exponential smoothing business forecasting

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Extrapolation Methods

Exponential smoothing

  • Most managers have to forecast the inventories or sales of many product each day, week or month.
  • While sophisticated forecast method can be used, some simple method of time series smoothing will do the job in this case.
  • In smoothing method we use a form of weighted average of the past data to smooth (eliminate) the short term fluctuations.

Exponential smoothing

  • In smoothing models we assume that fluctuation in the past values are random values and, once identified, can be extrapolated into the future to create a forecast

Exponential smoothing

  • We examine .the following smoothing methods:
  • Simple exponential smoothing
  • Holt’s exponential smoothing
  • Winter’s exponential smoothing
  • Adaptive-response –rate single exponential smoothing will not be covered

Extrapolation methods

  • Extrapolation methods of forecast focus on a single time series to identify the past patterns in the historical data.
  • This pattern are then extrapolated to map out the like future path of the series.

Exponential smoothing

Simple Smoothing

  • Simple smoothing forecast is used when there is no trend or seasonality present in the data.

Simple smoothing

Chart1

45
42
40
50
55
60
54
52
64
70
90
100
65
45
60
Actual

Sheet1

There are two categories of statisticals tools for forecasting :
causal and time series. Causal techniques link the forecast values
of a variable (dependent variable) to one or more anticipated causes
(independent variables). For example a change in inventory level
causes sales to change. regression method cab be used to forecast
this type of relationships.
Time-series techniques link future movements in the forecasted
varaible to pattern revealed by historical movement in the same
variable. The moving average , exponential smoothing, time series
time series regressions are some of this techniques that we will
study.
&A
Page &P

simple

Simple Smoothing
•Lt = a( Yt ) +(1- a ) L t-1
Ft+h = Lt
a= 0.9999
` Month Actual Lt Forecast Error E squared
1 Jan 45 45.0
2 Feb 42 42.0
3 Mar. 40 40.0 42.0 -2.0 4.00
4 Apr. 50 50.0 40.0 10.0 100.00
5 May. 55 55.0 50.0 5.0 25.01
6 June 60 60.0 55.0 5.0 25.01
7 July 54 54.0 60.0 -6.0 35.99
8 Aug 52 52.0 54.0 -2.0 4.00
9 Sep. 64 64.0 52.0 12.0 144.00
10 Oct. 70 70.0 64.0 6.0 36.01
11 Nov. 90 90.0 70.0 20.0 400.02
12 Dec. 100 100.0 90.0 10.0 100.04 Sum squared Error 2724.1
13 Jan. 65 65.0 100.0 -35.0 1224.93 Mean squared error 209.5
14 Feb. 45 45.0 65.0 -20.0 400.14 Root mean squared error 14.4756883729
15 Mar. 60 60.0 45.0 15.0 224.94
April 60.0 60.0
May 60.0 60.0
Simple smoothing
Forecast
alpha 0.8
RMSE
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simple
Actual

Holt trend
Actual
42.0

holt's

we us e the Holt model when there is a trend in the data
Lt+1 = aYt + (1-a)(Lt + T t) level
Tt+1 = b(Lt+1 –Lt) + (1-b)Tt Trend
start at
L3 = (Y1 + Y2 +Y3 )/3 + (Y3 - Y1 )/2 a 0.99999
T3 = (Y3- Y1 )/2 b 0.1177464406
F t+h = Lt+1 + hTt+1 Forecast trend
` Month Actual L T Forecast Error E squared
1 Jan 45
2 Feb 42
3 Mar. 40 39.8 -2.5
4 Apr. 50 40.0 -2.2 37.8 12.2 148.4996
5 May. 55 50.0 -0.8 49.2 5.8 33.0773
6 June 52 55.0 -0.1 54.9 -2.9 8.5613
7 July 54 52.0 -0.4 51.6 2.4 5.8490
8 Aug 52 54.0 -0.1 53.9 -1.9 3.4829
9 Sep. 64 52.0 -0.4 51.6 12.4 152.6079
10 Oct. 70 64.0 1.1 65.1 4.9 24.0006
11 Nov. 66 70.0 1.7 71.7 -5.7 32.2383
12 Dec. 60 66.0 1.0 67.0 -7.0 49.1322
13 Jan. 75 60.0 0.2 60.2 14.8 219.5102 TSS or SSE 904.9016
14 Feb. 80 75.0 1.9 76.9 3.1 9.4347 MSE 53.2295
15 Mar. 76 80.0 2.3 82.3 -6.3 39.5665 RMS 7.2958555729
16 April 85 76.0 1.5 77.5 7.5 55.5078
17 May 83 85.0 2.4 87.4 -4.4 19.5961
18 June 84 83.0 1.9 84.9 -0.9 0.8202
19 July 76 84.0 1.8 85.8 -9.8 96.0198
20 Aug 74 76.0 0.6 76.6 -2.6 6.9975
21 Sep. 74.0 0.3 74.3
22 Oct. 74.7
TSS 904.9016

holt's

Holt_Winter

Holt's Linear Trend Model
a= 1 Lt+1 = aYt + (1-a)(Lt + T t) level
g= 0 Tt+1 = b(Lt+1 –Lt) + (1-b)Tt Trend
time Xt Ft+1 Tt forecast Error E squared start at
1 1813 L3 = (Y1 + Y2 +Y3 )/3 + (Y3 - Y1 )/2
2 1650 T3 = (Y3- Y1 )/2
3 1822
4 1778 F t+h = Lt+1 + hTt+1 Forecast
5 1520
6 1103
7 1266
8 1478
9 1431
10 1767
11 2162
12 2337
13 2608
14 2518
15 2641
16 2178
17 1928
18 1911
19 1991
20 1788
21 1693
22 1871
23 1899
24 1693
25 1633
26 1666
27 1575
28 1395
29 1389
30 1297
0.0
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Page &P

Holt_Winter
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Xt
forecast
Xt
The Holt-Winters Algorithm For Seasonal TIME Series
t Xt Ft Tt St forecast Error E square
1 897
2 476
3 376
4 509
5 967
6 529
7 407
8 371
9 884
10 407
11 310
12 338
13 900
14 448
15 344
16 274
17 740
18 261
19 289
20 319
21 1036
22 602
23 536
24 349
25 1050
26 633
27 435
28 415
29
30
31
32
33
34
35
36
MSE= 0
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ss
Xt
forecast
time
sale
Holt_Winter forecasting

Simple Smoothing

  • Forecast value at any time is a weighted average of all available previous data. the weights decline geometrically as you go back in time.

  • Ft+1 = a * Xt + a(1-a) * Xt-1 + a(1-a)2 * Xt-2 +

a(1-a)3 * Xt-3 +....

Simple smoothing

  • The above formula can be written in the following form:
  • Ft+1 = a( Xt ) +(1- a ) F t
  • Ft+1 = Forecast value for period t+1
  • a= smoothing constant (weight) (0< a <1)
  • X=actual value
  • Ft= Forecast value for period t

Simple Exponential smoothing

  • The weights a are made to decline geometrically with the age of observation to conform to the argument that the most recent observations contain the most relevant information.
  • The value of the smoothing constant a must be between 0 and 1. the value cannot be 0 or 1.

Simple smoothing

  • As a guide
  • select a value close to 0 if the series has a great deal of random variation.
  • select a value close to 1 if the series is smooth.
  • The root-mean-squared (RMSE) is often used as a criteria for choosing the appropriate value of a.

Starting value

  • Before we start calculation the first F has to be determined. In another words, the model has to be initialized. The initial number of F is arbitrary. One way to start the model is by making the (F) equal to the first observation X:
  • F1 = X1

Forecast with simple smoothing

  • In the absence of new data the forecast for two time ahead or three time ahead period will be equal to he forecast of value X in future (Ft+h) is equal to the last estimated level Ft
  • Ft+1+h = Ft+1
  • Where

h=1 if we want to forecast one period ahead,

h=2 if we want to forecast two period ahead

Simple Exponential smoothing
Example

Sheet1

Simple Smoothing
The current level of a series at time t is estimated as a weighted average of
the past observations. Most weight is given to the most recent observation,
with weights decreasing for more distant observations through a system of
exponentially decreasing weights.
The estimated level of the time series made at time t is , then,
Ft = a*Xt + a(1-a)Xt-1 + a(1-a)^2Xt-2 + a(1-a)^3 Xt-3 +..........
The sum of weights are equal to one.
`
The above model can be simplified to
FT+1 = a Xt + (1-a) FT where F1=X1
forecast Ft+m = Ft+1
a= 0.8800
Forecast
` Actual Error E squared
1 106.6
2 110.4
3 106.5
4 108.7
5 106.5
6 105.6
7 105.2
8 104.4
9 100.9
10 97.4
11 102.7
12 100.5
13 103.9
14 108.1
15 105.7
16 104.6
17 106.8
18 107.3
19 106
20 104.5
21 107.2
22 103.2
23 107.2
24 105.4
25 112
26 111.3
27 107.1
28 109.2
29 110.7
30 106.4
31 108.3
32 107.3
33 106.8
34 105.8
35 107.6
36 98.4
37 94.7
38 90.6
39 91.5
40 88.4
41 92
42 92.6
43 92.4
44 91.5
45 81.8
46 82.7
47 83.9
48 88.8
49 93
50 90.7
51 95.7
52 93
53 96.9
54 92.4
55 88.1
56 87.6
57 86.1
58 80.6
59 84.2
60 86.7
61 82.4
62 79.9
63 77.6
64 86
65 92.1
66 89.7
67 90.9
68 89.3
69 87.7
70 89.6
71 93.7
72 92.6
73 103.8
74 94.4
75 95.8
76 94.2
77 90.2
78 95.6
79 96.7
80 95.9
81 94.2
82 91.4
83 92.8
84 97.1
85 95.5
86 94.1
87 92.6
88 87.7
89 86.9
90 96
91 96.5
92 89.1
93 76.9
94 74.2
95 81.6
96 91.5
97 91.2
98 86.7
99 88.9
100 87.4
101 79.1
102 84.9
103 84.7
Sum of squared error
Mean squared error
Root means squared error

consumer sentiment index

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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 106.6 110.4 106.5 108.7 106.5 105.6 105.2 104.4 100.9 97.4 102.7 100.5 103.9 108.1 105.7 104.6 106.8 107.3 106 104.5 107.2 103.2 107.2 105.4 112 111.3 107.1 109.2 110.7 106.4 108.3 107.3 106.8 105.8 107.6 98.4 94.7 90.6 91.5 88.4 92 92.6 92.4 91.5 81.8 82.7 83.9 88.8 93 90.7 95.7 93 96.9 92.4 88.1 87.6 86.1 80.599999999999994 84.2 86.7 82.4 79.900000000000006 77.599999999999994 86 92.1 89.7 90.9 89.3 87.7 89.6 93.7 92.6 103.8 94.4 95.8 94.2 90.2 95.6 96.7 95.9 94.2 91.4 92.8 97.1 95.5 94.1 92.6 87.7 86.9 96 96.5 89.1 76.900000000000006 74.2 81.599999999999994 91.5 91.2 86.7 88.9 87.4 79.099999999999994 84.9 84.7 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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103

Holt’s Model

  • In some situations, the observed data will contain information that allows the anticipation of future upward or downward movements (trend).

  • In that case, rather than a constant forecast function, some trending function would be preferable. The simplest possibility of this sort is a linear trend forecast.

Holt’s Model

  • If the data series show a trend Holts model must be selected.
  • Holt’s two-parameter exponential smoothing method is an extension of simple exponential smoothing.
  • It adds a growth factor to the smoothing equation as a way of adjusting for the trend.

Holts Model

Holt’s Model

  • In Holt’s model we estimate the level (weighted average) and trend (slope) as follows:

Ft+1 = aXt + (1-a)(Ft + T t) level

Tt+1 = g(Ft+1 –Ft) + (1-g)Tt trend

  • Where

X t = actual value now

Ft = smoothed value (level)

Tt = Trend estimate

a = smoothing constant 0 < a < 1

g = smoothing constant for trend estimate 0 <b<1

Starting values

To start this forecasting procedure, we need starting values for level and trend as well as a and b.

start at

  • F3 = (Y1 + Y2 +Y3 )/3 + (Y3 - Y1 )/2
  • T3 = (Y3- Y1 )/2

Forecast with Holt’s Model

  • The forecast of X for m period ahead at time t is estimated by

H t+m = Ft+1 + mTt+1

Where:

m = number of period ahead to forecast

m=1 for forecast of one period ahead

m=2 for forecast of two period ahead

m=3 for forecast of three period ahead

H t+m = Holt’s forecast value for period t+m

Forecast with Holt’s Model
Example

Sheet1

Holt's Linear Trend Model
In some situations, the observed data will contain information that
allows the anticipation of future upward or downward movements.
In that case, rather than a constant forecast function, some trending
function would be preferable. The simplest possibility of this sort is
a linear trend forecast.
Just as in simple exponential smoothing, an estimate of the current
level of the time series is required
level of the time series is required. Also since upward or downward
movement is to be anticipated, we also need an estimate of the cur-
rent slope or change in the level, of the series.
Ft+1 = aXt + (1-a)(Ft + T t)
Tt+1 = g(Ft+1 -Ft) + (1-g)Tt
where we start at F2 = X2 and T2 = X2- X1
The forecast of X for h period ahead is estimated by
H t+m = Ft+1 + mTt+1
a= 0.5
g= 0.3
time Xt Ft+1 Tt forecast Error E squared
1 4757
2 4773
3 4792
4 4758
5 4738
6 4779
7 4800
8 4795
9 4875
10 4903
11 4951
12 5009
13 5027
14 5071
15 5127
16 5172
17 5230
18 5268
19 5305
20 5358
21 5367
22 5411
23 5458
24 5496
25 5544
26 5604
27 5640
28 5687
29 5749
30 5775
31 5870
32 5931
33 5996
34 6092
35 6165
36 6248
37 6311
38 6409
39 6476
40 6556
41 6661
42 6703
43 6768
44 6825
45 6853
46 6870
47 6900
48 7017
49 7042
50 7083
51 7123
52 7148
53 7184
54 7249
55 7352
56 7394
57 7475
58 7520
59 7585
60 7664
61 7709
62 7775
63 7852
64 7876
65 7961
66 8009
67 8063
68 8141
69 8215
70 8244
71 8302
72 8341

Xt 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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 4757 4773 4792 4758 4738 4779 4800 4795 4875 4903 4951 5009 5027 5071 5127 5172 5230 5268 5305 5358 5367 5411 5458 5496 5544 5604 5640 5687 5749 5775 5870 5931 5996 6092 6165 6248 6311 6409 6476 6556 6661 6703 6768 6825 6853 6870 6900 7017 7042 7083 7123 7148 7184 7249 7352 7394 7475 7520 7585 7664 7709 7775 7852 7876 7961 8009 8063 8141 8215 8244 8302 8341

Xt 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 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 4757 4773 4792 4758 4738 4779 4800 4795 4875 4903 4951 5009 5027 5071 5127 5172 5230 5268 5305 5358 5367 5411 5458 5496 5544 5604 5640 5687 5749 5775 5870 5931 5996 6092 6165 6248 6311 6409 6476 6556 6661 6703 6768 6825 6853 6870 6900 7017 7042 7083 7123 7148 7184 7249 7352 7394 7475 7520 7585 7664 7709 7775 7852 7876 7961 8009 8063 8141 8215 8244 8302 8341

seasonal data
Holt-Winter model multiplicative

  • For a great many products, sales have a strong Seasonal component, so that it is obviously desirable to extend exponential smoothing algorithms to allow for seasonality. Holt's algorithm was extended in this way by Winters(1960).

Holt-Winter model for seasonal data

Chart1

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
Xt
897
476
376
509
967
529
407
371
884
407
310
338
900
448
344
274
740
261
289
319
1036
602
536
349
1050
633
435
415

Sheet1

There are two categories of statisticals tools for forecasting :
causal and time series. Causal techniques link the forecast values
of a variable (dependent variable) to one or more anticipated causes
(independent variables). For example a change in inventory level
causes sales to change. regression method cab be used to forecast
this type of relationships.
Time-series techniques link future movements in the forecasted
varaible to pattern revealed by historical movement in the same
variable. The moving average , exponential smoothing, time series
time series regressions are some of this techniques that we will
study.
&A
Page &P

simple

Simple Smoothing
•Lt = a( Yt ) +(1- a ) L t-1
Ft+h = Lt
a= 0.9999
` Month Actual Lt Forecast Error E squared
1 Jan 45 45.0
2 Feb 42 42.0
3 Mar. 40 40.0 42.0 -2.0 4.00
4 Apr. 50 50.0 40.0 10.0 100.00
5 May. 55 55.0 50.0 5.0 25.01
6 June 60 60.0 55.0 5.0 25.01
7 July 54 54.0 60.0 -6.0 35.99
8 Aug 52 52.0 54.0 -2.0 4.00
9 Sep. 64 64.0 52.0 12.0 144.00
10 Oct. 70 70.0 64.0 6.0 36.01
11 Nov. 90 90.0 70.0 20.0 400.02
12 Dec. 100 100.0 90.0 10.0 100.04 Sum squared Error 2724.1
13 Jan. 65 65.0 100.0 -35.0 1224.93 Mean squared error 209.5
14 Feb. 45 45.0 65.0 -20.0 400.14 Root mean squared error 14.4756883729
15 Mar. 60 60.0 45.0 15.0 224.94
April 60.0 60.0
May 60.0 60.0
Simple smoothing
Forecast
alpha 0.8
RMSE
&A
Page &P

simple
Actual

Holt trend
Actual
42.0

holt's

we us e the Holt model when there is a trend in the data
Lt+1 = aYt + (1-a)(Lt + T t) level
Tt+1 = b(Lt+1 –Lt) + (1-b)Tt Trend
start at
L3 = (Y1 + Y2 +Y3 )/3 + (Y3 - Y1 )/2 a 0.99999
T3 = (Y3- Y1 )/2 b 0.1177464406
F t+h = Lt+1 + hTt+1 Forecast trend
` Month Actual L T Forecast Error E squared
1 Jan 45
2 Feb 42
3 Mar. 40 39.8 -2.5
4 Apr. 50 40.0 -2.2 37.8 12.2 148.4996
5 May. 55 50.0 -0.8 49.2 5.8 33.0773
6 June 52 55.0 -0.1 54.9 -2.9 8.5613
7 July 54 52.0 -0.4 51.6 2.4 5.8490
8 Aug 52 54.0 -0.1 53.9 -1.9 3.4829
9 Sep. 64 52.0 -0.4 51.6 12.4 152.6079
10 Oct. 70 64.0 1.1 65.1 4.9 24.0006
11 Nov. 66 70.0 1.7 71.7 -5.7 32.2383
12 Dec. 60 66.0 1.0 67.0 -7.0 49.1322
13 Jan. 75 60.0 0.2 60.2 14.8 219.5102 TSS or SSE 904.9016
14 Feb. 80 75.0 1.9 76.9 3.1 9.4347 MSE 53.2295
15 Mar. 76 80.0 2.3 82.3 -6.3 39.5665 RMS 7.2958555729
16 April 85 76.0 1.5 77.5 7.5 55.5078
17 May 83 85.0 2.4 87.4 -4.4 19.5961
18 June 84 83.0 1.9 84.9 -0.9 0.8202
19 July 76 84.0 1.8 85.8 -9.8 96.0198
20 Aug 74 76.0 0.6 76.6 -2.6 6.9975
21 Sep. 74.0 0.3 74.3
22 Oct. 74.7
TSS 904.9016

holt's

Holt_Winter

Holt's Linear Trend Model
a= 1 Lt+1 = aYt + (1-a)(Lt + T t) level
g= 0 Tt+1 = b(Lt+1 –Lt) + (1-b)Tt Trend
time Xt Ft+1 Tt forecast Error E squared start at
1 1813 L3 = (Y1 + Y2 +Y3 )/3 + (Y3 - Y1 )/2
2 1650 T3 = (Y3- Y1 )/2
3 1822
4 1778 F t+h = Lt+1 + hTt+1 Forecast
5 1520
6 1103
7 1266
8 1478
9 1431
10 1767
11 2162
12 2337
13 2608
14 2518
15 2641
16 2178
17 1928
18 1911
19 1991
20 1788
21 1693
22 1871
23 1899
24 1693
25 1633
26 1666
27 1575
28 1395
29 1389
30 1297
0.0
&A
Page &P

Holt_Winter
&A
Page &P
Xt
forecast
Xt
The Holt-Winters Algorithm For Seasonal TIME Series
t Xt Ft Tt St forecast Error E square
1 897
2 476
3 376
4 509
5 967
6 529
7 407
8 371
9 884
10 407
11 310
12 338
13 900
14 448
15 344
16 274
17 740
18 261
19 289
20 319
21 1036
22 602
23 536
24 349
25 1050
26 633
27 435
28 415
29
30
31
32
33
34
35
36
MSE= 0
&A
Page &P
&A
Page &P
ss
Xt
forecast
time
sale
Holt_Winter forecasting
Xt

Holt-Winter model for seasonal data
Multiplicative Seasonality

  • Retaining the notation of previous section, given observations Yt on a time series, estimates Ft and Tt of current level and slope are again required.
  • In addition, for each time period it is necessary to estimate a multiplicative seasonal factor s.

Holt-Winter model equations
seasonal data

The updating equations are:

  • Ft = a(Xt / St-p ) + (1-a)* ( Ft-1 +T t-1) Level
  • Tt = g*( Ft - Ft-1) + (1- g) Tt-1 Trend
  • St = b*( Xt/Ft ) + (1- b)* St-p seasonal index

Where

b = smoothing constant for seasonality estimate 0< b <1

P = is the number of periods per year, so that p=4 for quarterly data and p=12 for monthly data.

Holt-Winter model
Example

Sheet1

The Holt-Winters Algorithm For Seasonal TIME Series
•Ft = a(Xt / St-p ) + (1-a)* ( Ft-1 +T t-1)
•Tt = g*( Ft - Ft-1) + (1- g) Tt-1
•St = b*( Xt/Ft ) + (1- b)* St-p
The forecast of W t+h made at time t, is now obtained from
Wt+m = (Ft +mTt) St+m-p P=4 for quarterly and 12 for monthly data
The initial values for the level and slope and seasonal factors are:
t Xt
Mar-86 213
Jun-86 231
Sep-86 205
Dec-86 197
Mar-87 252
Jun-87 249
Sep-87 220
Dec-87 239
Mar-88 271
Jun-88 271
Sep-88 231
Dec-88 269
Mar-89 311
Jun-89 309
Sep-89 240
Dec-89 248
Mar-90 264
Jun-90 322
Sep-90 254
Dec-90 218
Mar-91 194
Jun-91 285
Sep-91 248
Dec-91 271
Mar-92 279
Jun-92 322
Sep-92 271
Dec-92 326
Mar-93 378
Jun-93 391
Sep-93 315
Dec-93 394
Mar-94 449
Jun-94 447
Sep-94 376
Dec-94 421
Mar-95 446
Jun-95 460
Sep-95 377
Dec-95 427
Mar-96 448
Jun-96 488
Sep-96 403
Dec-96 452
Mar-97 513
Jun-97 509
Sep-97 437
Dec-97 543
Mar-98 566
Jun-98 535
Sep-98 440
Dec-98 565
Mar-99 632
Jun-99 646

Xt 31472 31564 31656 31747 31837 31929 32021 32112 32203 32295 32387 32478 32568 32660 32752 32843 32933 33025 33117 33208 33298 33390 33482 33573 33664 33756 33848 33939 34029 34121 34213 34304 34394 34486 34578 34669 34759 34851 34943 35034 35125 35217 35309 35400 35490 35582 35674 35765 35855 35947 36039 36130 36220 36312 213 231 205 197 252 249 220 239 271 271 231 269 311 309 240 248 264 322 254 218 194 285 248 271 279 322 271 326 378 391 315 394 449 447 376 421 446 460 377 427 448 488 403 452 513 509 437 543 566 535 440 565 632 646

The seasonal indices

  • This data shows that in March the data is 4% above the average value for the quarter (adding all the quarters in a year and dividing by 4). The highest value is in quarter 2.
March 1.04
June 1.08
seep 0.89
Dec 0.97

Forecast
Holt-Winter model for seasonal data

  • Forecast for m period ahead
  • Wt+m = (Ft +mTt) St+m-p

where

  • Wt+m = Winter’s forecast for m period into future.
  • P = number of periods in the seasonal cycle.

Adaptive –Response Rate

  • Is not covered

How to forecast seasonal data

  • When data is seasonal, Winter’s model provide an easy way to include seasonality directly in to the forecast.
  • An alternative method however, is widely practiced as follows,
  • De-seasonalize the data by dividing each value by its corresponding seasonal index
  • Forecast the de-seasonalized data (simple, Holt)
  • Re-seasonalize the results by multiplying each de-seasonalized forecast by its corresponding seasonal index

How to forecast seasonal data

Sheet1

The Holt-Winters Algorithm For Seasonal TIME Series
•Ft = a(Xt / St-p ) + (1-a)* ( Ft-1 +T t-1)
•Tt = g*( Ft - Ft-1) + (1- g) Tt-1
•St = b*( Xt/Ft ) + (1- b)* St-p
The forecast of W t+h made at time t, is now obtained from March 1.04
June 1.08
Wt+m = (Ft +mTt) St+m-p P=4 for quarterly and 12 for monthly data sep 0.89
Dec 0.97
The initial values for the level and slope and seasonal factors are:
t Xt deceasonalize forecast of deseanalize reseasonalize
Mar-86 213
Jun-86 231
Sep-86 205
Dec-86 197
Mar-87 252
Jun-87 249
Sep-87 220
Dec-87 239
Mar-88 271
Jun-88 271
Sep-88 231
Dec-88 269
Mar-89 311
Jun-89 309
Sep-89 240
Dec-89 248
Mar-90 264
Jun-90 322
Sep-90 254
Dec-90 218
Mar-91 194
Jun-91 285
Sep-91 248
Dec-91 271
Mar-92 279
Jun-92 322
Sep-92 271
Dec-92 326
Mar-93 378
Jun-93 391
Sep-93 315
Dec-93 394
Mar-94 449
Jun-94 447
Sep-94 376
Dec-94 421
Mar-95 446
Jun-95 460
Sep-95 377
Dec-95 427
Mar-96 448
Jun-96 488
Sep-96 403
Dec-96 452
Mar-97 513
Jun-97 509
Sep-97 437
Dec-97 543
Mar-98 566
Jun-98 535
Sep-98 440
Dec-98 565
Mar-99 632
Jun-99 646

How to forecast data

  • Many forecaster finds the alternative method more accurate than using the winters’ model.

Advantage of exponential Smoothing Models

  • The major advantage of exponential smoothing algorithms is their ability to produce quite reliable forecasts relatively quickly for a large collection of time series.
  • This is particularly valuable as an input to inventory management, for which monthly, or perhaps quarterly sales forecasts are needed.

New product Forecasting

  • The new products lack the historical data, therefore most forecasting techniques does not produce the satisfying results for these products.
  • To overcome these difficulty, forecasters usually use growth model or S-curve method
  • We will use the logistic model to forecast these products.

Sales of new product

  • New products have life cycles that follow a common pattern.
  • A period of slow growth just after introduction of the product.
  • A period of rapid growth.
  • Slowing growth in a mature phase.
  • Decline

Sales of new product

Logistics curve

Forecasting sales

  • Suppose you have the data for the sales for 2001 2014.
  • The first 5 data are as follows

  • Suppose you think sales will not exceed $20 Million at the maturity phase.

Using E-views

Forecasts

  • The forecasts of the future sales are as follows:

Microsoft Excel

Worksheet

010002000300040005000600070008000900001020304050607080Xt

Microsoft Excel

Worksheet

Microsoft Excel

Worksheet

seasonal data

20011000

20022000

20033500

20045700

200512000

0

4,000

8,000

12,000

16,000

20,000

24,000

0102030405060708091011121314

XF

20011012.11557...

20021530.84190...

20034230.27908...

20047463.04107...

200510708.3909...

200613953.5978...

200717177.5342...

200819646.5892...

200919992.8476...

201019999.8843...

201119999.9981...

201219999.9999...

201319999.9999...

201419999.9999...