Improved method for correcting sample Mahalanobis distance without estimating population eigenvalues or eigenvectors of covariance matrix

Abstract

The recognition performance of the sample Mahalanobis distance (SMD) deteriorates as the number of learning samples decreases. Therefore, it is important to correct the SMD for a population Mahalanobis distance (PMD) such that it becomes equivalent to the case of infinite learning samples. In order to reduce the computation time and cost for this main purpose, this paper presents a correction method that does not require the estimation of the population eigenvalues or eigenvectors of the covariance matrix. In short, this method only requires the sample eigenvalues of the covariance matrix, number of learning samples, and dimensionality to correct the SMD for the PMD. This method involves the summation of the SMD's principal components (each of which is divided by its expectation obtained using the delta method), Lawley's bias estimation, and the variances of the sample eigenvectors. A numerical experiment demonstrates that this method works well for various cases of learning sample number, dimensionality, population eigenvalues sequence, and non-centrality. The application of this method also shows improved performance of estimating a Gaussian mixture model using the expectation-maximization algorithm.

Analysis of COVID-19 in different countries using Wigner energy distribution and clustering

“…This work also illustrates the causal effect of mortality rate in poverty traps and amortization of physical and human capital. Prior work [19] presents a correction method for the sample Mahalanobis distance from its population by using sample of Eigen values of covariance matrix to correct the sample distance and tests this algorithm for Gaussian mixture models using expectation maximization algorithm.…”

“…Lower bound on log likelihood is achieved by calculating free energy which is estimated by using KL-divergence as a limiting parameter. The free energy is given by (19) where q(s) is the cluster assignments. Due to the nonnegativity of KL divergence value Free energy becomes (20) The EM algorithm determines the optimal parameter by iterating with two steps for optimizing and separately to configure optimal energy likelihood value.…”