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Exponential Filter

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:


  • Math image is the output of the filter at time moment Math image ;
  • Math image is the output of the filter at the previous time moment Math image ;
  • Math image is the input of the filter;
  • Math image is the parameter of the filter.

In simple words, the output Math image of the exponential filter is the weighted sum of the previous output Math image (taken with weight Math image ) and the current input value Math image (taken with weight Math image ). The smaller the parameter Math image , 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 Math image , defining the contribution of the input values Math image to the output Math image , decline exponentially with Math image . The “exponential” here means that each previous input value Math image contributes Math image times smaller to the output Math image than Math image .

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.

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