1

BABS 502 Moving Averages, Decomposition and

Exponential SmoothingRevised March 11, 2014

0

50

100

150

200

250

300

350

400

450

500

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

© Martin L. Puterman – Sauder School of Business 2

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) ]

• 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

The closer is to 1, the more reactive the forecast is

to changes

© Martin L. Puterman – Sauder School of Business 3

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 4

Single Exponential Smoothing Example

Diagram 3.2: SES results with different smoothing parameters

140

160

180

200

220

240

260

280

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

Sal

es

Sales

Alpha = 0.1

Alpha = 0.7

© Martin L. Puterman – Sauder School of Business 5

Single Exponential SmoothingComponent Form

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 6

Simple Exponential SmoothingSpreadsheet 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 7

Single Exponential SmoothingNCSS Output

200.0

400.0

600.0

800.0

1000.0

0.9 23.9 46.9 69.9 92.9

Pulp_Price Forecast Plot

Time

Pulp

_P

ric

e

Variable Pulp_PriceNumber of Rows 84Mean 579.2857Pseudo R-Squared 0.798127Mean Square Error 4232.143Mean |Error| 44.28571Mean |Percent Error| 7.838659

Alpha Search Mean |Percent Error|Alpha 1Forecast 540

© Martin L. Puterman – Sauder School of Business 8

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 9

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 10

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

TH

S

© Martin L. Puterman – Sauder School of Business 11

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

TH

S

© Martin L. Puterman – Sauder School of Business 12

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, 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 13

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 14

Double Exponential Smoothing Example

5.6

5.2

4.8

6.0

6.4

1 25 49 73 97

Double Exponential Smoothing

Time

Wages

= 0.637 =0.020 L72 = 5.916 T72 = 0.013

F72(1) = 5.916 + 0.013 = 5.929 F72(2) = 5.916 + 0.013*2 = 5.942

© Martin L. Puterman – Sauder School of Business 15

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 = Xt + (1 - ) (Lt-1 + T t-1)

Trend Equation

Tt = (Lt - Lt-1) + (1 - ) Tt-1

Forecasting Equation

Implemented in R.

t

k

i

itt TLkF

1

)(

© Martin L. Puterman – Sauder School of Business 16

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 17

Seasonal Data Example1

40

.01

75

.02

10

.02

45

.02

80

.0

1 40 79 118 157

Plot of Power

Time

Pow

er

-1.0

00

-0.5

00

0.0

00

0.5

00

1.0

00

0 10 21 31 41

Autocorrelations of Power (0,0,12,1,0)

Time

Auto

corr

ela

tions

Monthly US Electric Power Consumption

© Martin L. Puterman – Sauder School of Business 18

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 19

The Holt-Winters Model tracks the seasonal pattern

200.0

400.0

600.0

800.0

1000.0

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 20

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 21

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 22

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 23

Holt-Winters’ Exponential Smoothing EquationsSeasonal Equation: It = (Current Value/Current Level)

+ (1-)(Seasonal Adjustment)t-p

It = (Xt/Lt) + (1-)It-pwhere 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 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

© Martin L. Puterman – Sauder School of Business 24

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 25

Holt-Winters’ Exponential Smoothing Example

200.0

450.0

700.0

950.0

1200.0

2000.9 2007.1 2013.4 2019.6 2025.9

Pulp_Price Forecast Plot

TimeP

ulp

_P

rice

Forecast Summary SectionVariable Pulp_PriceNumber of Rows 84Mean 579.2857Pseudo R-Squared 0.766036Mean Square Error 4904.916Mean |Error| 44.74108Mean |Percent Error| 7.992905

Forecast Method Winter's with multiplicative seasonal adjustment.Search Iterations 120Search Criterion Mean |Percent Error|Alpha 0.999787Beta 0.1984507Gamma 0.4674903

Intercept (A) -113.6628Slope (B) 7.878917Season 1 Factor 1.008922Season 2 Factor 0.9970459Season 3 Factor 0.9850978Season 4 Factor 1.008935

Initial values for forecasts

© Martin L. Puterman – Sauder School of Business 26

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 27

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 28

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.