A fast algorithm for the minimum covariance determinant estimator
A fast algorithm for the minimum covariance determinant estimator
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DOI：
10.2307/1270566
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发表时间：
1999-08-01
期刊：
影响因子：
2.5
通讯作者：
Van Driessen, K
中科院分区：
文献类型：
--
作者：
Rousseeuw, PJ;Van Driessen, K
最小协方差行列式（MCD）方法是一种高度稳健的多元位置和散布估计。它的目标是找到h个观测值（从n个中），其协方差矩阵具有最低的行列式。到目前为止，MCD的应用受到现有算法的计算时间的阻碍，这些算法仅限于几个维度中的几百个对象。我们讨论了两个重要的应用较大的尺寸，一个是关于生产过程在飞利浦与n = 677对象和p = 9变量，和一个数据集从天文学与n = 137，256对象和p = 27变量。为了处理这样的问题，我们已经开发了一种新的算法的MCD，称为快速MCD。其基本思想是一个涉及顺序统计量和行列式的不等式，以及我们称之为“选择迭代”和“嵌套扩展”的技术。“对于小数据集，FAST-MCD通常会找到精确的MCD，而对于较大的数据集，它会给出比现有算法更准确的结果，并且速度要快几个数量级。此外，FAST-MCD能够检测到一个精确的拟合，也就是说，一个超平面包含h或更多的观察。新算法使MCD方法成为分析多元数据的常规工具。我们还提出了距离-距离图（D-D图），它显示了基于MCD的鲁棒距离与马氏距离，并用一些例子来说明它。
The minimum covariance determinant (MCD) method of Rousseeuw is a highly robust estimator of multivariate location and scatter. Its objective is to find h observations (out of n) whose covariance matrix has the lowest determinant. Until now, applications of the MCD were hampered by the computation time of existing algorithms, which were limited to a few hundred objects in a few dimensions. We discuss two important applications of larger size, one about a production process at Philips with n = 677 objects and p = 9 variables, and a dataset from astronomy with n = 137,256 objects and p = 27 variables. To deal with such problems we have developed a new algorithm for the MCD, called FAST-MCD. The basic ideas are an inequality involving order statistics and determinants, and techniques which we call "selective iteration" and "nested extensions." For small datasets, FAST-MCD typically finds the exact MCD, whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders of magnitude. Moreover, FAST-MCD is able to detect an exact fit-that is, a hyperplane containing h or more observations. The new algorithm makes the MCD method available as a routine tool for analyzing multivariate data. We also propose the distance-distance plot (D-D plot), which displays MCD-based robust distances versus Mahalanobis distances, and illustrate it with some examples.