Initialization

for optimal values of a that can then be used for a while. In the case of a very large numbers of time series (e.g., cheap products sold in various markets) the continuous control of a lot of parameters can be fairly expensive and not worth the effort. This is why one might consider the so-called self-adaptzve methods that self-select the parameters according to the demand patterns.

In general, in a self-adaptive method, the value of a depends on the tracking signal, that measures the rate of change of demand. A possible choice is to set a = a

.

ITStl. that is a changes proportionally t o the absolute value of TSt. The parameter a is often set t o 1.

SIMPLE EXPONENTIAL SMOOTHING 135

110 120

100

80

60 40 20

n dtl

Ft

I 6 I 1 16 21 2 6 31 36 41 46 t

Fig. 3.1 7 Case of initial forecast equal to 0 with various levels of cy

are using a fit sample (see section 3 . 3 . 7 ) : that is. we are using demand d a t a t o initialize the forecasting process. We shall simply be careful and fair when we judge the performance of our forecasting method. When we measure the performance of our method, we shall use a test sample that does not contain any d a t a we have used to initialize the exponential smoothing technique.

Using demand Yt-~+l to set the initial estimate of the baseline demand

Bt-1 is totally acceptable. We must be careful not to use it to judge the quality of our forecast. Therefore, when we use this approach the first demand observation y t - ~ + l cannot be used to measure the accuracy and bias of our forecasting process. To put it in a different way. we use the initial value t o initialize the estimate of demand but we do not use it to forecast.

This second approach provides an initial estimate Bt-I that is not bla- tantly biased like in the former case. Xevertheless, it might significantly differ from the average demand since it is based on a single demand observation t ha t might be affected by noise (see figure 3.18).

3. A third approach is designed to partially fix the problems we have just highlighted. 'We can use the average of the first I periods to initialize

120 -

100 -

80

20

4

0~ , ,

1 6 ^{1 1 16 21 26 31 36 41 46 t}

Fig. 3.18 Case of initial forecast equal t o the first d e m a n d observation with various levels of a.

the estimate of demand level*O:

t-I+l Y,

Bt-1 = k t - 1 + 1

c ^7'

In this case, the initialization is based on 1 periods rather than a single one. Thus it can capture more accurately the long run average demand (see figure 3.19). However, this approach too has a side effect: We cannot use 1 periods t o judge the quality of the forecasting process. For these periods the demand forecast depends on (i.e., exploits the informa- tion about) the demand itself (the forecast depends on the initialization that in turn depends on the demand during the first 1 periods).

This is actually a minor problem, when one just wants t o generate a demand forecast in current period t. However, when one wants t o in-

20Notice th a t we use I periods but still initializeat period t-I, t h a t is we initialize as far back into the past as possible. Indeed, one could be tempted t o set B,-I+L =

x,=t-I+l

^{Yt/L~ or}

even worse Bt ^{= ~ ~ ; ~ +}

x/L.

^{~ +}Actually, the initialization procedure is just ^l a violation of the basic mechanics of this forecasting process t h a t is based on progressive updates of previous estimates of demand. The more t h e initialization is set far into t h e past, t h e more time the exponential smoothing has t o actuallyupdate demand and t o limit the effect of t h e initialization. On the contrary. if we set B,-I+l ⁼

zL=t--I+l

^{t-'+' K / l .} we increase t h e weight of t h e initialization by a factor l / d . Finally, if we set Bt =

C:=,'_+:+, x/L.

Basically, the first forecast Ft,h = Bt is not based on any sort of exponential moving average, b u t rather on a simple average of demand observations t h a t might not even be recent. Thus.

we would simply be using a different forecasting method rather t h a n t h e one we believe is appropriate for our forecasting problem.

t-1+1

SIMPLE EXPONENTlAL SMOOTHlNG 137

120 -,

I00 ^- 80 ^-

v

40

i

<

2o

!,

period during which checking forecast accuracy does not make sense

___

deinand

-Q- 0.=0.05 --t a=o.1

1

^{t 0.=0.25}

Fig 3 19

levels of a. Case of initialization equal to the averageof the first 10 periods with various vestigate the performance of various methods (or various sets of param- eters) to select the best one, we have to set aside a "test sample'' t o measure the forecasting errors. Thus we face a tradeoff between (i) the quality of the initialization and (ii) our ability to judge what is the best forecasting method (or the best set of parameters).

Figures 3.17-3.19 show the behavior of the smoothing algorithm under the three initialization policies. Figure 3.17 shows that setting the initial forecast to zero leads to a biased forecast during the first periods. The duration of this transient state depends on a: T h e higher the value of a. the more quickly the initial forecast loses weight and the forecast reaches steady state 1 alues.

Figure 3.18 shows t h a t . in the second case. initializationis no longer biased.

but it can be fairly inaccurate. as it is based on a single demand observation.

So. also this method for initialization can generate fairly inaccurate forecasts for the first few periods. especially in the case of low a.

Figure 3.19 shows t h a t the third choice usually guarantees a better initial- ization. This is particularly important in the case of low a . However. we also notice that we can measure the accuracy of the forecasting algorithm from period 11 onward. as the first 10 observations were used t o initialize the forecasting algorithm. In the second case. instead. one can start measuring accuracy in the second period.21 Hence. the third option tends t o provide a better initialization. but it ''consumes" a lot of d a t a and we might be left

21Kotice t h a t we shall be expecting larger errors in t h e early periods as the forecasting technique is basically drawing conclusions on very small samples. So in a n odd way even t h e second option might be misleading. as it might lead us t o prefer t h e option with lower d a t a

with a small test sample. So, in an odd way the error might be smaller but we have a limited ability t o properly quantify it.