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
• With time series regression models,
forecasting (prediction) is a natural step and
forecasting limits (intervals) can be
constructed
• 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
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.
yt   0   t
Forecasting scheme:
 T    yT  (1   )   T 1 ,
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 ?
Use a part (usually half) of the historical data to
estimate β0  ̂ 0
Set l 0=
̂ 0
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
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
150
sales
Year
145
140
1985
1990
1995
year
2000
Assume the model:
yt   0   t
Estimate β0 by calculating the mean value of the
first 8 observations of the series
ˆ
  0  (151  151  ...145)/8  146.75
Set l8 = ̂ 0 =146.75
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. Thumb rule: 0.05 < α < 0.3
E.g. Set α=0.1
Update the estimates of β0 using the next 8 values of the
historical data
 9  0.1  y9  0.9   8  0.1 141  0.9 146.75  146.175
 10  0.1  y10  0.9   9  0.1 143  0.9 146.175  145.8575
 11  0.1  y11  0.9   10  0.1 145  0.9 145.8575  145.772
 12  0.1  y12  0.9   11  0.1 138  0.9 145.772  144.995
 13  0.1  y13  0.9   12  0.1 147  0.9 144.995  145.1955
 14  0.1  y14  0.9   13  0.1 151  0.9 145.1955  145.776
 15  0.1  y15  0.9   14  0.1148  0.9 145.776  145.998
Forecasts
 16  0.1 y16  0.9   15  0.1148  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.
 T    yT  (1   )   T 1
for T=1,…, n and l0 = ̂ 0
Analysis of example data with MINITAB

MTB > Name c3 "FORE1" c4 "UPPE1" c5 "LOWE1"
MTB > SES 'Sales values';
SUBC>
Weight 0.1;
SUBC>
Initial 8;
SUBC>
Forecasts 3;
SUBC>
Fstore 'FORE1';
SUBC>
Upper 'UPPE1';
SUBC>
Lower 'LOWE1';
SUBC>
Title "SES alpha=0.1".
Single Exponential Smoothing for Sales values
Data
Sales values
Length
16
Smoothing Constant
Alpha
0.1
Accuracy Measures
MAPE
2.2378
MAD
3.2447
MSD
14.4781
Forecasts
Period
Forecast
Lower
Upper
17
146.043
138.094
153.992
18
146.043
138.094
153.992
19
146.043
138.094
153.992
MINITAB uses smoothing
from 1st value!
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 (A bit exaggerated)
Single Exponential Smoothing for Sales values
Data
Sales values
Length
16
Smoothing Constant
Alpha
0.5
Accuracy Measures
MAPE
1.9924
MAD
2.8992
MSD
13.0928
Forecasts
Period
Forecast
Lower
Upper
17
147.873
140.770
154.976
18
147.873
140.770
154.976
19
147.873
140.770
154.976
Slightly wider prediction intervals
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.
Single Exponential Smoothing for Sales values
--Smoothing Constant
SES optimal alpha
156
Alpha
Variable
Actual
Fits
Forecasts
95.0% PI
0.567101
Accuracy Measures
MAPE
1.7914
MAD
2.5940
MSD
12.1632
Sales values
152
Smoothing Constant
0.567101
Alpha
148
144
140
2
Forecasts
Period
Accuracy Measures
1.7914
MAPE
2.5940
MAD
12.1632
MSD
4
Forecast
Lower
Upper
17
148.013
141.658
154.369
18
148.013
141.658
154.369
19
148.013
141.658
154.369
6
8
10
Index
12
14
16
18
Yet, wider prediction
intervals
Exponential smoothing for times series with trend
and/or seasonal variation
• Double exponential smoothing (one smoothing
parameter) for trend
• Holt’s method (two smoothing parameters) for
trend
• Multiplicative Winter’s method (three smoothing
parameters) for seasonal (and trend)
• Additive Winter’s method (three smoothing
parameters) for seasonal (and trend)
Example: Real Estate Price Index for Weekend
Cottages in Sweden
REPI_C
1993
226
1994
241
1995
239
1996
240
1997
268
1998
303
1999
336
2000
414
2001
472
Time Series Plot of REPI_C
600
500
REPI_C
Year
400
300
200
2002
496
2003
505
2004
546
2005
591
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Year
Trend but no seasonal variation
Applying Holt’s method with MINITAB (denoted Double
exponential smoothing in Minitab)
2 smoothing
parameters, one for
level and one for trend.
Option to let Minitab
calculate optimal
parameters.
Smoothing parameters should
still be kept low (0.05,0.3)
Double Exponential Smoothing for REPI_C
Data
REPI_C
Length
13
Double Exponential Smoothing Plot for REPI_C
Variable
A ctual
Fits
Forecasts
95.0% PI
700
Smoothing Constants
600
0.2
Gamma (trend)
0.2
Smoothing C onstants
A lpha (lev el)
0.2
Gamma (trend)
0.2
500
REPI_C
Alpha (level)
Accuracy Measures
MAPE
9.78
MAD
30.15
MSD
1160.79
400
Accuracy Measures
300
MAPE
9.78
MAD
30.15
MSD
1160.79
200
100
1
2
Forecasts
Period
Forecast
Lower
Upper
14
611.411
537.537
685.286
15
646.167
570.753
721.581
3
4
5
6
7
8 9
Index
10 11 12 13 14 15
Example: Quarterly sales data
year
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
170
1993
1
126
160
1993
2
160
1993
3
181
140
1993
4
121
130
1994
1
131
120
1994
2
168
1994
3
189
110
Quarter Q1
Year 1991
1994
4
134
1995
1
133
1995
2
167
1995
3
195
1995
4
131
Time Series Plot of sales
200
190
sales
180
150
Q3
Q1
1992
Q3
Q1
1993
Q3
Q1
1994
Q3
Q1
1995
Q3
Applying Winter’s multiplicative method with MINITAB
3 smoothing parameters, one for level, one for trend an one for seasonal variation.
No option to calculate optimal parameters. Choices have do be based on visual
inspection of the times series
Winters' Method for sales
Multiplicative Method
Data
sales
Length
20
Winters' Method Plot for sales
Multiplicative Method
210
Variable
A ctual
Fits
Forecasts
95.0% PI
200
Smoothing Constants
190
0.2
Gamma (trend)
0.2
Delta (seasonal)
0.2
180
sales
Alpha (level)
Smoothing C onstants
A lpha (lev el)
0.2
Gamma (trend)
0.2
Delta (seasonal)
0.2
170
160
A ccuracy Measures
MA PE
2.6446
MA D
3.8808
MSD
23.7076
150
Accuracy Measures
140
MAPE
2.6446
130
MAD
3.8808
120
MSD
23.7076
Quarter
Year
Q3
2008
Forecasts
Period
Forecast
Lower
Upper
Q3-2013
135.625
126.117
145.133
Q4-2013
174.430
164.773
184.087
Q1-2014
194.667
184.844
204.490
Q2-2014
136.933
126.928
146.939
Q3
2009
Q3
2010
Q3
2011
Q3
2012
Q3
2013