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，使用核機器推展切片逆迴歸法為一非線性維 度縮減法(稱為核化切片逆迴歸法, KSIR)已在Wu (2008) 及Yeh, Huang and Lee (2009)中被提 出與研究。這一個計畫我們將以Wu (2008)為基礎，推展核化切片逆迴歸法到多變量反應變 數迴歸問題，我們稱此一非線性維度縮減方法為多變量核化切片逆迴歸法(mKSIR)。我們將 比較多變量反應變數的幾個不同切片策略來實現mKSIR 並估計預測變數的非線性投影方 向。這些策略包含: 完全切片法、邊際切片法、合併邊際切片法及以群集為基礎的切片法。 除此，我們將在再生核希氏空間(RKHS)的架構中探討mKSIR 的理論根據並評估mKSIR 在區 別能力、低維度視覺化和迴歸問題中的表現。應用方面，我們將利用mKSIR 所找到的特徵 向量在分類問題上並和其它現存的多變量維度縮減方法的結果相比較。同時，我們也會討論 mKSIR 和其它多變量統計方法，例如核化典型相關分析(KCCA)之間的關連性。 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. The nonlinear extension of SIR using a kernel machine for univariate response regression (KSIR) has been explored in Wu (2008) and Yeh, Huang and Lee (2009). This proposal is based on the work by Wu (2008). We plan to extend KSIR for multivariate response regression which we call multivariate KSIR (mKSIR). Several realizations of mKSIR using different slicing schemes for estimation of nonlinear projection directions of the predictors will be proposed and compared. These schemes include the complete slicing, the marginal slicing, the pooled marginal slicing, and the clustering-based slicing. We will provide a theoretical description of the mKSIR algorithm within the framework of reproducing kernel Hilbert space (RKHS). We will also evaluate the performance of mKSIR for discriminative, visualization, and regression purposes. We will apply mKSIR for classification problems and compare the results with those obtained with several existing dimension reduction techniques for multivariate response. The connections between mKSIR and other multivariate statistical techniques such as the kernel canonical correlation analysis (KCCA) will be also discussed.
