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21

weighted average, in which the oldest values have less importance, reduces this effect because the high-volatility data slowly become a smaller part of the result before being dropped off. Exponential smoothing, discussed next is immune from this problem by its very nature

EXPONENTIAL SMOOTHING

Rxponential smoothing may appear to be more complex than other techniques, but it is only another form of a weighted moving average. It has the added advantage of being simpler to calculate than any other method discussed: only the last exponentially smoothed

value , artel «he smoothing constant a are necessary to compute the new value. Thcr ttch-niqu-- of exponential -moothinft was developed durinK World VCar II for tracking aircraft aiHl

proieetin ctieir pusitiun-ttw; imincdiacc past is used to pretJict the immediate future

The seumecrie pruresslun a, a», . . , *("- applied to the terms

lighted moving; :

J be 50% .smoothed, the

1, - , . . . . (i)"-

-1 of a fjcometrie

where P, is the most rei:ent price and S e S 1. It can be seen that 1O096 of the combined value of past prices is distributed such (hat a >c I0096 goes to the previous exponential rrttjv-Ifig average and tbe balance to the most recent price If = VO, the current price f, will receive a weiting of 3096 of the total moving average An equally popular form that reverses thcr w<.-ightin}$ notations and has one less calculation, but gives identical results, is:

Xhe smoothing process can be started by letting , = P, and calculate the next value

The smoothing constant, * , will determine the number of calculations necessary to wind up the data A longer-term smoothing, where <a is closer to O. will take more data for Ihe smoothed line, to reach a stable value- The interpretation of this last equation can be seen in Figure 4-3 as

New exponential value = prior e

>f (todays price - prior exponential value)

FIGURE 4-3 Exponential smoothing.

"t

An important feature of the exponentially smoothed moving average is that all data previously used are ahvaj-s part



of the new result, although with diminishing significance, in general

E. e <J»,+ (1 -3), , -I- - a-)"M\ +--->

For example;, if tJie smoothing constani; a = .lO, add of the new dlETerence to the

csld aveE-ae.

£.=£. , + .10 < -£-, ,>

Eucia i-iew calculation i-eiluees all daca from points 1 throub t - 1 \yy 1U%. the next cal culation for t will cause th claca from t chrouKh 1 t-- be -du«-*l aain by 10% Therefore, at any ilm t th Impact of data used t a previ tim is based on tht number of clays elapsed, t k, and the smoothinK constant a. Let the sienifieancc - , = = Then on day Jfe we have

ka -

O* - - 1- --- +e-- - 0*>

= « - . - « * twr.tten in siimiruiiion form>

This shows that the signiflcance of the data on day Jfe foes to rem a-s eet>i .nfinitety laree Consider the folIowJMTg example. An investor makes a deal to buy 1096 of the outstanding shares of stock in a corporation In whi h there are 1 investors The corporation decides to talte in another investor and gyve chat Investor 10% of the total outstandins shares-There are now « -b 1 investors, and the original share owners are all diluted by 10%; Che investor t who bought TO% now owns S»%. Another investor buys in at 10% of the total; there are now i -1 2 investors, and the investor * has X«i of the stock (lOWi Iess> As more

investors are added, ts stockholding dwindles to 7-29° o, 6-56Po, and 55049° o. Even though the number of shares that are held remain the same, eadi previous owner holds a less significant part of the whole- No matter how many investors are added at 10° o, the original shareholders will alwajs have some small percent of the whole- In exactly the same way, the original price used in an exponentially smoothed moving average alwajs retains some relevance-, with a standard moving average of n dajs, the (n + l)th day is dropped off and ceases to have any impact-Smoothing and Restoring the Lag

As a trend continues in its direction, the exponentially smoothed moving average will lag farther behind- By selecting a smoothing constant nearer to 1, the magnitude of this tag will be lessened but it will still increase, just as using a smaller number of dajs, or shorter period, will limit the lag of a simple moving average- If the lag is considered the predictive error e, in the exponential smoothing calculation, then

where E, is the exponential smoothing approximation of the price P,. The same exponential smoothing technique can be applied to the pattem of increasing or decreasing errors to get

err, = err, , -i- a(e, - err,

and then add the difference between the original smoothing value and this second-order smoothing back into the approximation:

EE, = -b ERR.

By measunng the new error, the difference between the first-order exponential E, and the secondorder EE, it can be seen whether there has been an improvement in the forecast- This process can be continued to tbird-order smoothing by appljing the same method to tiie difference between the current price and the second-order trend EE,- A comparison of this metiiod, restoring the forecast error, is compared with other exponential smoothing techniques in Figure 4-4 and Table 4-4-

Double Smoothing



[ "" smoother, the period of a movmg average or the ex-ponential smoothing constant is

nomy mcieased. This succeeds in reducing the short-term maikel noise al tte cost of increasiJ the It hasn sZoS s:S 1 ".w Z" *"ble-smootbed, that is, the trend vals can memseV"

smoothed. This will slow down the trendhne. but gives weight to the previous values in a way that may be unexpected , 1,7 " ""S ""Se. MA. would take the original three moving average values

and average them to get a double-smoothed moving average. DMA: s s



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