Fuzzy Time Series
Chronological sequences of imprecise data are considered as time series with fuzzy data. A time series with fuzzy data is referred to as fuzzy time series τ. A fuzzy time series is defined as a sequence of fuzzy variables τ ordered by time τT, at which T is a finite set of equidistant points in time. Fuzzy time series are regarded as realizations of fuzzy random processes. Fig. 1 gives illustration of such a realization.
Figure 1: Fuzzy time series
Analysis and forecast of fuzzy time series is a challenging problem. It is solved by specifying an underlying fuzzy random process with the aid of generally applicable numerical methods. For modeling of fuzzy time series fuzzy ARMA processes or fuzzy artificial neural networks are applied. Fig. 2 shows an empirical fuzzy time series and the computed optimal long running forecast. By analysis and forecast of fuzzy time series e.g. prediction of structural responses can be obtained.
Figure 2: Optimal long running forecast of a fuzzy time series
- Möller, B, Beer, M, and Reuter, U (2005) Theoretical Basics of Fuzzy Randomness - Application to Time Series with Fuzzy Data, In: Safety and Reliability of Engineering Systems and Structures - Proceedings of the 9th Int. Conference on Structural Safety and Reliability, edited by G. Augusti and G.I. Schueller and M. Ciampoli. Millpress, Rotterdam, pages 1701--1707.