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  \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
  \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textsc{DEPARTMENTAL SEMINAR}}
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  4:15 p.m., Tuesday, March 4, 2008\\
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  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Peter Hoff} \\
  Departments of Statistics and Biostatistics \\
  University of Washington
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  \subsection*{Hierarchical eigenmodels for matrix-valued data}
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\noindent

Matrix decomposition models are a popular way to represent % and summarize
matrix-valued data. For example, the variation among the entries of 
an $n\times p$ data matrix $Y$ is often expressed using a
singular value decomposition model $Y\sim U D V^T +E$, where 
  $U$ and $V$ are orthonormal matrices and $D$ is a diagonal matrix. 
In this work I consider pooling information across multiple such 
data matrices $Y^{(1)},\ldots Y^{(K)}$ for situations in which 
common columns  across matrices
represent repeated  measurements under a common 
set of conditions. 
For example, 
suppose individuals from several different groups have expression 
levels measured across a common set of genes. 
If $y_{i,j}^{(k)}$ is the expression level of 
gene $j$ for patient $i$ in group $k$,
 then it might be beneficial 
to share information about the column variation across the different
groups. 
I propose doing this by estimating the parameters in a 
model for the variability among the orthonormal eigenvector matrices  
$V^{(1)},\ldots, V^{(K)}$
of the $K$ data matrices. 
The model is based on a variation of the matrix Bingham  
distribution, for which estimation is accomplished primarily 
with Gibbs sampling. 
The methodology is applied to the analysis of 
several multiple multivariate datasets. 


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