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Statistical Methods for Fuzzy Data by Reinhard Viertl

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24.4 Components model

For this section it is assumed that all data of the time series (xt*)tT have characterizing function with compact support. For practical applications this is usually fulfilled. Therefore this is no restriction.

In classical descriptive time series analysis the observed data (xt)tT are assumed to be generated by the following model:

(24.2)Numbered Display Equation

The component mt is called a trend and describes the long term behavior of the time series. The component st is called a seasonal component, and describes the cyclic behavior with more or less constant time period. The last component εt, the so-called error term, is modelling the stochastic influences. The time period of the seasonal component is denoted by p. For so-called stable seasonal behavior it is assumed that st = st+kp, k.

In the case of fuzzy data (xt*)tT the components of model (24.2) are also fuzzy. The model takes the following form:

Unnumbered Display Equation

For the seasonal component st* it is also assumed that st* = st+kp*, k. Contrary to standard time series analysis, for fuzzy data xt* no exact decomposition into trend, seasonal ...

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