Exponential smoothing: SES – DES – Holt-Winter

SES: x StepAhead. a:minimize Rmse. F0:mean. OutSample. Slacked. DES: a&B. holt Linear: small period ahead. Damped T.
TES: Winters: Seasonality: SI visually or by Acf. Additive/multiplicative. Deseason. MultiSeasonality. a B ¥. Outliers!

We normally use Multiplicative. We only use additive qhen the data set has unstable variability, e.g. Hetero.

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)

Ft(1) = Ft-1(1) + a [ Xt – Ft-1(1) ]

  • I.e. previous forecast plus a constant times previous forecast error

To apply this we need to choose the smoothing weight α

◦The closer a is to 1, the more reactive the forecast is to changes

Recursive function:

◦F[t](1) = αX[t]+ (1-α) F[t-1](1),

◦F[t-1](1) = αX[t]-1+ (1-α) F[t-2](1) ,  etc

Backward substitute:

◦Ft(1) = αX[t] + (1-α)αX[t]-1 + (1-α)2 αX[t-2] + (1-α)3 αX[t-3] +…

image14
Simple Exponential Smoothing

Today’s level = α* Today’s value +(1-a)*Yesterday’s Level
Tomorrow’s forecast = Today’s level
L[t] = α X[t] + (1- α) L[t]-1
F[t](k) = L[t]  for all k
The level represents the systematic part of the series

image9

Simple Exponential Smoothing
Simple Exponential Smoothing

Starting Values – need F0(1) to start process. Possible Choices:
◦Data Mean
◦Backcasting
SES is identical to ARIMA(0,1,1) model.
Parameter is chosen to minimize either the root mean square (RMS), mean absolute or mean absolute percentage one step (F(1)) ahead forecast error.

Out of Sample Testing: training data is called In-Sample data, while the test data is called Out-of-Sample data.

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.


Double Exponential Smoothing
 Double Exponential Smoothing factors-in the trends in the so it models trending data better; but only as the trend remains, so it ages out quickly.

Double Exponential Smoothing
Importance of Double Exponential Smoothing

Linear Trend Model (regression) Yt=b0+b1t is inflexible.  Assumes  a constant  trend b1 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 b.

}Referred to as Holt’s Linear Trend Model

}Trend dominates after a few periods in forecasts so forecasts are only good for a short term.

Double Exponential Smoothing
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

Damped Trend Models

}Problem with a trend model is that trend dominates forecast in a couple of periods.

}Approach – introduce trend damping parameter  f

Level Equation

Lt =  aXt + (1 – a) (Lt-1 + fT t-1)

Trend Equation

Tt =  b(Lt  – Lt-1) + (1 –  b) fTt-1

Forecasting Equation  

◦Available in SAS ETS, R, Textbook’s forecast package for Excel and http://www.peerforecaster.com

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

Double Exponential smoothing (with trend factor) doesn’t model seasonality

Double Exponential Smoothing with Seasonality
Double Exponential Smoothing with Seasonality

The Holt-Winters Model: Factors-in seasonality:

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

 

}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

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

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  for k=1,2, …, p

}Ft(k) = (Lt + kTt)It-2p+k  for k=p+1,p+2, …, 2p

 

}Lt = a(Xt/It-p) + (1-a)(Lt-1 + Tt-1)     Level Equation

}Tt = b(Lt – Lt-1) + (1-b)Tt-1     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

}

}

}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”(chapter 5)

}Different computer packages give different estimates and forecasts.

}Early reference:  Chatfield and Yar “Holt-Winters forecasting: some practical issues”, The Statistician, 1988, 129-140.

Holt-Winters Exponential Smoothing with Trend and Seasonality
Holt-Winters Exponential Smoothing with Trend and Seasonality

}

}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)

}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.

 

SES is ARIMA(0,1,1):
short forecast lags behind actual
DES plot fix ses trend bad>dump.
Winters (TES): plot/Acf deseasonlize

(anual 4cast)? Why? Damped too.

Addtv L +T + I vs multiplicative: L +T * I. can decide b4 modeling?
Missing values and outliers to replace with estimts.

Procedure: Plot & acf dtct trnd outlir set params1 fit calc rmse recalibrate by diff params & models.

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