Robust Multivariate Statistics: Beyond Ellipticity and Affine Equivariance

在过去的30年中，仿射等方差和椭圆对称分布的概念在稳健的多元统计方法的发展中发挥了核心作用。在椭圆分布类上稳健的统计方法比仅基于多元正态分布的方法适用更广泛。然而，椭圆分布类只代表了一小类多变量模型。不属于这一类的重要情况是混合模型和独立成分模型。仿射等方差是一个有用的属性，因为拥有它的方法在不同的协方差结构上表现得同样好。然而，在许多情况下，人们对某些类型的协方差结构感兴趣，例如因子分析模型。因此，研究者的研究目标是双重的。首先，研究者要进一步发展和研究“不变坐标选择”。“这是一种新的多变量方法，最近由研究人员引入，非常适合探索非椭圆模型。特别地，它可以用来发现混合模型的Fisher线性判别子空间时，组标识是未知的，可以用来发现独立分量模型中的独立分量。其次，研究了某些非仿射等变方法的性质，如正交等变M-估计。这里的主要目标是更好地理解协方差结构的类型，这种方法可能是有利的，也可能不是。分析多变量数据的需求出现在许多不同的学科，如计算机科学，心理学，气象学，社会学，生物学，计量经济学和工程。对这些数据的主要兴趣通常不是单独了解每个变量，而是变量之间的相互关系或不可测量的“潜在”变量。在这些领域中采用的许多常用方法都是基于多元正态模型，现在众所周知，如果正态模型不成立，则其表现很差。特别是，数据中的一些错误或模型中的轻微偏差可能会严重影响实验或数据集的解释，有时会带来灾难性的后果。这对于高维数据（即，由许多变量组成的数据）尤其成问题，因为每当坏数据点或与模型的偏差不与仅一个变量相关联而是与多个变量相关联时，它们可能难以检测。因此，多变量的方法，这是不是很大的影响，这些问题是至关重要的，这样的数据进行适当的分析。研究人员预计，预期的研究将产生重要的影响，不仅在强大的统计数据的研究方向，而且在许多学科内使用的方法，经常处理多变量数据。

The concepts of affine equivariance and elliptically symmetric distributions have played a central role in the development of robust multivariate statistical methods over the past 30 years. Statistical methods which are robust over the class of elliptical distributions are more widely applicable than methods based solely on the multivariate normal distribution. The class of elliptical distributions, though, represents only a small class of multivariate models. Important cases which do not fall within this class are mixture models and independent components models. Affine equivariance is a useful property since methods possessing it behave equally well over different covariance structures. However, there are many cases where interest lies in certain type of covariance structures, e.g. factor analysis models. The investigator's research goals are thus two-fold. First, the investigator is to further develop and study "invariant coordinate selection." This is a new multivariate method, recently introduced by the investigator, which is well suited for exploring non-elliptical models. In particular, it can be used to uncover Fisher's linear discriminant subspace for mixture models when the group identifications are unknown, and can be use to uncover the independent components in independent components models. Second, the investigator is to study the properties of certain non-affine equivariant methods, such as orthogonal equivariant M-estimates. The primary goal here is to achieve a better understanding of the type of covariance structures for which such methods may or may not be advantageous.The need to analyze multivariate data arises in many diverse disciplines, such as computer science, psychology, meteorology, sociology, biology, econometrics and engineering. The primary interest in such data typically is not with an understanding of each variable separately, but rather with the interrelationships among the variables or with unmeasurable "latent" variables. Many common methods employed in these areas are based upon the multivariate normal model, which are now well known to perform poorly if the normal model does not hold. In particular, only a few errors in the data or a slight deviation in the model can highly influence the interpretation of an experiment or a data set, sometimes with disastrous consequences. This is particularly problematic with high dimensional data, i.e. data consisting of many variables, since bad data points or deviations from the model can be difficult to detect whenever they are associated not with just one variable but with a number of the variables. Thus, multivariate methods which are not greatly affected by such problems are crucial to a proper analysis of such data. The investigator anticipates that the intended research will have an important impact not only on steering the direction of research within robust statistics, but also on the methodology used within the many disciplines that routinely deal with multivariate data.