Sensitivity analysis and variable selection in statistical models

1. 核多元方法的敏感性分析：提出了核多元主成分分析的敏感性分析方法。基本思想是用有限维欧几里德空间近似核函数定义的变换后的希尔伯特空间，并将SA的一般过程应用于普通多元方法。我们将我们的方法应用于代谢基因网络数据，并提出了发现在生物系统中发挥重要作用的基因的可能性（见Yamanishi， Y. & Tanaka, 2006; Tanaka & Yamanishi, 2006）。同样的思想可以用于开发核典型相关分析（CCA）中的SA。功能数据分析中的SA：我们提出了功能PCA、回归分析（RA）和CCA的敏感性分析方法。在主成分分析中的SA中，我们讨论了我们的方法与通过从功能数据中采样获得的多变量数据的普通主成分分析中的SA的关系（Yamanishi & Tanaka, 2005）。在功能性RA中的SA中，我们发现Cook’s D可以近似地表示为……与普通RA的情况一样，更多地是残差和杠杆的函数（Harasawa, Fueda & Tanaka, 2006）。Cox比例风险模型中的SA：我们导出了与Cox比例风险模型相关的影响函数，并提出了一种SA方法，不仅包括单病例诊断，还包括多病例诊断（Sung & Tanaka, 2004）。通过对一些数值算例的分析，说明了所提方法的有效性。因子分析中的稳健SA：我们进行了模拟研究，并表明联合应用Tanaka & Watadani（1994）提出的稳健SA的一般方法和Atkinson的前向搜索方法提供了观测影响的精确信息（Yang, Tanaka & Nakaya, 2006）。我们提出了一种用于空间预测的协变函数模型的敏感性分析方法，并研究了将其应用于数值示例的有效性（Choi, Beum & Tanaka, 2006）。PCA中的变量选择与其他无外部变量的多元方法：我们对PCA中各种变量选择方法、因子分析和对应分析的性能进行了比较研究。在主成分分析中，我们发现基于Tanaka & Mori（1997）改进的主成分分析的程序比其他方法更能揭示潜在结构（Mori et al., 2005,2007）。少

1. Sensitivity analysis (SA) in kernel multivariate methods : We proposed a method of SA in kernel PCA. Basic idea is to approximate the transformed Hilbert space defined by the kernel function with finite dimensional Euclidean space and apply our general procedure of SA in ordinary multivariate methods. We applied our method to metabolic gene network data and suggested a possibility of finding genes which played important role in the biological system (see, Yamanishi, Y. ＆ Tanaka, 2006; Tanaka ＆ Yamanishi, 2006). The same idea can be used for developing SA in kernel canonical correlation analysis (CCA).2. SA in functional data analysis : We proposed methods of sensitivity analysis in functional PCA, regression analysis (RA) and CCA. In SA in PCA we discussed the relation of our method with SA in ordinary PCA of the multivariate data obtained by sampling from the functional data (Yamanishi ＆ Tanaka, 2005). In SA in functional RA we found that Cook's D can be expressed approximately as … More a function of residuals and leverages as in the case of ordinary RA (Harasawa, Fueda ＆ Tanaka, 2006)3. SA in Cox's proportional hazards model : We derived influence functions related to Cox's proportional hazards models with tied observations and proposed a method of SA including not only single-case but also multiple-case diagnostics (Sung ＆ Tanaka, 2004). We showed usefulness of our method with the analysis of some numerical examples.4. Robust SA in factor analysis : We performed a simulation study and showed that joint application of our general method of robust SA proposed by Tanaka ＆ Watadani (1994) and Atkinson's forward search method provided precise information of the influence of observations (Yang, Tanaka ＆ Nakaya, 2006).5. We proposed a method of sensitivity analysis in covariogram model, which is used in spatial prediction, and studied its usefulness in applying it to a numerical example (Choi, Beum ＆ Tanaka, 2006).6. Variable selection in PCA and other multivariate methods without external variables : We performed comparative studies on the performances of various methods of variable selection in PCA, factor analysis and correspondence analysis. In PCA we found that the procedure based on Tanaka ＆ Mori (1997)'s modified PCA reveals the latent structure better than other methods (Mori et al., 2005, 2007). Less