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\begin{document}

\title{CMMSE: Joint analysis of Structured orthogonal families}



\author[1]{Cristina Dias}%
\author[2]{Carla Santos}%
\author[3]{Nuno M. Brites}%
\author[1]{João Mexia}%
\affil[1]{Universidade Nova de Lisboa Centro de Matematica e Aplicacoes}%
\affil[2]{Instituto Politecnico de Beja Departamento de Tecnologias e Ciencias Aplicadas}%
\affil[3]{ISEG Lisbon School of Economics and Management}%


\vspace{-1em}



  
  \date{May 17, 2024}


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\begin{abstract}
The models for symmetric stochastic matrices that we consider in this
study are developed using the spectral analysis of the respective mean
matrices. The adjustment and validation of the models require the usage
of the estimated structure vectors. The information enclosed in these
matrices can be condensed into the pair consisting of the estimated
structure vectors and the sum of squares of residuals. The results
obtained allow for cross-sectional and longitudinal inference. For
models of degree greater than one, it is also considered the possibility
of truncating the model when eigenvalues are much higher than the
others. A direct consequence of the adopted methodology is the
application of the degree-one models to cross-product matrices and
Hilbert-Schmidt scalar product matrices. In addition to these models,
structured family models were also considered. The models of these
families are associated with the treatments of a base design. The action
of the factors considered in the base design on the structure vectors is
also analyzed. In structured families with orthogonal base design, the
designs are associated to partitions, and the hypotheses formulated are
associated with the spaces of these partitions. We carry out ANOVA-like
analysis for the action of the factors in the base design, on homolog
components on estimated structure vectors, considering that the
estimator's structure vectors have, approximately, the same covariance
matrix. To apply our results, we assume the factors in the base design
to have fixed effects and that the base design has orthogonal structure.
The action of factors in the base design is studied. An application is
given, using a data set from a breeding program of durum wheat (
\emph{Triticum turgidum L}., Durum Group) conducted in Portugal. The
results show that our methodology is fully applicable to complete and
incomplete data sets, often observed in multi-environmental trials.%
\end{abstract}%



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\textbf{Hosted file}

\verb`CMMSE_Joint_analysis_of_of Structured orthogonal families.docx`  available at \url{https://authorea.com/users/661945/articles/664826-cmmse-joint-analysis-of-structured-orthogonal-families}

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