Advance forecasting
Forecasting by identifying patterns in the past data
Chapter outline:
1.
Extrapolation from the past
•
Cause and effect relationships
•
Trend analysis
- Regression analysis
- Simple linear regression analysis
- Multiple linear regression analysis
- Quadratic regression analysis
3.
Cyclical and seasonal issues
Seasonal decomposition of time series data
•
Type of seasonal variation
•
Computing Multiplication seasonal indices
•
Using seasonal indices to forecast
•
A caution regarding seasonal indices
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
1
Extrapolation from the past
Cause-and-effect Relationships
-
Causal forecasting seeks to identify specific cause-effect relationships
that will influence the pattern of future data. Causes appear as
independent variables, and effects as dependent , response variables in
forecasting models.
Independent variable
Dependent, response variable
Price
demand
Decrease in population
decrease in demand
Number of teenager
demand for jeans
-
Causal relationships exist even when there is no specific time series
aspect involved.
-
The most common technique used in causal modeling is least squares
regression.
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
2
Extrapolation from the past
Linear Trend analysis
D
Its noticed from
this figure that
there is a growth
trend influencing
the demand, which
should be
extrapolated into
the future.
D
D
P =
P =
D
D
D
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
3
Extrapolation from the past
Linear Trend analysis
The linear trend model or sloping line rather than horizontal line.
The forecasting equation for the linear trend model is
Y = +X
or
Y = a + bX
Where X is the time index (independent variable). The
parameters alpha and beta ( a and b) (the “intercept” and “slope”
of the trend line) are usually estimated via a simple regression in
which Y is the dependent variable and the time index X is the
independent variable.
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
4
Extrapolation from the past
Linear Trend analysis
Although linear trend models have their uses, they are often
inappropriate for business and economic data.
Most naturally occurring business time series do not behave as
though there are straight lines fixed in space that they are trying
to follow: real trends change their slopes and/or their intercepts
over time.
The linear trend model tries to find the slope and intercept that
give the best average fit to all the past data, and unfortunately its
deviation from the data is often greatest near the end of the time
series, where the forecasting action is.
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
5
Extrapolation from the past
Linear Trend analysis
Forecasting using three data items
Current
Intercept:
Current
Slope:
Period
1
2
3
42
Using a data table (what if analysis )
to determine the best-fitting straight
line with the lowest MSE
8
Straight Line
Demand
Forecast
50
50
60
58
64
66
Sums of
Squares:
MSE:
Squared
Deviaton
0
4
4
8
1.63
Table of MSE
Slope
Intercept
1.632993
38
40
42
44
46
48
50
52
BIS APPLICATION 2002-2003
4
12.33
10.39
8.49
6.63
4.90
3.46
2.83
3.46
5
10.23
8.29
6.38
4.55
2.94
2.16
2.94
4.55
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6
8.16
6.22
4.32
2.58
1.63
2.58
4.32
6.22
7
6.16
4.24
2.45
1.41
2.45
4.24
6.16
8.12
8
4.32
2.58
1.63
2.58
4.32
6.22
8.16
10.13
9
2.94
2.16
2.94
4.55
6.38
8.29
10.23
12.19
10
2.83
3.46
4.90
6.63
8.49
10.39
12.33
14.28
6
Extrapolation from the past
Linear Trend analysis
Simple linear Regression Analysis
Regression analysis is a statistical method of taking one or more variable
called independent or predictor variable- and developing a mathematical
equation that show how they relate to the value of a single variable- called
the dependent variable.
Regression analysis applies least-squares analysis to find the best-fitting line,
where best is defined as minimizing the mean square error (MSE) between
the historical sample and the calculated forecast.
Regression analysis is one of the tools provided by Excel.
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
7
Simple linear Regression Analysis
Quarters
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
BIS APPLICATION 2002-2003
Demand
3.47
3.12
3.97
4.50
4.06
6.90
3.60
6.47
4.27
5.24
6.39
5.45
5.88
8.99
4.12
6.68
9.44
7.75
9.91
9.14
14.25
14.89
14.22
15.56
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.866
0.749
0.738
1.986
24
ANOVA
df
Regression
Residual
Total
Intercept
Quarters
1
22
23
SS
MS
259.031 259.031
86.750 3.943
345.782
F
Significance F
65.691
0.000
Coefficients Standard Error t Stat P-value
1.495
0.837 1.787
0.088
0.475
0.059 8.105
0.000
Slope
MANAGEMENT INFORMATION SYSTEM
Lower 95% Upper 95%
-0.240
3.231
0.353
0.596
Intercept
8
Quarters Demand
1
3.47
2
3.12
3
3.97
4
4.50
5
4.06
6
6.90
7
3.60
8
6.47
9
4.27
10
5.24
11
6.39
12
5.45
13
5.88
14
8.99
15
4.12
16
6.68
17
9.44
18
7.75
19
9.91
20
9.14
21
14.25
22
14.89
23
14.22
24
15.56
25
26
27
28
BIS APPLICATION 2002-2003
Fitted
Demand Difference
1.97
2.24
2.45
0.45
2.92
1.11
3.40
1.23
3.87
0.04
4.35
6.54
4.82
1.48
5.30
1.39
5.77
2.26
6.25
1.01
6.72
0.11
7.20
3.03
7.67
3.22
8.15
0.71
8.62
20.25
9.10
5.83
9.57
0.02
10.05
5.26
10.52
0.38
11.00
3.45
11.47
7.74
11.95
8.70
12.42
3.24
12.90
7.09
13.37
13.85
14.32
14.80
MANAGEMENT INFORMATION SYSTEM
Intercept
Slope
MSE
1.495
0.475
1.901
20.00
15.00
10.00
5.00
0.00
1
5
9
13
17
21
25
9
Extrapolation from the past
Linear Trend analysis
Multiple linear Regression Analysis
Simple linear regression analysis use one variable (quarter number)
as the independent variable in order to predict the future value. In
many situations, it is advantageous to use more than one independent
variable in a forecast.
BIS APPLICATION 2002-2003
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10
Multiple linear Regression Analysis
Hours
Before
Breakdown
205
236
260
176
245
123
176
150
148
265
200
45
110
216
176
90
176
112
230
280
BIS APPLICATION 2002-2003
Age
59
48
25
39
20
66
40
62
70
20
52
75
75
25
63
75
69
65
30
23
Number of
Computer
Controls
1
1
0
0
1
2
0
0
0
0
1
0
0
0
1
0
2
0
0
1
Two factors that control the
frequency of breakdown. So they
are the independent variables.
Y = a + bX1 + cX2
Intercept
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Slope 1
Slope2
11
Multiple linear Regression Analysis
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.905
0.818
0.797
28.651
20
ANOVA
df
Regression
Residual
Total
2
17
19
SS
MS
62,920.044 31,460.022
13,954.906
820.877
76,874.950
Coefficients Standard Error
Intercept
308.451
17.552
Age
-2.800
0.325
No of Computer Controls
25.232
9.631
Intercept
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
F
Significance F
38.325
0.000
t Stat
P-value Lower 95% Upper 95%
17.573
0.000
271.419
345.484
-8.622
0.000
-3.485
-2.115
2.620
0.018
4.912
45.551
Slope 1
Slope 2
12
Hours
Before
Breakdown
205
236
260
176
245
123
176
150
148
265
200
45
110
216
176
90
176
112
230
280
Age
59
48
25
39
20
66
40
62
70
20
52
75
75
25
63
75
69
65
30
23
BIS APPLICATION 2002-2003
Number of
Computer
Controls
1
1
0
0
1
2
0
0
0
0
1
0
0
0
1
0
2
0
0
1
Hourse to
breakdown Difference
169
1332
199
1347
238
464
199
541
278
1069
174
2616
196
419
135
229
112
1261
252
157
188
141
98
2861
98
133
238
505
157
349
98
72
166
105
126
210
224
31
269
115
MANAGEMENT INFORMATION SYSTEM
Intercept
Age
No of Computer Controls
MSE
308.451
-2.800
25.232
26.41487
300
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
10 11 12
13 14 15 16
17 18 19 20
13
Extrapolation from the past
Linear Trend analysis
Quadratic Regression Analysis
Quadratic regression analysis fits a second-order curve of the form
Y = a + bX + cX2
Quadratic regression is prepared by adding the squared value of the
time periods. The coefficients in the quadratic formula are calculated
again using regression, where time periods and the squared time
periods are the independent variables and the demand remains the
dependent variable.
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
14
Quadratic Regression Analysis
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.927
0.859
0.846
1.524
24
ANOVA
df
Regression
Residual
Total
2
21
23
Intercept
Quarters
Quarters Squared
BIS APPLICATION 2002-2003
SS
MS
297.037 148.518
48.745 2.321
345.782
F
Significance F
63.984
0.000
Coefficients Standard Error t Stat P-value
4.685
1.017 4.609
0.000
-0.261
0.187 -1.395
0.178
0.029
0.007 4.046
0.001
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Lower 95% Upper 95% Upper 95.0%
2.571
6.799
6.799
-0.651
0.128
0.128
0.014
0.045
0.045
15
Quadratic Regression Analysis
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
Intercept
Slope 1
Slope 2
MSE
3.500
0.000
0.019
1.494
20.00
Forecast
15.00
10.00
5.00
28
25
22
16
13
10
7
19
Demand
0.00
4
Demand
3.47
3.12
3.97
4.50
4.06
6.90
3.60
6.47
4.27
5.24
6.39
5.45
5.88
8.99
4.12
6.68
9.44
7.75
9.91
9.14
14.25
14.89
14.22
15.56
Fitted
Demand Difference
3.52
0.00
3.58
0.21
3.67
0.09
3.80
0.49
3.98
0.01
4.18
7.39
4.43
0.69
4.72
3.09
5.04
0.60
5.40
0.03
5.80
0.35
6.24
0.61
6.71
0.70
7.22
3.10
7.78
13.36
8.36
2.83
8.99
0.20
9.66
3.63
10.36
0.20
11.10
3.85
11.88
5.63
12.70
4.83
13.55
0.45
14.44
1.24
15.38
16.34
17.35
18.40
1
Quarters
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Quarters
Squared
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
729
784
16
Extrapolation from the past
Cyclical and Seasonal Issues
The fundamental approach to including cyclical or seasonal factors
is to break the forecast into two components:
(1)
The underlying growth component
(2)
The seasonal variations
To prepare a forecast model:
-
Use a method to fit a growth curve to the historical record
-
Determine the pattern of the seasonal variability
In general, two sets of parameters to be estimated:
( the coefficients in the trend line, and the percents in the seasonal
patterns )
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Extrapolation from the past
Cyclical and Seasonal Issues
Basically two things must be done:
1- determine the trend line
2- take the trend line out ( calculate deviations from the trend)
3- create a pie, radar, or polar chart of the average period value
1
3
2
43,000
2
2,500
3
4
1
2
1
1,000
2
500
3
2
2
1
1,500
4
1
3
4
2,000
1
4
3
3
4
0
3
2
4
4
3
1
2
1
1
4
2
3
3
2
2
3
1
4
3
2
1
4
4
1
BIS APPLICATION 2002-2003
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Cyclical and Seasonal Issues
Seasonal Decomposition of Time Series Data
Time series data are usually considered to consist of six component :
1. Average demand: is simply the long-term mean demand
2. Trend component : is how rapidly demand is growing or shrinking
3. Autocorrelation: is simply a statement that demand next period is
related to demand this period
4. Seasonal component: is that portion of demand that follows a shortterm pattern
5. Cyclical component: is much like the seasonal component, only its
period is much longer.
6. Random component is the unpredictable component of demand
BIS APPLICATION 2002-2003
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Cyclical and Seasonal Issues
Type of Seasonal Variation
There are two types of seasonal variation:
Additive seasonal variation :
Occurs when the seasonal effects are the same regardless
of the trend.
Multiplication seasonal variation :
Occurs when the seasonal effects vary with the trend
effects. It’s the most common type of seasonal variation
BIS APPLICATION 2002-2003
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20
Cyclical and Seasonal Issues
Computing Multiplicative Seasonal Indices
Steps of Multiplicative Time Series Model:
1. Decide that the data is seasonal in nature.
2. Then realized that the seasonal variation is quarterly
3. If the variation of the data is larger to the right, then that
seasonal variation is multiplicative.
4. Seasonal indices is needed to produce the seasonal forecast
model.
BIS APPLICATION 2002-2003
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Cyclical and Seasonal Issues
Computing Multiplicative Seasonal Indices
1.
Computing seasonal indices requires data that match the seasonal
period. If the seasonal period is monthly, then monthly data are
required. A quarterly seasonal period requires quarterly data.
2.
Calculate the centered moving averages (CMAs) whose length matches
the seasonal cycle. The seasonal cycle is the time required for one cycle
to be completed. Quarterly seasonality requires a 4-period moving
average, monthly seasonality requires a 12-period moving average and
so on.
3.
Determine the Seasonal-Irregular Factors or components. This can be
done by dividing the raw data by the corresponding depersonalized
value.
4.
Determine the average seasonal factors. In this step the random and
cyclical components will be eliminated by averaging them.
BIS APPLICATION 2002-2003
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Cyclical and Seasonal Issues
Computing Multiplicative Seasonal Indices
Step 1
Quarter
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Seasonal
Four Period
Irregular
Data Moving Average Component
560
990
1,100
0.90000
1,740
1,120
1.55357
1,110
1,088
1.02069
640
1,133
0.56512
860
1,080
0.79630
1,920
1,090
1.76147
900
1,150
0.78261
680
1,163
0.58495
1,100
1,190
0.92437
1,970
1,198
1.64509
1,010
1,198
0.84342
710
1,263
0.56238
1,100
1,313
0.83810
2,230
1,275
1.74902
1,210
1,363
0.88807
560
1,393
0.40215
1,450
1,475
0.98305
2,350
1,573
1.49444
1,540
1,525
1.00984
950
1,648
0.57663
1,260
1,575
0.80000
2,840
1,250
BIS APPLICATION 2002-2003
MANAGEMENT INFORMATION SYSTEM
Seasonal
Index
0.87364
1.64072
0.90893
0.53825
Step 4
=AVERAGE(D3,D7,D11,D15,D19,D23)
=AVERAGE(D4,D8,D12,D16,D20)
=AVERAGE(D5,D9,D13,D17,D21)
=AVERAGE(D6,D10,D14,D18,D22)
Step 2
= AVERAGE(B2:B5)
Step 3
= B3/C3
23
Cyclical and Seasonal Issues
Using Seasonal Indices to Forecast
To forecast using seasonal indices
1- Compute the forecast using an annual values. Any forecasting techniques
can be used.
2- Use the seasonal indices to share out the annual forecast by periods
Year
1
2
3
4
5
6
7
7
Data
4,400
4,320
4,760
5,250
5,900
6,300
6,754
BIS APPLICATION 2002-2003
Forecast
Including
Trend Forecast
4,125
4,000
4,498
4,290
4,545
4,391
4,893
4,674
5,433
5,107
6,179
5,713
6,754
6,252
1
912
2
1469
3
2769
4
1537
MANAGEMENT INFORMATION SYSTEM
Trend
125
208
154
219
326
466
502
MAD
275
178
215
357
467
121
Q1
Q2
Q3
Q4
Alpha
Delta
MAD
0.6
0.5
269
0.54
0.87
1.64
0.91
24
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MANAGEMENT INFORMATION SYSTEM
25