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BABS 502 Moving Averages, Decomposition and Exponential Smoothing Revised March 6, 2009. Moving Averages. F t (1) is average of last m observations Issue is to choose m Most appropriate if series is random variation around a mean This is the case if all autocorrelations are near zero - PowerPoint PPT Presentation
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BABS 502 Moving Averages, Decomposition and
Exponential SmoothingRevised March 6, 2009
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© Martin L. Puterman – Sauder School of Business 2
Moving Averages Ft(1) is average of last m observations Issue is to choose m Most appropriate if series is random variation
around a mean This is the case if all autocorrelations are near zero
Not intended as a forecasting method - best for smoothing a series and determining patterns
Lags behind an increasing series Calculated in a spreadsheet using Average
function or using the MAV transformation in NCSS
© Martin L. Puterman – Sauder School of Business 3
Moving Average Example
A B C1 Period Value Forecast for next period... 1 - 79 ...81 80 23082 81 8383 82 1184 83 76 average(b81:84)=10085 84 220 average(b82:85) =97.586 85 49 average(b83:b86) = 89
© Martin L. Puterman – Sauder School of Business 4
Decomposition Method Represent series Additively as Yt = Tt + St + Ct + It Multiplicatively as Yt = Tt St Ct It whereTt is the trend component at tSt is the seasonal component at tCt is the cyclical component at tIt is the irregular or noise component at t
© Martin L. Puterman – Sauder School of Business 5
Decomposition Methods Some comments
Cyclical components not usually included since they cannot be forecasted and are hard to determine
A plausible approach for understanding time series behavior
Suggest the following general forecasting approach;- Deseasonalize data – use a forecasting method for
stationary or trending series on the deseasonalized data and then reseasonalize.
- This may be sub-optimal since the two effects can be estimated simultaneously
Multiplicative version available in NCSS approach is ad hoc See help file
© Martin L. Puterman – Sauder School of Business 6
Single Exponential Smoothing One-step ahead forecast is the weighted average of
current value and past forecast Ft(1) = Current Value)+ (1-) Past Forecast =
Xt+ (1-) Ft-1(1) Alternative representation Ft(1) = Ft-1(1) + Xt - Ft-1(1) ] To apply this we need to choose the smoothing weight
The closer is to 1, the more reactive the forecast
is to changes
© Martin L. Puterman – Sauder School of Business 7
Single Exponential SmoothingRecursive function:
Ft(1) = Xt+ (1-) Ft-1(1), Ft-1(1) = Xt-1+ (1-) Ft-2(1), etc
Backward substitute: Ft(1) = Xt + (1-)Xt-1 + (1-)2 Xt-2 + (1-)3 Xt-3 +…
When 0.3 this becomes Ft(1) = .3Xt+ .7*.3 Xt-1 + (.7)2 *Xt-2 + (.7)3 Xt-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/which is the mean of the geometric distribution.
© Martin L. Puterman – Sauder School of Business 8
Single Exponential Smoothing Example
Diagram 3.2: SES results with different smoothing parameters
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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
Time
Sale
s
Sales
Alpha = 0.1
Alpha = 0.7
© Martin L. Puterman – Sauder School of Business 9
Single Exponential Smoothing
Today’s level = Today’s value + (1-)Yesterday’s Level
Tomorrow’s forecast = Today’s levelLt = Xt + (1- ) Lt-1
Ft(k) = Lt for all kThe level represents the systematic part
of the series
© Martin L. Puterman – Sauder School of Business 10
Simple Exponential SmoothingSpreadsheet Example
© Martin L. Puterman – Sauder School of Business 11
Single Exponential SmoothingNCSS Output
Batting Averages
0.32
00.
350
0.38
00.
410
0.44
0
1 27 52 78 104Time
avg
Batting Average =.24
© Martin L. Puterman – Sauder School of Business 12
Some Comments on Exponential Smoothing (Gardner, 1985) Starting Values - need F0(1) to start process.
Possible Choices Data Mean Backcasting
It is identical to an ARIMA(0,1,1) model. In inventory applications can choose to
minimize replenishment costs. Can let vary with t and control it adaptively. Parameter is chosen to minimize one step ahead
forecast error.
© Martin L. Puterman – Sauder School of Business 13
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 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 14
Double Exponential Smoothing In a trending series, single
exponential smoothing lags behind the series
340000.0
360000.0
380000.0
400000.0
420000.0
0.9 11.6 22.4 33.1 43.9
BIRTHS Forecast Plot
Time
BIR
THS
© Martin L. Puterman – Sauder School of Business 15
Double Exponential Smoothing Double Exponential Smoothing
tracks trending data better; but forecasts may not be good after a few periods
300000.0
330000.0
360000.0
390000.0
420000.0
0.9 9.6 18.4 27.1 35.9
BIRTHS Forecast Plot
Time
BIR
THS
© Martin L. Puterman – Sauder School of Business 16
Double Exponential Smoothing The model: Separate smoothing equations for level
and trend Level Equation Lt = (Current Value)
+ (1 - ) (Level + Trend Adjustment)t-1
Lt = Xt + (1 - ) (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 17
Double Exponential Smoothing Linear Trend Model Yt=0+1t is too inflexible.
Requires a constant trend. Basic idea - introduce a trend estimate that
changes over time Similar to single exponential smoothing Issue is to choose two smoothing rates, and Referred to as Holt’s Linear Trend Model in NCSS Trend dominates after a few periods in forecasts
so forecasts are only good for a short term.
© Martin L. Puterman – Sauder School of Business 18
Double Exponential Smoothing Example
5.6
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Double Exponential Smoothing
Time
Wages
= 0.637 =0.020 L72 = 5.916 T72 = 0.013
F72(1) = 5.916 + 0.013 = 5.929 F72(1) = 5.916 + 0.013*2 = 5.942
© Martin L. Puterman – Sauder School of Business 19
Damped Trend Models Problem with a trend model is that trend dominates forecast
in a couple of periods. Approach - introduce trend damping parameter
Level EquationLt = Xt + (1 - ) (Lt-1 + T t-1)
Trend Equation Tt = (Lt - Lt-1) + (1 - ) Tt-1 Forecasting Equation
Available in SAS ETS and at Rob Hyndman’s website where he has R and Excel implementations of all exponential smoothing methods.
t
k
i
itt TLkF
1
)(
© Martin L. Puterman – Sauder School of Business 20
Seasonality A persistent pattern that occurs at regularly
spaced time intervals quarterly, monthly, weekly, daily
Data may exhibit several levels of seasonality
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 21
Seasonal Data Example14
0.0
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021
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Plot of Power
Time
Pow
er
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.500
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500
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Autocorrelations of Power (0,0,12,1,0)
Time
Aut
ocor
rela
tions
Monthly US Electric Power Consumption
© Martin L. Puterman – Sauder School of Business 22
Exponential Smoothing with Trend and Seasonality
Exponential Smoothing with trend does not track or forecast seasonal data well
200.0
375.0
550.0
725.0
900.0
0.9 7.9 14.9 21.9 28.9
sales Forecast Plot
Time
sale
s
© Martin L. Puterman – Sauder School of Business 23
The Holt-Winters Model tracks the
seasonal pattern
200.0
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1994.9 1996.6 1998.4 2000.1 2001.9
sales Forecast Plot
Time
sale
s
Exponential Smoothing with Trend and Seasonality
© Martin L. Puterman – Sauder School of Business 24
Holt-Winters’ Exponential Smoothing Equations
Level Equation: Lt = (Current Value/Seasonal
Adjustmentt-p) + (1-)(Levelt-1 + Trendt-1)
Lt = (Deseasonalized Current Value) + (1-)(Levelt-1 + Trendt-1)
Lt = (Xt/It-p) + (1-)(Lt-1 + Tt-1)where It-p = Seasonal component
© Martin L. Puterman – Sauder School of Business 25
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 26
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 27
Holt-Winters’ Exponential Smoothing EquationsSeasonal Equation: It = (Current Value/Current Level)
+ (1-)(Seasonal Adjustment)t-p
It = (Xt/Lt) + (1-)It-p where p is the length of the seasonality (i.e. p months)
Forecasting equations: Ft(k) = (Lt + kTt)It-p+k for k=1,2, …, p Ft(k) = (Lt + kTt)It-2p+k for k=p+1,p+2, …, 2p
© Martin L. Puterman – Sauder School of Business 28
Holt-Winters’ Exponential Smoothing Equations Summary Lt = (Xt/It-p) + (1-)(Lt-1 + Tt-1) Level Equation Tt = (Lt - Lt-1) + (1-)Tt-1 Trend Equation It = (Xt/Lt) + (1- )It-p Seasonal Factor
Equation
Forecasting equations: Ft(k) = (Lt + kTt)It-p+k for k=1,2, …, p Ft(k) = (Lt + kTt)It-2p+k for k=p+1,p+2, …, 2p
© Martin L. Puterman – Sauder School of Business 29
Holt-Winters’ Exponential Smoothing Example
Smoothing parameter estimates. = 0.239, = .012 and = 0.287. The seasonal factorswere:
Season 1 Factor 0.822 Season 2 Factor 0.742 Season 3 Factor 0.764Season 4 Factor 0.708 Season 5 Factor 0.748 Season 6 Factor 0.814Season 7 Factor 0.887 Season 8 Factor 0.888 Season 9 Factor 0.772Season 10 Factor 0.736 Season 11 Factor 0.724 Season 12 Factor 0.809.
Note some programs normalize these factors so that their average is one (NCSS does not).
Forecasts for 1991 are:
Jan 246.86 Feb 223.23 Mar 230.17 Apr 213.48May 225.94 June 246.41 July 268.88 Aug 269.45Sept 234.69 Oct 224.17 Nov 220.62 Dec 246.93
© Martin L. Puterman – Sauder School of Business 30
Holt-Winters’ Exponential Smoothing Example
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Year1 4 8 11 15
Hol t -W in te rs ’ app l i ed to powergenera t i on da ta
Power F or ec a s t
© Martin L. Puterman – Sauder School of Business 31
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. Excellent reference: Chatfield and Yar “Holt-Winters
forecasting: some practical issues”, The Statistician, 1988, 129-140.
© Martin L. Puterman – Sauder School of Business 32
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
© Martin L. Puterman – Sauder School of Business 33
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.