Statistical Glossary
Linear Filter:
A linear filter is the filter whose output is a linear function of the input. Any output value of a linear filter is the weighted mean of input values. In other words, to form one element 
of the output at time 
, it is necessary to multiply the input values for time moments adjacent to by coefficients and to sum up the products.
Mathematically, the output of a linear filter may be described by the expression
 |
where
-
is the output of the filter (the result of filtering); -
is the input of the filter (the original time series to be filtered);
-
is the size of the "window" of the filter - the number of the input values affecting one output value; -
are "weights" that completely describe any linear filter.
is the input of the filter (the original time series to be filtered); 
is the size of the "window" of the filter - the number of the input values affecting one output value; 
are "weights" that completely describe any linear filter. In contrast to nonlinear filter , there is a well developed and conceptually rich mathematical theory of linear filters.
All the linear filters are subdivided into two broad categories - nonrecursive filters and recursive filters .
Examples of a linear filters are: the rectangular filter , the triangular filter , the Gaussian filter , the exponential filter , Kalman filter .
See also: Smoother (Smoothing Filter) , predicting filter