In many statistical problems, the estimation of a (d × d) orthogonal matrix Q is involved . The orthonormality constraints on Q often makes this estimation difficult. To cope with this problem, we use the well-known PLU decomposition , which factorizes any invertible (d × d) matrix as the product of a (d × d) permutation matrix P, a (d × d) unit lower triangular matrix L, and a (d × d) upper triangular matrix U. Thanks to the QR decomposition , we find the formulation of U when the PLU decomposition is applied to Q. We call the result as PLR decomposition; it produces a one-to-one correspondence between Q and the d (d − 1) /2 entries below the diagonal of 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 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 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.

Bagnato, L., Punzo, A., A New Decomposition of Orthogonal Matrices with Application to Common Principal Components, Abstract de <<17th Conference of the International Federation of Classification Societies>>, (Porto, Portogallo, 19-23 July 2022 ), Printed in Portugal by Instituto Nacional de Estatística, Porto 2022: 1-320 [http://hdl.handle.net/10807/214888]

### A New Decomposition of Orthogonal Matrices with Application to Common Principal Components

#### Abstract

In many statistical problems, the estimation of a (d × d) orthogonal matrix Q is involved . The orthonormality constraints on Q often makes this estimation difficult. To cope with this problem, we use the well-known PLU decomposition , which factorizes any invertible (d × d) matrix as the product of a (d × d) permutation matrix P, a (d × d) unit lower triangular matrix L, and a (d × d) upper triangular matrix U. Thanks to the QR decomposition , we find the formulation of U when the PLU decomposition is applied to Q. We call the result as PLR decomposition; it produces a one-to-one correspondence between Q and the d (d − 1) /2 entries below the diagonal of 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 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 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.
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Classification and Data Science in the Digital Age - Book of Abstract IFCS 2022
17th Conference of the International Federation of Classification Societies
Porto, Portogallo
19-lug-2022
23-lug-2022
978-989-98955-9-1
Printed in Portugal by Instituto Nacional de Estatística
Bagnato, L., Punzo, A., A New Decomposition of Orthogonal Matrices with Application to Common Principal Components, Abstract de <<17th Conference of the International Federation of Classification Societies>>, (Porto, Portogallo, 19-23 July 2022 ), Printed in Portugal by Instituto Nacional de Estatística, Porto 2022: 1-320 [http://hdl.handle.net/10807/214888]
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/10807/214888`
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