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BABS 502 Moving Averages, Decomposition and Exponential Smoothing Revised March 6, 2009

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

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BABS 502 Moving Averages, Decomposition and Exponential Smoothing Revised March 6, 2009

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  1. BABS 502 Moving Averages, Decomposition and Exponential SmoothingRevised March 6, 2009

  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

  3. Moving Average Example

  4. Decomposition Method • Represent series Additively as Yt = Tt + St + Ct + It Multiplicatively as Yt = Tt St Ct It where Tt is the trend component at t St is the seasonal component at t Ct is the cyclical component at t It is the irregular or noise component at t

  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

  6. 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) ] • 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

  7. 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.

  8. Single Exponential Smoothing Example

  9. Single Exponential Smoothing 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

  10. Simple Exponential SmoothingSpreadsheet Example

  11. BattingAverage a=.24 0.440 Batting Averages 0.410 avg 0.380 0.350 0.320 1 27 52 78 104 Time Single Exponential SmoothingNCSS Output

  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.

  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.

  14. Double Exponential Smoothing • In a trending series, single exponential smoothing lags behind the series

  15. Double Exponential Smoothing • Double Exponential Smoothing tracks trending data better; but forecasts may not be good after a few periods

  16. 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 = b(Lt - Lt-1) + (1 - b) Tt-1 • Forecasting Equation • Ft(k) = Lt + k Tt

  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, a and b. • 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.

  18. Double Exponential Smoothing Example a = 0.637 b=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

  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 Equation Lt = aXt + (1 - a) (Lt-1 + T t-1) • Trend Equation • Tt = b(Lt - Lt-1) + (1 - b) 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.

  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

  21. Seasonal Data Example Monthly US Electric Power Consumption

  22. Exponential Smoothing with Trend and Seasonality • Exponential Smoothing with trend does not track or forecast seasonal data well

  23. Exponential Smoothing with Trend and Seasonality • The Holt-Winters Model tracks the seasonal pattern

  24. 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

  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

  26. Holt-Winters’ Exponential Smoothing Equations Trend Equation: • Same as double exponential smoothing method • Tt = b(Change in level in the last period) + (1 - b) (Trend Adjustment)t-1 • Tt = b(Lt - Lt-1) + (1 - b) Tt-1

  27. 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) Forecasting equations: • Ft(k) = (Lt + kTt)It-p+kfor k=1,2, …, p • Ft(k) = (Lt + kTt)It-2p+kfor k=p+1,p+2, …, 2p

  28. Holt-Winters’ Exponential Smoothing Equations Summary • Lt = a(Xt/It-p) + (1-a)(Lt-1 + Tt-1) Level Equation • Tt = b(Lt - Lt-1) + (1-b)Tt-1Trend Equation • It = g(Xt/Lt) + (1- g)It-pSeasonal Factor Equation Forecasting equations: • Ft(k) = (Lt + kTt)It-p+kfor k=1,2, …, p • Ft(k) = (Lt + kTt)It-2p+kfor k=p+1,p+2, …, 2p

  29. Holt-Winters’ Exponential Smoothing Example

  30. Holt-Winters’ Exponential Smoothing Example

  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.

  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

  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.

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