Taylor and Bi-local Piecewise Approximations with Neuro-Fuzzy Systems
1 Institute of Computer Science of the Romanian Academy, Iaşi Branch,
11, Carol I Blvd., 700505, Iaşi, Romania
2 Gheorghe Asachi Technical University of Iasi,
67, Dimitrie Mangeron Blvd., 700050, Iaşi, Romania,
Abstract: A fuzzy neuron (linear combiner of fuzzy systems) with piecewise polynomial characteristic function is defined and analyzed. The linear fuzzy neuron uses a pre-specified type of fuzzy logic systems with complementary pairs of input membership functions. Taylor local approximations are built using the described fuzzy neuron. Moreover, local Taylor approximations are obtained using single fuzzy systems. Hence, the linear combiner fuzzy neurons are universal local approximators implementing truncated Taylor series. Moreover, they represent continuous piecewise approximators. Bi-local approximation with fuzzy logic systems are also introduced and demonstrated.
Keywords: Local approximation, TS system, fuzzy neuron, universal local approximator, approximation algorithm.
CITE THIS PAPER AS:
H.-N. TEODORESCU, Taylor and Bi-local Piecewise Approximations with Neuro-Fuzzy Systems, Studies in Informatics and Control, ISSN 1220-1766, vol. 21 (4), pp. 367-376, 2012.
Methods of local approximations based on truncated Taylor series are well understood; several methods to compute bounds for the approximation errors are available (Christensen and Christensen, 2006), (Powell, 1981), (Shahriari, 2006). The use of Taylor approximations is convenient, among others, because methods to determine the approximation accuracy are well established. Taylor approximations are widely used in solving differential equations (Kloeden, Platen, 1995), (Tachev, 2009), control applications (Hedjar et al., 2005), circuit modeling (Jridi and Alfalou, 2009), data processing, image generation, signal prediction (Hedjar et al., 2005), and economic modeling (Judd, 1998).
Another class of widely used and well understood approximation methods is based on piecewise polynomials, see for example (Powel, Chapter 18, p. 212). The use of piecewise polynomials approximations is convenient, because methods to determine the approximation accuracy are well established. Proving that some kind of fuzzy systemsare identical with the class of polynomials on a specified interval is important because polynomials constitute a core family of functions that stand at the basis of numerous mathematical constructs and applications.
In addition, because Taylor approximations and piecewise polynomials approximations are so widely used, showing that some kind of fuzzy system may implement such approximating functions is both of intellectual and practical interest.
It is well known that fuzzy logic systems (FLS) endowed with defuzzification perform a mapping from the input Rn spaceto the output space, the latter being typically the real line R. When the fuzzy system has a single input taking values in R, the mapping performedis R R. The input-output functionsare also named characteristic functions.
Recall that a Sugeno fuzzy system, also named TSK (standing for Takagi-Sugeno-Kang) or T-S fuzzy system, of order zero, is defined by the input-output function
where Si Rare called singletons and : R [0, 1] are the corresponding membership functions. In case of TSK systems with several inputs, each fuzzified independently, with max-type inference and weighted sum defuzzification, the output is
while for product-type inference the characteristic function is (Tanaka & Wang, 2001), (Yu and Li, 2004),
with the membership functions related to the ith rule denoted by , i = 1..n, the index h denoting the hth input variable. TSK fuzzy systems of higher orders are defined in a similar way, but instead of conclusions represented by constants, they have polynomial conclusions. The rules describing a TSK system of order n have the form If input x is A, then output is Pn(x), where Pn(x) is a polynomial of order n.
Bikdash (Bikdash, 1999), (Bikdash et al., 2001) have introduced and applied a modified Sugeno (TSK) systems, named Interpretable Sugeno Approximators (ISA), which use outputs in the form uk = b0k + b1k(x – r1k) + b2k(x – r2k) + … + bnk(x- rnk), “where rk is the rule center, i.e., centers of the membershipfunctions of all inputs tested by the kth rule, […] and the coefficients b are interpreted as Taylor series coefficients.”
One of the first papers to use neuro-fuzzy systems in relation to Taylor approximation was (Yu and Li, 2004). However, that approach endeavored only to approximate a Taylor approximating function, not to implement the Taylor approximant. Herera et al. (Herrera et al., 2004)(Herrera et al., 2005a,b) analyzed TSK systems with second order polynomial consequent and connected them to Taylor approximations. (Wang, Li, Niemann, & Tanaka, 2000) provided a detailed analysis of the TSK systems with linear consequent and their approximation power, for the general case. (Sonbol and Fadali 2002) and (Sonbol and Fadali, 2006) also provided an interesting approach for Taylor-like approximation with TSK systems, which contains some of the ideas presented in this paper, while (Fadali, 2002) furthers the analysis for general polynomial approximations.
We show that a constructive approach is possible that allowsbuilding a direct, true Taylor approximation using either a simpler neuro-fuzzy system or a modified TSK system. Also, bi-local approximations are introduced, as well as neuro-fuzzy systems for implementing them. A specific choice of the input membership functions allows us the direct implementation of the polynomial approximations, with zero-order TS systems, thus simplifying computations in the implementation of the approximators.
For brevity, we discuss only the case of single input single output (SISO) FLS. Also, to keep the paper focused, we deal here only with Taylor mono- or bi-local approximations, but the methods presented herein are readily extensible to Tchebychev and to other polynomial approximations. These will be dealt with elsewhere.
The flow of the paper is as follows. In the second section we introduce the linear combiner fuzzy neuron (LCFN) and the piecewise polynomial approximations using LCFNs. The piecewise Taylor approximant is derived as a particular case of the piecewise polynomial approximants. The notions of bi-local approximation and piecewise approximant are defined in Section 3 in relation to representations of the approximant by fuzzy logic systems. The fourth section is devoted to a specific class of fuzzy systems able to implement Taylor approximants and bi-local approximants. The general outline of an algorithm for building Taylor-like approximations with fuzzy systems is presented in the fifth section. The final section is devoted to a general discussion and to conclusions.
A preliminary, partial version of this paper was presented in (Teodorescu, 2010).
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