24.2 Filtering

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.

Definition 24.1:

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.

Remark 24.1:

By the following rules for fuzzy numbers x*, y*, z* ∈ () and real numbers α, β, γ ∈ ,

and

for two fuzzy time series (xt*)t∈T and (yt*)t

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