Download Exponential smoothing

Exponential smoothing
This is a widely used
forecasting technique in
retailing, even though it has
not proven to be especially
accurate.
Why is exponential smoothing so popular?
It's easy—the exotic term notwithstanding.
Data storage requirements are minimal (even
though this is not the problem it once was due to
plunging memory prices).
It is very cost effective when forecasts must be
made for a large number of items--hence it has
extensive use in retailing.
The basic algorithm
Lt  Xt  (1   ) Lt
1
(1)
Where:
•Lt is the forecast for the current period;
•Xt is the most recent observation of the time series
variable—such as, for example, sales last month of part
#000897
•Lt-1 is the most recent forecast; and
• is the smoothing constant, where 0 <  < 1
Equation (1)
can be
written as
follows:
New Forecast = (New Data) + (1 - )Most Recent Forecast
Exponential smoothing is weighted moving average
process
To demonstrate, let
Lt  1  Xt  1  (1   ) Lt  2
Substitute (2) into (1):
Lt  Xt  (1   )[Xt  1  (1   ) Lt  2] 
Xt   (1   ) Xt  1  (1   ) 2 Lt  2
But notice that:
Lt  2  Xt  2  (1   ) Lt  3
(4)
Substitute (4) into (3) to obtain:
Lt  Xt   (1   ) Xt  1  (1   ) 2 Lt  2  (1   )3 Lt  3
If we continue to substitute recursively, we get:
Lt  Xt   (1   ) Xt  1   (1   ) 2 Xt  2   (1   )3 Xt  3  
Notice that
 , (1   ),  (1   ) 2 , (1   )3 , (1   ) 4 ,
are the weights attached to past values of X. Since
 < 1, the weights attached to earlier or more
remote observations of X are diminishing.
You don’t have to go
through this recursive
process each time you do
a forecast. The process is
summarized in the most
recent forecast.
Selecting the smoothing constant ()
•The range of possible values is zero and one.
Sales of part #56
•If you select a value of  close to 1, that means you are attaching a
large weight to the most recent observation. This is not indicated if
your series is very erratic (swings widely from period to period). For
example, suppose you were forecasting the demand for part #56 in
month t.
If you attached too
much weight to the
observation for t-1,
you will have a large
forecast error for
month t.
t-2
t-1
t
Month
We will now forecast
sales of liquor and
floor covering using
this technique. We
have monthly data for
each variable
beginning in January
1999 and running
through July of 2007.
Exponential Smoothing Demonstration
Millions of Dollars
5000
4000
3000
7000
6500
2000
6000
1000
5500
5000
4500
4000
99
00
01
02
03
04
Year/Month
Parts, Accessories, Tires
05
06
07
Beer, Wine, Liquor
Beer, Wine,
Liquor
Mean
Standard Error
Median
Mode
Standard
Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Parts, Accessories, Tires
2653.35922
49.798155
2567
2232
Mean
Standard Error
Median
Mode
5568.1068
54.5247066
5546
5613
505.396076
255425.193
1.88359717
1.19777269
2770
1818
4588
273296
103
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
553.365335
306213.194
-0.2735442
0.23599223
2411
4503
6914
573515
103
s
Amplitude 
X
Beer, Wine, Liquor = 0.1904
Parts, Tires, etc. = 0.099
The ratio of the standard
deviation to the mean
gives us a nice measure of
the amplitude or volatility
of a series month-to-month
(or day-to-day, quarter-toquarter, as the case may
be).
Selecting the
smoothing constant
•Pricey time series forecasting software, such
as EViews, use an algorithm to select the value
of the smoothing constant that minimizes mean
square error for in-sample forecasts.
•If you lack this software, you can use a trial
and error process.
Beer, Wine, and Liquor Sales, Smoothed (Alpha = 0.1280)
5000
4500
4000
3500
3000
2500
2000
1500
99
00
MSE  $405.35
01
02
03
04
Year/Month
Actual
05
Smoothed
06
07
Sales of Auto Parts, Accessories, and T ires, Smoothed (Alpha = 0.69)
millions of dollars
7000
6500
6000
5500
5000
4500
4000
99
00
MSE  $347.56
01
02
03
04
Year/Month
Actual
05
Smoothed
06
07
Auto Parts, Accessories, and Tires (Alpha = .69)
Year
2006
2006
2006
2006
2006
2006
2007
2007
2007
2007
2007
2007
2007
Month
7
8
9
10
11
12
1
2
3
4
5
6
7
Actual
6493
6914
6245
6419
6072
5900
5628
5526
6608
6144
6702
6619
6538
Smoothed
6642.08
6539.21
6797.82
6416.37
6418.19
6179.32
5986.59
5739.16
5592.08
6293.06
6190.21
6543.34
6595.55
Beer, Wine, and Liquor (Alpha = .1280)
Year
2006
2006
2006
2006
2006
2006
2007
2007
2007
2007
2007
2007
2007
Month
7
8
9
10
11
12
1
2
3
4
5
6
7
Actual
3322
3228
3212
3120
3359
4588
2710
2748
3176
3037
3459
3578
3547
Smoothed
2994.10
3036.07
3060.64
3080.01
3085.13
3120.19
3308.08
3231.52
3169.63
3170.44
3153.36
3192.48
3241.83
Forecasts for August, 2007
Remember our basic algorithm
Lt  Xt  (1   ) Lt
1
Hence to parts, accessories, and tire sales (PAT) for
August, 2007:
PATAUG07  [(.69)(6,538)]  [(1  .69)(6,595.55)]  $6,555.84
To forecast beer, wine, and liquor sales (BWL):
BWL AUG07  [(.1280)(3,547)]  [(1  .1280)(3,241.83)]  $3,280.89