Download Truck Forecasting with Time Series Analysis: A Case Study of the

TRUCK FORECASTING
WITH TIME SERIES
ANALYSIS: A CASE STUDY
OF THE BLUE WATER
BRIDGE
University of Wisconsin – Milwaukee
Paper No. 11-5
National Center for Freight & Infrastructure Research & Education
College of Engineering
Department of Civil and Environmental Engineering
University of Wisconsin, Madison
Authors: Jing Mao and Alan J. Horowitz
Center for Urban Transportation Studies
University of Wisconsin – Milwaukee
Principal Investigator: Alan J. Horowitz
Professor, Civil Engineering and Mechanics Department, University of Wisconsin – Milwaukee
December 22, 2011
Truck Forecasting with Time Series Analysis: A Case Study of the Blue Water
Bridge
INTRODUCTION
This document contains images of all slides in a course module about the use of time
series techniques for truck forecasting. The techniques are illustrated with data from the Blue
Water Bridge between Michigan and Ontario. This presentation is available upon request to
Alan Horowitz, [email protected].
12/22/2011
Truck Forecasting with Time Series
Analysis: A Case Study of the Blue
Water Bridge
Prepared by
Jing Mao
Alan J. Horowitz
Outline
•
•
•
•
Introduction
Data Collection
Methodology
Conclusion
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Blue Water Bridge
Location
The Blue Water Bridge spans the Saint Clair River, and carries
international traffic between Port Huron, Michigan and Point
Edward and Sarnia, Ontario. Located near interchange of I-94
and I-69, the bridge forms a critical gateway linking Canada
and the United States.
Blue Water Bridge
Lane characteristics
The original Blue Water Bridge, opened in 1938 and
renovated in 1999, is a three-lane westbound bridge.
The second Blue Water Bridge, which carries three lanes
of eastbound traffic, is an impressive modern bridge
opened in 1997.
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Purpose of the Study
• Forecasting the eastbound and westbound
monthly truck volume of blue water bridge from
2011 to 2013
• Applying the different time series models and
select the best one for forecasting
Data Source
• Dependent variables
Blue water bridge eastbound/ westbound truck volume
• Independent variables
o
o
o
o
Michigan population (why freight moves)
Ontario population (why freight moves)
U.S. GDP and population (why freight moves)
Approximate Michigan GDP (derived from U.S. GDP
by the proportion of Michigan population and U.S.
population)
o U.S. all grades all formulations retail gasoline fuel
price (major cost of freight)
o North American Free Trade Agreement (NAFTA)
(why freight moves)
o September 11 attacks (why freight does not move)
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Descriptive Statistic of Dependent
Variables
• Westbound and eastbound truck volume data from Jan
1984 to Dec 2010 by each month
Westbound
Eastbound
30000
30000
25000
25000
20000
20000
15000
15000
10000
Westbound
5000
Eastbound
10000
5000
Oct-87
May-87
Jul-86
Dec-86
Apr-85
Feb-86
Sep-85
Nov-84
Jan-84
Westbound truck volume
from Jan 1984 to Dec 1987
Jun-84
0
Oct-87
May-87
Jul-86
Dec-86
Feb-86
Apr-85
Sep-85
Nov-84
Jan-84
Jun-84
0
Eastbound truck volume
from Jan 1984 to Dec 1987
Descriptive Statistic of Independent
Variables
• Michigan and Ontario population data from Jan 1984 to
Dec 2010 by each month
Michigan
Population(million)
10.200
10.000
9.800
9.600
9.400
9.200
9.000
8.800
8.600
8.400
Ontario Population(million)
16.0000
14.0000
12.0000
10.0000
Ontario
Populati
on(mil…
8.0000
Michigan
6.0000
4.0000
Jan-10
Jan-08
Jan-06
Jan-04
Jan-02
Jan-00
Jan-98
Jan-96
Jan-94
Jan-92
Jan-90
Jan-88
Jan-86
0.0000
Jan-84
Jan-09
Jul-06
Jan-04
Jul-01
Jan-99
Jul-96
Jul-91
Jan-94
Jul-86
Jan-89
Jan-84
2.0000
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Descriptive Statistic of Independent
Variables
• US GDP data from Jan 1984 to Dec 2010 by each month
and yearly U.S. population data
US GDP(billion)
U.S. Population (million)
6000.000
350
300
4000.000
250
200
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
0
1988
Sep-08
Jul-02
Aug-05
Jun-99
Apr-93
50
May-96
100
0.000
Mar-90
1000.000
Jan-84
Population
150
1986
US GDP(billion)
2000.000
1984
3000.000
Feb-87
US GDP
5000.000
Approximate Michigan GDP
• Computing the ratio of Michigan population and U.S. population
• Computing Michigan GDP by applying the ratio to Michigan GDP
and U.S. GDP
Michigan GDP = Ratio * U.S. GDP
Ratio = Michigan population / U.S. population
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Descriptive Statistic of Independent
Variables
• Michigan GDP data and U.S. all grades all formulations
retail gasoline fuel price data from Jan 1984 to Dec 2010
by each month
U.S. All Grades All Formulations Retail
Gasoline fuel Price (Dollar per Gallon
based current year)
Michigan GDP(billion)
180
4.500
140
4.000
120
3.500
100
3.000
80
Michigan
GDP(billion)
60
40
Fuel price
160
2.500
U.S. All Grades
All Formulations
Retail Gasoline
fuel Price…
2.000
1.500
1.000
20
Sep…
Mar…
Nov…
Jan-…
Jul-…
May…
Sep…
Nov…
Jan-…
Jul-…
Mar…
May…
Sep…
Nov…
Jul-06
Jan-09
Jul-01
Jan-04
Jul-96
Jan-99
Jul-91
Jan-94
Jul-86
Jan-89
Jan-84
0.000
Jan-…
0.500
0
Other Data
• NAFTA
o The North American Free Trade Agreement or NAFTA is
an agreement signed by the governments
of Canada, Mexico, and the United States, creating a
trilateral trade bloc in North America. The agreement
came into force on January 1, 1994. It superseded
the Canada – United States Free Trade
Agreement between the U.S. and Canada.
• September 11
o The September 11 attacks could also be a factor to
influence the truck volume within that month.
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Time Series Models
•
•
•
•
•
Central Moving Average
Growth Factor
Exponential Smoothing
Linear Regression
ARIMA
o Box-Cox Transformation
Central Moving
Average
Growth Factor
Exponential
Smoothing
Linear Regression
ARIMA
Central Moving Average
• Central moving average is a moving average such that
time period is at the center of the N time periods used
to determine which values to average.
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Central Moving Average
• Westbound and Eastbound moving average data from
Jul 1984 to Aug 2010
100000
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
Eastbound Moving Average
90000
80000
Series1
Series2
Truck Volume
70000
60000
50000
40000
Series1
30000
Series2
20000
Apr-06
R square 0.971
Sep-08
Nov-03
Jan-99
Jun-01
Aug-96
Oct-91
Mar-94
Jul-84
May-89
R square 0.956
0
Dec-86
Nov-07
Jan-02
Dec-04
Feb-99
Apr-93
Mar-96
May-90
Jul-84
10000
Jun-87
Truck Volume
Westbound Moving
Average
Seasonal adjustment factor
• Seasonal adjustment factor can be used to improve
accuracy of truck volume forecasts
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Growth Factor
Linear growth:
F(n) = Constant + AGF * (n)
F(n): forecast volume
AGF: average growth factor
n: the number of months from the first
observation
Growth Factor
• Determination of constant and AGF from the linear
regression
o
o
o
o
Tools / Add-ins / Analysis Tool Park
Tools / Data Analysis / Regression
Independent variable: month
Dependent variable: westbound truck volume (partial
initial data)
Constant:13767
AGF: 119
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Growth Factor Forecasting
F(n)=13767+119*n
Partial results with growth factor forecasting
Forecasting Results with All Initial Data
Westbound Forecasting
Truck Volume
100000
80000
60000
40000
Actual data
20000
Forecasted data
R Square 0.915
Jun-10
Dec-06
Sep-08
Jun-03
Mar-05
Dec-99
Sep-01
Jun-96
Mar-98
Sep-94
Jun-89
Mar-91
Dec-92
Sep-87
Mar-84
Dec-85
0
R Square 0.919
Actual data
Forecasted data
Mar-84
Dec-85
Sep-87
Jun-89
Mar-91
Dec-92
Sep-94
Jun-96
Mar-98
Dec-99
Sep-01
Jun-03
Mar-05
Dec-06
Sep-08
Jun-10
Truck Volume
Eastbound Forecasting
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
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Growth Factor Forecasting with
Central Moving Average Series
F(n)=14291+230*n
Partial central moving average truck volume
Partial central moving
average series
Forecasting Results with All Smoothed
Data
Westbound Forecasting
Truck Volume
100000
80000
R Square 0.935
60000
40000
Moving Average
20000
Forecasting
Aug-11
Oct-08
Mar-10
May-07
Jul-04
Dec-05
Feb-03
Apr-00
Sep-01
Jan-96
Jun-97
Nov-98
Aug-94
Oct-91
Mar-93
May-90
Jul-87
Dec-88
Feb-86
Sep-84
0
R Square 0.940
Moving Average
Jan-10
May-11
Sep-08
May-07
Jan-06
Sep-04
May-03
Jan-02
Sep-00
May-99
Jan-98
Sep-96
May-95
Jan-94
Sep-92
May-91
Jan-90
Sep-88
Jan-86
May-87
Forecasting
Sep-84
Truck Volume
Eastbound Forecasting
80000
70000
60000
50000
40000
30000
20000
10000
0
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Comparison
• Comparing R Square of Growth factor with initial
data and moving average(smoothed) data
R Square
Initial data
Moving average
data
Westbound
0.915
0.935
Eastbound
0.919
0.940
Growth Factor
• Compound growth
F(n)=Constant*AGF(n)
o
If there are two years
AGF = ⎛⎜ F2 ⎞⎟
⎝ F1 ⎠
1
Y2 − Y1
Where F1 is the freight flow in year Y1, F2 is the
freight flow in year Y2
o If there are more than two years, AGF can be found
from the linear regression
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Growth Factor
• Determination of constant and AGF
from the linear regression
Tools / Add-ins / Analysis Tool Park
Tools / Data Analysis / Regression
Independent variable: month
Dependent variable: westbound truck
volume expressed as natural logarithm
(partial initial data)
o Constant = EXP (intercept)
AGF = EXP (x-variable coefficient)
o
o
o
o
Constant:13745
AGF: 1.008
Growth Factor Forecasting
F(n) = 13745*1.008n
Partial results with compound regression forecasting
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Exponential Smoothing
• Model formulation:
where
St: exponentially smoothed value for time period t
St-1: exponentially smoothed value for time period t-1
xt-1 : actual time series value for time period t
α: the smoothing factor, and 0 < α < 1
Example
α = 0.7
St = 0.7xt-1 + (1-0.7)St-1
0.7*12878+0.3*13253
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Exponential Smoothing
• Smoothing factor
o The larger α is, the closer the smoothed value will
track the original data value. The smaller α is, the
more fluctuation is smoothed out.
• The determination of smoothing factor
o Graph fitting
o Mean squared error (MSE)
• Smoothing factor assumed (0.3, 0.5, 0.7)
100000
80000
Alpha=0.3
60000
Initial data
40000
20000
Exponential
smoothing(alpha=0.3)
Feb-09
May-10
Aug-06
Nov-07
Feb-04
May-05
Nov-02
Feb-99
Aug-01
May-00
Aug-96
Nov-97
Feb-94
May-95
Aug-91
Nov-92
Feb-89
May-90
Aug-86
Nov-87
Feb-84
May-85
0
MSE=27875534
100000
Alpha=0.5
80000
60000
40000
Initial data
20000
Oct-10
Jun-09
Oct-06
Feb-08
Jun-05
Oct-02
Feb-04
Jun-01
Oct-98
Feb-00
Jun-97
Feb-96
Oct-94
Jun-93
Oct-90
Feb-92
Jun-89
Oct-86
Feb-88
Jun-85
Feb-84
0
MSE=25483163
100000
80000
60000
Initial data
40000
20000
Feb-09
Nov-07
May-10
Aug-06
May-05
Feb-04
Aug-01
Nov-02
Feb-99
May-00
Aug-96
Nov-97
Feb-94
Nov-92
May-95
Aug-91
May-90
Feb-89
Aug-86
Nov-87
Feb-84
0
May-85
Alpha=0.7
Exponential
smoothing(alpha=0.7)
MSE=24306420
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Westbound Truck Volume Forecasting
Alpha
0.3
0.5
0.7
MSE
27875534
25483163
24306420
When alpha=0.7, MSE is smallest, so we select alpha=0.7 to
forecast westbound truck volume
100000
90000
80000
70000
60000
50000
Initial data
40000
Forecasting
30000
20000
10000
Oct-09
Dec-10
Jun-07
Apr-06
Aug-08
Feb-05
Oct-02
Dec-03
Jun-00
Aug-01
Apr-99
Feb-98
Oct-95
Dec-96
Jun-93
Aug-94
Apr-92
Feb-91
Oct-88
Dec-89
Jun-86
Aug-87
Apr-85
Feb-84
0
Linear Regression
• Model
Y = f(x1,x2,…,xn) = b0 + b1x1 + b2x2 + … + bnxn
• Dataset with initial truck volume(partial):
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Initial Analysis
Westbound truck & Michigan
population scatterplot
Westbound truck & Ontario
population scatterplot
Initial Analysis (cont’d)
Westbound truck & US GDP
scatterplot
Westbound truck & Fuel price
scatterplot
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Initial Analysis (cont’d)
Westbound truck & NAFTA
scatterplot
Westbound truck & September 11
scatterplot
Regression Model Establishment with
SPSS
• Select Analysis-Regression-Linear
• Put “Westbound Truck” as dependent variable and other
variables as independent variables
• Select “Stepwise” as
analysis mode
•
Click OK
18
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Results Analysis
US GDP NAFTA,
Sep.11
The order of
regression equation
The name of entered variables
The name of removed variables
The basis of entering and
removing variables
Common Statistic
Adjusted R Square=0.942 indicates
model 3 should be selected as
regression model
19
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Analysis of Coefficients
• Regression equation:
Y=−652201.511+83171.959x1−10177.394x2+4889.856x3
x1: Michigan population(million)
x2: Ontario population(million)
x3: Fuel price(current dollar)
Independent Variables Forecasting
• Forecasting the independent variables (Ontario population,
US GDP, Fuel price ) with linear regression, respectively.
US GDP
Forecasting(billion)
Ontario population
forecasting(million)
6000
5000
4000
3000
2000
1000
0
Jul-04
Dec-07
Feb-01
Apr-94
Sep-97
Nov-90
Jan-84
Forecasting
Fuel price forecasting(Dollar
per Gallon)
3
2
1
Forecasting
Sep-09
Jan-06
Nov-07
Mar-04
Jul-00
May-02
Sep-98
Jan-95
Nov-96
Mar-93
Jul-89
Sep-87
May-91
Jan-84
0
Nov-85
Jan-06
Oct-08
Jul-00
Apr-03
Oct-97
Jul-89
Jan-95
Apr-92
Jan-84
Oct-86
Forecasting
Jun-87
14
12
10
8
6
4
2
0
20
12/22/2011
Michigan Population and GDP Forecasting
• Results of the 2010 headcount show the number of
Michigan residents fell by 0.6 percent since 2000
=9.88*0.94
Forecasting the Michigan GDP
⎛ Michigan Population ⎞
⎟⎟ * US GDP
US Population
⎝
⎠
Michigan GDP = ⎜⎜
Independent Variables Forecasting Results
• Independent variables forecasting results from
Feb 2011 to Dec 2013
Partial forecasting results
21
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Regression Forecasting
• Westbound truck forecasting results from Feb
2011 to Dec 2013 with the multilinear regression
model
Partial forecasting results
Regression Model Establishment with
EXCEL
• Inputting the dataset
• Click Anova Tools – Regression
y = -652211.746 +83173.846*Michigan -10178.161*Ontario
+4890.563*Fuel Price
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Forecasting Results
• Plot of initial value and forecasting value
100000
90000
80000
70000
60000
50000
Initial data
40000
Forecasting
30000
20000
10000
Jan-84
Feb-85
Mar-86
Apr-87
May-88
Jun-89
Jul-90
Aug-91
Sep-92
Oct-93
Nov-94
Dec-95
Jan-97
Feb-98
Mar-99
Apr-00
May-01
Jun-02
Jul-03
Aug-04
Sep-05
Oct-06
Nov-07
Dec-08
Jan-10
Feb-11
0
Linear Regression with Smoothed
Series(Central Moving Average Series)
• Select Analysis-Regression-Linear
• Put Westbound Truck as Dependent variable and
other variables as independent variables
• Select Stepwise
as analysis mode
• Click OK
23
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Regression Model Selection
Linear Regression Equation
Y = -562356.809 +75631.264*Michigan -13266.173*Ontario
+8.828*US GDP(billion) -1680.692*NAFTA
24
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Forecasting with EXCEL
• Plot of westbound smoothed truck volume
and forecasting volume
90000
80000
70000
60000
50000
40000
Westbound
30000
Forecasting
20000
10000
May-11
Jan-09
Mar-10
Nov-07
Jul-05
Sep-06
May-04
Jan-02
Mar-03
Jul-98
Sep-99
Nov-00
May-97
Jan-95
Mar-96
Nov-93
Jul-91
Sep-92
May-90
Jan-88
Mar-89
Nov-86
Jul-84
Sep-85
0
Regression Forecasting with Seasonal
Adjustment Factor
Seasonal adjustment
factor data is derived
from central moving
average series
25
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Comparison
• Comparing R Square of linear regression with
smoothed data and unsmoothed data
R Square
Regression with
unsmoothed data
0.942
Regression with smoothed 0.984
data
• Linear regression with smoothed data has a
higher R square, this model can better fit the
forecasted and actual value
Linear Regression
• Michigan GDP as one of the independent
variables instead of Michigan population and
U.S. GDP
26
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Forecasting
• Y=155753-20609*Ontario population(million) -13133*Fuel
price+13033*NAFTA+1294*Michigan GDP(billion)
R Square: 0.989
Comparison
• Comparing R Square of linear regression with
the independent variable of Michigan GDP and
the independent variables without Michigan GDP
R Square
Regression with Michigan
GDP
0.984
Regression without
Michigan GDP
0.989
• Linear regression with Michigan GDP has a higher
R square, this model can better fit the forecasted
and actual value
27
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ARIMA
• ARIMA(p,d,q)
o Auto-regressive model
p is the number of autoregressive terms
o Integrated model
d is the number of nonseasonal differences
o Moving average model
q is the number of lagged forecast errors in the
prediction equation
Auto-Regressive Model
• The definition of autoregressive(AR) model
o Takes advantage of autocorrelation
• 1st order - correlation between consecutive values
• 2nd order - correlation between values 2 periods apart
• pth order Autoregressive models
Yi = A 0 + A1Yi-1 + A 2 Yi-2 + L + A p Yi-p + δi
Random Error
28
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Autoregressive Modeling Example
• Develop the second order Autoregressive model
Excel Output
Coefficients
Intercept
13273
X Variable 1
-0.25
X Variable 2
0.128
Ŷi = 13273 − 0.25Yi −1 +0.128Yi − 2
ACF and PACF
• ACF
γ
covariance at lag k
ρk = k =
variance
γ0
=
∑ (Y − Y )(Y − Y )
∑ (Y − Y )
t +k
t
2
t
ρ k : The ACF at lag k.
• PACF
o The Partial Autocorrelation Function (PACF) is similar to the
ACF, however it measures correlation between observations
that are k time periods apart, after controlling for correlations
at intermediate lags.
29
12/22/2011
Autocorrelation
•
Using initial truck series to make autocorrelation
with SPSS
o Putting transformed data as variable
o Click “Autocorrelation” and “Partial autocorrelation”
and input maximum number of lags as 24
o Click “OK”
Autocorrelation of Initial Truck Series
30
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Identifying the Order of Differencing
• A stationary series
o Constant mean
o Constant variance
o Constant autocorrelation structure
Nonstationary
The variability is changing
Oct-07
Dec-10
Mar-06
May-09
Jan-03
Aug-04
Jun-01
Apr-98
Nov-99
Sep-96
Jul-93
Feb-95
Dec-91
Oct-88
May-90
Mar-87
Jan-84
Series1
Aug-85
Truck Volume
Westbound truck volume
100000
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
Identifying the Order of Differencing
• Making the first differencing (d=1)
Differenced Data
25000
20000
10000
5000
Series1
Feb-09
May-10
Nov-07
Aug-06
Feb-04
May-05
Nov-02
Aug-01
Feb-99
Nov-97
May-00
Aug-96
Feb-94
May-95
Nov-92
Aug-91
May-90
Feb-89
Nov-87
Aug-86
5000
10000
May-85
0
Feb-84
Truck Volume
15000
15000
20000
The series appears stationary, So we decide the d=1
31
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Removing the Changing of Variability
• Although the series appear stationary by
differencing once, the variability is still changing
over time. Transformation is considered for
series in which variance changes over time and
differencing does not stabilize the variance.
Box-Cox Transformation
• Transformation formulation
Where λ is transform parameter
32
12/22/2011
Transform with EXCEL
• Inputting the westbound truck volume (undifferenced)
in EXCEL
• Click QI Macros – Anova Tools – Box cox
• Inputting the transform parameter lambda with -0.7, 0.5, -0.3, 0.1, 0.3, 0.5, respectively
• Dividing the transformed data into 3 components by
time and computing the standard deviation of every
component
Order of components
Time interval
1
Jan 84-Apr 93
2
May 93-Aug 02
3
Sep 02-Jan 11
Identifying Lambda
0
1
2
3
0.001
0
1
Series1
1
0.5
0
1
2
3
Stand Deviation
Stand Deviation
2
1.5
2
3
0.006
0.1
Lambda=0.5
Lambda=0.3
2.5
Series1
0.0005
Stand Deviation
Series1
0.0001
0.0015
Stand Deviation
0.0002
Lambda=-0.3
Lambda=-0.5
Stand Deviation
Stand Deviation
Lambda=-0.7
0.0003
26
25
24
23
22
21
20
Series1
1
2
3
0.004
Series1
0.002
0
1
2
3
Lambda=0.1
0.08
0.06
Series1
0.04
0.02
0
1
2
3
• When lambda is 0.3, the stand deviation of the three parts
is most smoothing, we select lambda as 0.3.
33
-0.5
-1
Oct-09
Sep-10
Jan-84
Sep-10
May-09
Jan-08
Sep-06
May-05
Jan-04
Sep-02
May-01
Jan-00
Sep-98
May-97
Jan-96
Sep-94
May-93
Jan-92
Sep-90
May-89
Jan-88
Sep-86
May-85
Truck Volume
15
Nov-08
Dec-07
Jan-07
Feb-06
Mar-05
Apr-04
May-03
Jun-02
Jul-01
Aug-00
Sep-99
Oct-98
Nov-97
Dec-96
Jan-96
Feb-95
Mar-94
Apr-93
May-92
Jun-91
Jul-90
Aug-89
Sep-88
Oct-87
Nov-86
Dec-85
Jan-85
Feb-84
12/22/2011
Transforming with Undifferenced Data
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Transformed data
30
25
20
10
Series1
5
0
Transforming with Differenced Data
2.5
Transformed and differenced data
1.5
2
0.5
1
0
Series1
-1.5
-2.5
-2
-3
Stationary
Seasonal
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12/22/2011
Identifying AR(p)
• Identify the numbers of AR by looking at the
autocorrelation function (ACF) and partial
autocorrelation (PACF) plots of differenced series
AR(p)
ACF
Tails off
PACF
Cuts after p
Moving Average Model
• The definition of moving average(MA) model
: the correlation coefficient
: the series under investigation
: the residual
• MA(2) model
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12/22/2011
Identifying MA(q) Model
• Identify the numbers of MA by looking at the
autocorrelation function (ACF) and partial
autocorrelation (PACF) plots of differenced
series
MA(q)
ACF
Cuts after q
PACF
Tails off
Differencing=0
ACF falls to zero very slowly
indicates that non-seasonal
differencing is required
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12/22/2011
Differencing=1
ACF cuts off to zero at
lag 2 sharply, no nonseasonal differencing is
needed
PACF falls to zero slowly
and ACF cuts off to zero
after lag 1,indicating
MA(1) is needed
Identification of ARIMA(p,d,q)
P=0, d=1,q=1
The potential model is ARIMA(0,1,1)
(1−B)Yt=(1−θ1B)at
t: indexes time
B: the backshift operator, BYt= BYt-1
θ1: the nonseasonal moving average coefficient
at: the random error
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12/22/2011
Model with Seasonal Components
• Seasonal model:
where s is seasonal cycle
P: the order of seasonal AR model
D: the order of seasonal differencing
Q: the order of seasonal MA model
Identification of P,D,Q
Differencing: 0
ACF falls to zero slowly at lags 12
and 24, indicating seasonal
differencing is needed
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12/22/2011
ACF and PACF of Seasonal Differenced
Series
Differencing : 1
• ACF cuts off to zero at lags 12 and 24 indicates no more
seasonal differencing is needed.
• Both ACF and PACF do not show to cut off to zero at lags 12
and 24, indicating no seasonal AR model and seasonal MA
model are needed.
Identification of Final ARIMA(p,d,q)(P,D,Q)s
p=0,d=1,q=1
P=0,D=1,Q=0
ARIMA(0,1,1)(0,1,0)12
(1−B)(1−B12)Yt=(1−θ1B)at
t: indexes time
B: the backshift operator, BYt= BYt-1
B12: the seasonal backshift operator, B12Yt= BYt-12
θ1: the nonseasonal moving average coefficient
at: the random error
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12/22/2011
Model Building with SPSS
R square : 0.971
Residual Graph
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12/22/2011
Forecasting
Forecasting Results (Partial)
Forecasts of truck volume
derived from the forecasts
of transformed series
Forecasts of transformed series
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12/22/2011
Conclusions
•
•
•
•
Growth factor
Exponential smoothing
Linear regression
ARIMA
Conclusions (cont’d)
• Growth factor
o Advantages
• Simple and easy to understand
• Considering the linear and nonlinear trend of the
historical data
o Disadvantages
• Neglecting the effects of cyclical or seasonal
components
• Increasing the time
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12/22/2011
Conclusions (cont’d)
• Exponential smoothing
o Advantages
• Including both linear and nonlinear
• Structural view of the data that include level, trend,
seasonality, and events
o Disadvantage
•
The forecast is constant for all future values
Conclusions (cont’d)
• Linear regression
o Advantage
• Considering the characteristic of independent
variables and the relationship between independent
variables(economical and political factors) and
dependent variables
o Disadvantage
• Difficult to determine the trends of every
independent variable
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12/22/2011
Conclusions (cont’d)
• ARIMA
o Advantage
• The comprehensiveness of the family of models
o Disadvantages
• ARIMA identification is difficult and time consuming
• ARIMA may be difficult to explain to others
• Models that perform similarly on the historical data
may yield quite different forecasts
• Empirical
o Although there are more disadvantages than
advantages, the advantages may still outweigh
the disadvantages.
Reference
1. Daniel Beagan, Michael Fischer, Arun Kuppam.
“Quick Response Freight Manual II” (2007).
2. Alan Pankratz. “Forecasting With Univariate
Box-Jenkins Models” (1983).
3. R.M.SAKIA. “The Box-Cox transformation
technique: a review” (1992).
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