Geometric Interpretation of Vector Variance

Abstract

Multivariate dispersion is difficult to measure, and thus to manage, because of the complexity of covariance structure. There is no single measure that can properly represent the whole structure. The most popular and widely used measure is the generalized variance. Unfortunately, it has some serious limitations. An alternative measure that features good properties is the vector variance. However, its geometric interpretation in terms of random sample is still vague. This paper is to clarify the geometric meaning of vector variance which will ensure the proper application of this measure in practice. For that purpose we use Escoufier's operator, an operator representation of random vector, to show that sample vector variance is equal to the squared Frobenius norm of that operator in random sample setting. Keywords: Escoufier's operator; Frobenius norm; generalized variance; multivariate dispersion; vector variance 2010 Mathematics Subject Classification 62H10; 62H25; 62H86.