Docstoc

Forecasting

Document Sample
Forecasting Powered By Docstoc
					               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
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
Accuracy Measures                                                                               A lpha  0.567101



                          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

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