24.4 Components model
For this section it is assumed that all data of the time series (xt*)t∈T 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)t∈T are assumed to be generated by the following model:
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*)t∈T the components of model (24.2) are also fuzzy. The model takes the following form:
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 ...