Effects of observations on the eigensystem of a sample covariance matrix

Subspace Clustering with Active Learning

“…Similarly, the correspondence can be made for the second order coefficients. We use a first order approximation for our purpose from now on, as it has been shown to be sufficiently accurate [18].…”

“…As for the coefficients in the approximations for the eigenvalues and eigenvectors, Lemma 2 in [18] provides us with the following results…”

Further theoretical results and a comparison between two methods for approximating eigenvalues of perturbed covariance matrices

“…The eigenvalues of S play an important role in the statistical analysis of data including estimation and hypotheses testing. It has been recognized that one or few observations can exert an undue influence on the eigenvalues of S. In recent years, 0960-3174 , 9 1993 Chapman & Hall much attention has been given to the problem of relating the eigenvalues of S and those of T S(I ) : X(i)g (1), (1.2) where X(I) is the matrix X without a subset of observations indexed by L See, for example, Radhakrishnan and Kshirsagar (1981), Dorsett et al (1983), Kempthorne (1985), Mason and Gunst (1985), Hadi (1986Hadi ( , 1988, Chatterjee and Hadi (1988), Wells (1990, 1992), Belsley (1991), Wang and Liski (1990) and Wang and Nyquist (1991).…”

“…Wang and Nyquist (1991) give an alternative method for approximating the eigenvalues of S(i), that is, when the subset I contains only one observation i. This method was also extended by Wang and Liski (1990) to the case where I contains m > 1 observations. In this paper we improve on these two methods and give some additional theoretical results that may give further insight into the problem.…”

Eigenstructure-Based Angle for Detecting Outliers in Multivariate Data

“…If th is an outlier, therefore v j will change when ith observation is deleted from the sample data matrix, X. Let θ j(i) be the angle between the jth eigenvectors of S for the given data X and the j(i)th eigenvectors when the ith observation is deleted in X (i.e., X (i) ), then one has the formulae of θ j(i) by Wang and Nyquist (1991) as cos(θ j(i) ) = or it can be re-written as a function of eigenvalues and eigenvectors by θ j(i) = cos are given by Wang and Liski (1993). Supposing that one only deletes ith observation and considers the maximum eigenvalue, replacing j = 1 in (1) leads to (2) Next, one can apply the angle, θ 1(i) to identify the outlier in the data set; note that there are a few criteria that will control θ j(i) value: First, consider λ j ≥ λ j(i) and λ j(i) ≥ λ k+1 where j, k = 1, 2, …, p. One finds that the θ j(i) value is dominated by the first component of the denominator, i.e.…”