ForecastingForecasting

Exponential SmoothingExponential SmoothingForFor

Stationary ModelsStationary Models

• The Last Period method uses only one period (the last) and the n-Period Moving Average and Weighted Moving methods use only the last n periods to make forecasts – the rest of the data the rest of the data is ignoredis ignored.

• Exponential SmoothingExponential Smoothing uses allall the time series values to generate a forecast with lesser weights given to the observations further back in time.

Exponential SmoothingExponential Smoothing

Basic ConceptBasic Concept• Exponential smoothing is actually a way of

“smoothing” out the data by eliminating much of the “noise” (random effects).

• At each period t, an exponentially smoothed exponentially smoothed level, Llevel, Ltt,, is calculated which updates the previous level, Lt-1, as the best current

estimate of the unknown constant level, estimate of the unknown constant level, ββ00, of the time series by the following formula:

Lt = αyt + (1-α)Lt-1

Revised Estimate of the Level at time t

Weight placed on current time series value

Weight placed on last estimate for the Level

Current time series value

Last estimate for the Level

αα in in Exponential Smoothing Exponential Smoothing• The idea behind “smoothing” the data is to

get a more realistic idea about what is “really going on”.– The value of the smoothing constant, smoothing constant, αα,, is

selected by the modeler.• Higher values of α allow the time series to be

swayed quickly by the most recent observation.• Lower values keep the smoothed time series

“flatter” as not that much weight will be given to the most recent observation.– Usual values of α are between about .1 and .7– See graphs for α = .1 and α = .7 later in this module.

– The value (1-(1-αα)) is called the damping factor.damping factor.

Using Exponential Smoothing to Prepare Using Exponential Smoothing to Prepare Forecasts in Stationary ModelsForecasts in Stationary Models

• The Level, Lt, calculated at time period t is the best estimate at time t for the unknown constant, β0.

• Since that is the best estimate of β0, it will be the forecast for the next data value of the time series, Ft+1.

• Since the model is stationary, it will be the forecast for all future time periods until more time series data is observed.

Ft+1 = Lt

• Once a value of α has been selected, the Level (or smoothed value) at time t depends on only two values --– The current period’s actual value (yt) with weight of .

– The forecast value for the current period (which is the level at the previous period, Lt-1) with weight of 1-1-.

• Calculations then, for Lt (and hence for Ft+1) are very simple.

• Initialization Step –– There is no L0. So we cannot calculate L1 by αy1+ (1-α )L0

– Since y1 is the only value known after period 1, set:

Exponential Smoothing TechniqueExponential Smoothing Technique

Initialization StepInitialization Step

LL11 = y = y11

Sample Calculations for First Four Sample Calculations for First Four Periods of Yoho Data Periods of Yoho Data

• The first four values of the time series for the Yoho yoyo time series were:

415, 236, 348, 272

• Suppose we have selected to use a smoothing constant of αα = .1 = .1.

Initialization – Period 1L1 = y1 = 415 -- the level for week 1 is 415

F2 = L1 = 415 -- the forecast for week 2 is 415

ContinuedContinued

Week 2L2 = .1y2 + .9L1 = .1(236) + .9(415) = 397.1

The smoothed (leveled) value for week 2 is 397.1

F3 = L2 = 397.1 The forecast for week 3 is 397.1

Week 3L3 = .1y3 + .9L2 = .1(348) + .9(397.1) = 392.19

The smoothed (leveled) value for week 3 is 392.19

F4 = L3 = 392.19 The forecast for week 4 is 392.19

Week 4L4 = .1y4 + .9L3 = .1(272) + .9(392.19) = 380.171

The smoothed (leveled) value for week 4 is 380.171

F5 = L4 = 380.171 The forecast for week 5 is 380.171

Excel – Exponential SmoothingExcel – Exponential Smoothing

Note:Rows 8-43are hidden

=B2 =.1*B3+.9*C2

=D54

Drag C3 down to C53

Drag D3 down to D54

Drag D55 down to D56

=C3

How Exponential Smoothing Uses How Exponential Smoothing Uses All Previous Time Series ValuesAll Previous Time Series Values

• Recall that the recursive formula used is:

Lt = αyt + (1-α)Lt-1

• This means:Lt-1 = αyt-1 + (1-α)Lt-2

Lt-2 = αyt-2 + (1-α)Lt-3

Lt-3 = αyt-3 + (1-α)Lt-4

Etc.

• Substituting, Lt = αyt + (1-α)Lt-1 = αyt + (1-α)(αyt-1 + (1-α)Lt-2) =

= αyt + α(1-α)yt-1 + (1-α)2Lt-2 =

= αyt + α(1-α)yt-1 + α(1-α)2yt-2 + (1-α)3Lt-3

= αyt + α(1-α)yt-1 + α(1-α)2yt-2 + α(1-α)3yt-3 + (1-α)4Lt-4

Etc.• Thus all all time series values, yt, yt-1, yt-2, yt-3, etc. will be included with

successive weights reduced (dampened) by a factor of (1-α).

Exponential Smoothing (.1)

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60

Period

Exponential Smoothing (α = .1)

How Much Smoothing Is There?How Much Smoothing Is There?

• We said the lower the value of α, the more “smooth” the time series will become.

Actual Data

Smoothed time series with α = .1A “flat” smoothed series

What About Larger Values of What About Larger Values of αα??

• Here is the “smoothed” series for α = .7:Exponential Smothing ( .7)

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60

Period

Exponential Smoothing (α = .7)

Actual Data

Smoothed time series with α = .7Very sensitive to most recent time

series value – not much smoothing

What Value of What Value of αα Should Be Used? Should Be Used?

• Up to the modeler• If the modeler is considering several

values of α, a forecast using each value could be prepared. – Only consider values of α that would give

useful results (not α = 0, for instance)

• Then a performance measure (MSE, MAD, MAPE, LAD) could be used to determine which of the values of α that are being considered have the lowest value of the selected performance measure.

ReviewReview• Exponential smoothing is a way to take some of

the random effects out of the time series by using all time series values up to the current period.

• The smoothed value (Level) at time period t is: αα(current value) + (1-(current value) + (1-αα)(last smoothed value))(last smoothed value)

• Forecast for period t+1= Smoothed Value at t• Initialization:

First smoothed value = first actual time series valueFirst smoothed value = first actual time series value • The smaller the value of α, the less movement in

the time series.• Excel approach to exponential smoothing