Exponential Smoothing with Trend
• As we move toward medium-range
forecasts, trend becomes more important.
• Incorporating a trend component into
exponentially smoothed forecasts is called
double exponential smoothing.
– The estimate for the average and the estimate
for the trend are both smoothed.
Exponential Smoothing with Trend Adjustment
Forecast including trend (FITt )OR Adjusted Forecast (AFt) =
exponentially smoothed forecast (Ft)
+ exponentially smoothed trend (Tt)
That is,
AFt = Ft + Tt
We need to compute both Ft and Tt
Exponential Smoothing with Trend Adjustment – (contd.)
or
Ft = Last period’s forecast
+ D (Last period’s actual – Last period’s forecast)
Ft = Ft-1 + D (At-1 – Ft-1)
Tt = E(This period’s Forecast - last period’s Forecast)
+ (1-E) (Trend estimate last period)
or
Tt = E(Ft - Ft-1) + (1- E) Tt-1 for all t •
Ft = exponentially smoothed forecast of the data series in period t
Tt = exponentially smoothed trend in period t
At = actual demand in period t
α = smoothing constant for the average
β = smoothing constant for the trend
Adjusted Exponential Smoothing Example
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
37
40
41
37
45
50
43
47
56
52
55
54
Adjusted Exponential Smoothing Example
Per
1
2
3
4
5
6
7
8
9
10
11
12
13
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Dem
37
40
41
37
45
50
43
47
56
52
55
54
-
Ft+1
37.00
37.00
38.50
39.75
38.37
41.68
45.84
44.42
45.71
50.85
51.42
53.21
53.61
Tt+1
0.00
0.45
0.69
0.07
1.04
1.97
0.95
1.05
2.28
1.76
1.77
1.36
AFt+1
37.00
38.95
40.44
38.44
42.73
47.82
45.37
46.76
58.13
53.19
54.98
54.96
F3=F2+0.50(A2-F2) = 37+0.50*3 = 38.5
T3 = E(F3 - F2) + (1 - E) T2
= (0.30)(38.5 - 37.0) + (0.70)(0)
= 0.45
AF3 = F3 + T3
=38.5 + 0.45 = 38.95
T13 = E(F13 - F12) + (1 - E) T12
=(0.30)(53.61 - 53.21) + (0.70)(1.77)
=1.36
AF13 = F13 + T13 = 53.61 + 1.36 = 54.96
Forecast (D = 0.50)
Adjusted Exponential Smoothing Forecasts
70 –
Adjusted forecast (D = 0.50; E = 0.30))
60 –
Actual
Demand
50 –
40 –
Forecast (D = 0.50)
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
|
6
7
Period
|
8
|
9
|
10
|
11
|
12
|
13
Seasonal Adjustments
• Repetitive increase/decrease in demand
– Use seasonal factor to adjust forecast
D
i
Seasonal factor = Si = D
∑
Where
Di is the sum of demands of the period i in the time series data
6Dis net sum of demands of the entire period in the time series data
Example: Seasonal Adjustment [1]
Year
2004
2005
2006
Total
Si
Demand (1000’s per quarter)
1
2
3
4
12.6
8.6
6.3
17.5
14.1
10.3
7.5
18.2
15.3
10.6
8.1
19.6
42.0
29.5
21.9
55.3
0.28
0.20
0.15
0.37
Total
45.0
50.1
53.6
148.7
Computed trend line for data y = 40.97 + 4.30 X [Given to you]
2006 (year 4) forecast = 40.97 + 4.30 (4) = 58.17
Forecasted demand after seasonal adjustment for the year 2006 is
2006
16.28
11.63
8.73
21.53
---------------------------------------------------------------------------------Details
58.17 x 0.28 = 16.28;
58.17 x 0.20 = 11.63;
58.17 x 0.15 = 8.73;
58.17 x 0.37 = 21.53
--------------------------------------------------------------------------------
S1 =
D1 = 42.0 = 0.28
∑ D 148.7
SF1 = (S1) (F5)
= (0.28)(58.17) = 16.28
SF2 = (S2) (F5)
= (0.20)(58.17) = 11.63
SF3 = (S3) (F5)
= (0.15)(58.17) = 8.73
SF4 = (S4) (F5)
= (0.37)(58.17) = 21.53
Example: Seasonal Adjustment [2]
Quarter
1
2
3
4
Total
Average
Year 1
Year 2
Year 3
Year 4
45
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1000
250
1200
300
1800
450
2200
550
Example: Seasonal Adjustment [2]
Quarter
1
2
3
4
Total
Average
Year 1
Year 2
Year 3
Year 4
45
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1000
250
1200
300
1800
450
2200
550
Seasonal Index =
Actual Demand
Average Demand
Example: Seasonal Adjustment [2]
Quarter
1
2
3
4
Total
Average
Year 1
Year 2
Year 3
Year 4
45
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1000
250
1200
300
1800
450
2200
550
Seasonal Index =
45
250
= 0.18
Example: Seasonal Adjustment [2] – Contd.
Quarter
1
2
3
4
Year 1
Year 2
Year 3
Year 4
45/250 = 0.18
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1200
300
1800
450
2200
550
Total
Average
1000
250
Seasonal Index =
45
250
= 0.18
Example: Seasonal Adjustment [2] – Contd.
Quarter
1
2
3
4
Year 1
Year 2
45/250 = 0.18
335/250 = 1.34
520/250 = 2.08
100/250 = 0.40
70/300 = 0.23
370/300 = 1.23
590/300 = 1.97
170/300 = 0.57
Year 3
Year 4
100/450 = 0.22 100/550 = 0.18
585/450 = 1.30 725/550 = 1.32
830/450 = 1.84 1160/550 = 2.11
285/450 = 0.63 215/550 = 0.39
Example: Seasonal Adjustment [2] – Contd.
Quarter
1
2
3
4
Year 1
Year 2
45/250 = 0.18
335/250 = 1.34
520/250 = 2.08
100/250 = 0.40
70/300 = 0.23
370/300 = 1.23
590/300 = 1.97
170/300 = 0.57
Year 3
Year 4
100/450 = 0.22 100/550 = 0.18
585/450 = 1.30 725/550 = 1.32
830/450 = 1.84 1160/550 = 2.11
285/450 = 0.63 215/550 = 0.39
Quarter
Average Seasonal Index
1
2
3
4
(0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
Example: Seasonal Adjustment [2] – Contd.
Quarter
1
2
3
4
Year 1
Year 2
45/250 = 0.18
335/250 = 1.34
520/250 = 2.08
100/250 = 0.40
70/300 = 0.23
370/300 = 1.23
590/300 = 1.97
170/300 = 0.57
Year 3
Year 4
100/450 = 0.22 100/550 = 0.18
585/450 = 1.30 725/550 = 1.32
830/450 = 1.84 1160/550 = 2.11
285/450 = 0.63 215/550 = 0.39
Quarter
Average Seasonal Index
1
2
3
4
(0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
(1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30
(2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00
(0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50
Example: Seasonal Adjustment [2] – Contd.
Quarter
1
2
3
4
Year 1
Year 2
Year 3
Year 4
45/250 = 0.18
70/300 = 0.23 100/450 = 0.22 100/550 = 0.18
Projected
Annual
Demand = 2600 [Given]
335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.32
520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.11
Average Quarterly Demand = 2600/4 = 650
100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39
Quarter
Average Seasonal Index
1
2
3
4
(0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
(1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30
(2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00
(0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50
Forecast
Example: Seasonal Adjustment [2] – Contd.
Quarter
1
2
3
4
Year 1
Year 2
Year 3
Year 4
45/250 = 0.18
70/300 = 0.23 100/450 = 0.22 100/550 = 0.18
Projected
Annual
Demand = 2600
335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.32
Average
Demand
= =2600/4
= 650= 2.11
520/250
= 2.08 Quarterly
590/300 = 1.97
830/450
1.84 1160/550
100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39
Quarter
Average Seasonal Index
1
2
3
4
(0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
(1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30
(2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00
(0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50
Forecast
650(0.20) = 130
Example: Seasonal Adjustment [2] – Contd.
Quarter
1
2
3
4
Year 1
Year 2
45/250 = 0.18
335/250 = 1.34
520/250 = 2.08
100/250 = 0.40
70/300 = 0.23
370/300 = 1.23
590/300 = 1.97
170/300 = 0.57
Year 3
Year 4
100/450 = 0.22 100/550 = 0.18
585/450 = 1.30 725/550 = 1.32
830/450 = 1.84 1160/550 = 2.11
285/450 = 0.63 215/550 = 0.39
Quarter
Average Seasonal Index
Forecast
1
2
3
4
(0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
(1.34 + 1.23 + 1.30 + 1.32)/4 = 1.30
(2.08 + 1.97 + 1.84 + 2.11)/4 = 2.00
(0.40 + 0.57 + 0.63 + 0.39)/4 = 0.50
650(0.20) =
650(1.30) =
650(2.00) =
650(0.50) =
130
845
1300
325
Seasonalised Time Series Regression Analysis
1. Select a representative historical data set.
2. Develop a seasonal index for each season.
3. Use the seasonal indexes to De-Seasonalise the data.
4. Perform linear regression analysis on the De-Seasonalised data.
5. Use the regression equation to compute the forecasts.
6. Use the seasonal indexes to reapply the seasonal patterns to the
forecasts.
Example: Computer Products Corp.
Seasonalized Times Series Regression Analysis
An analyst at CPC wants to develop next year’s
quarterly forecasts of sales revenue for CPC’s line
of Epsilon Computers. She believes that the most
recent 8 quarters of sales (shown on the next slide)
are representative of next year’s sales.
Example: Computer Products Corp.
• Seasonalised Times Series Regression Analysis
– Representative Historical Data Set
Year Qtr. ($mil.)
1
1
1
1
1
2
3
4
7.4
6.5
4.9
16.1
Year Qtr. ($mil.)
2
2
2
2
1
2
3
4
8.3
7.4
5.4
18.0
Example: Computer Products Corp.
– Compute the Seasonal Indexes
Year
1
2
Totals
Qtr. Avg.
Seas.Ind.
Quarterly Sales
Q1
Q2 Q3
Q4
Total
7.4
6.5 4.9 16.1 34.9
8.3
7.4 5.4 18.0 39.1
15.7 13.9 10.3 34.1 74.0
7.85 6.95 5.15 17.05 9.25
.849 .751 .557 1.843 4.000
7.85 / 9.25
Example: Computer Products Corp.
Time series data:
Quarterly Sales
Year
Q1
Q2
Q3
Q4
1
7.4
6.5
4.9
16.1
2
8.3
7.4
5.4
18.0
Seasonal Index
0.849 0.751 0.557 1.843
De-Seasonalised data for Q1 = { Actual Q1 sales / Seas. Index }
De-Seasonalise the Data
Quarterly Sales
Year Q1
Q2
Q3
1
8.72
8.66
8.80
2
9.78
9.85
9.69
Yt = Tt x St x Ct x Rt
Q4
8.74
9.77
Assum. There is no Ct & Rt
Yt = Tt x St
Yt (dese.)= (Yt/St)
Example: Computer Products Corp.
– Perform Regression on De-seasonalized Data
Yr.
Qtr.
x
y
x2
1
1
1
1
2
2
2
2
xy
1
2
3
4
1
2
3
4
1
2
3
4
5
6
7
8
8.72
8.66
8.80
8.74
9.78
9.85
9.69
9.77
1
4
9
16
25
36
49
64
8.72
17.32
26.40
34.96
48.90
59.10
67.83
78.16
Totals
36
74.01
204
341.39
204(74.01) − 36( 341.39 )
a =
= 8.357
2
8( 204 ) − ( 36 )
8(341.39 ) − 36(74.01)
b=
= 0.199
2
8(204 ) − (36 )
Y = 8.357 + 0.199X
Example: Computer Products Corp.
Compute the De-Seasonalised Forecasts
MODEL : Y = 8.357 + 0.199X
Y9
Y10
Y11
Y12
= 8.357 + 0.199(9) = 10.148
= 8.357 + 0.199(10) = 10.347
= 8.357 + 0.199(11) = 10.546
= 8.357 + 0.199(12) = 10.745
Note: Average sales are expected to increase by
.199 million (about $200,000) per quarter.
Example: Computer Products Corp.
Seasonalised the Forecasts
Seas.
Yr. Qtr. Index
3
3
3
3
1
2
3
4
.849
.751
.557
1.843
De-seas.
Forecast
Seas.
Forecast
10.148
10.347
10.546
10.745
8.62
7.77
5.87
19.80
Time Series Models & Classical
Decomposition
• Decomposition time series models:
• Multiplicative:
• Additive:
Y=TxCxSxe
Y=T+C+S+e
T = Trend component
C = Cyclical component
S = Seasonal component
e = Error or random component
Time Series Models & Classical
Decomposition
• Classical decomposition is used to isolate trend,
seasonal, and other variability components from a
time series model
Classical Decomposition Explained
Basic Steps:
1. Determine seasonal indexes using the ratio to
moving average method
2. Deseasonalize the data
3. Develop the trend-cyclical regression equation
using deseasonalized data
4. Multiply the forecasted trend values by their
seasonal indexes to create a more accurate
forecast
•
Start with multiplicative model…
Y = TCSe
•
Then
Se = (Y/TC)
Classical Decomposition:
Illustration
• Gem Company’ s operations department
has been asked to deseasonalize and
forecast sales for the next four quarters
of the coming year
• The Company has compiled its past
sales data in Table 1
• An illustration using classical
decomposition will follow
Table 1: Gem Company’s Sales Data
Original
Year Quarter Period Sales
t
Y
1
1
1
55
2
2
47
3
3
65
4
4
70
2
1
5
65
2
6
58
3
7
75
4
8
80
3
1
9
65
2
10
62
3
11
80
4
12
85
4
1
13
70
2
14
65
3
15
85
4
16
90
5
1
17
2
18
3
19
4
20
-
Forecasted
Sales
TS
?
?
?
?
Classical Decomposition Illustration:
Step 1
• (a) Compute the fourquarter simple
moving average
Ex: simple MA at end
of Qtr 2 and
beginning of Qtr 3
(55+47+65+70)/4 =
59.25
Moving
Year Quarter Period Sales Average
t
Y
55
1
1
1
2
2
47
59.25
3
3
65
61.75
70
64.50
4
4
2
1
5
65
67.00
2
6
58
69.50
3
7
75
69.50
4
8
80
70.50
3
1
9
65
71.75
2
10
62
73.00
3
11
80
74.25
4
12
85
75.00
4
1
13
70
76.25
2
14
65
77.50
3
15
85
4
16
90
Classical Decomposition
Illustration: Step 1
• (b) Compute the twoquarter centered
moving average
Ex: centered MA at
middle of Qtr 3
(59.25+61.25)/2
= 60.500
Table 2: Four-Quarter Moving Average
Simple Centered
Moving Moving
Year Quarter Period Sales Average Average
t
Y
TC
1
1
1
55
59.25
2
2
47
3
3
65
61.75
60.500
4
4
70
64.50
63.125
2
1
5
65
67.00
65.750
2
6
58
69.50
68.250
3
7
75
69.50
69.500
4
8
80
70.50
70.000
3
1
9
65
71.75
71.125
2
10
62
73.00
72.375
3
11
80
74.25
73.625
4
12
85
75.00
74.625
4
1
13
70
76.25
75.625
2
14
65
77.50
76.875
3
15
85
4
16
90
Classical Decomposition
Illustration: Step 1
• (c) Compute the
seasonal-error
component (percent
MA)
Ex: percent MA at Qtr
3
(65/60.500)
= 1.074
Table 2: Four-Quarter Moving Average
Simple Centered
Moving Moving
Year Quarter Period Sales Average Average
t
Y
TC
1
1
1
55
2
2
47
59.25
3
3
65
61.75
60.500
4
4
70
64.50
63.125
2
1
5
65
67.00
65.750
2
6
58
69.50
68.250
3
7
75
69.50
69.500
4
8
80
70.50
70.000
3
1
9
65
71.75
71.125
2
10
62
73.00
72.375
3
11
80
74.25
73.625
4
12
85
75.00
74.625
4
1
13
70
76.25
75.625
2
14
65
77.50
76.875
3
15
85
4
16
90
Percent
Moving
Average
Se=Y/(TC)
1.074
1.109
0.989
0.850
1.079
1.143
0.914
0.857
1.087
1.139
0.926
0.846
Classical Decomposition
Illustration: Step 1
• (d) Compute the unadjusted seasonal index using the seasonalerror components from Table 2
Ex (Qtr 1): [(Yr 2, Qtr 1) + (Yr 3, Qtr 1) + (Yr 4, Qtr 1)]/3
= [0.989+0.914+0.926]/3 = 0.943
Table 3: Seasonal Index Computation
Quarter
1
2
3
4
Average
(0.989+0.914+0.926)/3
(0.850+0.857+0.846)/3
(1.074+1.079+1.087)/3
(1.109+1.143+1.139)/3
=
=
=
=
Unadjusted
Seasonal
Index
0.943
0.851
1.080
1.130
4.004
x
x
x
x
Adjusting
Factor
(4.000/4.004)
(4.000/4.004)
(4.000/4.004)
(4.000/4.004)
=
=
=
=
Adjusted
Seasonal
Index
0.942
0.850
1.079
1.129
4.000
Classical Decomposition
Illustration: Step 1
• (e) Compute the adjusting factor by dividing the number of
quarters (4) by the sum of all calculated unadjusted seasonal
indexes
= 4.000/(0.943+0.851+1.080+1.130) = (4.000/4.004)
Table 3: Seasonal Index Computation
Quarter
1
2
3
4
Average
(0.989+0.914+0.926)/3
(0.850+0.857+0.846)/3
(1.074+1.079+1.087)/3
(1.109+1.143+1.139)/3
=
=
=
=
Unadjusted
Seasonal
Index
0.943
0.851
1.080
1.130
4.004
x
x
x
x
Adjusting
Factor
(4.000/4.004)
(4.000/4.004)
(4.000/4.004)
(4.000/4.004)
=
=
=
=
Adjusted
Seasonal
Index
0.942
0.850
1.079
1.129
4.000
Classical Decomposition
Illustration: Step 1
• (f) Compute the adjusted seasonal index by multiplying the
unadjusted seasonal index by the adjusting factor
Ex (Qtr 1): 0.943 x (4.000/4.004) = 0.942
Table 3: Seasonal Index Computation
Quarter
1
2
3
4
Average
(0.989+0.914+0.926)/3
(0.850+0.857+0.846)/3
(1.074+1.079+1.087)/3
(1.109+1.143+1.139)/3
=
=
=
=
Unadjusted
Seasonal
Index
0.943
0.851
1.080
1.130
4.004
x
x
x
x
Adjusting
Factor
(4.000/4.004)
(4.000/4.004)
(4.000/4.004)
(4.000/4.004)
=
=
=
=
Adjusted
Seasonal
Index
0.942
0.850
1.079
1.129
4.000
Classical Decomposition
Illustration: Step 2
• Compute the
deseasonalized sales by
dividing original sales
by the adjusted seasonal
index
Ex (Yr 1, Qtr 1):
(55 / 0.942)
= 58.386
Table 4: Deseasonalizing Sales
Adjusted
Original Seasonal Deseasonalized
Year Quarter Period Sales
Index
Sales
t
Y
S
TCe
55
0.942
58.386
1
1
1
2
2
47
0.850
55.294
3
3
65
1.079
60.241
4
4
70
1.129
62.002
2
1
5
65
0.942
69.002
2
6
58
0.850
68.235
3
7
75
1.079
69.509
4
8
80
1.129
70.859
3
1
9
65
0.942
69.002
2
10
62
0.850
72.941
3
11
80
1.079
74.143
4
12
85
1.129
75.288
4
1
13
70
0.942
74.310
2
14
65
0.850
76.471
3
15
85
1.079
78.777
4
16
90
1.129
79.717
Classical Decomposition
Illustration: Step 3
• Compute the trend-cyclical
regression equation using
simple linear regression
Tt = a + bt
t-bar = 8.5
T-bar = 69.6
b
= 1.465
a
= 57.180
Tt = 57.180 + 1.465t
Table 5: Regression Equation Values
Deseasonalized
Year Quarter Period
Sales
t
TCe = (Y/S)
1
1
1
58.386
2
2
55.294
3
3
60.241
4
4
62.002
2
1
5
69.002
2
6
68.235
3
7
69.509
4
8
70.859
3
1
9
69.002
2
10
72.941
3
11
74.143
4
12
75.288
4
1
13
74.310
2
14
76.471
3
15
78.777
4
16
79.717
136
1114.176
t( Y/S)
58.386
110.588
180.723
248.007
345.011
409.412
486.562
566.873
621.019
729.412
815.570
903.454
966.030
1070.588
1181.650
1275.465
9968.750
t2
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
1496
Classical Decomposition
Illustration: Step 4
• (a) Develop trend sales
Tt = 57.180 + 1.465t
Ex (Yr 1, Qtr 1):
T1 = 57.180 + 1.465(1)
= 58.645
Table 6: Trend Sales
Year Quarter Period
t
1
1
1
2
2
3
3
4
4
2
1
5
2
6
3
7
4
8
3
1
9
2
10
3
11
4
12
4
1
13
2
14
3
15
4
16
5
1
17
2
18
3
19
4
20
Original Deseasonalized
Sales
Sales
Y
TCe = (Y/S)
55
58.386
47
55.294
65
60.241
70
62.002
65
69.002
58
68.235
75
69.509
80
70.859
65
69.002
62
72.941
80
74.143
85
75.288
70
74.310
65
76.471
85
78.777
90
79.717
Trend
Sales
T
58.645
60.110
61.575
63.040
64.505
65.970
67.435
68.900
70.365
71.830
73.295
74.760
76.225
77.690
79.155
80.620
82.085
83.550
85.015
86.480
Classical Decomposition
Illustration: Step 4
• (b) Forecast sales for
each of the four quarters
of the coming year
Ex (Yr 5, Qtr 1):
0.942 x 82.085
= 77.324
Table 7: Forecasted Sales
Year Quarter Period
t
1
1
1
2
2
3
3
4
4
2
1
5
2
6
3
7
4
8
3
1
9
2
10
3
11
4
12
4
1
13
2
14
3
15
4
16
5
1
17
2
18
3
19
4
20
Seasonal
Index
S
0.942
0.850
1.079
1.129
0.942
0.850
1.079
1.129
0.942
0.850
1.079
1.129
0.942
0.850
1.079
1.129
0.942
0.850
1.079
1.129
Trend
Sales
T
58.645
60.110
61.575
63.040
64.505
65.970
67.435
68.900
70.365
71.830
73.295
74.760
76.225
77.690
79.155
80.620
82.085
83.550
85.015
86.480
Forecasted
Sales
TS
77.324
71.018
91.731
97.636
Classical Decomposition
Illustration: Graphical Look
Graph 1: Comparison of Trend, Original, and Deseasonalized Sales
100
90
Sales ($)
80
(Y/S) = TCe
Deseasonalized
70
Y
Original
T
Trend
60
50
40
0
2
4
6
8
10
Quarter
12
14
16
18
The Classical DecompositionProcedure
• Smooth the time series to remove
random effects and seasonality.
• Calculate moving averages.
• Determine “period factors” to
isolate the (seasonal)•(error)
factor.
• Calculate the ratio yt/MAt.
• Determine the “unadjusted
seasonal factors” to eliminate the
random component from the period
factors
• Average all the yt/MAt that
correspond to the same season.
The Classical DecompositionProcedure – Contd.
• Determine the “adjusted
seasonal factors”.
Calculate:
[Unadjusted seasonal factor]
[Average seasonal factor]
•
Determine “Deseasonalized data
values”.
Calculate:
•
Determine a deseasonalized trend
forecast.
Use linear regression on the
deseasonalized time series.
•
Determine an “adjusted seasonal
forecast”.
Calculate:
(yt/Mat) •[Adjusted seasonal forecast].
yt
[Adjusted seasonal factors]t
Monitoring and Controlling Operations Forecasts
• Reasons for out-of-control forecasts
– change in trend
– appearance of cycle
– politics
– weather changes
– promotions
Monitoring and Controlling a Forecasting Model
Forecasts can be monitored using
either Tracking Signal (TS)
or
Control Charts
• Why track the forecast?
– To plan around the error in the future
– To measure actual demand versus forecasts
– To improve our forecasting methods
Monitoring and Controlling a Forecasting Model
• Tracking Signal (TS)
– The TS measures the cumulative forecast error over n
periods in terms of MAD
n
TS =
∑(Actual demand
i=1
i
- Forecast demand i )
MAD
– If the forecasting model is performing well, the TS
should be around zero
– TS indicates the direction of the forecasting error
• if the TS is positive -- increase the forecasts,
• if the TS is negative -- decrease the forecasts.
Monitoring and Controlling a Forecasting Model
• Tracking Signal
– The value of the TS can be used to automatically
trigger new parameter values of a model, thereby
correcting model performance.
– If the limits are set too narrow, the parameter values will
be changed too often.
– If the limits are set too wide, the parameter values will
not be changed often enough and accuracy will suffer.
Tracking Signal Computation
Mo Fcst
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
TS
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
Error = Actual - Forecast
= 90 - 100 = -10
TS
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
-10
RSFE = 6 Errors
= NA + (-10) = -10
TS
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
-10
10
Abs Error = |Error|
= |-10| = 10
TS
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
-10
10
TS
10
Cum |Error| = 6 |Errors|
= NA + 10 = 10
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
-10
10
10 10.0
MAD = 6 |Errors|/n
= 10/1 = 10
TS
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
-10
10
10 10.0
TS = RSFE/MAD
= -10/10 = -1
TS
-1
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
2
100
95
-5
3
100 115
4
100 100
5
100 125
6
100 140
-10
10
10 10.0
Error = Actual - Forecast
= 95 - 100 = -5
TS
-1
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
2
100
95
-5
-15
3
100 115
4
100 100
5
100 125
6
100 140
10
10 10.0
RSFE = 6 Errors
= (-10) + (-5) = -15
TS
-1
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
10 10.0
Abs Error = |Error|
= |-5| = 5
TS
-1
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
10 10.0
15
Cum Error = 6 |Errors|
= 10 + 5 = 15
TS
-1
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
10 10.0
15
MAD = 6 |Errors|/n
= 15/2 = 7.5
7.5
TS
-1
Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
TS
10 10.0
-1
15
-2
TS = RSFE/MAD
= -15/7.5 = -2
7.5
Plot of a Tracking Signal
Signal exceeded limit
MAD
+
Upper control limit
0
-
Tracking signal
Acceptable range
Lower control limit
Time
3
160
140
120
100
80
60
40
20
0
2
Forecast
1
Actual demand
0
Tracking Signal
-1
-2
-3
0
1
2
3
4
Time
5
6
7
Tracking Singal
Actual Demand
Tracking Signals
NOTE on TS
¾ The cumulative forecast error reflects the bias in forecasts, which
is the persistent tendency for forecasts to be greater or
less than the actual values of a time series.
¾ Tracking signal values are compared to predetermined limits
based on judgment and experience. They often range from
r3 to r8; for the most part, we shall use limits of ±4, which
are roughly comparable to three standard deviation limits.
¾ Values within the limits suggest – but do not guarantee – that the
forecast is performing adequately.
Statistical Control Charts
The control chart approach involves setting upper and lower limits for
individual forecast errors (instead of cumulative errors, as in the case with
a tracking signal). The limits are multiples of the “square root of MSE”
(The square root of MSE is used in practice as an estimate of the standard
deviation, V, of the distribution of errors).
V=
¦(Dt - Ft)2
n-1
9 This methods assumes (a) Forecast errors are randomly
distributed around a mean of zero and (b) The
distribution of errors is normal.
9 Using V we can calculate statistical control limits for the
forecast error
Statistical Control Charts (Contd.)
9
Recall that for a ND, approximately 95% of the values (errors in
this case) can be expected to fall within limits of 0 r 2V, and
approximately 99.7% of the values can be expected to fall
within r 3V of zero.
9
9
Hence, if the forecast is “in control”, 99.7% or 95% of
the errors should fall within the limits, depending upon
whether r 3V or r 2V limits are used.
Points that fall outside these limits should be regarded as
evidence that corrective action is needed [that is the forecast is
not performing adequately).
Statistical Control Charts
18.39 –
12.24 –
Errors
6.12 –
0–
-6.12 –
-12.24 –
-18.39 –
|
0
|
1
|
2
|
3
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
Statistical Control Charts
18.39 –
UCL = +3V
12.24 –
Errors
6.12 –
0–
-6.12 –
-12.24 –
-18.39 –
|
0
LCL = -3V
|
1
|
2
|
3
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
Ranging Forecasts
• Forecasts for future periods are only
estimates and are subject to error.
– One way to deal with uncertainty is to develop
best-estimate forecasts and the ranges within
which the actual data are likely to fall.
• The ranges of a forecast are defined by the upper
and lower limits of a confidence interval.
Ranging Forecasts
• The ranges or limits of a forecast are estimated by:
Upper limit = Y + t(syx)
Lower limit = Y - t(syx)
where:
Y = best-estimate forecast
t = number of standard deviations from the mean
of the distribution to provide a given probability of
exceeding the limits through chance
syx = standard error of the forecast
Ranging Forecasts
• The standard error (deviation) of the
forecast is computed as:
s yx =
2
y
∑ - a ∑ y - b ∑ xy
n -2
Example: Railroad Products Co.
• Ranging Forecasts
Recall that linear regression analysis
provided a forecast of annual sales for RPC
in year 8 equal to $20.55 million.
Set the limits (ranges) of the forecast so
that there is only a 5 percent probability of
exceeding the limits by chance.
Example: Railroad Products Co.
• Ranging Forecasts
Step 1: Compute the standard error of the
forecasts, syx.
1287.5 − .528(93) − .0801(15, 440)
= .5748
syx =
7−2
Step 2: Determine the appropriate value for t.
n = 7, so degrees of freedom = n – 2 = 5.
Area in upper tail = .05/2 = .025
Statistical Table shows t = 2.571.
Example: Railroad Products Co.
• Ranging Forecasts
– Step 3: Compute upper and lower limits.
Upper limit = 20.55 + 2.571(.5748)
= 20.55 + 1.478
= 22.028
Lower limit = 20.55 - 2.571(.5748)
= 20.55 - 1.478
= 19.072
We are 95% confident that the actual sales for year
8 will be between $19.072 and $22.028 million.
Criteria/factor to be considered for Selecting a
Forecasting Method
•
•
•
•
•
•
Cost
Accuracy
Data available
Time span
Nature of products and services
Impulse response and noise dampening
Criteria for Selecting a Forecasting Method
• Cost and Accuracy
– There is a trade-off between cost and accuracy;
generally, more forecast accuracy can be obtained at a
cost.
– High-accuracy approaches have disadvantages:
•
•
•
•
Use more data
Data are ordinarily more difficult to obtain
The models are more costly to design, implement, and operate
Take longer to use
- Low/Moderate-Cost Approaches – statistical models,
historical analogies, executive-committee consensus
- High-Cost Approaches – complex econometric models,
Delphi, and market research
Criteria for Selecting a Forecasting Method
• Availability of historical data
– Is the necessary data available or can it be
economically obtained?
• If the need is to forecast sales of a new product,
then a customer survey may not be practical;
instead, historical analogy or market research may
have to be used.
Criteria for Selecting a Forecasting Method
• Time Span
– What operations resource is being forecast and
for what purpose?
– Short-term staffing needs might best be
forecast with moving average or exponential
smoothing models.
– Long-term factory capacity needs might best
be predicted with regression or executivecommittee consensus methods.
Criteria for Selecting a Forecasting Method
• Nature of Products and Services
– Is the product/service high cost or high
volume?
– Where is the product/service in its life cycle?
– Does the product/service have seasonal
demand fluctuations?
Criteria for Selecting a Forecasting Method
• Impulse Response and Noise Dampening
– An appropriate balance must be achieved
between:
• How responsive we want the forecasting model to
be to changes in the actual demand data
• Our desire to suppress undesirable chance
variation or noise in the demand data
Reasons for Ineffective Forecasting
• Not involving a broad cross section of people
• Not recognizing that forecasting is integral to
business planning
• Not forecasting the right things
• Not selecting an appropriate forecasting method
• Not tracking the accuracy of the forecasting
models
• Not recognizing that forecasts will always be
wrong
Forecasting in Small Businesses
and Start-Up Ventures
• Forecasting for these businesses can be difficult for the
following reasons:
– Not enough personnel with the time to forecast
– Personnel lack the necessary skills to develop good
forecasts
– These businesses are not data-rich environments
– Forecasting for new products/services is always
difficult, even for the experienced forecaster
Sources of Forecasting Data and Help
• Government agencies at the local, regional,
state, and federal levels
• Industry associations
• Consulting companies
Some Specific Forecasting Data
•
•
•
•
•
•
•
•
Consumer Confidence Index
Consumer Price Index (CPI)
Gross Domestic Product (GDP)
Index of Leading Economic Indicators
Personal Income and Consumption
Producer Price Index (PPI)
Purchasing Manager’ s Index
Retail Sales
NOTE
The wise decision maker does not limit forecasting decisions to a
single technique but combines the subjective and objective methods.
Furthermore, the approximate way of defining forecast could be
Forecast = Projection r Judgment
Good Forecasting has to be determined with the tool : DSS