Transcript Exponential Smoothing Lecture Notes

BABS 502
Moving Averages, Decomposition
and Exponential Smoothing
Revised March 11, 2014
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Single Exponential Smoothing
 One-step ahead forecast is the weighted average of
current value and past forecast
 Ft(1) = a(Current Value)+ (1-a) Past Forecast
aXt+ (1-a) Ft-1(1)
 Alternative representation
Ft(1) = Ft-1(1) + a [ Xt - Ft-1(1) ]
=
• This is previous forecast plus a constant times previous forecast
error
 Text also gives a component form representation
 To apply this we need to choose the smoothing weight a
 The closer a is to 1, the more reactive the forecast is
to changes
© Martin L. Puterman – Sauder School of Business
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Single Exponential Smoothing
Recursive function:
 Ft(1) = aXt+ (1-a) Ft-1(1),
 Ft-1(1) = aXt-1+ (1-a) Ft-2(1), etc
Backward substitute:
 Ft(1) = aXt + (1-a)aXt-1 + (1-a)2 aXt-2 + (1-a)3 aXt-3 +…
When a = 0.3 this becomes
 Ft(1) = .3Xt+ .7*.3 Xt-1 + (.7)2 *.3Xt-2 + (.7)3 .3Xt-3 + …
= .3Xt+ .21 Xt-1 + .147 Xt-2 + .1029 Xt-3 + …
This is the justification for the name “exponential”
smoothing. “Age” of data is about 1/a which is
the mean of the geometric distribution.
© Martin L. Puterman – Sauder School of Business
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Single Exponential Smoothing
Example
Diagram 3.2: SES results with different smoothing parameters
280
260
Sales
240
220
200
180
Sales
Alpha = 0.1
160
Alpha = 0.7
140
1
4
7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76
Tim e
© Martin L. Puterman – Sauder School of Business
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Single Exponential Smoothing
Component Form
Today’s level = a * Today’s value +
(1-a)*Yesterday’s Level
Tomorrow’s forecast = Today’s level
Lt = a Xt + (1- a) Lt-1
Ft(k) = Lt for all k
The level represents the systematic part of the
series
© Martin L. Puterman – Sauder School of Business
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Simple Exponential Smoothing
Spreadsheet Example
Easy to use excel optimizer to choose alpha to minimize
mean absolute percentage out of sample forecast error.
© Martin L. Puterman – Sauder School of Business
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Single Exponential Smoothing
NCSS Output
Variable
Number of Rows
Mean
Pseudo R-Squared
Mean Square Error
Mean |Error|
Mean |Percent Error|
Pulp_Price
84
579.2857
0.798127
4232.143
44.28571
7.838659
Alpha Search Mean |Percent Error|
Alpha
1
Forecast
540
Pulp_Price Forecast Plot
1000.0
Pulp_Price
800.0
600.0
400.0
200.0
0.9
23.9
46.9
69.9
92.9
Time
© Martin L. Puterman – Sauder School of Business
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Some Comments on Exponential
Smoothing (Gardner, 1985)
 Starting Values - need F0(1) to start process.
Possible Choices
 Data Mean
 Backcasting
 Simple exponential smoothing is identical to
ARIMA(0,1,1) model.
 Parameter is chosen to minimize either the root
mean square, mean absolute or mean absolute
percentage one step ahead forecast error.
 R chooses to maximize liklehood.
© Martin L. Puterman – Sauder School of Business
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Some Comments on Out of Sample
Testing
 When comparing methods out of sample
be sure to check how the out of sample
forecast is computed and what
information is assumed known.
 In some automatic programs –
exponential smoothing is applied one step
ahead out of sample so that it uses more
data than other methods.
© Martin L. Puterman – Sauder School of Business
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Double Exponential Smoothing
In a trending series, single exponential
smoothing lags behind the series
BIRTHS Forecast Plot
400000.0
380000.0
B
I R
T
H S
420000.0
360000.0
340000.0
0.9
11.6
22.4
33.1
43.9
Time
© Martin L. Puterman – Sauder School of Business
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Double Exponential Smoothing
Double Exponential Smoothing tracks
trending data better; but forecasts may
not be good after a few periods
BIRTHS Forecast Plot
390000.0
360000.0
B
I R T
H S
420000.0
330000.0
300000.0
0.9
9.6
18.4
27.1
35.9
Time
© Martin L. Puterman – Sauder School of Business
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Double Exponential Smoothing
 Linear Trend Model Yt=0+1t is inflexible. Assumes a
constant trend 1 per period throughout the data.
 Basic idea - introduce a trend estimate that changes
over time.
 Similar to single exponential smoothing but two
equations.
 Issue is to choose two smoothing rates, a and .
 Referred to as Holt’s Linear Trend
 Trend dominates after a few periods in forecasts so
forecasts are only good for a short term.
© Martin L. Puterman – Sauder School of Business
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Double Exponential Smoothing
 The model: Separate smoothing equations for level and
trend
 Level Equation
Lt = a(Current Value)
+ (1 - a) (Level + Trend Adjustment)t-1
Lt = aXt + (1 - a) (Lt-1 + T
t-1)
 Trend Equation
Tt = (Lt - Lt-1) + (1 - ) Tt-1
 Forecasting Equation
Ft(k) = Lt + k Tt
© Martin L. Puterman – Sauder School of Business
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Double Exponential Smoothing Example
Double Exponential Smoothing
6.4
W 6.0
a
g
e 5.6
s
5.2
4.8
1
a = 0.637
25
=0.020
F72(1) = 5.916 + 0.013 = 5.929
49
73
L72 = 5.916
Time 97
T72 = 0.013
F72(2) = 5.916 + 0.013*2 = 5.942
© Martin L. Puterman – Sauder School of Business
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Damped Trend Models
 Problem with a trend model is that trend dominates
forecast in a couple of periods.
 Approach - introduce trend damping parameter 
 Level Equation
Lt = aXt + (1 - a) (Lt-1 + T
t-1)
 Trend Equation
Tt = (Lt - Lt-1) + (1 - ) Tt-1
 Forecasting Equation
k
Ft (k ) = Lt    iTt
 Implemented in R.
i =1
© Martin L. Puterman – Sauder School of Business
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Seasonality
 A persistent pattern that occurs at regularly
spaced time intervals
 quarterly, monthly, weekly, daily
 Data may exhibit several levels of seasonality
simultaneously
 May be modeled as multiplicative or additive
 Should be included in systematic part of
forecasting model
 Detected visually or through ACF
© Martin L. Puterman – Sauder School of Business
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Seasonal Data Example
40
79
118
157
0.500
0.000
-0.500
Autocorrelations
1
-1.000
210.0
175.0
140.0
Power
245.0
1.000
Autocorrelations of Power (0,0,12,1,0)
280.0
Plot of Power
0
10
Time
21
31
41
Time
Monthly US Electric Power
Consumption
© Martin L. Puterman – Sauder School of Business
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Exponential Smoothing with Trend and
Seasonality
Exponential Smoothing with trend does not
track or forecast seasonal data well
sales Forecast Plot
900.0
s a le s
725.0
550.0
375.0
200.0
0.9
7.9
14.9
21.9
28.9
Time
© Martin L. Puterman – Sauder School of Business
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Exponential Smoothing with Trend
and Seasonality
The Holt-Winters Model tracks the seasonal
pattern
sales Forecast Plot
1000.0
s a le s
800.0
600.0
400.0
200.0
1994.9
1996.6
1998.4
2000.1
2001.9
Time
© Martin L. Puterman – Sauder School of Business
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Holt-Winters’ Exponential
Smoothing Equations
Level Equation:
 Lt = a(Current Value/Seasonal Adjustmentt-p)
+ (1-a)(Levelt-1 + Trendt-1)
 Lt = a(Deseasonalized Current Value)
+ (1-a)(Levelt-1 + Trendt-1)
 Lt = a(Xt/It-p) + (1-a)(Lt-1 + Tt-1)
where It-p = Seasonal component
© Martin L. Puterman – Sauder School of Business
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Holt-Winters’ Exponential
Smoothing
 Generalizes Double Exponential Smoothing by including
(multiplicative) seasonal indicators.
 Separate smoothing equations for level, trend and
seasonal indicators.
 Allows trend and seasonal pattern to change over time
 Must estimate three smoothing parameters
 Equations more complicated but implemented with
software
 One of the best methods for short term seasonal
forecasts
© Martin L. Puterman – Sauder School of Business
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Holt-Winters’ Exponential
Smoothing Equations
Trend Equation:
 Same as double exponential smoothing method
 Tt = (Change in level in the last period)
+ (1 - ) (Trend Adjustment)t-1
 Tt = (Lt - Lt-1) + (1 - ) Tt-1
© Martin L. Puterman – Sauder School of Business
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Holt-Winters’ Exponential
Smoothing Equations
Seasonal Equation:
 It = g(Current Value/Current Level)
+ (1-g)(Seasonal Adjustment)t-p
 It = g(Xt/Lt) + (1-g)It-p
where p is the length of the seasonality (i.e. p months)
so that t-p is the same season in the previous year.
Note this model assumes the same g for every season.
Forecasting equations:
 Ft(k) = (Lt + kTt)It-p+k
 Ft(k) = (Lt + kTt)It-2p+k
for k=1,2, …, p
for k=p+1,p+2, …, 2p
© Martin L. Puterman – Sauder School of Business
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Holt-Winters’ Exponential
Smoothing Equations Summary
 Lt = a(Xt/It-p) + (1-a)(Lt-1 + Tt-1) Level Equation
 Tt = (Lt - Lt-1) + (1-)Tt-1
Trend Equation
 It = g(Xt/Lt) + (1- g)It-p
Seasonal Factor Equation
Forecasting equations:
 Ft(k) = (Lt + kTt)It-p+k
 Ft(k) = (Lt + kTt)It-2p+k
for k=1,2, …, p
for k=p+1,p+2, …, 2p
© Martin L. Puterman – Sauder School of Business
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Holt-Winters’ Exponential
Smoothing Example
Pulp_Price Forecast Plot
1200.0
950.0
Pulp_Price
Forecast Summary Section
Variable
Pulp_Price
Number of Rows
84
Mean
579.2857
Pseudo R-Squared
0.766036
Mean Square Error 4904.916
Mean |Error|
44.74108
Mean |Percent Error| 7.992905
700.0
450.0
200.0
2000.9
2007.1
2013.4
2019.6
2025.9
Time
Forecast Method
Winter's with
multiplicative seasonal adjustment.
Search Iterations
120
Search Criterion
Mean |Percent Error|
Alpha
0.999787
Beta
0.1984507
Gamma
0.4674903
Initial values for forecasts
Intercept (A)
Slope (B)
Season 1 Factor
Season 2 Factor
Season 3 Factor
Season 4 Factor
-113.6628
7.878917
1.008922
0.9970459
0.9850978
1.008935
© Martin L. Puterman – Sauder School of Business
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Holt-Winters Further Comments
 Can add damped trend to this model too.
 Additive version also available but multiplicative model is
preferable. Note the HW model combines additive trend
with multiplicative seasonality.
 Missing values cannot be skipped, they must be
estimated.
 Outliers have a big impact and could be handled like
missing values
 This is a special case of a “state space model”.
 Different computer packages give different estimates
and forecasts.
 Early reference: Chatfield and Yar “Holt-Winters
forecasting: some practical issues”, The Statistician,
1988, 129-140.
© Martin L. Puterman – Sauder School of Business
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Applying Exponential Smoothing
Models
 Plot data
 determine patterns
- seasonality, trend, outliers
 Fit model
 Check residuals
 Any information present?
- Plots or ACF functions
 Adjust
 Produce forecasts
 Calibrate on hold out sample
 Multiple one step ahead
 k-step ahead (where is k is the practical forecast horizon)
© Martin L. Puterman – Sauder School of Business
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Using Exponential Smoothing in Practice
 Important issue is how frequently to recalibrate
the model
 Possible choices
- Every period
- Quarterly
- Annually
 The point here is that the model can be
determined by analysts, programmed into a
forecasting system with fixed parameters and
recalibrated as needed.
© Martin L. Puterman – Sauder School of Business
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