Regularized multivariate analysis

统计理论到目前为止主要是由渐近理论。然而，我们经常需要处理小样本数据。在这种情况下，最大似然估计（MLE），证明最佳的大样本，可能不是最佳的。我建议在这种情况下使用正则化估计。正则化是指在数据分析中利用先验知识的任何过程，包括所有这些过程，这些过程被称为惩罚，平滑，收缩，软约束等。更具体地说，我建议在冗余分析中加入岭型以及其他类型的收缩估计（降秩回归）、多组典型相关分析以及多变量分析（MVA）的其他技术。已知这种类型的估计提供与较小的均方误差（真实参数值与估计值之间的平均差异）相关联的参数估计，特别是在小样本中。我们证明了正则化（收缩估计）在这些技术中的有效性，使用真实的数据和模拟方法。我们计划编写一系列用户友好的计算机程序，并将其公开，使正则化MVA成为多元数据分析的常规部分。

The theory of statistics has so far largely been dominated by asymptotic theories. However, we often have to deal with small sample sized data. In such situations, the maximum likelihood estimation (MLE), proven optimal for large samples, may not be optimal. I propose to use regularized estimation in such situations. The regularization means any processes that utilize prior knowledge in data analysis, and includes all such processes that are veriously called penalizing, smoothing, shrinking, soft constraining, etc. More specifically, I propose to incorporate a ridge type as well as other types of shrinkage estimation in redundancy analysis (reduced rank regression), multiple-set canonical correlation analysis as well as other techniques of multivariate analysis (MVA). This type of estimation is known to provide parameter estimates associated with smaller mean square errors (average discrepancies between true parameter values and the estimates) particularly in small samples. We demonstrate the effectiveness of regularization (shrinkage estimation) in these techniques, using both real data and simulation methods. We plan to write a series of user friendly computer programs and make them publicly available to make regularized MVA a routine part of multivariate data analysis.