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Maximum Likelihood Estimation in the Tensor Normal Model with a Structured Mean
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. University of Rwanda, PO.Box 3900 Kigali, Rwanda.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. bDepartment of Energy and Technology, Swedish University of Agricultural Sciences, SE–750 07 Uppsala, Sweden..
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.ORCID iD: 0000-0001-9896-4438
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 [en]
growth curve model, Kronecker product structure, maximum likelihood estimators, multi-way data, tensor normal model, trilinear regression
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-117512ISRN: LiTH-MAT-R--2015/08--SEOAI: oai:DiVA.org:liu-117512DiVA: diva2:808970
Available from: 2015-04-30 Created: 2015-04-30 Last updated: 2015-05-21
In thesis
1. Bilinear and Trilinear Regression Models with Structured Covariance Matrices
Open this publication in new window or tab >>Bilinear and Trilinear Regression Models with Structured Covariance Matrices
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:nbn:se:liu:diva-118089 (URN)10.3384/diss.diva-118089 (DOI)978-91-7519-070-9 (ISBN)
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

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Maximum Likelihood Estimation in the Tensor Normal Model with a Structured Mean(1045 kB)421 downloads
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Nzabanita, Josephvon Rosen, DietrichSingull, Martin

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