1. BABS 502 Moving Averages, Decomposition and Exponential Smoothing Revised 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 Xt 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 Smoothing Spreadsheet Example

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

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 = b=0.020 L72 = T72 = 0.013
F72(1) = = F72(1) = *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. Monthly US Electric Power Consumption
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)
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+k for k=1,2, …, p
Ft(k) = (Lt + kTt)It-2p+k for 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 Trend Equation
It = g(Xt/Lt) + (1- g)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

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,

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.