Forecasting

Document Sample

```					               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 < 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
y17  146.2
ˆ
y18  146.2
ˆ
y19  146.2
ˆ
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

ˆ
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
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
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
Accuracy Measures                                                                               A lpha  0.567101

Sales values
148                                                   Accuracy Measures
MAPE     1.7914
MSD     12.1632
MAPE      1.7914                         144
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
Accuracy Measures                                                                                      MSD     1160.79
300

MAPE      9.78                       200

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
Accuracy Measures                           140                                                         MSD     23.7076

MAPE     2.6446                             130
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

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 7 posted: 8/8/2012 language: English pages: 35