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
To forecast a time series we will separately forecast its trend yt and the corresponding residua rt. To forecast trend, we use two methods:
- 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 
The residua are forecast by special method developed in IRAFM. Details can be found in .
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.
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