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 ,
yT T
ˆ
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
ˆ
estimate β0 0
Set l 0= ˆ
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
Year Sales values
1985 151
1986 151
1987 147
150
1988 149
1989 146
1990 142
sales
145
1991 143
1992 145
1993 141 140
1994 143
1995 145 1985 1990 1995 2000
1996 138 year
1997 147
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
ˆ
0 (151 151 ...145)/8 146.75
ˆ
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 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 Variable
Actual
Alpha 0.567101 Fits
Forecasts
152 95.0% PI
Smoothing C onstant
A lpha 0.567101
Accuracy Measures
Sales values
148 Accuracy Measures
MAPE 1.7914
MAD 2.5940
MSD 12.1632
MAPE 1.7914 144
MAD 2.5940
MSD 12.1632 140
2 4 6 8 10 12 14 16 18
Forecasts Index
Period Forecast Lower Upper
17 148.013 141.658 154.369 Yet, wider prediction
18 148.013 141.658 154.369 intervals
19 148.013 141.658 154.369
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
Year REPI_C
1993 226
Time Series Plot of REPI_C
1994 241
600
1995 239
1996 240 500
1997 268
REPI_C
1998 303 400
1999 336
300
2000 414
2001 472
200
2002 496 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Year
2003 505
2004 546
2005 591 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
700 Actual
Smoothing Constants Fits
Forecasts
600 95.0% PI
Alpha (level) 0.2 Smoothing Constants
Alpha (lev el) 0.2
500 Gamma (trend) 0.2
Gamma (trend) 0.2 REPI_C
Accuracy Measures
400 MAPE 9.78
MAD 30.15
Accuracy Measures MSD 1160.79
300
MAPE 9.78 200
MAD 30.15
100
MSD 1160.79 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Index
Forecasts
Period Forecast Lower Upper
14 611.411 537.537 685.286
15 646.167 570.753 721.581
Example: Quarterly sales data
year quarter sales
1991 1 124
1991 2 157
1991 3 163
1991 4 126 Time Series Plot of sales
200
1992 1 119
190
1992 2 163
180
1992 3 176
1992 4 127 170
1993 1 126 sales 160
1993 2 160 150
1993 3 181 140
1993 4 121 130
1994 1 131 120
1994 2 168 110
1994 3 189 Quarter Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3
Year 1991 1992 1993 1994 1995
1994 4 134
1995 1 133
1995 2 167
1995 3 195
1995 4 131
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 Winters' Method Plot for sales
Multiplicative Method
Length 20
210 Variable
Actual
200 Fits
Smoothing Constants Forecasts
190 95.0% PI
Alpha (level) 0.2 180 Smoothing Constants
Alpha (lev el) 0.2
Gamma (trend) 0.2 170 Gamma (trend) 0.2
sales
Delta (seasonal) 0.2
Delta (seasonal) 0.2 160
Accuracy Measures
150 MAPE 2.6446
MAD 3.8808
Accuracy Measures 140 MSD 23.7076
MAPE 2.6446 130
MAD 3.8808 120
MSD 23.7076 Quarter Q3 Q3 Q3 Q3 Q3 Q3
Year 2008 2009 2010 2011 2012 2013
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