Filtering or filtration of time series are procedures to transform the values of a time series. Moving averages are special forms of filtering procedures.
24.2.1 Linear filtering
Moving averages are examples of linear transformations of fuzzy time series (xt*)t∈T to a smoothed time series (yt*). A general method from classical time series analysis can be adapted for fuzzy time series. This generalization is given in the following definition.
A transformation L = (a−q, … , as) ∈ s+q+1 of a time series (xt*)t∈T to a time series (yt*)t=q+1(1)N−s by
is called a linear filter.
A linear filter L with is called a moving average.
If s = q with ai = (2q + 1)−1, i = −q (1) q it is called an ordinary moving average.
By the following rules for fuzzy numbers x*, y*, z* ∈ () and real numbers α, β, γ ∈ ,
for two fuzzy time series (xt*)t∈T and (yt*)t