1. To estimate the current value of a time series (X(t), t = 1,2, ...) , which is not directly observable, from observed values of another time series (Y(t), t=1,2,...) , related to the time series X(t).
2. To predict the next value Y(t+1) of the observed time series Y from the current value Y(t) and previous values Y(t-1),Y(t-2), ... .
Consider a simple example - a time series x(i) observed in the presence of additive noise
where y(i) are observed values, x(i) are non-observable values, and ni are values of additive random noise. the goal is to estimate x(i) from y(i) . A filter of order N can be specified, for example, by function FN() of N+1 arguments:
Filters are also used for processing random process es. In this case filters are specified by integral operators or differential equations.