In the simple exponential smoothing technique. the current level of demand is estimated through a weighted average of the last demand observation yt and the previous estimate of the demand level Bt-1. This method increases the previous estimate Bt-l when actual demand yt is greater than we had estimated, while it reduces the previous estimate Bt-1 when demand
U,
turnsout to be lower than we thought:ls
(3.13) Also, given the assumption of a stationary demand over time, the forecast generated at a given point in time t is the same for all forecasting horizons h:
Notice that, just like in the case of moving average, we keep on updating the estimate of the demand level Bt. So Ft,h really depends on t but does not depend on h. Given equations (3.13) and (3.6.2), we can also write
In this forecasting model, a is a parameter between 0 and 1 that determines the reactivity (i.e.. promptness) of the model. Indeed, as Q changes we change the weight of the most recent demand observation yt and of the previous expectation of demand Bt-1. If Q is 1, the smoothing algorithm behaves just like a moving average with a unit time window ( k = 1) and thus reacts very promptly to any change in demand.
If Q is set t o zero, then the previous estimate Bt-1 is not affected by the last demand observation U, and thus Bt = Bt-1. This clearly makes the forecast- ing technique extremely stable. Also, noise has no influence whatsoever on future forecasts. However, this brings the forecasting technique to a standstill and the model cannot adapt to any change in demand.
I7As we already discussed. for a truly stationary process t h e best estimate of the expected demand is the simple mean of all observations. If this was t h e case; taking only t h e last k observations would not make sense. Also, it would not make sense t o give more recent observations a greater weight. If demand is really stationary, all observations are equally relevant and thus have the same weight.
18Notice t h a t we assume t h a t we update our forecast a t each period. If the forecast is reviewed less frequently (say every j periods), we simply take the weighted average of t h e average of the last j demand observations and the level of demand at time t - j .
SIMPLE EXPONENTIAL SMOOTHING 129
20 -
40 6o
1
time
0 1 I
-+ demand -*- U.=0.5
Fig 3.10 Behavior of exponential smoothing: cy = 0.5, stationary demand.
40
20
1
-la + demand a =0.1 time0 1 ’
1 3 5 7 9 I 1 13 15 17 19 21 23 25 27 29 Fig. 3.11 Behavior of exponential smoothing: CY = 0.1. stationary demand.
T h e parameter cy plays a role t h a t is very similar t o the role of k in the moving average technique. Figures 3.10-3.15 show t h a t the pattern we get with a lorn cu resembles the one we get with a large k and vice versa. Figures with cu = 0.1 show a rather inertial behavior, but also a great ability to filter noise. just ltke in the case of k = 6 for the moving average. Figures with Q = 0.5 reseinble the ones with k = 2 . as both techniques show a good reactivity. but a poor ability t o filter noise.
fig. 3.12 Behavior of exponential smoothing: a: = 0.5, demand featuring a pulse.
f i g . 3.13 Behavior of exponential smoothing: a: = 0.1, demand featuring a pulse.
SIMPLE EXPONENTIAL SMOOTHING 131
YJFt 450 - 400 - 350 - 300 - 250 - 200 -
i '
100 I5O
I
i
i
i?
I
' i
-+- demand
?.- a =0.5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Fig. 3.14 Behavior of exponential smoothing: cy = 0.5, step demand.
Yt'F, 450 - 400 - 350 - 300 - 250 - 200 - 150 -
-e demand
+ a =0.1
50
i
time1 3 5 7 9 1 1 13 15 17 19 21 23 25 27 29 Fig. 3.15 Behavior of exponential smoothing: a = 0.1, step demand.
We can elaborate on equations (3.13) and (3.14) t o derive two formulations that provide interesting insights. A first reformulation is
Ft,h = Bt =
QV, +
( I - a)Bt-l = Bt-1+
a(V, - Bt-1) Vh. (3.15) In other words, the new forecast generated a t time t is equal t o the previous one generated a t time t - 1 (Ft-l,h = Bt-l) plus a term smoothed through the parameter a, which can be interpreted as the error we made while attempting t o forecast demandut
at time t - 1. Indeed, as the formula holds for all h we can set h = 1 and read the smoothed factor asV,
- Ft-l,l. Thus exponential smoothing can be interpreted as a method that tends t o correct the error by reducing the forecast when errors are positive and by increasing it when errors are negative.We can provide a second reading by exploiting the recursiveness of equation (3.13) :
By substituting in equation (3.13), we obtain
Bt-1 = 0yt-1
+
(1 - ~ ) B t - 2 .Bt = aU,
+
a(1- a)%-1+ (1 - a)2Bt-2= aV,
+
cY(1- C2)yt-l+
a(1- ayYt-2+
(1 - 4 3 B t - 3QV, +
a(1 - Q ) Y t - l f a(1- a ) * I L+
a(1- a ) 3 V , - 3+
(1 - a ) *B t - 4 --
This formulation shows t ha t exponential smoothing gives past demands a weight t ha t decreases with the time elapsed since the demand observation.
The weight of the demand observation a t time t - i is a decreasing function of a. Figure 3.16 shows the pattern of these weights with various levels of a.
For low Q the weight of observation yt is very similar to the weight of observation
X-1.
and so on. On the contrary, for high values of a the weight of observation Yt-1 is significantly lower than for the latest observationV,.
Also. we can use the properties of geometric series to show that the sum of all weights is just 1. as one would intuitively expect.lg This property also suggests that all demand observations prior t o t - 20 have an overall weight that is equal t o 1 minus the sum of weights of all demands from period t - 20 to t. Figure 3.16 shows t h at in case of a very small a. the weight of the
"remote past" is fairly relevant (see "other periods" in the figure).
3.6.3 Setting the parameter
The above analysis suggests that high values of a enjoy reactivity, that is the ability t o promptly react to changes in average demand. whereas low values of Q filter noise very effectively. This is why in real-life situations the choice
I g I n case we sum t h e weights of a n infinite number of demand observations.
SIMPLE EXPONENTIAL SMOOTHING 133
o,6 weights
0.5
I
Y other t-20 t-19 t-18 t-17 1-16 1-15 t-14 t-13 t-12 1-11 1-10 t-9 t-8 t-7 t-6 t-5 t-4 t-3 t-2 t - I t periods
Fig. 3.16 Weights of the demand observations with various levels of a.
of a: (and more generally all smoothing parameters t h a t are presented in this chapter) should be dynamically adapted t o the changes in demand. We shall increase a!, as demand is going through a period of changes. while we shall reduce it when we expect demand to be rather stable and we only observe random fluctuations around the mean demand.
To support the choice of the appropriate level of a:, we can use the tracking szgnal (TSt):
The tracking signal is basically a smoothed average of most recent errors.
The logic behind this tool is t h a t if expected demand is relatively stable. the demand forecast is unbiased, however inaccurate it might be. Thus. errors are positive in some periods and negative in other periods: They tend to cancel out and the tracking signal tends t o be close to zero. On the contrary, if demand starts growing (or decreasing), exponential smoothing generates conservative (optimistic) forecasts and errors tend to be positive (negative).
Thus errors tend t o add up rather t h a n cancel out, and the tracking signal (TSt ) significantly (differs from zero.
TSt signals the tendency of demand t o increase (decrease) as it significantly differs from zero. So it can be used to decide when to choose large values of a (tracking signal differs from zero) and when to choose small ones (tracking signal close to zero).
The choice of the appropriate values of Q (and more generally the pa- rameters of the smoothing algorithms) is a key lever t o control and improve the forecasting proress. So, in general, it requires some managerial attention.
Most software (and even Excel. if managed properly) can automatically search
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