切片逆迴歸法於符號型資料分析之研究

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Title:	切片逆迴歸法於符號型資料分析之研究
Other Titles:	Sliced Inverse Regression for Symbolic Data Analysis
Authors:	吳漢銘
Contributors:	淡江大學數學學系
Keywords:	資料視覺化;逆迴歸法;充份維度縮減法;符號型資料分析法;符號型主成份分析法;Data visualization;Inverse regression;Sufficient dimension reduction;Symbolic data analysis;Symbolic principal component analysis
Date:	2012-08
Issue Date:	2015-05-12 15:33:10 (UTC+8)
Abstract:	Li (1991)所提出的切片逆迴歸法(SIR)，目的在找出有效的維度縮減方向來探索高維度資料的內在結構。針對單一反應變數迴歸問題，SIR 已發展並應用在不同類型的資料型態上，例如: 存活資料(Li, Wnag and Chen, 1999; Riggs, 2010)、時間序列資料(Becker and Fried, 2003)、函數型資料(Ferre and Yao, 2003)及縱向資料(Li and Yin, 2009)等等。本計畫中，我們將推展SIR 到符號型資料分析(symbolic data analysis, SDA)。符號型資料可略分為以下三型: (1) 區間資料(interval-valued); (2)多值資料(multi-valued)及(3)模態資料(modal-valued)。本計畫中，我們首先利用頂點法(vertices method)或中心法(centers method)將區間資料做轉換，再應用傳統SIR 的演算法在轉換後的資料上。實際資料分析初步結果顯示，不同的切片策略會產生不同的維度縮減方向及呈現不同的低維度視覺化結果。因此找出合適的切片策略有助於正確地分析這類型高維度資料所隱含的結構與資訊，這結果促使我們採用以群集為基礎 (cluster-based)的切片逆迴歸法(Kuentz and Saracco, 2010)來進一步修正及實現symbolic SIR。除此，我們也將探討傳統SIR 的假設及理論(例如: linearity condition)對於符號型資料是否成立或需做適當的修正。同時。我們將比較symbolic SIR 和其它現存的符號型維度縮減方法(例如: 符號型主成份分析法)，評估各方法在區別能力、低維度視覺化和迴歸問題中的表現。我們也會根據在區間資料的研究結果，將symbolic SIR 再進一步的推展到多值型及模態型的符號型資料。 Sliced inverse regression (SIR) was introduced by Li (1991) to find the effective dimension reduction directions for exploring the intrinsic structure of high-dimensional data. For univariate response regression, SIR has been extended and applied to different data types. Examples were the cases of the survival data (Li, Wnag and Chen, 1999; Riggs, 2010), the time series data (Becker and Fried, 2003), the functional data (Ferre and Yao, 2003) and the longitudinal data (Li and Yin, 2009). This proposal intends to develop SIR for symbolic data analysis (SDA). The symbolic data can be roughly divided into the following three categories: (1) the interval-valued data; (2) the multi-valued data and (3) the modal-valued data. In this study, we firstly transform the interval-valued data using the vertices method or the centers method and then the classical SIR algorithm can be directly applied to the transformed data. The initial results of the real data analysis shown that using different slicing schemes produced different projection directions and different lower-dimensional visualization results. A suitable slicing scheme is then needed for correctly investigating the embedded structure and information of the high-dimensional symbolic data in the lower-dimensional plots. This result motivated us to adopt the clustered-based SIR (Kuentz and Saracco, 2010) to improve the implementation of the symbolic SIR. Besides, we will discuss the assumption and the theory (e.g., linearity conditions) of SIR to see whether the modification is needed when the symbolic data were investigated. Moreover, we will compare and evaluate the results with those obtained with several existing symbolic dimension reduction techniques (such as the symbolic principal component analysis) for discriminative, visualization, and regression purposes. Based on these finding, we will further extend the proposed method to the other types of symbolic data such as the multi-valued and the modal-valued data sets.
Appears in Collections:	[Graduate Institute & Department of Mathematics] Research Paper

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