Principal components and orthogonal regression based on robust scales
Principal components and orthogonal regression based on robust scales
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DOI：
10.1198/004017005000000166
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发表时间：
2005-08-01
期刊：
影响因子：
2.5
通讯作者：
Maronna, R
中科院分区：
文献类型：
--
作者：
Maronna, R
主成分分析（PCA）和正交回归都处理寻找p维线性流形，使m维数据点到流形的正交距离的尺度最小化。主要的概念差异是，在主成分分析中，p是从数据中估计的，以获得一小部分无法解释的变异，而在正交回归中，p等于m - 1。鲁棒PCA的两种主要方法是使用鲁棒协方差矩阵的特征向量和搜索最大化或最小化鲁棒（单变量）离散度量的投影。这篇文章更接近于第二种方法。但是，而不是找到一个接一个的组件，我们直接承担的问题，找到一个给定的p，一个p维线性流形最小化的数据点到流形的正交距离的强大规模。比例尺可以是平滑的M比例尺或“修剪”比例尺。一个迭代算法的开发，收敛到局部最小值。基于随机搜索的策略用于近似全局最小值。该程序被证明是快于其他高击穿点的竞争对手，特别是对于大m。p = m-I的情况产生正交回归。对于PCA，给出了一种计算效率高的选择p的方法。基于模拟和真实的数据的比较表明，所提出的方法是更强大的比它的竞争对手。
Both principal components analysis (PCA) and orthogonal regression deal with finding a p-dimensional linear manifold minimizing a scale of the orthogonal distances of the m-dimensional data points to the manifold. The main conceptual difference is that in PCA p is estimated from the data, to attain a small proportion of unexplained variability, whereas in orthogonal regression p equals m - 1. The two main approaches to robust PCA are using the eigenvectors of a robust covariance matrix and searching for the projections that maximize or minimize a robust (univariate) dispersion measure. This article is more akin to second approach. But rather than finding the components one by one, we directly undertake the problem of finding, for a given p, a p-dimensional linear manifold minimizing a robust scale of the orthogonal distances of the data points to the manifold. The scale may be either a smooth M-scale or a "trimmed" scale. An iterative algorithm is developed that is shown to converge to a local minimum. A strategy based on random search is used to approximate a global minimum. The procedure is shown to be faster than other high-breakdown-point competitors, especially for large m. The case whereas p = m - I yields orthogonal regression. For PCA, a computationally efficient method to choose p is given. Comparisons based on both simulated and real data show that the proposed procedure is more robust than its competitors.