lecture3_08
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• With Classical decomposition, forecasting may be done, but estimation of accuracy lacks and no forecasting limits are produced
• Classical decomposition is usually combined with Exponential smoothing methods
143
1992
145
1993
141
140
1994
143
1995
145
1996
138
1997
147
1985
1990
1995
year
2000
1998
151
1999
148
2000
148
Assume the model:
yt 0 t
Estimate β0 by calculating the mean value of the first 8 observations of the series
Set α to be relatively small. This means that the latest observation plays a less role than the history in the forecasts. Thumb rule: 0.05 < α < 0.3
E.g. Set α=0.1
statistical theory
Single exponential smoothing
• Assume historical values y1,y2,…yT • Assume data contains no trend, i.e.
yt 0 t
Forecasting scheme:
T yT (1 ) T 1,
11 0.1 y11 0.9 10 0.1145wk.baidu.com 0.9 145.8575 145.772
12 0.1 y12 0.9 11 0.1138 0.9 145.772 144.995
13 0.1 y13 0.9 12 0.1147 0.9 144.995 145.1955
14
0.1
y14
0.9
13
0.1151
0.9 145.1955
145.776
15 0.1 y15 0.9 14 0.1148 0.9 145.776 145.998
Forecasts
16 0.1 y16 0.9 15 0.1148 0.9 145.998 146.2 yˆ17 146.2 yˆ18 146.2 yˆ19 146.2 etc.
Alternative
In Bowerman/O’Connell/Koehler the updates of estimates of β0 are done on all historical data i.e.
• Common sense and mathematics in a good combination produces ”optimal” forecasts
• With time series regression models, forecasting (prediction) is a natural step and forecasting limits (intervals) can be constructed
ˆ0 (151151...145)/8146.75
Set l8 = ˆ0 =146.75
Assume first that the sales are very stable, i.e. during the period the mean value β0 is assumed not to change
l T which can be used to forecast yT+τ
Example: Sales of everyday commodities
Year
Sales values
1985
151
1986
151
1987
147
150
1988
149
1989
146
sales
1990
142
145
1991
Use a part (usually half) of the historical data to
estimate β0 ˆ0
Set l 0= ˆ0
Update the estimates of β0 for the rest of the historical data with the recursion formula
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
yˆT T
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 ?
Update the estimates of β0 using the next 8 values of the historical data
9 0.1 y9 0.9 8 0.1141 0.9 146.75 146.175
10 0.1 y10 0.9 9 0.1143 0.9 146.175 145.8575
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
• Classical decomposition is usually combined with Exponential smoothing methods
143
1992
145
1993
141
140
1994
143
1995
145
1996
138
1997
147
1985
1990
1995
year
2000
1998
151
1999
148
2000
148
Assume the model:
yt 0 t
Estimate β0 by calculating the mean value of the first 8 observations of the series
Set α to be relatively small. This means that the latest observation plays a less role than the history in the forecasts. Thumb rule: 0.05 < α < 0.3
E.g. Set α=0.1
statistical theory
Single exponential smoothing
• Assume historical values y1,y2,…yT • Assume data contains no trend, i.e.
yt 0 t
Forecasting scheme:
T yT (1 ) T 1,
11 0.1 y11 0.9 10 0.1145wk.baidu.com 0.9 145.8575 145.772
12 0.1 y12 0.9 11 0.1138 0.9 145.772 144.995
13 0.1 y13 0.9 12 0.1147 0.9 144.995 145.1955
14
0.1
y14
0.9
13
0.1151
0.9 145.1955
145.776
15 0.1 y15 0.9 14 0.1148 0.9 145.776 145.998
Forecasts
16 0.1 y16 0.9 15 0.1148 0.9 145.998 146.2 yˆ17 146.2 yˆ18 146.2 yˆ19 146.2 etc.
Alternative
In Bowerman/O’Connell/Koehler the updates of estimates of β0 are done on all historical data i.e.
• Common sense and mathematics in a good combination produces ”optimal” forecasts
• With time series regression models, forecasting (prediction) is a natural step and forecasting limits (intervals) can be constructed
ˆ0 (151151...145)/8146.75
Set l8 = ˆ0 =146.75
Assume first that the sales are very stable, i.e. during the period the mean value β0 is assumed not to change
l T which can be used to forecast yT+τ
Example: Sales of everyday commodities
Year
Sales values
1985
151
1986
151
1987
147
150
1988
149
1989
146
sales
1990
142
145
1991
Use a part (usually half) of the historical data to
estimate β0 ˆ0
Set l 0= ˆ0
Update the estimates of β0 for the rest of the historical data with the recursion formula
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
yˆT T
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 ?
Update the estimates of β0 using the next 8 values of the historical data
9 0.1 y9 0.9 8 0.1141 0.9 146.75 146.175
10 0.1 y10 0.9 9 0.1143 0.9 146.175 145.8575
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