Document Zbl 1159.65013

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Comput. Appl. Math.</a> <a href="/?q=in%3A243418" title="Articles in this Issue">223, No. 2, 1028-1042 (2009)</a>. </div> <div class="abstract">It is well known that despite the big progress seen in the past years, a statistical side (unlike the probabilistic side) of a fuzzy approach to data analysis still suffers from an absence of a firm basement. The main objective of this paper is to define, within the fuzzy setting, counterparts of such important statistical notions as are, e.g. skewness or kurtosis, as well as other higher moments. These definitions are based on an $f$-weighted interval values possibilistic mean. Of interest are also new alternative justifications to the definition of the $f$-weighted interval values possibilistic mean of fuzzy numbers.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A3746">Jaromir Antoch (Praha)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5486405">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5486405">Cited in <strong>22</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td class="mono"> 65C60 </td> <td class="space"> Computational problems in statistics (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26E50" title="MSC2020">26E50</a> </td> <td class="space"> Fuzzy real analysis </td> </tr><tr> <td class="mono"> 62-07 </td> <td class="space"> Data analysis (statistics) (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afuzzy+number">fuzzy number</a>; <a href="/?q=ut%3Anearest+weighted+interval">nearest weighted interval</a>; <a href="/?q=ut%3Anearest+weighted+point">nearest weighted point</a>; <a href="/?q=ut%3A%5C%28f%5C%29-weighted+interval+valued+possibilistic+mean">$f$-weighted interval valued possibilistic mean</a>; <a href="/?q=ut%3Adata+analysis">data analysis</a>; <a href="/?q=ut%3Askewness">skewness</a>; <a href="/?q=ut%3Akurtosis">kurtosis</a>; <a href="/?q=ut%3Ahigher+moments">higher moments</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1159.65013" data-ciurl="/ci/05486405" data-biburl="/bibtex/05486405.bib" data-amsurl="/amsrefs/05486405.bib" data-xmlurl="/xml/05486405.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05486405.pdf" title="Zbl 1159.65013 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.cam.2008.03.045" aria-label="DOI for “The possibilistic moments of fuzzy numbers and their applications”" title="10.1016/j.cam.2008.03.045">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bustince, H.; Burillo, P., Correlation of interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 74, 237-244 (1995) · <a href="/0875.94156" class="nowrap">Zbl 0875.94156</a></td> </tr><tr> <td>[2]</td> <td class="space">Buckley, J. J., Fuzzy hierarchical analysis, Fuzzy Sets and Systems, 17, 233-247 (1985) · <a href="/0602.90002" class="nowrap">Zbl 0602.90002</a></td> </tr><tr> <td>[3]</td> <td class="space">Carlsson, C.; Fullér, R., On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems, 122, 315-326 (2001) · <a href="/1016.94047" class="nowrap">Zbl 1016.94047</a></td> </tr><tr> <td>[4]</td> <td class="space">Carlsson, C.; Fullér, R.; Majlender, P., On possibilistic correlation, Fuzzy Sets and Systems, 155, 425-445 (2005) · <a href="/1085.94028" class="nowrap">Zbl 1085.94028</a></td> </tr><tr> <td>[5]</td> <td class="space">Civanlar, M.; Trussel, H., Constructing membership functions using statistical data, Fuzzy Sets and Systems, 18, 1-13 (1986) · <a href="/0617.94017" class="nowrap">Zbl 0617.94017</a></td> </tr><tr> <td>[6]</td> <td class="space">Dubois, D.; Prade, H., Operations on fuzzy numbers, International Journal of Systems Science, 9, 626-631 (1978) · <a href="/0383.94045" class="nowrap">Zbl 0383.94045</a></td> </tr><tr> <td>[7]</td> <td class="space">Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Application (1980), Academic Press: Academic Press New York · <a href="/0444.94049" class="nowrap">Zbl 0444.94049</a></td> </tr><tr> <td>[8]</td> <td class="space">Dubois, D.; Prade, H., Possibility Theory: An Approach to Computrized Processing of Uncertainty (1988), Plenum Press: Plenum Press New York · <a href="/0703.68004" class="nowrap">Zbl 0703.68004</a></td> </tr><tr> <td>[9]</td> <td class="space">Dubois, D.; Prade, H., Unfair coins and necessity measures: Towards a possibilistic interpretation of histogram, Fuzzy Sets and Systems, 18, 15-20 (1983) · <a href="/0515.60005" class="nowrap">Zbl 0515.60005</a></td> </tr><tr> <td>[10]</td> <td class="space">Diamond, P.; Kloeden, P., Metric spaces of fuzzy setes, Fuzzy Sets and Systems, 35, 241-249 (1990) · <a href="/0704.54006" class="nowrap">Zbl 0704.54006</a></td> </tr><tr> <td>[11]</td> <td class="space">Fullér, R.; Majlender, P., On weighted possibilisitic mean and variance of fuzzy numbers, Fuzzy Sets and Systems, 136, 363-374 (2003) · <a href="/1022.94032" class="nowrap">Zbl 1022.94032</a></td> </tr><tr> <td>[12]</td> <td class="space">Grzegorzewski, P., Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97, 83-94 (1998) · <a href="/0930.03073" class="nowrap">Zbl 0930.03073</a></td> </tr><tr> <td>[13]</td> <td class="space">Grzegorzewski, P., Nearest interval approximation of a fuzzy number, Fuzzy Sets and Systems, 130, 321-330 (2002) · <a href="/1011.03504" class="nowrap">Zbl 1011.03504</a></td> </tr><tr> <td>[14]</td> <td class="space">Goetschel, R.; Voxman, W., Elementary calculus, Fuzzy Sets and systems, 18, 31-43 (1986) · <a href="/0626.26014" class="nowrap">Zbl 0626.26014</a></td> </tr><tr> <td>[15]</td> <td class="space">T. Gerstenkorn, J. Mańko, Correlation of intuitionistic fuzzy sets, Fuzzy Sets and Systems 44 (191) 39-43; T. Gerstenkorn, J. Mańko, Correlation of intuitionistic fuzzy sets, Fuzzy Sets and Systems 44 (191) 39-43 · <a href="/0742.04008" class="nowrap">Zbl 0742.04008</a></td> </tr><tr> <td>[16]</td> <td class="space">Ghotb, F.; Warren, L., A casestudy comparoson of the analytic hierarchy process and a fuzzy decision methodology, The Engineering Economist, 40, 233-246 (1995)</td> </tr><tr> <td>[17]</td> <td class="space">Hung, W.; Wu, J., Correlation of intuitionistic fuzzy sets by centroid method, Information Sciences, 144, 219-225 (2002) · <a href="/1013.03067" class="nowrap">Zbl 1013.03067</a></td> </tr><tr> <td>[18]</td> <td class="space">Liu, B., Uncertainty Theory An Introduction to its Axiomatic Foundations (2004), Springer-Verlag: Springer-Verlag Berlin · <a href="/1072.28012" class="nowrap">Zbl 1072.28012</a></td> </tr><tr> <td>[19]</td> <td class="space">Mikhailov, L.; Tsvetinov, P., Evaluation of services using a fuzzy analytic hierarchy process, Applied Soft Computing, 5, 23-33 (2004)</td> </tr><tr> <td>[20]</td> <td class="space">Matarazzo, B.; Munda, G., New approaches for the comparisson of L-R fuzzy numbers: A theoretical and operational analysis, Fuzzy Sets and Systems, 118, 407-418 (2001) · <a href="/0972.03055" class="nowrap">Zbl 0972.03055</a></td> </tr><tr> <td>[21]</td> <td class="space">Sladoje, N.; Lindblad, J., Represention and reconstruction of fuzzy disks by moments, Fuzzy Sets and Systems, 158, 517-543 (2007) · <a href="/1111.68143" class="nowrap">Zbl 1111.68143</a></td> </tr><tr> <td>[22]</td> <td class="space">Saaty, T. L., A scaling method for priorities in hierarchical strucheres, Journal of Mathematical Psychology, 15, 234-281 (1977) · <a href="/0372.62084" class="nowrap">Zbl 0372.62084</a></td> </tr><tr> <td>[23]</td> <td class="space">Saaty, T. L., The Analytical Hierarchy Process (1980), McGraw-Hill: McGraw-Hill New York · <a href="/1176.90315" class="nowrap">Zbl 1176.90315</a></td> </tr><tr> <td>[24]</td> <td class="space">Sachs, L., Angewandte Statistik (1978), Springer: Springer Berlin · <a href="/0388.62002" class="nowrap">Zbl 0388.62002</a></td> </tr><tr> <td>[25]</td> <td class="space">Thavaneswaran, A.; Thiagarajah, K.; Appadoo, S. S., Fuzzy cofficient volatility (FCV) models with applications, Mathematical and computer Modelling, 45, 777-786 (2007) · <a href="/1165.91415" class="nowrap">Zbl 1165.91415</a></td> </tr><tr> <td>[26]</td> <td class="space">Van Laarhoven, P. J.M.; Pedrycz, W., A fuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems, 11, 229-241 (1983) · <a href="/0528.90054" class="nowrap">Zbl 0528.90054</a></td> </tr><tr> <td>[27]</td> <td class="space">Wang, Ying-Ming, On lexicographic goal programming method for generating weighting from inconsistent interval comparison matrices, Applied Mathematics and Computation, 173, 985-991 (2006) · <a href="/1137.90607" class="nowrap">Zbl 1137.90607</a></td> </tr><tr> <td>[28]</td> <td class="space">Wang, Ying-Ming; Yang, Jian-Bo; Xu, Dong-Ling, A two-stage logarithmic goal programing method for generating weights from interval comparison matrices, Fuzzy Sets and Systems, 152, 475-498 (2005) · <a href="/1114.90493" class="nowrap">Zbl 1114.90493</a></td> </tr><tr> <td>[29]</td> <td class="space">Ying-Ming Wang, Kwai-Sang Chin, An eigenvector method for generating normalized interval and fuzzy weights, Applied Mathematics and Computation (in press) (2006); Ying-Ming Wang, Kwai-Sang Chin, An eigenvector method for generating normalized interval and fuzzy weights, Applied Mathematics and Computation (in press) (2006) · <a href="/1102.90359" class="nowrap">Zbl 1102.90359</a></td> </tr><tr> <td>[30]</td> <td class="space">Yu, C., Correlation of fuzzy numbers, Fuzzy Sets and Systems, 55, 303-307 (1993) · <a href="/0782.94028" class="nowrap">Zbl 0782.94028</a></td> </tr><tr> <td>[31]</td> <td class="space">Yen, C. L., Using fuzzy sets in developing mathematical learning-progress indicator, Proceedings of the Natational Science Council, Republic of China, Part D, 6, 57-64 (1996)</td> </tr><tr> <td>[32]</td> <td class="space">Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · <a href="/0139.24606" class="nowrap">Zbl 0139.24606</a></td> </tr><tr> <td>[33]</td> <td class="space">Zadeh, L. A., Fuzzy sets as basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28 (1978) · <a href="/0377.04002" class="nowrap">Zbl 0377.04002</a></td> </tr><tr> <td>[34]</td> <td class="space">Zimmermann, H. J., Fuzzy Sets Theory and its Application (1991), Kluwer Academic Press: Kluwer Academic Press Dordrecht · <a href="/0719.04002" class="nowrap">Zbl 0719.04002</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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