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A taxonomy of exponential smoothing methodsExponential smoothing methods are not restricted to those we have presented so far. By

considering variations in the combination of the trend and seasonal components, fifteen

exponential smoothing methods are possible, listed in Table 7.7. Each method is labelled

by a pair of letters (T,S) defining the type of Trend! and Seasonal! components. "or

example, (#,$) is the method with an additive trend and m%ltiplicative seasonality&

($,') is the method with m%ltiplicative trend and no seasonality& and so on.

Seasonal ComponentTrend N A MComponent (None) (Additive) (Multiplicative)N (None) (N,N) (N,A) (N,M)A (Additive) (A,N) (A,A) (A,M)

Ad(Additive damped) (Ad,N) (Ad,A) (Ad,M)M (Multiplicative) (M,N) (M,A) (M,M)Md(Multiplicative damped) (Md,N) (Md,A) (Md,M)

Some of these methods we have already seen

(N,N) = simple exponential smoothing(A,N) = Holts linear method(M,N) = Exponential trend method(Ad,N) = additive damped trend method

(Md,N) = multiplicative damped trend method(A,A) = additive Holt-Winters method(A,M) = multiplicative Holt-Winters method(Ad,M) = Holt-Winters damped method

This type of classification was proposed by egels (*++). -t was later extended by

ardner (*+/0) to incl%de methods with additive damped trend and by Taylor (1223) to

incl%de methods with m%ltiplicative damped trend.

Table 7./ gives the rec%rsive form%lae for applying all possible fifteen exponential

smoothing methods. Each cell incl%des the forecast e4%ation for generatingh5step5

ahead forecasts and the smoothing e4%ations for applying the method. -n Table 7.+ we

present some strategies for selecting initial val%es for some of the most commonly

applied exponential smoothing methods. 6e do not recommend that these strategies be

%sed directly& rather, they are %sef%l in providing starting val%es for the optimiation

process.

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Table

7.8: Formulae for recursive calculations and point forecasts. In each case,tdenotes the series level

at timet,btdenotes the slope at timet,stdenotes the seasonal component of the series at timet,

andmdenotes the number of seasons in a year;,,andare smoothing

parameters,h=+2++handh+m=(h1) modm+1. (lic! the table for a

larger version."

Method Initial values

(N,N) 0=y1

(A,N) (Ad,N) 0=y1,b0=y2y1

(M,N) (Md,N) 0=y1,b0=y2/y1

(A,A) (Ad,A) 0=1m(y1++ym)

b0=1m[ym+1y1m++ym+mymm]

s0=ym0,s1=ym10, ,sm+1=y10

(A,M) (Ad,M) 0=1m(y1++ym)

b0=1m[ym+1y1m++ym+mymm]

s0=ym/0,s1=ym1/0, ,sm+1=y1/0