{"created":"2023-06-20T13:20:44.443873+00:00","id":776,"links":{},"metadata":{"_buckets":{"deposit":"e31a3eef-e0a1-411d-bf68-98d750da9c92"},"_deposit":{"created_by":1,"id":"776","owners":[1],"pid":{"revision_id":0,"type":"depid","value":"776"},"status":"published"},"_oai":{"id":"oai:ir.soken.ac.jp:00000776","sets":["2:429:17"]},"author_link":["0","0","0"],"item_1_creator_2":{"attribute_name":"著者名","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"謝, 剛強"}],"nameIdentifiers":[{"nameIdentifier":"0","nameIdentifierScheme":"WEKO"}]}]},"item_1_creator_3":{"attribute_name":"フリガナ","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"チァ, カンキャン"}],"nameIdentifiers":[{"nameIdentifier":"0","nameIdentifierScheme":"WEKO"}]}]},"item_1_date_granted_11":{"attribute_name":"学位授与年月日","attribute_value_mlt":[{"subitem_dategranted":"2006-09-29"}]},"item_1_degree_grantor_5":{"attribute_name":"学位授与機関","attribute_value_mlt":[{"subitem_degreegrantor":[{"subitem_degreegrantor_name":"総合研究大学院大学"}]}]},"item_1_degree_name_6":{"attribute_name":"学位名","attribute_value_mlt":[{"subitem_degreename":"博士（統計科学）"}]},"item_1_description_12":{"attribute_name":"要旨","attribute_value_mlt":[{"subitem_description":"　　In data analysis and engineering applications, one often comes across unknown<br />densities which are complex and multimodal. In such situations, it is natural and<br />intuitive to break up the original density into a mixture of simpler, structurally less<br />complex densities, so as to facilitate analysis and modelling. In this thesis, we demonstrate<br />that it is possible to decompose a multimodal density into simpler densities via<br />the novel concept of M-decomposability. The letter M derives from “multimodal”<br />or “mixture”.<br />    <br />　　For clarity of presentation, this thesis is divided into two parts. Part one consists<br />of Chapters 1 to 4, and solely considers densities in one-dimension. In Chapter 2,<br />we introduce the notion of M-decomposability in one-dimension. We say that a density<br />f is M-decomposable if it is possible to rewrite f as a mixture of two densities<br />g and h such that the sum of the standard deviations of g and h is less than the<br />standard deviation of f. If f does not satisfy the above condition, we say that f<br />is M-undecomposable. To clarify matters, we then provide examples to illustrate<br />the concept of M-decomposability. We also derive a theorem that states that “All<br />uniform densities in one-dimension areM-undecomposable” (Theorem 2.1). In Chapter<br />3, we demonstrate that unimodal densities in one-dimension can be approximated<br />to an arbitrary level of accuracy using a specially constructed mixture of uniform<br />densities. In Chapter 4, we make use of Theorem 2.1 and the representation in Chapter<br />3 to derive a theorem which states that “All symmetric unimodal densities in<br />one-dimension are M-undecomposable” (Theorem 4.1).<br /><br />　　The second part of the thesis builds up on the results derived in the first and<br />extends to d-dimensions. To avoid confusion of notation, we provide a fresh set of<br />notations in Chapter 5 and a list of theorems and definitions to apply to the second<br />part of the paper. In Chapters 6 and 7, we provide the theoretical aspects<br />of M-decomposability in d-dimensions. In Chapter 6, we define the uniform density<br />in d-dimensions to be the elliptical uniform. To extend the definition of M-undecomposability<br />to apply d-dimensions, the “standard deviation” that appears<br />in the first part is replaced by the “square-root of the determinant of covariance”<br />of the underlying density. This step is crucial to the future development of M-decomposability<br />in d-dimensional. We derive a theorem that says that “All elliptical<br />uniform densities in d-dimension are M-undecomposable”. In Chapter 7, we extend<br />Theorem 4.1 derived in Chapter 4 to d-dimensions, i.e., “All elliptical unimodal<br />densities in d-dimension are M-undecomposable” (Theorem 7.2).<br />     <br />　　In Chapter 8, we derive a theorem which links M-decomposability with Kullback-<br />Leibler divergence. This provides justification of using M-decomposability in a number<br />of statistical applications, namely clustering and density estimation. Simulation<br />examples of both clustering and density estimation are provided in the chapter. On<br />top of that, we also demonstrate the application of M-decomposability to real data<br />cluster analysis, using the Iris dataset as test data. The results not only show that <br />M-decomposability can be used to improve cluster analysis and density estimation, but<br />also suggest that M-decomposability is a viable criterion for cluster discrimination.<br />Concluding remarks are given in Chapter 9.","subitem_description_type":"Other"}]},"item_1_description_7":{"attribute_name":"学位記番号","attribute_value_mlt":[{"subitem_description":"総研大甲第993号","subitem_description_type":"Other"}]},"item_1_select_14":{"attribute_name":"所蔵","attribute_value_mlt":[{"subitem_select_item":"有"}]},"item_1_select_8":{"attribute_name":"研究科","attribute_value_mlt":[{"subitem_select_item":"複合科学研究科"}]},"item_1_select_9":{"attribute_name":"専攻","attribute_value_mlt":[{"subitem_select_item":"15 統計科学専攻"}]},"item_1_text_10":{"attribute_name":"学位授与年度","attribute_value_mlt":[{"subitem_text_value":"2006"}]},"item_creator":{"attribute_name":"著者","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"CHIA, Kang-Kiang Nicholas","creatorNameLang":"en"}],"nameIdentifiers":[{"nameIdentifier":"0","nameIdentifierScheme":"WEKO"}]}]},"item_files":{"attribute_name":"ファイル情報","attribute_type":"file","attribute_value_mlt":[{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2016-02-17"}],"displaytype":"simple","filename":"甲993_要旨.pdf","filesize":[{"value":"221.6 kB"}],"format":"application/pdf","licensetype":"license_11","mimetype":"application/pdf","url":{"label":"要旨・審査要旨","url":"https://ir.soken.ac.jp/record/776/files/甲993_要旨.pdf"},"version_id":"7ad5ad6c-79e0-4201-8e49-2b4a282d286d"},{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2016-02-17"}],"displaytype":"simple","filename":"甲993_本文.pdf","filesize":[{"value":"3.6 MB"}],"format":"application/pdf","licensetype":"license_11","mimetype":"application/pdf","url":{"label":"本文","url":"https://ir.soken.ac.jp/record/776/files/甲993_本文.pdf"},"version_id":"dd47886a-66f0-4b44-b650-d5632247cbdb"}]},"item_language":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"eng"}]},"item_resource_type":{"attribute_name":"資源タイプ","attribute_value_mlt":[{"resourcetype":"thesis","resourceuri":"http://purl.org/coar/resource_type/c_46ec"}]},"item_title":"M-Decomposability and Elliptical Unimodal Densities","item_titles":{"attribute_name":"タイトル","attribute_value_mlt":[{"subitem_title":"M-Decomposability and Elliptical Unimodal Densities"},{"subitem_title":"M-Decomposability and Elliptical Unimodal Densities","subitem_title_language":"en"}]},"item_type_id":"1","owner":"1","path":["17"],"pubdate":{"attribute_name":"公開日","attribute_value":"2010-02-22"},"publish_date":"2010-02-22","publish_status":"0","recid":"776","relation_version_is_last":true,"title":["M-Decomposability and Elliptical Unimodal Densities"],"weko_creator_id":"1","weko_shared_id":1},"updated":"2023-06-20T15:59:52.386121+00:00"}