O'Reilly logo

Statistical Methods for Fuzzy Data by Reinhard Viertl

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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*)tT. 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.

Unnumbered Display Equation

and application of the extension principle in the case of fuzzy data xt*. For that, first the fuzzy numbers xtq*, … , 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

Unnumbered Display Equation

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

Unnumbered Display Equation

The result can be given in short by

Unnumbered Display Equation

For fuzzy observations (xt*)tT with approximately linear behavior, ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required