Abstract
Many statistical problems involve the estimation of a left( dtimes dright) orthogonal matrix varvec{Q}. Such an estimation is often challenging due to the orthonormality constraints on varvec{Q}. To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible left( dtimes dright) matrix as the product of a left( dtimes dright) permutation matrix varvec{P}, a left( dtimes dright) unit lower triangular matrix varvec{L}, and a left( dtimes dright) upper triangular matrix varvec{U}. Thanks to the QR decomposition, we find the formulation of varvec{U} when the PLU decomposition is applied to varvec{Q}. We call the result as PLR decomposition; it produces a one-to-one correspondence between varvec{Q} and the dleft( d-1right) /2 entries below the diagonal of varvec{L}, which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to varvec{L}. For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of varvec{Q} in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.
Highlights
With the term orthogonal matrix we refer to a (d × d) matrix Q whose columns are mutually orthogonal unit vectors
Thanks to the QR decomposition, we find the formulation of U when the PLU decomposition is applied to Q
We apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group
Summary
With the term orthogonal matrix we refer to a (d × d) matrix Q whose columns are mutually orthogonal unit vectors (i.e., orthonormal vectors). A similar problem is bumped into when a (d × d) positive-definite matrix , often encountered in statistics in the form of a covariance matrix, needs to be estimated In this case, the Cholesky decomposition allows to map the d (d + 1) /2 independent parameters of with the d (d + 1) /2 real-valued elements of a (d × d) lower triangular matrix (Pourahmadi 1999, 2000; Pourahmadi et al 2007). 71), under non-normal distributions it may provide an orthogonal matrix which does not maximize the likelihood Motivated by this consideration, we assume groups having a leptokurtic-normal distribution and use the PLR decomposition to allow Q to be estimated by any standard unconstrained maximization routine.
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