Weighted Mean - Exponentially Decreasing Weights

Exponentially Decreasing Weights

In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction at each time step. Setting we can define normalized weights by

,

where is the sum of the unnormalized weights. In this case is simply

,

approaching for large values of .

The damping constant must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then the following properties of exponentially decreasing weights are useful in making a suitable choice: at step, the weight approximately equals, the tail area the value, the head area . The tail area at step is . Where primarily the closest observations matter and the effect of the remaining observations can be ignored safely, then choose such that the tail area is sufficiently small.