Exponential Filter:
The exponential filter is the simplest linear recursive filter . Exponential filters are widely used in time series analysis , especially for forecasting time series (see the short course Time Series Forecasting ).
The exponential filter is described by the following expression:
 |
where
-
is the output of the filter at time moment 
; -
is the output of the filter at the previous time moment 
; -
is the input of the filter; -
is the parameter of the filter.
In simple words, the output of the exponential filter is the weighted sum of the previous output (taken with weight 
) and the current input value 
(taken with weight 
). The smaller the parameter , the longer the "memory" of the exponential filter and the greater the degree of smoothing .
The term "exponential" stems from the fact that, if to try to realize an equivalent nonrecursive filter , then the weights 
, defining the contribution of the input values 
to the output , decline exponentially with 
. The "exponential" here means that each previous input value 
contributes times smaller to the output than .
This exponential character of the decline of weights means that, if to try to implement an equivalent filter as a nonrecursive filter, then an infinite number of preceding input values should be taken into account (and this is, strictly speaking, computationally impossible). This feature illustrates a major advantage of recursive filters over nonrecursive filters - computational simplicity.