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    Title: 累積切片估計的非線性維度縮減法於非線性流形學習之研究
    Other Titles: Nonlinear Dimension Reduction through Cumulative Slicing Estimation for Nonlinear Manifolds Learning
    Authors: 吳漢銘
    Contributors: 淡江大學數學學系
    Keywords: 累積切片估計法;等軸距特徵映射;非線性維度縮減;非線性流形;秩二橢圓排序;切片逆迴歸法;Cumulative slicing estimation;isometric feature mapping;nonlinear dimension reduction;nonlinear manifold;rank-two ellipse seriation;sliced inverse regression
    Date: 2013-08
    Issue Date: 2015-04-21 14:14:28 (UTC+8)
    Abstract: 運用切片逆迴歸法可以找出有效的維度縮減方向來探索高維度資料的内在 結構。等軸距切片逆迴歸法是傳統切片逆迴歸法的非線性推展。它利用K-均値法在事先計算的資料等軸距距離矩陣上,使得傳統切片逆迴歸法可直接 應用。研究已証明等軸距切片逆迴歸法可以找到非線性流形資料(例如瑞士 捲資料)内隱的維度和低維度幾何結構。然而,使用K-均値法,等軸距切片 逆迴歸法乎略了切片内及切片間,反應變數觀察値的順序資訊。而此順序資 訊是非線性流形資料其中一項很重要的特徵。在這一個計畫中,我們採用累 積切片估計法來解決此問題。所提的方法中,首先先計算兩兩資料點等軸距 距離矩陣,然後以秩二橢圓排序法排序這個距離矩陣,使得傳統的累積切片 估計法可以被應用。我們針對瑞士捲資料做了一個初探性研究,結果顯示所 提的方法可在低維度空間中揭示資料的幾何結構,而且可和等軸距切片逆迴 歸法的結果相媲美。我們將討論所提方法的統計性質,以及當反應變數無法 觀察到時,如何估計它以用來表示資料的排序結構。同時我們將研究非線性 降維後的資料在分類、分群及迴歸上的應用。説明的例子會有一般的實際資 料及微陣列基因表現資料。所提的方法也會和其它現存的幾個非線性維度縮 減方法相比較。
    Sliced inverse regression (SIR) was developed to find the effective dimension reduction directions for exploring the intrinsic structure of high-dimensional data. The isometric SIR (ISOSIR), a nonlinear extension of SIR, employed K-means on the pre-calculated isometric distance matrix of the data set so that the classical SIR algorithm can be applied. It has been shown that ISOSIR can recover the embedded dimensionality and the geometric structure of the nonlinear manifolds data sets such as the Swiss roll. However, by using K-means, ISOSIR ignored the ordering information of response ys both within and between the slices where the ordering information was one of the most important characteristics of the nonlinear manifold data sets. In this proposal, we are motivated to settle this problem by using the cumulative slicing estimation. First, the proposed method computes the isometric distance between data points; the resulting distance matrix is then sorted by the rank-two ellipse seriation method, and the classical cumulative slicing estimation algorithm is applied. We conducted a pilot study and shown that the proposed method can reveal the geometric structure of a nonlinear manifold data set such as the Swiss roll and the results were comparable to ISOSIR. We will then discuss the statistical properties of the proposed method and address how to obtain the response as an estimation of the ordering structure of the data when it was not available. We will also investigate the further applications of the found features for the classification, clustering and regression problems to the real world data and microarray gene expression data. The comparisons with those obtained with several existing nonlinear dimension reduction techniques will be also examined.
    Appears in Collections:[數學學系暨研究所] 研究報告

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