Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling)
In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have functions A1(x)≥0,...,An≥0, with ∑i=1nAi(x)=1, we can then represent each function f(x) by the coefficients Fi=(∫f(x)⋅Ai(x)dx)/(∫Ai(x)dx). Once we know the coefficients Fi, we can (approximat...
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| Format: | Article |
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Wiley
2011-01-01
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| Series: | Advances in Fuzzy Systems |
| Online Access: | http://dx.doi.org/10.1155/2011/719256 |
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| author | Irina Perfilieva Vladik Kreinovich |
| author_facet | Irina Perfilieva Vladik Kreinovich |
| author_sort | Irina Perfilieva |
| collection | DOAJ |
| description | In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have functions A1(x)≥0,...,An≥0, with ∑i=1nAi(x)=1, we can then represent each function f(x) by the coefficients Fi=(∫f(x)⋅Ai(x)dx)/(∫Ai(x)dx). Once we know the coefficients Fi, we can (approximately) reconstruct the original function f(x) as ∑i=1nFi⋅Ai(x). The original motivation for this transformation came from fuzzy modeling, but the transformation itself is a purely mathematical transformation. Thus, the empirical successes of this transformation suggest that this transformation can be also interpreted in more traditional (nonfuzzy) mathematics as well. Such an interpretation is presented in this paper. Specifically, we show that the 2002 probabilistic interpretation of fuzzy modeling by Sánchez et al. can be modified into a natural probabilistic explanation of fuzzy transform formulas. |
| format | Article |
| id | doaj-art-8ecacbc403e34fa4b5b7b7855958286c |
| institution | Kabale University |
| issn | 1687-7101 1687-711X |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Fuzzy Systems |
| spelling | doaj-art-8ecacbc403e34fa4b5b7b7855958286c2025-02-03T05:43:45ZengWileyAdvances in Fuzzy Systems1687-71011687-711X2011-01-01201110.1155/2011/719256719256Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling)Irina Perfilieva0Vladik Kreinovich1Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 701 03 Ostrava, Czech RepublicDepartment of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USAIn many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have functions A1(x)≥0,...,An≥0, with ∑i=1nAi(x)=1, we can then represent each function f(x) by the coefficients Fi=(∫f(x)⋅Ai(x)dx)/(∫Ai(x)dx). Once we know the coefficients Fi, we can (approximately) reconstruct the original function f(x) as ∑i=1nFi⋅Ai(x). The original motivation for this transformation came from fuzzy modeling, but the transformation itself is a purely mathematical transformation. Thus, the empirical successes of this transformation suggest that this transformation can be also interpreted in more traditional (nonfuzzy) mathematics as well. Such an interpretation is presented in this paper. Specifically, we show that the 2002 probabilistic interpretation of fuzzy modeling by Sánchez et al. can be modified into a natural probabilistic explanation of fuzzy transform formulas.http://dx.doi.org/10.1155/2011/719256 |
| spellingShingle | Irina Perfilieva Vladik Kreinovich Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling) Advances in Fuzzy Systems |
| title | Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling) |
| title_full | Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling) |
| title_fullStr | Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling) |
| title_full_unstemmed | Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling) |
| title_short | Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling) |
| title_sort | towards an even more natural probabilistic interpretation of fuzzy transforms and of fuzzy modeling |
| url | http://dx.doi.org/10.1155/2011/719256 |
| work_keys_str_mv | AT irinaperfilieva towardsanevenmorenaturalprobabilisticinterpretationoffuzzytransformsandoffuzzymodeling AT vladikkreinovich towardsanevenmorenaturalprobabilisticinterpretationoffuzzytransformsandoffuzzymodeling |