How do you derive the exponential smoothing factor?

Answer 1

Lets define exponential smoothing first...

Exponential smoothing, or exponential moving average, is a running average of a set of observations, where the weight of each observation is inversely exponentially weighted as a function of how old it is. It is a relatively simple thing to do.

Given a set of observations O1, O2, O3, ... ON the running exponential moving average A1, A2, A3, ... AN can be calculated in real time, at each time N, with the expression ...

AN = AN-1 (1 - X) + ON X

... where X is a weighting factor that determines that amount of smoothing. For instance, if X were zero, then the smoothing is infinite, and O does not contribute at all to A, and if X were one, then smoothing is zero, and A follows O with no smoothing at all. In a more useful example, if X were 0.2, then the smoothing would be five, and A would follow O with a time constant of five iterations, i.e. after five iterations we would be at about 63% of one step change and after 25 iterations we would be at about 95% of one step change.

Some people swap the position of X and (1 - X) in the above equation. Its their choice, but the discussion that follows will have to change accordingly.

X is the smoothing factor. It is simply the number of iterations that you want for your time constant. If you were to model this as an electronic circuit, for instance, with a capacitor and a resistor, the exponential curve would be in the form ...

e-T/RC

... where RC was your time constant. The same thing applies here. If you evaluated the first equation once per second, with an X value of 0.2, you would have a time constant of 5 seconds. If you, on the other hand, evaluated it 100 times per second, with X being 0.002, you would still have a time constant of 5 seconds, but it would much more closely approximate the second equation, which is a continuous equation, rather than a discrete equation.

In summary, then, the smoothing factor, or X, is one over the number of iterations that you want to be your time constant.