24.1 Moving averages

In order to obtain an overview concerning the long time behavior of a time series it is useful to eliminate the random oscillations of an observed time series (xt*)t∈T. This elimination is possible by local approximation. A simple way to do this is by smoothing the values of the time series by a local arithmetic mean. This smoothing can be taken from classical time series analysis, i.e.

and application of the extension principle in the case of fuzzy data xt*. For that, first the fuzzy numbers xt−q*, … , xt+q* have to be combined to a fuzzy vector x* ∈ with vector-characterizing function ξx*(·, … , ·). From this the fuzzy value yt* and its characterizing function ξyt*(·) is given by its values ξyt*(·)(y) for all y ∈ by

The δ-cuts Cδ(yt*) are given by theorem 3.1

The result can be given in short by

For fuzzy observations (xt*)t∈T with approximately linear behavior, ...

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