Forecasting
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Forecasting • Purpose is to forecast, not to explain the historical pattern • Models for forecasting may not make sense as a description for ”physical” beaviour of the time series • Common sense and mathematics in a good combination produces ”optimal” forecasts
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Forecasting. Purpose is to forecast, not to explain the historical pattern Models for forecasting may not make sense as a description for ”physical” beaviour of the time series Common sense and mathematics in a good combination produces ”optimal” forecasts. Exponential smoothing. - PowerPoint PPT Presentation
Transcript of Forecasting
Forecasting
• Purpose is to forecast, not to explain the historical pattern
• Models for forecasting may not make sense as a description for ”physical” beaviour of the time series
• Common sense and mathematics in a good combination produces ”optimal” forecasts
Exponential smoothing
• Use the historical data to forecast the future• Let different parts of the history have
different impact on the forecasts• Forecast model is not developed from any
statistical theory
Single exponential smoothing
• Assume historical values y1,y2,…yT
• Assume data contains no trend, i.e.
tty 0
Forecasting scheme:
TT
TTT
yy
ˆ
,)1( 1
where is a smoothing parameter between 0 and 1
• The forecast procedure is a recursion formula
• How shall we choose α?• Where should we start, i.e. Which is the
initial value l0 ?
Use a part (usually half) of the historical data to estimate β0
Set l 0=
0̂
0̂
Update the estimates of β0 for the rest of the historical data with the recursion formula
l T which can be used to forecast yT+τ
Example: Sales of everyday commodities
Year Sales values
1985 151
1986 151
1987 147
1988 149
1989 146
1990 142
1991 143
1992 145
1993 141
1994 143
1995 145
1996 138
1997 147
1998 151
1999 148
2000 148
Time series graph
1985 1990 1995 2000
140
145
150
year
sale
s
Assume the model:
tty 0
Estimate β0 by calculating the mean value of the first 8 observations of the series
75.146...145)/8151151(ˆ0
Set l8 = =146.750̂
Assume first that the sales are very stable, i.e. during the period the mean value β0 is assumed not to change
Set α to be relatively small. This means that the latest observation plays a less role than the history in the forecasts.
E.g. Set α=0.1
Update the estimates of β0 using the next 8 values of the historical data
998.145776.1459.01481.09.01.0776.1451955.1459.01511.09.01.0
1955.145995.1449.01471.09.01.0995.144772.1459.01381.09.01.0772.1458575.1459.01451.09.01.0
8575.145175.1469.01431.09.01.0175.14675.1469.01411.09.01.0
141515
131414
121313
111212
101111
91010
899
yyyyyyy
Forecasts
.2.146ˆ2.146ˆ2.146ˆ
2.146998.1459.01481.09.01.0
19
18
17
151616
etcyyy
y
Alternative
In Bowerman/O’Connell/Koehler instead the updates of estimates of β0 are done on all historical data i.e.
1)1( TTT y
for T=1,…, n and l0 = 0̂
Analysis of example data with MINITAByear sales lT yT – lT forecasts
1985 151 146,750 4,25000 *
1986 151 147,175 3,82500 *
1987 147 147,558 -0,55750 *
1988 149 147,502 1,49825 *
1989 146 147,652 -1,65158 *
1990 142 147,486 -5,48642 *
1991 143 146,938 -3,93778 *
1992 145 146,544 -1,54400 *
1993 141 146,390 -5,38960 *
1994 143 145,851 -2,85064 *
1995 145 145,566 -0,56557 *
1996 138 145,509 -7,50902 *
1997 147 144,758 2,24188 *
1998 151 144,982 6,01770 *
1999 148 145,584 2,41593 *
2000 148 145,826 2,17433 *
146,043
146,043
146,043
146,043
Actual
Predicted
Forecast Actual PredictedForecast
0 10 20
138
143
148
153
sale
s
Time
Smoothing Constant
Alpha:
MAPE:
MAD:
MSD:
0,100
2,2378
3,2447
14,4781
Single Exponential Smoothing
Assume now that the sales are less stable, i.e. during the period the mean value β0 is possibly changing
Set α to be relatively large. This means that the latest observation becomes more important in the forecasts.
E.g. Set α=0.5
Analysis with MINITAB
Actual
Predicted
Forecast Actual PredictedForecast
0 10 20
140
145
150
155
sale
s
Time
Smoothing ConstantAlpha:
MAPE:
MAD:MSD:
0,500
1,9924
2,899213,0928
Single Exponential Smoothing
We can also use some adaptive procedure to continuosly evaluate the forecast ability and maybe change the smoothing parameter over time
Alt. We can run the process with different alphas and choose the one that performs best. This can be done with the MINITAB procedure.
Automatic selection of smoothing parameter with MINITAB
Actual
Predicted
Forecast Actual PredictedForecast
0 10 20
140
145
150
155
sale
s
Time
Smoothing Constant
Alpha:
MAPE:
MAD:
MSD:
0,567
1,7914
2,5940
12,1632
Single Exponential Smoothing
Exponential smoothing for times series with trend and/or seasonal variation
• Double exponential smoothing (one smoothing parameter)
• Holt-Winter’s method (two smoothing parameters)
• Multiplicative Winter’s method (three smoothing parameters)
• Additive Winter’s method (three smoothing parameters)
Example: Quarterly sales datayear quarter sales
1991 1 124
1991 2 157
1991 3 163
1991 4 126
1992 1 119
1992 2 163
1992 3 176
1992 4 127
1993 1 126
1993 2 160
1993 3 181
1993 4 121
1994 1 131
1994 2 168
1994 3 189
1994 4 134
1995 1 133
1995 2 167
1995 3 195
1995 4 131
5 10 15 20
120
130
140
150
160
170
180
190
200
Index
sale
s
Applying Winter’s multiplicative method with MINITAB
Actual
PredictedActual Predicted
0 10 20
120
145
170
195
sale
s
Time
Smoothing ConstantsAlpha (level):Gamma (trend):Delta (season):
MAPE:MAD:MSD:
0,2000,2000,200
2,6446 3,880823,7076
Winters' Multiplicative Model for sales
