{"_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": ["17"]}, "author_link": ["0", "0", "0"], "item_1_biblio_info_21": {"attribute_name": "書誌情報（ソート用）", "attribute_value_mlt": [{"bibliographicIssueDates": {"bibliographicIssueDate": "2006-09-29", "bibliographicIssueDateType": "Issued"}, "bibliographic_titles": [{}]}]}, "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_1": {"attribute_name": "ID", "attribute_value_mlt": [{"subitem_description": "2006513", "subitem_description_type": "Other"}]}, "item_1_description_12": {"attribute_name": "要旨", "attribute_value_mlt": [{"subitem_description": "　　In data analysis and engineering applications, one often comes across unknown\u003cbr /\u003edensities which are complex and multimodal. In such situations, it is natural and\u003cbr /\u003eintuitive to break up the original density into a mixture of simpler, structurally less\u003cbr /\u003ecomplex densities, so as to facilitate analysis and modelling. In this thesis, we demonstrate\u003cbr /\u003ethat it is possible to decompose a multimodal density into simpler densities via\u003cbr /\u003ethe novel concept of M-decomposability. The letter M derives from “multimodal”\u003cbr /\u003eor “mixture”.\u003cbr /\u003e    \u003cbr /\u003e　　For clarity of presentation, this thesis is divided into two parts. Part one consists\u003cbr /\u003eof Chapters 1 to 4, and solely considers densities in one-dimension. In Chapter 2,\u003cbr /\u003ewe introduce the notion of M-decomposability in one-dimension. We say that a density\u003cbr /\u003ef is M-decomposable if it is possible to rewrite f as a mixture of two densities\u003cbr /\u003eg and h such that the sum of the standard deviations of g and h is less than the\u003cbr /\u003estandard deviation of f. If f does not satisfy the above condition, we say that f\u003cbr /\u003eis M-undecomposable. To clarify matters, we then provide examples to illustrate\u003cbr /\u003ethe concept of M-decomposability. We also derive a theorem that states that “All\u003cbr /\u003euniform densities in one-dimension areM-undecomposable” (Theorem 2.1). In Chapter\u003cbr /\u003e3, we demonstrate that unimodal densities in one-dimension can be approximated\u003cbr /\u003eto an arbitrary level of accuracy using a specially constructed mixture of uniform\u003cbr /\u003edensities. In Chapter 4, we make use of Theorem 2.1 and the representation in Chapter\u003cbr /\u003e3 to derive a theorem which states that “All symmetric unimodal densities in\u003cbr /\u003eone-dimension are M-undecomposable” (Theorem 4.1).\u003cbr /\u003e\u003cbr /\u003e　　The second part of the thesis builds up on the results derived in the first and\u003cbr /\u003eextends to d-dimensions. To avoid confusion of notation, we provide a fresh set of\u003cbr /\u003enotations in Chapter 5 and a list of theorems and definitions to apply to the second\u003cbr /\u003epart of the paper. In Chapters 6 and 7, we provide the theoretical aspects\u003cbr /\u003eof M-decomposability in d-dimensions. In Chapter 6, we define the uniform density\u003cbr /\u003ein d-dimensions to be the elliptical uniform. To extend the definition of M-undecomposability\u003cbr /\u003eto apply d-dimensions, the “standard deviation” that appears\u003cbr /\u003ein the first part is replaced by the “square-root of the determinant of covariance”\u003cbr /\u003eof the underlying density. This step is crucial to the future development of M-decomposability\u003cbr /\u003ein d-dimensional. We derive a theorem that says that “All elliptical\u003cbr /\u003euniform densities in d-dimension are M-undecomposable”. In Chapter 7, we extend\u003cbr /\u003eTheorem 4.1 derived in Chapter 4 to d-dimensions, i.e., “All elliptical unimodal\u003cbr /\u003edensities in d-dimension are M-undecomposable” (Theorem 7.2).\u003cbr /\u003e     \u003cbr /\u003e　　In Chapter 8, we derive a theorem which links M-decomposability with Kullback-\u003cbr /\u003eLeibler divergence. This provides justification of using M-decomposability in a number\u003cbr /\u003eof statistical applications, namely clustering and density estimation. Simulation\u003cbr /\u003eexamples of both clustering and density estimation are provided in the chapter. On\u003cbr /\u003etop of that, we also demonstrate the application of M-decomposability to real data\u003cbr /\u003ecluster analysis, using the Iris dataset as test data. The results not only show that \u003cbr /\u003eM-decomposability can be used to improve cluster analysis and density estimation, but\u003cbr /\u003ealso suggest that M-decomposability is a viable criterion for cluster discrimination.\u003cbr /\u003eConcluding 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", "download_preview_message": "", "file_order": 0, "filename": "甲993_要旨.pdf", "filesize": [{"value": "221.6 kB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_11", "mimetype": "application/pdf", "size": 221600.0, "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", "download_preview_message": "", "file_order": 1, "filename": "甲993_本文.pdf", "filesize": [{"value": "3.6 MB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_11", "mimetype": "application/pdf", "size": 3600000.0, "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"], "permalink_uri": "https://ir.soken.ac.jp/records/776", "pubdate": {"attribute_name": "公開日", "attribute_value": "2010-02-22"}, "publish_date": "2010-02-22", "publish_status": "0", "recid": "776", "relation": {}, "relation_version_is_last": true, "title": ["M-Decomposability and Elliptical Unimodal Densities"], "weko_shared_id": 1}