Theory and methods of fuzzy approximation

Fuzzy approximation is a special phenomenon which explains how complicated functions can be described by a set of fuzzy IF-THEN rules. In many cases, the way of designing the concrete set of these rules is left unexplained, or rather exhaustive technique based on neural networks or genetic algorithms is used. On the other hand, there is a series of theoretical papers unified by the key words “universal approximation” where the problem of representation of a continuous function by a so called fuzzy system (which can be regarded as an interpretation of fuzzy IF-THEN rules) is considered.

Our approach is based on three stable constructions which we use to interpret a set of fuzzy IF-THEN rules. These are normal forms in predicate fuzzy logic (see [1], [2]). These formulas generalize the respective classical forms and (on the other hand) generalize formulas using for interpolation or approximation of continuous functions. They can be called formulas for the universal approximation.

2. Focus of our research

To summarize, we use the following three formal representations before we construct the resulting approximating function to a given continuous function:

• Fuzzy IF-THEN rules,
• Logical formula in one of normal forms,
• Algebraic formula in a chosen algebra.
  

The resulting approximating function is represented by the algebraic formula which originates from the respective logical one. Our methods allow us to achieve the desired quality of approximation. It is worth mentioning that the latter notion can be expressed in the language of formal (fuzzy) logic, and general properties relating its behavior can be proved formally ([3], [4], [6]). Moreover, we are able to construct best approximations in each respective approximating space.

Special attention is paid to approximation by functions represented by additive normal forms. We elaborated a unique methodology of fuzzy transform which results in approximating function in the additive normal form. The fuzzy transform (shortly, F-transform) has two phases, namely direct and inverse, and it produces the approximating function (the inverse F-transform) as a function represented by the inversion formula. This powerful technique (introduced and elaborated by I. Perfilieva) continues the line of other known transforms: Fourier, Laplace etc. On the other hand, the fuzzy transform differs from the other transforms by its origin which comes from a fuzzy partition of a universe. The entire technique is more robust than its classical counterparts and, moreover, fulfills such sophisticated requirements as uniform convergence of a sequence of approximating functions, the weighted least square mean criterion for the components of the direct F-transform, precise numeric integration etc.

The inversion formula is used instead of precise representation of the original function in complex computations. However, when solving many problems (e.g., computation of a definite integral, solution of differential equations, etc.) we operate with the direct F-transform and not with the original function (see [5] for the details). By this trick, the problem can be transformed into a respective problem in a finite-dimensional vector space and then solved using methods of linear algebra. After the computation is finished, the result can be brought back to the space of continuous functions by the inverse F-transform.

3. Description of the main results

The idea of F transform turned out to be very general and powerful. The following list of applications justifies this claim:

In [5], [7], [13] we showed how ordinary and partial differential equations can be approximately solved with the help of the F-transform.

In [10] a property of removing noise from an origin function has been proved for the inverse F-transform.

The technique of F-transform is a part of the, so called, smooth perception-based deduction (see [8]) -- a method enabling to simulate “human-like” understanding and reasoning.

The F-transform can be used for data compression (see [9], [11]). A comparison (see [11]) shows that the quality index PSNR obtained with the usage of F-transform is higher than the PSNR determined either with fuzzy relation equations method or in the DCT one and it is close to the PSNR determined in JPEG method for small values of the compression rate.

Last, but not least, the F-transform can be used for data analysis (see [12]) and time series forecasting. In the International Competition (NN3) in the year 2007 our method showed better results (in the category “seasonal time series”) than four standard benchmarks including ARIMA (known also as Box-Jenkins method).

Literature

[1] PERFILIEVA I.: Normal Forms for fuzzy logic functions and their approximation ability. Fuzzy Sets and Systems 124 (2001), 371-384.

[2] PERFILIEVA I.: Normal forms in BL and LP algebras of functions, Soft Computing 8(2004), 291–298.

[3] PERFILIEVA, I. Logical Approximation. In Soft Computing. 2. 2002, vol.7, 7, pp.73-78, ISSN 1432-7643.

[4] DAŇKOVÁ, M., I. PERFILIEVA: Logical Approximation II. Soft Computing 7 (2003), 228 – 233.

[5] 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.

[6] PERFILIEVA, I. Normal Forms in BL-algebra of functions and their contribution to universal approximation. In Fuzzy Sets and Systems. 2004, vol.143, pp.111-127, ISSN 0165-0114.

[7] PERFILIEVA, I., E. KHALDEEVA, Fuzzy Transformation. Proc. of IFSA'2001 World Congress, Vancouver, Canada, 2001.

[8] NOVÁK, V., I. PERFILIEVA, On Semantics of Perception-Based Fuzzy Logic Deduction, Int. Journal of Intelligent Systems 19 (2004) 1007--1031.

[9] PERFILIEVA, I. Fuzzy Transforms and Their Applications to Data Compression. In: Proc. Int. Conference FUZZ-IEEE 2005, Reno, Nevada, USA, May 22-25, 2005, pp. 294--299.

[10] PERFILIEVA, I., R. Valášek, Fuzzy Transforms in Removing Noise. In: Reusch, B. (Ed.): Computational Intelligence, Theory and Applications, 2005, Springer, Heidelberg, 225--234.

[11] DI MARTINO F., SESSA S., LOIA V., PERFILIEVA, I., An image coding/decoding method based on direct and inverse fuzzy transforms, IJAR, to appear

[12] PERFILIEVA, I., NOVÁK, V., DVOŘÁK, A., Fuzzy Transform in the Analysis of Data, IJAR, to appear

[13] ŠTĚPNIČKA, M., VALÁŠEK, R. Numerical Solution of Partial Differential Equations with Help of Fuzzy Transform. In FUZZ-IEEE2005 The international Conference on FUZZY Systems. 22.5.-25.5. Reno, Nevada, USA. Reno, Nevada : Fuzz-IEEE 2005, 2005. pp. 1104-1109.