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Bilinear and Trilinear Regression Models with Structured Covariance Matrices
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis focuses on the problem of estimating parameters in bilinear and trilinear regression models in which random errors are normally distributed. In these models the covariance matrix has a Kronecker product structure and some factor matrices may be linearly structured. The interest of considering various structures for the covariance matrices in different statistical models is partly driven by the idea that altering the covariance structure of a parametric model alters the variances of the model’s estimated mean parameters.

Firstly, the extended growth curve model with a linearly structured covariance matrix is considered. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into appropriate orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

Secondly, we study a bilinear regression model with matrix normally distributed random errors. For those models, the dispersion matrix follows a Kronecker product structure and it can be used, for example, to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimating equations, a flip-flop relation, are established.

At last, the models based on normally distributed random third order tensors are studied. These models are useful in analyzing 3-dimensional data arrays. In some studies the analysis is done using the tensor normal model, where the focus is on the estimation of the variance-covariance matrix which has a Kronecker structure. Little attention is paid to the structure of the mean, however, there is a potential to improve the analysis by assuming a structured mean. We formally introduce a 2-fold growth curve model by assuming a trilinear structure for the mean in the tensor normal model and propose an estimation algorithm for parameters. Also some extensions are discussed.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. , 36 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1665
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:liu:diva-118089DOI: 10.3384/diss.diva-118089ISBN: 978-91-7519-070-9 (print)OAI: oai:DiVA.org:liu-118089DiVA: diva2:813054
Public defence
2015-06-11, BL32, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2015-05-21 Created: 2015-05-21 Last updated: 2015-05-21Bibliographically approved
List of papers
1. Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix
Open this publication in new window or tab >>Estimation of parameters in the extended growth curve model with a linearly structured covariance matrix
2012 (English)In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 16, no 1, 13-32 p.Article in journal (Refereed) Published
Abstract [en]

In this paper the extended growth curve model with two terms and a linearly structured covariance matrix is considered. We propose an estimation procedure that handles linear structured covariance matrices. The idea is first to estimate the covariance matrix when it should be used to define an inner product in a regression space and thereafter reestimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into three orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed and numerical examples are given.

Keyword
Extended growth curve model, estimation, linearly structured covariance matrix, residuals
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-73218 (URN)
Available from: 2011-12-22 Created: 2011-12-22 Last updated: 2017-12-08Bibliographically approved
2. Extended GMANOVA Model with a Linearly Structured Covariance Matrix
Open this publication in new window or tab >>Extended GMANOVA Model with a Linearly Structured Covariance Matrix
2015 (English)Report (Other academic)
Abstract [en]

In this paper we consider the extended generalized multivariate analysis of variance (GMANOVA) with a linearly structured covariance matrix. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into m + 1 orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2015. 17 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2015:07
Keyword
estimation, extended growth curve model, GMANOVA, linearly structured covariance matrix, residuals
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-117508 (URN)LiTH-MAT-R--2015/07--SE (ISRN)
Available from: 2015-04-29 Created: 2015-04-29 Last updated: 2015-05-21Bibliographically approved
3. Bilinear regression model with Kronecker and linear structures for the covariance matrix
Open this publication in new window or tab >>Bilinear regression model with Kronecker and linear structures for the covariance matrix
2015 (English)In: Afrika Statistika, ISSN 2316-090X, Vol. 10, no 2, 827-837 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, the bilinear regression model based on normally distributed random matrix is studied. For these models, the dispersion matrix has the so called Kronecker product structure and they can be used for example to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimating equations in a flip-flop relation are established and the consistency of estimators is studied.

Keyword
Bilinear regression, estimating equations, flip-flop algorithm, Kronecker product structure, linear structured covariance matrix, maximum likelihood estimation
National Category
Mathematics Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-118084 (URN)10.16929/as/2015.827.77 (DOI)
Available from: 2015-05-21 Created: 2015-05-21 Last updated: 2016-04-13Bibliographically approved
4. Maximum Likelihood Estimation in the Tensor Normal Model with a Structured Mean
Open this publication in new window or tab >>Maximum Likelihood Estimation in the Tensor Normal Model with a Structured Mean
2015 (English)Report (Other academic)
Abstract [en]

There is a growing interest in the analysis of multi-way data. In some studies the inference about the dependencies in three-way data is done using the third order tensor normal model, where the focus is on the estimation of the variance-covariance matrix which has a Kronecker product structure. Little attention is paid to the structure of the mean, though, there is a potential to improve the analysis by assuming a structured mean. In this paper, we introduce a 2-fold growth curve model by assuming a trilinear structure for the mean in the tensor normal model and propose an algorithm for estimating parameters. Also, some direct generalizations are presented.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2015. 16 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2015-:08
Keyword
growth curve model, Kronecker product structure, maximum likelihood estimators, multi-way data, tensor normal model, trilinear regression
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-117512 (URN)LiTH-MAT-R--2015/08--SE (ISRN)
Available from: 2015-04-30 Created: 2015-04-30 Last updated: 2015-05-21

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