Showing papers on "Covariance mapping published in 1995"

Statistical analysis of observations of increasing dimension

Abstract: List of Basic Notations and Assumptions. Introduction to the English Edition. 1: Introduction to General Statistical Analysis. 2: Limit Theorems for the Empirical Generalized Variance. 3: The Canonical Equations C1,...,C3 for the Empirical Covariance Matrix. 4: Limit Theorems for the Eigenvalues of Empirical Covariance Matrices. 5: G2-Estimator for the Stieltjes Transform of the Normalized Spectral Function of Covariance Matrices. 6: Statistical Estimators for Solutions of Systems of Linear Algebraic Equations. References. Index.

Point reconstruction from noisy images

Abstract: In this paper we treat the problem of determining optimally (in the least-squares sense) the 3D coordinates of a point, given its noisy images formed by any number of cameras of known geometry The optimality criterion is determined by the covariance matrices associated with the images of the point The covariance matrices are not restricted to be positive definite but are allowed to be singular Thus, image points constrained to lie along straight lines can be handled as well Estimation of the covariance of the reconstructed point is provided

Covariance estimation for classifying high dimensional data

Abstract: When classifying data with the Gaussian maximum likelihood classifier, the mean vector and covariance matrix of each class usually are not known and must be estimated from training samples. For p-dimensional data, the sample covariance estimate is singular, and therefore unusable, if fewer than p+1 training samples from each class are available, and it is a poor estimate of the true covariance unless many more than p+1 samples are available. Since inaccurate estimates of the covariance matrix lead to lowered classification accuracy and labeling training samples can be difficult and expensive in remote sensing applications, having too few training samples is a major impediment in using the Gaussian maximum likelihood classifier with high dimensional remote sensing data. In the paper, a new covariance estimator is presented that selects an appropriate mixture of the sample covariance and the common covariance estimates. The mixture deemed appropriate is the one that provides the best fit to the training samples in the sense that it maximizes the average likelihood of training samples not used in the estimates. When the number of training samples is limited or when the covariance matrices of the classes are similar, this estimator tends to select an estimate close to the common covariance, otherwise it favors the sample covariance estimate. Since it is non-singular whenever the common covariance estimate is non-singular, the new estimator can be used even when some of the sample covariance matrices are singular.

Inverting covariance matrices in unbalanced hierarchical models

Abstract: Explicit expression for the inverse of the covariance matrix in a linear experiment with unbalanced multiway hierarchical classification of the random effects is obtained.

Canonical spectral equation for empirical covariance matrices

Abstract: We study asymptotic properties of normalized spectral functions of empirical covariance matrices in the case of a nonnormal population. It is shown that the Stieltjes transforms of such functions satisfy a socalled canonical spectral equation.