The proposed methodology consists of two phases: analysis of a time series and its prediction. In the first phase, a time series is decomposed into two components, namely its trend and residua. Trend is represented either by a vector of fuzzy transform components, or by the inverse fuzzy transform. By residuum we understand the difference between original value of the time series and the corresponding trend value.

In the second phase, both trend as well as residua are predicted and then put together. For prediction we use one of three possibilities: second order fuzzy transform, extrapolation of the inverse fuzzy transform, or perception-based logical deduction. Prediction of the residua is obtained by linear combination of previous residua using optimization. A number of parameters are involved in this methodology. They are obtained by training on the basis of each time series. The best combination of parameters and prediction are taken for the final prediction.

The time series is assumed in the following form

*y*=

_{t}**y**

_{t}+

*r*

_{t}, t= 1, ..., T*y*is trend and

_{t}*r*

_{t}=**y**

*is residuum. Let us remark that we do not consider a more detailed decomposition, e.g., into seasonal and/or periodical component.*

_{t}- y_{t}To forecast a time series we will separately forecast its trend yt and the corresponding residua

*r*. To forecast trend, we use two methods:

_{t}- Second-order fuzzy transform, i.e. second fuzzy transform applied to components of the first one,

- perception-based logical deduction from linguistic description learned on the basis of known estimation of trend. The learning method is based on the idea presented in [1]

The residua are forecast by special method developed in IRAFM. Details can be found in [7].

The results of prediction of one of the time series are demonstrated in the following figure.

We can see the prediction using F-transform and PbLD. In this specific case, PbLD gave better results but this was not always the case.

REFERENCES

[1] BĚLOHLÁVEK, R., NOVÁK, V. Learning rule base of the linguistic expert systems. Soft Computing 7(2002), 79-88.

[2] NOVÁK, V. Perception-Based Logical Deduction. In Computational Intelligence, Theory and Applications. Berlin : Springer, 2005. ISBN 3-540-22807-1. pp. 237-250.

[3] NOVÁK, V., PERFILIEVA, I. On the Semantics of Perception-Based Fuzzy Logic Deduction. In Journal of Intelligent Systems. 2004, 19, pp.1007-1031, ISSN 0884-8173.

[4] PERFILIEVA, I.: Fuzzy Transform: Application to Reef Growth Problem. In Fuzzy Logic in Geology. Amsterdam : Academic Press, 2003. ISBN 0-12-415146-9. pp. 275-300.

[5] PERFILIEVA, I. Fuzzy Transforms: Theory and Applications. In Fuzzy Sets and Systems. 2006, 157, pp.993-1023, ISSN 0165-0114.

[6] PERFILIEVA, I., KHALDEEVA, E. Fuzzy Transformation. In 9th IFSA World Congress & 20th NAFIPS Int. Conf.. 20010725-20010728 Vancouver, Canada. Vancouver : IEEE, 2001. pp. 1946-1948. ISBN 0-7803-7079-1.

[7] PERFILIEVA, I., NOVÁK, V., PAVLISKA, V., DVOŘÁK, A., ŠTĚPNIČKA, M.: Prediction of Time Series by Soft Computing Methods. Proc. 10th Czech-Japan Seminar on data Analysis and Decision Making under Uncertainty. Liblice. Praha: VŠE v Praze, Nakladatelství Oeconomica, 2007 (to appear).