**Fuzzy Time Series Estimation and Prediction: Criticism, Suitable New Methods and Experimental Evidence**

**Vasile GEORGESCU**

University of Craiova

13 A.I.Cuza, Craiova, 200585, Romania

**Abstract**: This paper is devoted to exploring suitable methods for modelling, estimating and forecasting fuzzy time series, when facing the problem of non-invertibility of the standard Minkovsky addition and multiplication in a fuzzy framework. Some generalized versions of Hukuhara difference, which allow the fuzzy estimation problem to be handled in some L2-type metric space, are first examined from a critical viewpoint. This leads us to propose a new estimation procedure, where the monolithic fuzzy model is broken in several more tractable crisp estimation sub-problems, based upon a partial decoupling principle. Our aim is to produce fuzzy estimations with non-negative spreads, capable not only to help decomposing, but also to make the process invertible, by recomposing a non-stationary fuzzy time series from its components, such as trend, cycle, seasonality and the simulated residuals, all of them properly defined as LR-fuzzy sets. Computational Intelligence techniques such as wavelet decomposition and de-noising or nonlinear model fitting with wavelet networks are also addressed. Finally, the proposed methods are exemplified for a fuzzy time series with fuzzy daily temperatures (minimum, average and maximum values).

**Keywords**: Fuzzy time series estimation and prediction, Generalized Hukuhara difference, Projection cones vs. projection subspaces, Wavelet decomposition and de-noising, Nonlinear fitting with wavelet networks.

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**CITE THIS PAPER AS**:

Vasile, GEORGESCU, **Fuzzy Time Series Estimation and Prediction: Criticism, Suitable New Methods and Experimental Evidence**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 19 (3), pp. 229-242, 2010.

**1. Introduction**

The study of fuzzy time series has attracted great interest and is expected to expand rapidly. Fuzzy time series have an inherent fuzzy and random nature.

We consider an extension of the probability space by the dimension of fuzziness, i.e., by introducing a membership scale. This enables the consideration of imprecise observations as fuzzy realizations of each elementary event . We will restrict attention to the class of normal convex fuzzy sets on , whose -level sets are in the class of nonempty compact real intervals.

A fuzzy random variable is the fuzzy result of the uncertain mapping , such that for each and , the -level intervals , generated by the mapping , are random sets. In other words, are Borel-measurable w.r.t. the Borel -field generated by the topology associated with a suitable metric on , usually the Hausdorff metric .

A fuzzy random process is defined as a family of fuzzy random variables over the space of the time coordinate . A fuzzy time series is a realization of a fuzzy random process and consists of a temporally ordered sequence of fuzzy variables , each one assigned to each discrete observation time.

For extending both the classical estimation theory and some computational intelligence techniques like wavelet analysis and wavelet networks to time series models with fuzzy data, appropriate assumptions should be stated and suitable methods should be developed.

Square-integrable random variables are assumed, defined on a Hilbert space equipped with a suitable -metric that allows the projection theorem to be still valid. However, it cannot be properly applied as usually onto a subspace, but rather onto cones (i.e., subject to some constraints), due to the lack of a general additive inverse in the space of fuzzy variables, which is only a semi-linear space.

This may lead to distorted results such as obtaining fuzzy least squares estimates with negative spreads. Using Hukukara difference instead of fuzzy subtraction has been proposed to overcome the problem. Unfortunately, it does not always exist, and even if it exists, some distortions may still appear when applying least square estimation.

A criticism of the existing fuzzy estimation methods in the literature is first addressed and suitable new methods are then proposed, based upon a partial decoupling principle. It allows decomposing the monolithic fuzzy model into several crisp models, starting from that one corresponding to modal values () in fuzzy data, and then proceeding in a decremental way for left and right -level bounds, with progressively decreasing towards . The estimates of modal values are not subject to any constraints, thus being obtained by applying the Hilbert space projection theorem directly onto the corresponding subspace. However, the estimates for the left and right -level bounds can only be obtained by applying the projection theorem onto cones, in such a way to obtain least squares estimates without negative spreads. This leads to constrained quadratic programs, conveniently defined.

As an alternative to fuzzy estimation methods, computational intelligence techniques, based on wavelet decomposition and wavelet networks for nonlinear model fitting have been proposed to address fuzzy time series estimation and prediction.

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