Automatic Methods for Handling Nearly Singular Covariance Structures Using the Cholesky Decomposition of an Indefinite Matrix

TL;DR

This paper introduces a generalized Cholesky decomposition method for handling nearly singular covariance matrices in linear models, enabling improved statistical analysis in complex scenarios like Higgs boson decay detection.

Contribution

It presents a novel approach to decompose indefinite matrices using complex arithmetic, extending the applicability of ReML in singular covariance structures.

Findings

Effective suppression of background signals in Higgs decay analysis.

Method is robust against multiple interactions (pile up).

Improved vertex size determination for Higgs boson decay.

Abstract

Linear models have found widespread use in statistical investigations. For every linear model there exists a matrix representation for which the ReML (Restricted Maximum Likelihood) can be constructed from the elements of the corresponding matrix. This method works in the standard manner when the covariance structure is non-singular. It can also be used in the case where the covariance structure is singular, because the method identifies particular non-stochastic linear combinations of the observations which must be constrained to zero. In order to use this method, the Cholesky decomposition has to be generalized to symmetric and indefinite matrices using complex arithmetic methods. This method is applied to the problem of determining the spatial size (vertex) for the Higgs Boson decay in the Higgs -> 4 lepton channel. A comparison based on the Chi-Square variable from the vertex fit for Higgs signal and t-tbar background is presented and shows that the background can be greatly suppressed using the Chi-Square variable. One of the major advantages of this novel method over the currently adopted technique of b-tagging is that it is not affected by multiple interactions (pile up).