.6935 -0.2994 -0.4936 0.0817 0.2900 0.8568 -0.3653
-0.7923 -0.0465 -0.5556 -0.2343 0.8661 0.2658 -0.9185 -0.0165 -0.3367 0.6481
-0.4383 0.5638 -0.2955 0.4648 -0.3946 -0.7700 0.5356 0.3340 -0.8727 -0.5542
0.0759 -0.3463 -0.1862 -0.9984 -0.2188 -0.5141 0.7060 0.2972 -0.9402 -0.2873
0.4266 0.4748 0.0738 0.1115 -0.1253 0.3665 -0.1424 -0.1122 -0.0345 0.3118
-0.0085 0.8326 0.5058 -0.6702 0.0791 0.6774 0.6242 -0.0511 0.9090 0.1599
0.1457
-0.3860
0.1273
-0.6735
0.0185
0.7677
0.2216
0.7718
0.6336
0.9643
0.5422
-0.0460
0.7534
0.3091
0.4671
-0.0825
-0.7114
-0.0155
-0.1920
-0.1915
0.4048
0.3088
0.1604
0.2299
-0.7419
0.7806
-0.7848
0.1871
0.4994
0.2946
-0.8022
-0.1262
0.4338
-0.6027
-0.6282
0.3462
-0.3962
0.3971
-0.6612
-0.2806
0.0607
0.6229
net.LW{4,3} = [-0.2916 0.4238 -0.0932 0.1281 0.0077
0.0180 0.1889 -0.0356 -0.7170];
net.b{1} = [0.2532
-0.6654
0.5751
0.6534
-0.5557
0.6703
-0.3919
-0.2879];
net.b{2} = [ -0.7839
-0.5423
-0.1824
-0.9420
-0.0624
-0.0237
-0.7544
0.0924
0.5790
0.9473];
net.b{3} = [0.9031
0.6027
0.0509
-0.9412
-0.4695
-0.1232
0.2131
-0.6329
0.8309
-0.1721
0.4098
0.0188];
net.b{4} = [ -0.7555];
% set the training parameters
net.trainParam.show=100;
0.7158
0.1017
-0.6392
-0.4200
-0.1771
0.1633
0.6820
0.8914
0.3986
0.7341
-0.4737
0.6460
-0.3920
0.2116
-0.5327
-0.0280
-0.7218
0.4172];
0.0207 -0.0054 -0.0314 -
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.00000219;
% train the network
net=train(net,p,t);
% simulate the trained network
y=sim(net,p)
APPENDIX C(a)
Program for developing Hybrid MLR_ANN model for Rainfall
% Training for Hybrid MLR_ANN for Rainfall
% Creating the feed forward network
net = newff([0.0236 0.0432;0.0380 0.0980;0.0160 0.0920;0.0000 0.0185;
0.0007 0.0116],[8,10,12,1], {'tansig','tansig','tansig','logsig'},'trainscg');
% Enter the pattern and target
P = [0.0381 0.0740 0.0440 0.0122 0.0066
0.0347 0.0800 0.0520 0.0065 0.0069
…………………………………………..
0.0328 0.0930 0.0650 0.0041 0.0088
0.0320 0.0940 0.0640 0.0034 0.0078];
P = P';
T = [0.0056
0.0738
………
0.0272
0.0112];
T = T';
% set initial weights & biases
net.IW{1,1} = [-0.0165 -0.5602 -0.9254 0.5428 -0.0347
0.9585 0.4071 0.9759 -0.2952 -0.4694
-0.5800 0.0012 0.6181 0.6248 -0.2739
-0.4363 0.9256 -0.5187 -0.6159 -0.3227
-0.6768 -0.9767 0.6415 0.5018 -0.7421
0.8588 -0.5151 -0.5123 0.3539 -0.2847
0.6738 -0.8447 -0.4710 -0.8824 0.6447
0.2823 0.0128 -0.0896 0.0977 0.0253];
net.LW{2,1} =
[ 0.4033 0.8710 0.0954 0.3346 0.0840 -0.4327 -0.8064 -0.4690
-0.2987 -0.0491 0.8702 0.1368 0.4376 0.8301 0.3531 -0.7863
0.8230 -0.9564 -0.8084 0.3470 -0.3655 -0.0281 -0.0353
0.1396
0.7490 -0.3869 -0.0854 0.8928 0.1007 0.2776 -0.1711
0.5790
0.5934 -0.5088 0.0043 0.6617 0.0539 0.8461 0.2708 -0.2750
0.7333 -0.7863 0.5775 0.8317 0.7859 -0.2347 -0.3770 -0.7176
-0.5908 0.8086 -0.0282 0.5866 -0.2306 0.9521 0.5944
0.0665
-0.4093 0.5681 0.5001 -0.0451 0.1344
0.9377 0.0803 -0.6780
-0.7031 -0.3047 -0.9889 0.5810 -0.1917 -0.4595 0.3169 -0.2686
0.3083 0.0619 0.2205 0.2461 -0.5910 -0.0581 0.4316 -0.4020];
net.LW{3,2} =
[-0.2992 0.0736 0.1809 -0.1987 0.0344 0.3314 0.7406 -0.3810 0.0014 -0.3096
0.0585 -0.8555 0.8627 -0.8211 -0.0035 0.7772 -0.5794 -0.5452 -0.9230 0.2821
-0.9921 0.5174 0.3439 -0.4363 0.1567 -0.5887 0.2269 0.1921 -0.7470 0.2161
-0.2624 -0.1125 -0.3643 -0.5726 -0.6557 -0.4331 0.3500 -0.8120 -0.3295 -0.8066
0.8511 -0.2858 0.8399 -0.4140 -0.8231 -0.9964 0.9448 -0.1758 -0.5992 0.5170
0.6006 0.1218 0.2986 0.0855 -0.1872 0.8996 0.2479 0.5829 -0.5938 -0.0209
0.1483
-0.4478
0.3409
-0.0821
0.8429
-0.2655
-0.1589
0.2656
0.3838
0.6305
0.9359
-0.3821
0.5353
-0.0969
-0.9974
0.5099
0.1330
-0.4520
-0.4556
-0.3269
-0.7623
0.2730
0.1193
-0.0292
0.0892
0.0018
-0.5218
-0.7267
-0.1069
-0.7265
0.5796 -0.3841
-0.9472 -0.1770
0.7049 0.3017
0.2925 0.1331
0.8959 -0.9942
-0.2095 0.0033
-0.4574
0.7348
-0.2188
0.4866
0.5246
0.6487
net.LW{4,3} = [ 0.9979 0.5627 -0.5817 0.9911 -0.4803 -0.7350
0.4904 0.1651 0.5294 0.0905];
net.b{1} = [0.5913
-0.7489
-0.1915
0.0567
0.0964
0.5604
0.1472
-0.0661];
net.b{2} = [-0.0586
0.3628
-0.7196
0.0973
0.6545
-0.3077
0.1530
-0.3279
-0.1812
0.9241];
net.b{3} = [0.0867
0.1228
0.8659
0.0213
0.2901
0.7938
-0.1097
0.9280
-0.0195
0.2999
0.9186
-0.5942];
net.b{4} = [-0.6442];
% set the training parameters
net.trainParam.show = 100;
-0.0395
0.9646
-0.1944
-0.2774
0.5098
0.9488
0.0063
-0.0366
0.4547
0.2978
0.6319
0.3724
0.0321];
0.5415
net.trainParam.epochs = 100000;
% net.trainParam.time = 115;
net.trainParam.goal = 0.003;
% train the network
net = train(net,p,t);
% simulate the trained network
y = sim(net,p)
APPENDIX C(b)
Program for developing Hybrid ANN model for Minimum Temperature
% Training for Hybrid MLR_ANN for Minimum temperature
% Creating the feed forward network
net = newff([0.0236 0.0432;0.038 0.098; 0.016 0.092; 0.0007 0.0116],[8,10,12,1],
{'tansig','tansig','tansig','tansig'},'trainscg');
% Enter the pattern and target
P = [0.0381 0.074 0.044 0.0066
0.0347 0.080 0.052 0.0069
………………………………..
0.0328 0.093 0.065 0.0088
0.0320 0.094 0.064 0.0078];
P = P';
T = [0.0265
0.0233
..…….
0.0240
0.0229];
T = T';
% set initial weights&biases
net.IW{1,1} = [0.1089 -0.6727 -0.7924 -0.5323
0.2135 0.5008 0.7948 0.4625
-0.1546 0.2521 0.3830 -0.5170
0.2101 0.2383 0.2221 0.0331
-0.3497 0.8230 -0.5889 -0.5643
-0.8754 0.4537 0.1024 -0.7260
0.0623 -0.2027 0.3841 0.5354
-0.4644 -0.5981 -0.9038 0.2432];
net.LW{2,1} = [-0.4277 0.0179 0.9611 -0.2847 0.1105 -0.1778 -0.3126 -0.8705
0.4252 -0.0500 -0.8309 0.2823 0.0191 -0.7340 0.2093 0.9023
-0.3982 0.3143 0.6544 0.6092 -0.6228 -0.3388 0.8481 0.8864
0.0745 -0.1689 0.8537 -0.1139 -0.7820 0.1127 -0.8242 0.9426
-0.4584 0.9795 -0.6219 -0.5359 0.6711 -0.8649 -0.2676 0.5663
-0.3195 -0.1308 -0.4303 -0.6655 -0.6043 -0.7324 0.2152 -0.2900
-0.1622 0.4071 0.0196 0.3368 -0.7206 0.9249 0.7834 -0.0955
-0.1597 -0.2529 -0.9409 0.0042 -0.2452 0.3254 -0.5294 0.7569
-0.5632 -0.8681 -0.0340 0.7844 0.2597 -0.6888 -0.5017 -0.5120
-0.3300 -0.7983 0.9042 -0.5364 -0.2758 -0.3137 0.0577 -0.0872];
net.LW{3,2} =
[0.0216 0.6615 -0.5375 0.0296 -0.0124 0.3568 -0.6221 -0.4996 0.2910 0.7199
0.0868 0.7066 -0.1379 0.0388 0.0877 0.4924 -0.6638 -0.5075 -0.6423 0.0039
0.0775 -0.4218 -0.6704 0.8214 0.1289 0.3250 -0.2822 0.2744 -0.7502 -0.5108
0.6607 0.0993 0.8013 0.4702 -0.4041 0.2922 -0.4714 0.2923 -0.2316 -0.6579
-0.8421 -0.3035 0.4866 0.4739 -0.3395 -0.1389 -0.4692 -0.2069 -0.2946 0.3367
0.4515 -0.1562 -0.5675 0.3154 0.6925 -0.8764 -0.1842 -0.4420 0.8363 -0.2085
-0.5962
-0.7712
-0.0588
0.5991
-0.3036
0.6456
-0.4865
-0.6244
-0.6184
-0.9430
-0.8884
0.1542
-0.1099
-0.9410
0.8017
0.4101
0.3107
0.0703
-0.4632
0.9070
-0.4986
-0.4603
0.5624
0.5252
0.4839
0.7567
0.1131
-0.7504
0.6066
0.8308
-0.0587
-0.4974
-0.9043
0.6578
0.5451
0.2982
0.4288
0.2797
-0.1239
0.0244
0.6421
0.7434
net.LW{4,3} = [-0.2025 0.2587 0.5378 0.1469 0.2400
0.4419 0.0444 0.6404 0.4148];
net.b{1} = [ -0.6335
-0.9164
0.6704
-0.8871
0.2425
-0.0358
0.4393
0.6078];
net.b{2} = [ 0.8546
-0.5453
0.9412
-0.0433
0.2023
-0.1785
-0.6180
-0.5442
0.1806
-0.9093];
net.b{3} = [ 0.1424
0.8841
0.1328
0.6196
0.8312
-0.2198
0.8906
-0.2929
0.8971
0.8391
0.3129
0.2420];
net.b{4} = [0.6705];
% set the training parameters
net.trainParam.show=100;
0.5822
-0.0453
-0.4225
-0.4394
0.7287
0.1186
0.5679
-0.9753
-0.6122
-0.2440
-0.6833
0.8350
-0.9318
0.3295
-0.0001
0.0673
0.1758
0.7662];
0.4669 -0.3847 -0.0810
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.00000170076;
% train the network
net = train(net,p,t);
% simulate the trained network
y = sim(net,p)
APPENDIX C(c)
Program for developing Hybrid ANN model for Pan Evaporation
% Training for Hybrid MLR_ANN for Pan Evaporation
% Creating the feed forward network
net = newff([0.0236 0.0432;0.0380 0.0980; 0.016 0.092; 0.0000 0.4432],[8,10,12,1],
{'tansig','tansig','tansig','logsig'},'trainscg');
% Enter the pattern and target
P = [0.0381 0.074 0.044 0.0056
0.0347 0.080 0.052 0.0738
…………………………………
0.0328 0.093 0.065 0.0272
0.0320 0.094 0.064 0.0112];
P = P';
T = [0.0122
0.0065
.……..
0.0041
0.0034];
T = T';
% set initial weights & biases
net.IW{1,1} = [ -0.7272 0.9862 0.5016 -0.7559
0.3100 -0.3557 0.3844 0.3400
0.1637 0.4781 0.0540 0.2945
0.3590 0.2991 0.0643 -0.4392
-0.2047 0.3757 0.1413 -0.0442
0.8556 -0.1522 -0.6192 -0.2727
-0.2320 0.8968 -0.3116 -0.7358
0.5899 -0.6269 0.5045 0.3510];
net.LW{2,1} = [ -0.2135 0.4473 -0.7220 -0.1285 0.0845 -0.7815 -0.2210 -0.0254
0.3368 0.8847 0.4520 -0.7329 0.8688 -0.6782 0.7876 0.0363
0.3701 -0.1084 0.6084 0.1106 -0.3238 0.2478 -0.1009 0.7651
0.0292 0.2022 0.4127 -0.4327 0.0904 0.2583 0.0766 -0.2817
-0.8854 0.6838 0.4883 -0.0478 -0.6398 -0.9361 0.8232 -0.3119
-0.6100 0.8362 0.3270 0.8323 -0.0021 0.7533 0.9206 -0.4554
-0.2608 -0.8379 0.7985 -0.5701 -0.2513 -0.9704 -0.3494 0.8006
-0.6165 0.8619 -0.7470 0.1843 0.5553 0.1230 -0.0332 0.4142
0.6901 0.1444 -0.1737 0.3201
0.2916 -0.9144 -0.0566 -0.7507
-0.2038 -0.7971 -0.5943 0.7631 0.8431 0.4336 -0.7287 0.0105];
net.LW{3,2} =
[0.1557 0.4094 0.4130 -0.0939 -0.7884 -0.6147 -0.9166 0.1717 -0.8233 -0.5406
-0.5574 -0.0167 0.6716 -0.1286 0.8273 0.0231 -0.4696 0.3720 0.1806 -0.0803
0.5380 -0.7225 0.1275 0.2300 -0.0864 0.3762 0.0812 -0.0660 -0.0953 0.0520
0.2769 -0.2214 0.7927 -0.5422 -0.1949 -0.8049 0.5569 0.1738 0.6784 -0.8245
-0.6488 -0.7821 0.4875 -0.7928 -0.2820 0.5986 -0.2970 0.8247 -0.3944 -0.0402
-0.7540 0.6614 -0.5384 0.5072 -0.4650 0.5057 0.6111 0.5349 -0.7210 -0.4863
-0.4538 0.8376 0.8278 0.0409 0.9084 0.4262
-0.8118 -0.5475 -0.6702 -0.6423 -0.3983 -0.0980
-0.6887 0.2576 0.3332 -0.5090 0.5357 -0.6282
0.7363 -0.7128 0.3516 -0.7168 -0.1879 0.6679
-0.7172 0.0069 0.7357 -0.4856 -0.4966 -0.9634
-0.0472 -0.4132 -0.1695 -0.2286 0.0003 -0.1881
-0.1011
0.1542
0.2805
0.3366
0.2556
0.4694
net.LW{4,3} = [0.3270 -0.6599 -0.6565 0.0350 0.1762
0.1506 -0.4282 0.8604 0.4187];
net.b{1} = [0.4896
-0.6720
-0.4822
-0.3595
0.0715
0.2632
0.8832
-0.1039];
net.b{2} = [0.7302
-0.4762
-0.9153
-0.6191
0.1788
-0.1960
-0.5998
-0.9473
0.4340
-0.7052];
net.b{3} = [-0.8028
0.5738
-0.0520
-0.8137
0.4453
0.1055
-0.1666
-0.4724
-0.9803
0.1444
-0.2031
-0.7824];
net.b{4} = [-0.3111];
% set the training parameters
net.trainParam.show=100;
-0.3536
0.1578
-0.5193
0.5135
0.6031
0.3752
-0.0742
0.0552
0.5750
0.1297
0.4909
0.5988
0.7012
0.2281
-0.7378
0.8156
0.6793
0.9887];
0.4022
0.4682 -0.1866
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.00000285;
% train the network
net = train(net,p,t);
% simulate the trained network
y = sim(net,p)
APPENDIX C(d)
Program for developing Hybrid ANN model for Bright Sunshine
% Training for Hybrid MLR_ANN for Bright Sunshine
% Creating the feed forward network
net = newff([0.0236 0.0432; 0.0172 0.0335; 0.038 0.098;0.016 0.0920;0.0000
0.4432],[8,10,12,1], {'tansig','tansig','tansig','purelin'},'trainscg');
% Enter the pattern and target
P = [0.0381 0.0265 0.074 0.044 0.0056
0.0347 0.0233 0.080 0.052 0.0738
…………………………………………
0.0328 0.0240 0.093 0.065 0.0272
0.0320 0.0229 0.094 0.064 0.0112];
P = P';
T = [0.0066
0.0069
……….
0.0088
0.0078];
T = T';
% set initial weights & biases
net.IW{1,1} = [ 0.2535 0.8032 0.4855 0.0653 -0.2864
-0.2039 0.2511 -0.6987 0.6960 -0.7380
0.3083 -0.6499 -0.3556 -0.1896 -0.8882
-0.9468 -0.8670 -0.8440 0.9455 -0.1885
-0.3951 0.6232 0.2924 0.3258 0.3214
0.5504 0.1191 -0.2252 -0.9820 0.9634
-0.5582 0.1118 0.9788 0.1732 0.3051
0.5210 -0.3703 -0.7575 -0.5050 0.6340];
net.LW{2,1} = [ -0.8838 0.4402 -0.8666 0.9213 -0.1844 -0.3731 0.4570 -0.6265
0.8433 -0.8236 0.1431 0.8929 -0.3520 0.4636 0.3173 -0.9237
-0.4854 -0.1954 -0.7909 -0.0581 0.1324 -0.7104 0.0966 0.8494
-0.0146 0.5240 -0.4160 0.4259 0.3998 0.2870 0.6868 -0.7486
-0.2532 0.8772 0.5227 0.1875 0.7124 0.5905 0.8099 0.4093
-0.9875 -0.4068 -0.9776 0.8645 0.0461 0.2457 -0.7013 0.3713
-0.2534 0.2298 0.8699 0.5598 -0.1372 0.8524 -0.1342 -0.4537
-0.4086 -0.5370 0.5958 0.7701 -0.8375 0.8347 -0.3304 0.1210
0.4934 0.0580 -0.8741 0.3820 -0.9255 0.4894 -0.0820 0.3495
0.1691 -0.8404 -0.9516 0.0078 -0.4245 0.1088 -0.8149 -0.9201];
net.LW{3,2} =
[ 0.3296 0.7735 0.0756 0.8328 0.7670 -0.4081 0.3805 -0.5196 0.4570 0.2741
-0.4748 0.0588 0.7263 -0.5607 0.2643 0.7994 0.5495 -0.7471 -0.3801 -0.1198
0.8962 -0.4212 -0.1303 0.6565 0.3993 -0.3729 -0.2272 0.7454 0.6997 0.8395
-0.5805 0.1323 -0.7514 0.4964 0.6184 0.1572 0.6246 0.5602 -0.5959 0.5630
-0.4586 -0.5891 0.6003 -0.5022 -0.6595 0.2273 -0.6526 -0.5330 -0.0464 -0.6340
-0.8230 0.8332 -0.5941 0.3858 0.3046 0.7668 0.1651 -0.1934 -0.7275 -0.0343
0.5300
-0.6276
0.2037
0.4591
0.5458
-0.1765
0.4139
-0.5261
0.8647
-0.6960
-0.6874
-0.6446
0.9723
-0.7997
-0.5703
-0.3138
-0.8361
0.8022
0.1301
-0.7174
0.2161
0.1939
0.7449
-0.8591
0.4944
-0.3420
-0.2602
-0.0357
0.1117
0.0230
-0.5189
-0.6313
0.1436
-0.5145
0.6738
-0.7721
-0.3001
-0.4725
0.3466
0.2635
0.1509
0.0777
net.LW{4,3} = [-0.0691 0.4847 0.0491 -0.4330 -0.4161
-0.0993 0.2504 0.3449 -0.1747];
net.b{1} = [0.9807
0.6875
0.5157
0.5344
-0.2808
-0.1161
-0.8841
0.6626];
net.b{2} = [0.1246
-0.4557
0.1187
0.7682
0.4334
0.0097
-0.5773
-0.2063
0.5837
0.9203];
net.b{3} = [-0.8444
0.6469
-0.0153
0.7519
0.3512
0.1348
0.2617
-0.4950
0.8987
0.1456
0.4628
-0.8149];
net.b{4} = [-0.3443];
% set the training parameters
0.5363
-0.5147
-0.2824
0.4528
-0.3140
-0.1528
0.7358
-0.5239
0.3372
-0.6164
-0.0875
0.4186
0.4828 0.0139
0.6723
0.1270
0.0417
0.1827
-0.7798
0.7594];
0.4180
net.trainParam.show=100;
net.trainParam.epochs=100000;
% net.trainParam.time=115;
net.trainParam.goal=0.00000219;
% train the network
net = train(net,p,t);
% simulate the trained network
y = sim(net,p)