# Smoothing constant in single exponential smoothing

I have some SKUs and I'd like to do a forecast using single exponential smoothing as a forecasting method, when should we go for small value of alpha (.05,.1,...) and when for bigger values(.8,.9,...)? Does it depend on the characteristics of the series?

The data will tell you what coefficient is appropriate for your assumed model. The SES model is just one model from an infinite set of models. Just simply estimate the optimal coefficient for that model. This will be sufficient IFF this is the best ARIMA model AND IFF there are no outliers/inliers/pulses AND no level/step shifts AND no Seasonal Pulses AND no Local Time Trends AND the parameter is constant over time and the error variance is constant over time. IFF all of these are true you should be good to go !

I second IrishStats advice. Method of analysis should be dependent on characteristics of the series. When doing time series analysis or statistical analysis in general the methodology you use should be dictated by (1) basic knowledge of the subject matter that suggests model form and (2) models that seem to fit well with data. Exponential smoothing used to be looked at a a general forecasting method and modifications to it were used to compensate when simple exponential smoothing did not work well.

In the first edition of their book Box and Jenkins pointed out that single exponential was just one example of the broad class of ARIMA models, namely the IMA(1,1) model. So rather than taking an approach that uses an extension of single exponential smoothing it might be better to pick a model based on the data out of the ARIMA class.

When I worked for the US Army in the early 1970s I showed that exponential smoothing did a lot better than what the supply depots were using. They did not base their forecasts based on a mathemtical formula that used historical data. So it was easy for any method based on patterns in historical data to make sense. The fact that it is a special case of Box-Jenkins Arima models was brought to my attention and so began my interest in time series analysis.

After that I began using the Box-Jenkins approach. When I took time series from Box and Tiao, George Tiao would mrntion that it was common for exponential smoothing to be the optimal form for the model. So as IrishStat mentioned, first identify the best form of your model. If it turns out that exponential smoothing is the best form for the model then the estimated moving average parameter determines the smoothing constant.

Exponential smoothing is also what is known as an exponential filter. It's estimating the current value, and you're then using that same current value estimate for the prediction (unlike, say, double exponential smoothing where you assume a linear trend -- a nonzero derivative). Thinking in terms of filtering provides insight.

If you give more weight to the past values (less weight to the newest values), that's called "heavy" filtering. The fundamental tradeoff is that the heavier the filtering, the more you reduce noise, but the slower your response to actual changes when they come.

The estimate/prediction/filtered value predicts that the input is in reality unchanging except for noise. That's almost never actually true. So the real question is how long you're willing to believe the steady value. A month? A year? 10 years? 100 years? You can think about this in terms of a step response -- how a one-time change in the new actual value affects the estimate.

This is formalized in an intuitive way by converting the filter constant into the equivalent time constant, called tau. The exponential filtering output y (estimate/prediction) y[k] at time k, in terms of the raw data input x[k] at time k is

y[k] = a y[k-1] + (1-a) x[k]


(This switches the roles of a and (1-a) common in the forecasting literature).

Then the time constant tau is:

tau = -T/log(a)   or equivalently a = exp (-T/tau)


where T is the fixed time step between samples. If a step change in the true value does occur, the time constant tells you the response over time. The response looks like this:

In the step response plot, the time is divided by the filter time constant tau so you can more easily predict the results for any time period, for any value of the filter time constant. After a time equal to the time constant, the filter output rises to 63.21% of its final value. After a time equal to 2 time constants, the value rises to 86.47% of its final value. The outputs after times equal to 3,4,and 5 time constants are 95.02%, 98.17%, and 99.33% of the final value, respectively. Since the filter is linear, this means that these percentages can be used for any magnitude of the step change, not just for the value of 1 used here.

Although the step response in theory takes an infinite time, from a practical standpoint, think of the exponential filter as 98% to 99% “done” responding after a time equal to 4 to 5 filter time constants.

Here are some selected values for conversion between tau and the filter constant a:

Tau  Alpha          Tau Alpha           Tau Alpha

1   0.367879441     16  0.939413063     31  0.968256677
2   0.60653066      17  0.942873144     32  0.969233234
3   0.716531311     18  0.945959469     33  0.970151504
4   0.778800783     19  0.94872948      34  0.971016552
5   0.818730753     20  0.951229425     35  0.971832875
6   0.846481725     21  0.953496955     36  0.972604477
7   0.8668779       22  0.955563036     37  0.973334935
8   0.882496903     23  0.957453368     38  0.974027453
9   0.894839317     24  0.959189457     39  0.974684914
10  0.904837418     25  0.960789439     40  0.975309912
11  0.913100716     26  0.962268714     41  0.975904795
12  0.920044415     27  0.963640444     42  0.976471687
13  0.925961079     28  0.964915944     43  0.977012518
14  0.93106278      29  0.966104997     44  0.977529046
15  0.935506985     30  0.9672161       45  0.978022872


You can use the step response in your judgment as to how slow you want the response to be to actual changes over time.