Wednesday , October 4 2023

Coordinate Fuzzy Transforms and Fuzzy Tent Maps
– Properties and Applications

Horia-Nicolai L. TEODORESCU1,2
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.

>>Full text<<
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.

  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), j}j=1,…,N (such as in ‘a tall man’); the second assigns (one-to-one) intervals of the line, Jj, to the linguistic labels, λj ↔ Jj, 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 ={μj}j=1,…,N is put into one-to-one correspondence with the set of intervals and respectively to the linguistic labels, μj ↔ Jj ↔ λj. In this way, to every point of the real line, x∈∪j Jj , one associates a subset of membership functions overlapping in x, by the condition μ(x) ≠ 0 . Let us denote this subset by Mx = {μj| μj(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 wj(x) = μj(x) . In this way, every value of the real line is endowed with the set of couples {(aj, μj(x))}j .

All aj points corresponding to m.f.s not in Mx have null weights. Next, assign to x the weighted average of the points aj, x'(x)=∑j aj μj(x)/∑j μj(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 ηh to the real line and next assigning to every μk at least one ηh (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.


  1. BODLAENDER, H. L., J. Van LEEUWEN, Simulation of Large Networks on Smaller Networks. Information and Control, vol. 71, 1986, pp. 143-180.
  2. CRAMPIN, M., B. HEAL, On the Chaotic Behaviour of the Tent Map. Teaching Mathematics Applications vol. 13(2), 1994, pp. 83-89.
  3. DAŇKOVÁ, M., M. ŠTĚPNIČKA, Fuzzy Transform as an Additive Normal Form. Fuzzy Sets and Systems, vol. 157(8), Apr. 2006, pp. 1024-1035
  4. DU, H., N. ZHANG, Time Series Prediction using Evolving Radial Basis Function Networks with New Encoding Scheme, Neurocomputing, vol. 71(7-9), March 2008, pp. 1388-1400.
  5. GEMAN, O., H. N. TEODORESCU, C. ZAMFIR, Nonlinear Analysis and Selection of Relevant Parameters in Assessing the Treatment Results of Reducing Tremor, using DBS Procedure. Proceedings IEEE International Joint Conference on Neural Networks (IJCNN), Budapest, Hungary, Jul 25-29, 2004, vol. 14, pp. 2461-2466.
  6. GRAVES, D., W. PEDRYCZ, Fuzzy Prediction Architecture using Recurrent Neural Networks. Neurocomputing, vol. 72(7-9), Mar 2009, pp. 1668-1678.
  7. GUEGAN, D., Chaos in Economics and Finance. Annual Reviews in Control, IFAC, vol. 33(1), 2009, pp. 89-93.
  8. HSIEH, D. A., Chaos and Nonlinear Dynamics: Application to Financial Markets. The Journal of Finance, Vol. 46, no. 5, Dec 1991, pp. 1839-1877.
  9. MELIN, P., J. SOTO, O. CASTILLO, J. SORIA, A New Approach for Time Series Prediction using Ensembles of ANFIS Models. Expert Systems with Applications, vol. 39, 2012, pp. 3494-3506.
  10. PERFILIEVA, I., Fuzzy Transforms. In J.F. Peters et al. (Eds.): Transactions on Rough Sets II, LNCS 3135, Springer, Berlin Heidelberg 2004, pp. 63-81.
  11. PERFILIEVA, I., Fuzzy Transforms: Theory and Applications. Fuzzy Sets and Systems, Vol. 157(8), 2006, pp. 993-1023.
  12. PERFILIEVA, I., V. NOVAK, V. PAVLISKA, A. DVORAK, M. STEPNICKA, Forecasting Time Series using Fuzzy Transform. http://www.neural-forecasting -competition. com/downloads/NN3/methods/11_NN3_Pe rfilieva_NN3.pdf Apr 3, 2015.
  13. PERFILIEVA, I., Fuzzy Transforms in Image Compression and fusion. Acta Mathematica Universitatis Ostraviensis, vol. 15(1), 2007, pp. 27-37.
  14. PERFILIEVA, I., Fuzzy Transforms and Their Applications to Image Compression. Fuzzy Logic and Applications. Lecture Notes in Computer Science Vol. 3849, 2006, pp. 19-31.
  15. ROGERS, T. D., D. C. WHITLEY, Chaos in the Cubic Mapping. Mathematical Modelling, vol. 4, no. 1, 1983, pp. 9-25.
  16. PALACIOS, A., Cycling Chaos in OneDimensional Coupled Iterated Maps. Int. J. Bifurcation and Chaos, Vol. 12, 2002, no. 8, pp. 1859-1868
  17. DE OLIVEIRA J. A., E. R. PAPESSO, E. D. LEONEL, Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation. Entropy, vol. 15, 2013, pp. 4310-4318.
  18.  TEODORESCU, H.-N., A. KANDEL, M. SCHNEIDER, Fuzzy Modeling and Dynamics, Fuzzy Sets and Systems Vol. 106, Aug 16 1999, no. 1, pp. 1-2.
  19. TEODORESCU, H.-N., Pattern Formation and Stability Issues in Coupled Fuzzy Map Lattices. Studies in Informatics and Control, ICI Publishing House, vol. 20, 2011, no. 4, pp. 345-354.
  20. TEODORESCU, H.-N., Characterization of Nonlinear Dynamic Systems for Engineering Purposes – A Partial Review. International Journal of General Systems, vol. 41, 2012, no. 8, pp. 805-825.
  21. TEODORESCU, H.-N., On Fuzzy Sequences, Fixed Points and Periodicity in Iterated Fuzzy Maps. International Journal on Computers Communications & Control, vol. 6, no. 4, 2011, pp. 752-763.
  22. VAIRAPPAN, C., H. TAMURA, S. GAO, Z. TANG, Batch type Local Search-based Adaptive Neuro-fuzzy Inference System (ANFIS) with Self-feedbacks for Timeseries Prediction. Neurocomputing, vol. 72, nos. 7-9, Mar 2009, pp. 1870-1877.
  23. VLAD, A., A. LUCA, O. HODEA, R. TATARU, Generating Chaotic Secure Sequences Using Tent Map and a Running-Key Approach. Proceedings Romanian Academy, Series A, Vol. 14, Special Issue 2013, pp. 295-302.
  24. YI, X., Hash Function based on Chaotic Tent Maps, Circuits and Systems II: Express Briefs, IEEE Trans., vol. 52(6), 2005, pp. 354-357.
  25. ZAVADSKAS, E. K., Z. TURSKIS, J. ANTUCHEVICIENE, Selecting a Contractor by Using a Novel Method for Multiple Attribute Analysis: Weighted Aggregated Sum Product Assessment with Grey Values (WASPAS-G), Studies in Informatics and Control, ISSN 12201766, vol. 24, no. 2, 2015, pp. 141-150