Download Geostatistical Analysis

UNIVARIENT & BIVARIENT GEOSTATISTICAL ANALYSIS
Mirza Muhammad Waqar
Contact:
[email protected]
+92-21-34650765-79 EXT:2257
RG712
Course: Special Topics in Remote Sensing & GIS
What is Statistics About?
 Statistics is the science of collecting, organizing,
analyzing and interpreting data in order to make
decisions
 Statistics is the science of data-based decision
making in the case of uncertainty
Statistical Analysis
Problem
Statistical
Cycle
Conclusion
Analysis
Plan
Data
1. Problem




"I wonder if there are differences between...“
What information will you need to answer the
question?
Identify two or more sub-groups of the
population to compare.
What variables are likely to show differences?
2. Plan



If collecting data you will need to plan a survey
of questionnaire.
Using available data sets is recommended
If using a data set decide what sub-groups of
data are needed and choose from the available
variables (choose carefully so you can answer
the problem
3. Data



Collect data by making a survey or
questionnaire, OR take a sample from large data
set. (at least 30 values)
For example, Census data
Clean the data set before continuing
4. Analysis



Analyze the data to find similarities and
differences.
You will need measures of central tendency
(mean, median, mode) AND measures of spread
(range, inter quartile range, standard deviation)
Use technology to calculate the statistics:
calculator, or EXCEL (using excel)
5. Conclusion





Remember that you are analysing and
comparing data from a SAMPLE from a
population
Is there a difference between the subgroups?
Comparisons made from a Box-and-Whisker
graph
Comparisons bases on measures of central
tendency
Comparisons made from measures of spread
Role of Statistics in GIS





To describe and summarize spatial data.
To make generalizations concerning complex
spatial patterns.
To use samples of geographic data to infer
characteristics for a larger set of geographic data.
To determine if the magnitude or frequency of
some phenomenon differs from one location to
another.
To learn whether an actual spatial pattern
matches some expected pattern.
What is Geostatistics?




Applies the theories of statistical inference to geographic
phenomena.
Methods of geostatistics are used in petroleum geology,
hydrogeology, hydrology, meteorology, oceanography,
geochemistry
A way of describing the spatial continuity as an essential
feature of natural phenomena.
Recognized to have emerged in the early 1980’s as a
hybrid of mathematics, statistics, and mining engineering.
Some Useful Definitions

Data – information coming from observations,
counts, measurements or responses.
 The
data you will be analyzing will almost always be
a sample form a population.


Population – the collection of all outcomes,
responses, measurements or counts that are of
interest.
Sample – a subset of a population.
 We
will almost always be dealing with samples and
hopping to make inference about the population.
Some Useful Definitions


Parameter – numerical description of a
characteristic of the population.
Statistic – a description of a characteristic of the
sample.
 We
will often wish to make inferences about
parameter based on statistics.
Some Useful Definitions


Descriptive Statistics – relate to organizing,
summarizing and displaying data.
Inferential Statistics – relate to using a sample
to draw conclusions about a population.
 Inferential
statistics involves drawing a conclusion
from some data.
Inferences vs. Descriptive

Consider:
 Average
length of females and males: 90cm and
100cm respectively.
 Descriptive statistics: the values.
 Inference: males are (in general) taller than females.
Descriptive Statistics

3 categories of descriptive statics in geostatistics
 Univariate
Descriptive Statistics
 Use to describe and summarize single data/variable
 Bivariate
Descriptive Statistics
 Use to describe relationship between two data/variable
 Spatial
Descriptive Statistics
 Describe data in term of space and time
Univariate Description


Describe and summarize single variable
Graphical methods
 Histogram
 Cumulative

Frequency
Numerical methods divides in three categories
 Measurement of
location
 Measurement of spread
 Measurement of shape
Univariate Description

Measurement of location

Measurement of center location




Measurement of other part




Qunatile
Quartile
percentile
Measurement of spread (variability)




Mean
Median
Mode
Variance
Standard Deviation
Inter-Quartile range
Measurement of shape (symmetry & length)


Coefficient of skewness
Coefficient of Variation
Frequency Table and Histogram

Histogram – is a bar graph that plots the
frequency of distribution of dataset.
 The
horizontal scale is representing classes/bin
 The vertical scale measures the frequencies of the
classes.
 Consecutive boundaries much touch
Ideal Histogram for Image Analysis
Frequency (f)
Vegetation
Urban Area
Soil
Water
Band A
Actual Histogram from Image Analysis
Frequency (f)
Vegetation
Urban Area
Soil
Water
Band A
Histogram from Image Analysis


Very informative tool for analysis.
Histogram define the contrast of satellite image.
 More
the BV’s range, more the contrast.
Low Contrast Histogram
High Contrast Histogram
Histogram from Image Analysis


We can also identify the largest land cover in
satellite image by histogram.
Rough quantification of landcovers can be made
using histogram.
 This
rough quantification
quantification.

leads
to
correct
Using histogram, range of a particular landcover
can be identified in aspect of BV.
Frequency Table


To develop a histogram a frequency table is
used.
Frequency table: records how often observed
values fall within certain intervals or classes.
Constructing a Frequency Distribution


Decide on the number of classes to include in the
frequency distribution.
Find the class width as follows:



Determine the range of the data
Divide the range by the number of classes and round up to
the next convenient number
Find the class limits:



Start with the lowest value as the lower limit of the first class,
add the class width to this to obtain the lower limit for the
second class, etc.
Place a mark in the row for the class corresponding to each
data point
Count the number of marks in each class.
Frequency Table
Cumulative Frequency Table and Histogram


Cumulative frequency of a class is the sum of the
frequency of that class and all previous classes.
The cumulative frequency for the last class is
always n.
Cumulative Frequency Tables
Cumulative Histogram
Measure of Location


It provide us the information about where
various part (information) of data lies
Center of data can be find by
 Mean
 Median
 Mode

Location of other parts of the data are given by
the quantiles
Mean Median Mode

Mean – average of all the data points in the
data/distribution




Median – middle value in an ordered array of number.


Unique and unbiased
Based on every data point in the dataset
Can be sensitive to outlaying observations
Unaffected by extremely large and extremely small values.
Mode – the most frequently occurring value in a dataset.

Unlike the mean and median, the mode is not always
uniquely defined.


Bimodal – two values having same number of instances in the
data
Multimodal – three or more values having same number of
occurrences
Univarient Statistics for Image Analysis

The histogram of satellite image can not be the
uni-mode data.
 Number
of mode represents how many land covers
exists in the satellite image.

We can’t make decision about transition zone
using histogram.
Univarient Statistics for Image Analysis
Frequency (f)
Vegetation
Urban Area
Soil
Water
Frequency (f)
Band A
Vegetation
Urban Area
Soil
Water
Band A
Which Measure is Best?

No clear answer to this question.
 The
mean can be influenced by outliers while the
mode may not be particularly “typical central value”.
 Statistical inference based on the median and the
mode is difficult.
Percentiles




Divide a group of data into 100 parts
At least n% od data live below the nth percentile,
and most (100-n)% of the data lie above the nth
percentile.
Example – 90th percentile indicates that at least
90% of the data lie below it, and at most 10% of the
data live above it.
The median and the 50% percentile have the same
value.
Percentiles (i): Computational Procedure





Organize the data into an ascending ordered
array.
Calculated percentile location i=
𝑃 (𝑛)
100
Determine the percentile’s location and its value.
If i is a whole number, the percentile is the
average of the value at the i and (i+1) positions.
If i is not a whole number, the percentile is at
(i+1) position in the order array.
Percentiles: Example




Raw Data: 14, 12, 19, 23, 5, 13, 28, 17
Order Array: 5, 12, 13, 14, 17, 19, 23, 28
Location of
30th
percentile i
30 (8)
=
= 2.4
100
The location index, i, is not a whole number;
i+1=2.4+1=3.4; the whole number portion is 3;
the 30th percentile is at the 30th location of the
array; the 30th percentile is 13.
Quartiles
Formulae in EXCEL







Calculating Means: Average(data)
Calculating Median: Median(data)
Calculating Mode: Mode(data)
Calculating Minimum: min(data)
Calculating Maximum: max(data)
Calculating Quartile: QUARTILE(data,quart)
Calculating Percentile: PERCENTILE(array,k)
Measure of Spread/Variation


Measure of variability describe the spread or the
dispersion of a dataset.
Common measures of variability
 Range
 Interquartile Range
 Mean
Absolute Deviation
 Variance
 Standard Deviation
 Coefficient of Variation
Range



The difference between the largest and the
smallest values in a set od data
Simple to compute
Ignore all data points except two extremes
 Range

= Maximum – Minimum
Range tells us about the spread of data.
 Some
time range provides us very
information when outliers exists in data
biased
Interquartile Range



Range of values between the first and third
quartiles
Less influenced by extremes
Interquartile Range = Q3 – Q1
Deviation, Variance and Standard Deviation

The deviation of a data entry x in a population
data set is the difference between x and
population mean µ, i.e.
Deviation of x = x - µ

The sum of the deviation over entries is zero.
Mean Absolute Deviation

Average of the absolute deviation from the mean
X
X-µ
|X - µ|
5
-8
8
9
-4
4
16
3
3
17
4
4
18
5
5
0
24
M.A.D. =
M.A.D. =
∑ |X − µ|
𝑁
24
5
= 4.8
Variance

The population variance is the sum of squared
deviation over all entries:
Population Variance = σ2 =
∑ (Xi − µ)2
𝑵
Population Variance

Average of squared
arithmetic mean
X
X-µ
(X - µ)2
5
-8
64
9
-4
16
16
3
9
17
4
16
18
5
25
0
130
deviation
σ2
=
from
∑ (Xi − µ)2
𝑵
M.A.D. =
130
5
= 26.0
Sample Variance
S2 =
∑ (Xi − µ)2
𝒏−𝟏
the
Variance for Image Analysis


For variance analysis, we go for comparative
analysis.
By comparing variance of all bands we come to
know that which band has more dispersion.
Band #
Variance
Band 1
572
Band 2
634
Band 3
93
Band 4
224
Band 5
336
Band 7
325
Variance for Image Analysis


Less the variance, it depicts
homogeneity of the data is high.
Outlier can disturb the variance.
that
the
Standard Deviation

The population standard deviation is the square
root of the population variance i.e.
σ = σ2 =
∑ (Xi − µ)2
𝑁
Standard Deviation

Square root of the variance
X
X-µ
(X - µ)2
5
-8
64
9
-4
16
16
3
9
17
4
16
18
5
25
0
130
σ=
σ=
∑ (Xi − µ)2
𝑵
130
=
5
26 = 5.1
Standard Deviation of Sample
∑ (Xi − µ)2
σ=
𝒏−𝟏
Empirical Rules

Data are normally distributed (or approximately
normally distributed)
Distance from the mean
% of values falling within distance
µ ± 1σ
68
µ ± 2σ
95
µ ± 3σ
99.7
Shape of Distribution - Systematic

A frequency distribution is systematic when a
vertical line can be drawn through the middle of
a graph of distribution and the resulting halves
are mirror images.
Shape of Distribution - Uniform

A frequency distribution is uniform when the
number of entries in each class is equal.
Shape of Distribution - Skewed

A frequency distribution is skewed right (or
positively skewed) if its tail extends to the right
(mode < median < mean)
Shape of Distribution - Skewed

A frequency distribution is skewed left (of
negatively skewed) if its tail extends to the left
(mode > median > mean)
Measure of Shape

Shape of the distribution is described by
 Coefficient of
skewness
 Coefficient of kurtosis
Coefficient of Skewness

Sknewness
Absence of symmetry
 Extreme values in one side of distribution






Symmetry measure for skewness =
𝐸 𝑥−µ
σ3
3
Where E is Expected value (mean)
If S<0, distribution is negatively skewed (skewed to
the left)
If S=0, distribution is symmetric (not skewed)
If S>0, distribution is positively skewed (skewed to
the right)
Skewness
Skewness
Kurtosis



Describes the shape of the curve about the mean
Kurtosis is based on the size of distribution’s tail
A measure of weather the curve of distribution
is:
 Bell Shaped
– normal distribution
 Peaked – large tail (Leptokurtic)
 Flat – small tail (Platykurtic)
Kurtosis & Skewness
Coefficient of Kurtosis

The following formula can be used to calculate
kurtosis:
Kurtosis =

∑ 𝑿−µ
𝑵𝝈𝟒
𝟒
-3
Kurtosis can be expressed as a number or value
A
value of kurtosis = 0 indicates symmetrical or no
kurtosis
 Positive value = leptokurtic
 Negative value = platykurtic
Multivariate Statistical Parameter
1.
2.
Covariance
Correlation
Covariance



How the two variables are varying with respect to
each other.
Bands having same information content has high
covariance and vice versa.
Optimum index factor (OIF) can be used to identify
those bands which contain distinct information
content.
Correlation

It is the measurement of linear relationship
between the variables.
Correlation-Covariance Matrix
*
Band 1
Band 2
Band 3
Band 4
Band 5
Band 7
Band 1
1
0.5
0.7
0.2
-0.4
0.9
Band 2
0.5
1
0.25
0.15
0.75
0.65
Band 3
0.7
0.25
1
0.29
-0.45
-0.1
Band 4
0.2
0.15
0.29
1
0.12
-0.25
Band 5
-0.4
0.75
-0.45
0.12
1
0.19
Band 7
0.9
0.65
-0.1
-0.25
0.19
1
Correlation Coefficient:
Correlation Coefficient:
Correlation Coefficient:
+1
0
-1
Direct Relationship
No Relationship
Indirect Relationship
Questions & Discussion