{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:44:28Z","timestamp":1740149068832,"version":"3.37.3"},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["HRD-0734825","1 T36 GM078000-01"],"award-info":[{"award-number":["HRD-0734825","1 T36 GM078000-01"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["HRD-0734825","1 T36 GM078000-01"],"award-info":[{"award-number":["HRD-0734825","1 T36 GM078000-01"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Fuzzy Systems"],"published-print":{"date-parts":[[2011]]},"abstract":"<jats:p>In many practical applications, it turns out to be useful to use the notion of<jats:italic>fuzzy transform:<\/jats:italic>once we have functions<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msub><mml:mi>A<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2265<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>...<\/mml:mn><mml:mo>,<\/mml:mo><mml:msub><mml:mi>A<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msub><mml:mo>\u2265<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><\/mml:mrow><\/mml:math>with<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mstyle displaystyle=\"true\"><mml:msubsup><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>i<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mi>n<\/mml:mi><\/mml:msubsup><mml:mrow><mml:msub><mml:mi>A<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mstyle><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:math>, we can then represent each function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>by the coefficients<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msub><mml:mi>F<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo>=<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:mo>\u222b<\/mml:mo><mml:mrow><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u22c5<\/mml:mo><mml:msub><mml:mi>A<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\/<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:mo>\u222b<\/mml:mo><mml:mrow><mml:msub><mml:mi>A<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mo stretchy=\"false\">).<\/mml:mo><\/mml:mrow><\/mml:math>Once we know the coefficients<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msub><mml:mi>F<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:math>, we can (approximately) reconstruct the original function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>as<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mstyle displaystyle=\"true\"><mml:msubsup><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>i<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mi>n<\/mml:mi><\/mml:msubsup><mml:mrow><mml:msub><mml:mi>F<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo>\u22c5<\/mml:mo><mml:msub><mml:mi>A<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mstyle><\/mml:mrow><\/mml:math>. 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 S\u00e1nchez et al. can be modified into a natural probabilistic explanation of fuzzy transform formulas.<\/jats:p>","DOI":"10.1155\/2011\/719256","type":"journal-article","created":{"date-parts":[[2011,8,2]],"date-time":"2011-08-02T15:01:23Z","timestamp":1312297283000},"page":"1-9","source":"Crossref","is-referenced-by-count":0,"title":["Towards an (Even More) Natural Probabilistic Interpretation of Fuzzy Transforms (and of Fuzzy Modeling)"],"prefix":"10.1155","volume":"2011","author":[{"given":"Irina","family":"Perfilieva","sequence":"first","affiliation":[{"name":"Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 701 03 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vladik","family":"Kreinovich","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2005.11.012"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/S1076-5670(07)47002-1"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijar.2007.06.008"},{"year":"1995","key":"5"},{"year":"2006","key":"10"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/S0888-613X(01)00063-9"},{"year":"2006","key":"6"},{"year":"2003","key":"7"},{"year":"1998","key":"8"},{"year":"1995","key":"9"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2006.09.004"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2009.01.001"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2006.10.004"},{"year":"1999","key":"11"},{"first-page":"117","volume-title":"Optimal strategy of switching reasoning methods in fuzzy control","year":"2005","key":"19"}],"container-title":["Advances in Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2011\/719256.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2011\/719256.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2011\/719256.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,8,4]],"date-time":"2016-08-04T12:26:33Z","timestamp":1470313593000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/afs\/2011\/719256\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011]]},"references-count":15,"alternative-id":["719256","719256"],"URL":"https:\/\/doi.org\/10.1155\/2011\/719256","relation":{},"ISSN":["1687-7101","1687-711X"],"issn-type":[{"type":"print","value":"1687-7101"},{"type":"electronic","value":"1687-711X"}],"subject":[],"published":{"date-parts":[[2011]]}}}