Horia-Nicolai L. TEODORESCU^1,2
1 Romanian Academy – Iasi Branch, Iasi, Romania
² “Gheorghe Asachi” Technical University of Iasi, 67, D. Mangeron, Iasi, Romania

Abstract: The paper introduces transforms of the coordinates in the Euclidean space based on fuzzy logic systems and investigates properties of these transforms. Tents maps modified using coordinate fuzzy transforms are introduced as a direct application of the fuzzy transforms. For brevity, only line and plane fuzzy transforms are considered.

Keywords: Fuzzy transformation, real line, Euclidean space, tent map, nonlinear transformation, histogram equalization.

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CITE THIS PAPER AS:
Horia-Nicolai L. TEODORESCU, Coordinate Fuzzy Transforms and Fuzzy Tent Maps – Properties and Applications, Studies in Informatics and Control, ISSN 1220-1766, vol. 24 (3), pp. 243-250, 2015. https://doi.org/10.24846/v24i3y201501

1. Introduction

This section recalls basic facts, sometimes seen from a different perspective. In defining fuzzy sets on the real axis, typically two steps are involved; the first connects the line or a segment of it with a finite set Λ of linguistic descriptions (labels), {λ_j}_j=1,…,N (such as in ‘a tall man’); the second assigns (one-to-one) intervals of the line, J_j, to the linguistic labels, λ_j ↔ J_j, where typically each interval is overlapping with at least two other intervals. To each interval Jj one assigns a function, named membership function (m.f.). The m.f.s are defined on the line, positive, and normalized (i.e., with maximal value equal to 1); in addition, they are null valued everywhere except (at most) the specified intervals. Thus, a set of functions M ={μ_j}_j=1,…,N is put into one-to-one correspondence with the set of intervals and respectively to the linguistic labels, μ_j ↔ J_j ↔ λ_j. In this way, to every point of the real line, x∈∪_j J_j , one associates a subset of membership functions overlapping in x, by the condition μ(x) ≠ 0 . Let us denote this subset by M_x = {μ_j| μ_j(x) ≠ 0} .

This entire construction corresponds to the fuzzification of the real line (or of an interval of it.) A second construction, which reverses the above one, is stereotypically as follows. Assume that all m.f.s satisfy the condition that they have value 1 at exactly one point; denote that point by aj. For every x, assign to the points aj a weight w_j(x) = μ_j(x) . In this way, every value of the real line is endowed with the set of couples {(a_j, μ_j(x))}_j .

All aj points corresponding to m.f.s not in M_x have null weights. Next, assign to x the weighted average of the points a_j, x'(x)=∑_j a_j μ_j(x)/∑_j μ_j(x). This is equivalent with defining a Takagi-SugenoKang (TSK) system from R to itself; the last operation corresponds to the defuzzification. The entire construction, composed of the above two constructions, when applied to R, i.e., when R is fuzzified and then mapped on itself with a TSK, will be named type I coordinate fuzzy transformation (CoFT) of the real line. The construction is easily extended to ndimensional Euclidean spaces. The transformation corresponds to the TSK fuzzy systems, where the output of the system is the same space as the input. The second type of fuzzy transformation of the space follows the idea of Mamdani, assigning a second set of functions η_h to the real line and next assigning to every μ_k at least one η_h (i.e., providing the ‘rules’ of the FLS). The ‘reverse’ construction is finalized according to defuzzification for Mamdani systems. The above discussion was meant to show why the name of fuzzy transformation of the real line is used for these constructions.

The investigation of the issue of fuzzy transforms is useful, among others, because any inverse model based controller can be seen as a space transform of the input space of the controller by the reverse model. The paper studies several examples of fuzzy space transforms and potential applications.

The second Section introduces the definitions and properties of the fuzzy transforms, while the third Sections presents fuzzy tent maps defined by means of the transforms. Section 4 shows an example of CoFT in prediction. The last Section concludes the paper. The following abbreviations are used throughout the paper: m.f.(s) – membership function(s), FL – fuzzy logic, FLS – fuzzy logic system, TSK – Takagi-Sugeno-Kang FLS, CoFT – coordinate fuzzy transform.

Section 3 reviews ACS algorithms and describes in detail the proposed ACS algorithm for the VRPSPD. Section 4 presents the benchmark used in this study. Obtained results are analysed at the end of this section. Finally, some conclusions are outlined in Section 5.