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Numerical solution of fuzzy relational equations based on smooth fuzzy norms

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Abstract

In this paper, we study and formulate a BP learning algorithm for fuzzy relational neural networks based on smooth fuzzy norms for functions approximation. To elaborate the model behavior more, we have used different fuzzy norms led to a new pair of fuzzy norms. An important practical case in fuzzy relational equations (FREs) is the identification problem which is studied in this work. In this work we employ a neuro-based approach to numerically solve the set of FREs and focus on generalized neurons that use smooth s-norms and t-norms as fuzzy compositional operators.

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References

  • Barajas SE, Reyes CA (2005) Your fuzzy relational neural network parameters optimization with a genetic algorithm. In: The 14th IEEE international conference on fuzzy systems. Reno, Nevada (USA), 22–25 May 2005

  • Blanco A, Delgado M, Requena I (1994) Solving fuzzy relational equations by max-min neural networks. In: IEEE world congress on computational intelligence, proceedings of the third IEEE conference on fuzzy systems, Orlando, FL, USA, 26–29 January 1994

  • Bourke M, Grant Fisher D (2000) Identification algorithms for fuzzy relational matrices, part 2: optimizing algorithms. In: Fuzzy sets and systems 109, 2000

  • Ciaramella A, Pedrycz W, Tagliaferri R, Di Nola A (2003) “Fuzzy relational neural network for data analysis. In: Proceedings of the WILF conference, Napoli, 9–11 October

  • Ciaramella A, Tagliaferri R, Pedrycz W, Di Nola A (2006) Fuzzy relational neural network. Int J Approx Reason 41:146–163

    Article  MATH  Google Scholar 

  • Crowder RS (1990) Predicting the Mackey-Glass time series with cascade-correlation learning. In: Proc. 1990 Connectionist Models Summer School. In: Touretzky D, Elman J, Hinton G, Sejnowski T (eds) San Mateo, pp 117–123

  • Di Nola A, Sessa S (1983) On the set of composite fuzzy relation equations. Fuzzy Sets Syst 9:275–285. doi:10.1016/S0165-0114(83)80028-1

    Article  MATH  Google Scholar 

  • Di Nola A, Pedrycz W, Sessa S, Sanchez E (1989) Fuzzy relation equations and their application to knowledge engineering. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Hirota K, Pedrycz W (1995) Logical filtering in solving fuzzy relational equations. In: International joint conference of the fourth IEEE international conference on fuzzy systems and the second international fuzzy engineering symposium, proceedings of 1995 IEEE international conference on fuzzy systems, Yokohama, Japan, 20–24 March 1995, vol 2, pp 679–684. doi:10.1109/fuzzy.1995.409757

  • Mackey M, Glass L (1977) Oscillation and chaos in a physiological control system. Science 297:197–287

    Google Scholar 

  • Pedrycz W (1991) Neurocomputations in relational systems. IEEE Trans Pattern Anal Mach Intell 13(3). doi:10.1109/34.75517

  • Pedrycz W (1994) A hybrid genetic learning in fuzzy relational equations. In: IEEE world congress on computational intelligence, proceedings of the third IEEE conference on fuzzy systems, Orlando, FL, USA, 26–29 June 1994, vol 2, pp 1354–1358. doi:10.1199/fuzzy.1994.343615

  • Salehfar H, Bengiamin N, Huang J (2000) A systematic approach to linguistic fuzzy modeling based on input–output data. In: Proceedings of the 2000 winter simulation conference. doi:10.1109/WSC.2000.899755

  • Sanchez E (1976) Resolution of composite fuzzy relational equations. Inf Contr 30:38–48. doi:10.1016/S0019-9958(76)90446-0

    Article  MATH  Google Scholar 

  • Valente de Oliveira J (1993) Neuron inspired learning rules for fuzzy relational structures. Fuzzy Sets Syst 57:41–53. doi:10.1016/0165-0114(93)90119-3

    Article  MathSciNet  Google Scholar 

  • Valente de Oliveira J, Miranda Lemos J (1993) System modeling and fuzzy relational identification. IEEE 2:1074–1078. doi:10.1109/fuzzy.1993.327356

    Google Scholar 

  • Wang LX (1993) Solving fuzzy relational equations through network training. IEEE International Conference on Fuzzy Systems, San Francisco 2:956–960

    Google Scholar 

  • Zadeh LA (1971) Similarity relations and fuzzy ordering. Inf Sci 3(2):177–200. doi:10.1016/S0020-0255(71)80005-1

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang X, Hang C, Tan S, Wang PZ (1994) The delta MLE and learning for min-max neural networks. In: Proc. IEEE international conference on neural networks June 28–July 2, vol 1, pp 38–43

Download references

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Authors and Affiliations

  1. Department of Electrical Engineering, Amirkabir University of Technology (AUT), Hafez Ave., P.O. Box 15914, Tehran, Iran

    Arya Aghili Ashtiani & Mohammad Bagher Menhaj

Authors
  1. Arya Aghili Ashtiani
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  2. Mohammad Bagher Menhaj
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Correspondence to Arya Aghili Ashtiani.

Appendix

Appendix

For the probabilistic sum we can define:

$$ \begin{array}{l} p^{1} : = s\left( {a,0} \right) = a \hfill \\ p^{2} : = s\left( {a,b} \right) = a + b - ab \hfill \\ p^{3} : = s\left( {a,b,c} \right) = s\left( {a,s\left( {b,c} \right)} \right) \hfill \\ \qquad = a + b + c - \left( {ab + ac + bc} \right) + abc \hfill \\ p^{n} : = s\left( {a_{1} ,a_{2} , \ldots ,a_{n} } \right) \hfill \\ \end{array} $$

Now the above functions can be expressed according to the following scheme:

$$ p_{{}}^{n} = P_{1}^{n} \left( {\left[ {a_{1} ,a_{2} , \ldots ,a_{n} } \right]^{T} } \right) $$
$$ P_{{k_{0} }}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \sum\limits_{{k = k_{0} }}^{n} {N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} ,\;N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \left( { - 1} \right)^{k + 1} M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right),\;M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \frac{1}{{k!\left( {n - k} \right)!}}L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) $$
$$ L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \left| \begin{gathered} \hfill \\ \;\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \;}}^{{k\;{\text{vectors}}}}\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \;}}^{{\left( {n - k} \right){\text{vectors}}}}\; \hfill \\ \hfill \\ \end{gathered} \right|_{ + } ;\;\;\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } = \left[ \begin{gathered} \lambda_{1} \hfill \\ \vdots \hfill \\ \lambda_{n} \hfill \\ \end{gathered} \right],\;\;\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} : = \left[ \begin{gathered} 1 \hfill \\ \vdots \hfill \\ 1 \hfill \\ \end{gathered} \right] $$
$$ \Rightarrow \;\;\;\;\;N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = \frac{{\left( { - 1} \right)^{k + 1} }}{{k!\left( {n - k} \right)!}}\left| \begin{gathered} \hfill \\ \;\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \;}}^{{k\;{\text{vectors}}}}\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \;}}^{{\left( {n - k} \right){\text{vectors}}}}\; \hfill \\ \hfill \\ \end{gathered} \right|_{ + } $$

In the next step we consider the derivatives as follows:

$$ \begin{gathered} \frac{{\partial p^{1} }}{\partial a} = 1,\;\frac{{\partial p^{2} }}{\partial a} = 1 - b,\;\frac{{\partial p^{3} }}{\partial a} = 1 - \left( {b + c} \right) + bc, \hfill \\ \frac{{\partial p^{4} }}{\partial a} = 1 - \left( {b + c + d} \right) + \left( {cd + bd + bc} \right) - bcd, \hfill \\ \end{gathered} $$

and so on.

In \( P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right), \) the dimension \( n \) is indeed the length of the vector \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \) and can be obtained by the argument \( \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right), \) hence we can omit the superscript for the sake of brevity. Also by omitting the subscript \( k_{0} , \) we mean the common value \( k_{0} = 1. \) Therefore the following notations are equal:

$$ P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = P_{1}^{{}} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) $$

Now, we look for the derivatives:

$$ \begin{aligned} Q\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right): & = \frac{{\partial P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \frac{\partial }{{\partial \lambda_{i} }}\left( {\sum\limits_{k = 1}^{n} {N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} } \right) = \sum\limits_{k = 1}^{n} {\frac{{\partial N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }}} \\ & = \sum\limits_{k = 1}^{n} {\frac{{\partial N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}\frac{{\partial M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }}} = \sum\limits_{k = 1}^{n} {\frac{{\left( { - 1} \right)^{k + 1} }}{{k!\left( {n - k} \right)!}}\frac{{\partial L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }}} \\ \end{aligned} $$
$$ \frac{{\partial L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \frac{\partial }{{\partial \lambda_{i} }}\left| \begin{gathered} \hfill \\ \;\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \;}}^{{k\;{\text{vectors}}}}\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} \;}}^{{\left( {n - k} \right){\text{vectors}}}}\; \hfill \\ \hfill \\ \end{gathered} \right|_{ + } = \frac{\partial }{{\partial \lambda_{i} }}\left| {\begin{array}{*{20}c} {\lambda_{1} } & \ldots & {\lambda_{1} } \\ \vdots & {} & \vdots \\ {\lambda_{i} } & \cdots & {\lambda_{i} } \\ \vdots & {} & \vdots \\ {\lambda_{n} } & \cdots & {\lambda_{n} } \\ \end{array} \;\;\;\begin{array}{*{20}c} 1 & \ldots & 1 \\ \vdots & {} & \vdots \\ 1 & \cdots & 1 \\ \vdots & {} & \vdots \\ 1 & \cdots & 1 \\ \end{array} } \right|_{ + } = k\;L_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) $$

In the above equation, \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime} \) is a \( \left( {n - 1} \right) \times 1 \) vector which is obtained from \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \) by omitting the \( i \)’th element.

$$ \begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime} = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & \cdots & {\lambda_{i - 1} } & {\lambda_{i + 1} } & \cdots & {\lambda_{n} } \\ \end{array} } \right]^{T} \hfill \\ \left\{ \begin{gathered} \frac{{\partial P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \sum\limits_{k = 1}^{n} {\frac{{\left( { - 1} \right)^{k + 1} k}}{{k!\left( {n - k} \right)!}}L_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right)} \hfill \\ M_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = \frac{1}{{\left( {k - 1} \right)!\left( {n - k} \right)!}}L_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) \hfill \\ \end{gathered} \right. \Rightarrow \frac{{\partial P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \sum\limits_{k = 1}^{n} {\left( { - 1} \right)^{k + 1} M_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right)} \hfill \\ \end{gathered} $$

To obtain a more compact form we can write:

$$ \begin{aligned} & \left\{ {\begin{array}{*{20}c} {P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = \sum\limits_{k = 1}^{n} {N_{k}^{n} = \sum\limits_{k = 1}^{n} {\left( { - 1} \right)^{k + 1} } } M_{k} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} \\ {Q\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{'} } \right) = \sum\limits_{k = 1}^{n} {\left( { - 1} \right)^{k + 1} M_{k - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{'} } \right)} } \\ \end{array} } \right. \\ & \Rightarrow Q\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = \sum\limits_{k = 0}^{n - 1} {\left( { - 1} \right)^{k} M_{k}^{{}} } \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = - \sum\limits_{k = 0}^{n - 1} {\left( { - 1} \right)^{k + 1} M_{k}^{{}} } \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = - \sum\limits_{k = 0}^{n - 1} {N_{k}^{{}} } \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = - P_{0}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) \hfill \\ & \Rightarrow \boxed{\frac{\partial }{{\partial \lambda_{i} }}\left( {P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} \right) = - P_{0}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right)} \hfill \\ \end{aligned} $$
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Aghili Ashtiani, A., Menhaj, M.B. Numerical solution of fuzzy relational equations based on smooth fuzzy norms. Soft Comput 14, 545–557 (2010). https://doi.org/10.1007/s00500-009-0425-1

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