Document Zbl 0909.68040

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Intell.</a> <a href="/?q=in%3A52857" title="Articles in this Issue">106, No. 1, 109-137 (1998)</a>. </div> <div class="abstract">Summary: The main statistics used in rough set data analysis, the approximation quality, is of limited value when there is a choice of competing models for predicting a decision variable. In keeping within the rough set philosophy of non-invasive data analysis, we present three model selection criteria, using information theoretic entropy in the spirit of the minimum description length principle. Our main procedure is based on the principle of indifference combined with the maximum entropy principle, thus keeping external model assumptions to a minimum. The applicability of the proposed method is demonstrated by a comparison of its error rates with results of C4.5, using 14 published data sets.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1238962">Cited in <strong>92</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68P05" title="MSC2020">68P05</a> </td> <td class="space"> Data structures </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Arough+set+model">rough set model</a>; <a href="/?q=ut%3Aminimum+description+length+principle">minimum description length principle</a>; <a href="/?q=ut%3Aattribute+prediction">attribute prediction</a></div> <div class="software"> <h3>Software:</h3><a href="/software/12176">C4.5</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 0909.68040" data-ciurl="/ci/01238962" data-biburl="/bibtex/01238962.bib" data-amsurl="/amsrefs/01238962.bib" data-xmlurl="/xml/01238962.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01238962.pdf" title="Zbl 0909.68040 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/S0004-3702(98)00091-5" aria-label="DOI for “Uncertainty measures of rough set prediction”" title="10.1016/S0004-3702(98)00091-5">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">Bazan, J. G., A comparison of dynamic and non-dynamic rough set methods for extracting laws from decision tables (1997), Institute of Mathematics, University of Rzeszow, Preprint</td> </tr><tr> <td>[2]</td> <td class="space">Bazan, J.; Skowron, A.; Synak, P., Dynamic reducts as a tool for extracting laws from decision tables, (Proceedings Symposium on Methodologies for Intelligent Systems. Proceedings Symposium on Methodologies for Intelligent Systems, Charlotte, NC. Proceedings Symposium on Methodologies for Intelligent Systems. Proceedings Symposium on Methodologies for Intelligent Systems, Charlotte, NC, Lecture Notes in Artificial Intelligence (1994), Springer: Springer Berlin), 346-355</td> </tr><tr> <td>[3]</td> <td class="space">Browne, C., Enhanced rough set data analysis of the Pima Indian diabetes data, (Proceedings 8th Ireland Conference on Artificial Intelligence. Proceedings 8th Ireland Conference on Artificial Intelligence, Derry (1997)), 32-39</td> </tr><tr> <td>[4]</td> <td class="space">Browne, C.; Düntsch, I.; Gediga, G., IRIS revisited: a comparison of discriminant and enhanced rough set data analysis, (Polkowski, L.; Skowron, A., Rough Sets in Knowledge Discovery, Physica, Heidelberg, Vol. 2 (1998)), 345-368</td> </tr><tr> <td>[5]</td> <td class="space">Catlett, J., On changing continuous attributes into ordered discrete attributes, (Kodratoff, Y., Proceedings European Working Session on Learning—EWSL-91 (1991), Springer: Springer Berlin), 164-178</td> </tr><tr> <td>[6]</td> <td class="space">Cohen, J., Things I have learned (so far), American Psychologist, 45, 1304-1312 (1990)</td> </tr><tr> <td>[7]</td> <td class="space">Dougherty, J.; Kohavi, R.; Sahami, M., Supervised and unsupervised discretization of continuous features, (Proceedings 12th International Conference on Machine Learning (1995), Morgan Kaufmann: Morgan Kaufmann San Francisco, CA), 194-202</td> </tr><tr> <td>[8]</td> <td class="space">Düntsch, I., A logic for rough sets, Theoret. Comput. Sci., 179, 427-436 (1997) · <a href="/0896.03050" class="nowrap">Zbl 0896.03050</a></td> </tr><tr> <td>[9]</td> <td class="space">Düntsch, I.; Gediga, G., The rough set engine GROBIAN, (Sydow, A., Proceedings 15th IMACS World Congress, Vol. 4 (1997), Wissenschaft und Technik: Wissenschaft und Technik Berlin), 613-618</td> </tr><tr> <td>[10]</td> <td class="space">Düntsch, I.; Gediga, G., <span class="zbmathjax-textsc">roughian</span>—rough information analysis (extended abstract), (Sydow, A., Proceedings 15th IMACS World Congress, Vol. 4 (1997), Wissenschaft und Technik: Wissenschaft und Technik Berlin), 631-636, Full paper available from</td> </tr><tr> <td>[11]</td> <td class="space">Düntsch, I.; Gediga, G., Statistical evaluation of rough set dependency analysis, International Journal of Human-Computer Studies, 46, 589-604 (1997)</td> </tr><tr> <td>[12]</td> <td class="space">Düntsch, I.; Gediga, G., Simple data filtering in rough set systems, Internat. J. Approx. Reason., 18, 93-106 (1998) · <a href="/0942.68031" class="nowrap">Zbl 0942.68031</a></td> </tr><tr> <td>[13]</td> <td class="space">Jaynes, E. T., Information theory and statistical mechanics, Physical Review, 106, 620-630 (1957) · <a href="/0084.43701" class="nowrap">Zbl 0084.43701</a></td> </tr><tr> <td>[14]</td> <td class="space">Jumarie, G., Relative Information (1990), Springer: Springer Berlin · <a href="/0721.93004" class="nowrap">Zbl 0721.93004</a></td> </tr><tr> <td>[15]</td> <td class="space">Konrad, E.; Orłowska, E.; Pawlak, Z., Knowledge representation systems—definability of informations, (ICS Research Report 433 (1981), Polish Academy of Sciences)</td> </tr><tr> <td>[16]</td> <td class="space">Konrad, E.; Orłowska, E.; Pawlak, Z., On approximate concept learning, (Tech. Report 81-7 (1981), Technische Universität Berlin)</td> </tr><tr> <td>[17]</td> <td class="space">Li, M.; Vitányi, P., An Introduction to Kolmogorov Complexity and Its Applications, (Texts and Monographs in Computer Science (1993), Springer: Springer Berlin) · <a href="/1185.68369" class="nowrap">Zbl 1185.68369</a></td> </tr><tr> <td>[18]</td> <td class="space">Nguyen, H. S.; Nguyen, S. H., Discretization methods in data mining, (Polkowski, L.; Skowron, A., Rough Sets; in Knowledge Discovery, Physica, Heidelberg (1998)) · <a href="/0940.68139" class="nowrap">Zbl 0940.68139</a></td> </tr><tr> <td>[19]</td> <td class="space">Pagliani, P., Rough sets theory and logic-algebraic structures, (Orłowska, E., Incomplete Information—Rough Set Analysis, Physica, Heidelberg (1997)), 109-190</td> </tr><tr> <td>[20]</td> <td class="space">Pawlak, Z., Mathematical foundations of information retrieval, (ICS Research Report 101 (1973), Polish Academy of Sciences) · <a href="/0325.68054" class="nowrap">Zbl 0325.68054</a></td> </tr><tr> <td>[21]</td> <td class="space">Pawlak, Z., Rough sets, Internat. J. Comput. Inform. Sci., 11, 341-356 (1982) · <a href="/0501.68053" class="nowrap">Zbl 0501.68053</a></td> </tr><tr> <td>[22]</td> <td class="space">Pawlak, Z., Rough sets: theoretical aspects of reasoning about data, System Theory, (Knowledge Engineering and Problem Solving, Vol. 9 (1991), Kluwer: Kluwer Dordrecht) · <a href="/0758.68054" class="nowrap">Zbl 0758.68054</a></td> </tr><tr> <td>[23]</td> <td class="space">Pawlak, Z.; Grzymała-Busse, J. W.; Słowiński, R.; Ziarko, W., Rough sets, Comm. ACM, 38, 89-95 (1995)</td> </tr><tr> <td>[24]</td> <td class="space">Pawlak, Z.; Słowiński, R., Rough set approach to multi-attribute decision analysis, (ICS Research Report 36 (1993), Warsaw University of Technology) · <a href="/0805.90069" class="nowrap">Zbl 0805.90069</a></td> </tr><tr> <td>[25]</td> <td class="space">(Polkowski, L.; Skowron, A., Rough Sets in Knowledge Discovery, Physica, Heidelberg (1998)), (to appear) · <a href="/0910.00028" class="nowrap">Zbl 0910.00028</a></td> </tr><tr> <td>[26]</td> <td class="space">Quinlan, R., Improved use of continuous attributes in C4.5, Journal of Artificial Intelligence Research, 4, 77-90 (1996) · <a href="/0900.68112" class="nowrap">Zbl 0900.68112</a></td> </tr><tr> <td>[27]</td> <td class="space">Rissanen, J., Modeling by the shortest data description, Automatica, 14, 465-471 (1978) · <a href="/0418.93079" class="nowrap">Zbl 0418.93079</a></td> </tr><tr> <td>[28]</td> <td class="space">Rissanen, J., Minimum-description-length principle, (Kotz, S.; Johnson, N. L., Encyclopedia of Statistical Sciences (1985), Wiley: Wiley New York), 523-527</td> </tr><tr> <td>[29]</td> <td class="space">Teghem, J.; Benjelloun, M., Some experiments to compare rough sets theory and ordinal statistical methods, (Słowiński, R., Intelligent Decision Support: Handbook of Applications and Advances of Rough Set Theory, System Theory, Knowledge Engineering and Problem Solving, Vol. 11 (1992), Kluwer: Kluwer Dordrecht), 267-284 · <a href="/0820.68001" class="nowrap">Zbl 0820.68001</a></td> </tr><tr> <td>[30]</td> <td class="space">Wong, S. K.M.; Ziarko, W.; Ye, R. L., Comparison of rough-set and statistical methods in inductive learning, Internat. J. Man-Mach. Stud., 24, 53-72 (1986) · <a href="/0634.68088" class="nowrap">Zbl 0634.68088</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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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