Madhuchhanda Bhattacharjee, Reader in Statistics in the Department of Mathematics, University of Manchester.
Title: Part-A: A brief and selective review of applications of RMT; Part B: Estimation of rectangular random matrices: Applications to high-dimensional biological data
Abstract: Part-A: A brief and selective review of applications of RMT
Random Matrices have been encountered in Statistics, Mathematics, Physics, and
other disciplines for nearly a century now. Literature on the theoretic developments
in this field has grown substantially since then. With that there have been wide range
of applications of such theories. We discuss a few examples involving, the more
popular, square random matrices and a few where the matrix is rectangular.
Part B: Estimation of rectangular random matrices: Applications to high-dimensional biological data
It is commonly thought that for a high-dimensional data one of the key objectives
would be to study it at a lower dimensional level. Contrary to this, there are many
examples in real life where it might be necessary to study it at a latent finer (or further
higher dimensional) level.
Consider such an observed (random) vector, say Y. then the task would be to explain
Y by means of a collection of random vectors, say presented in a rectangular random
matrix X, where X is not observed.
In the two applications we consider here, we have a partially observed (random)
binary matrix, say B, and a completely unobserved (random) “signal” matrix, say S,
such that X can be viewed as a (Hadamard) product of B and S.
Our two examples are from neural coding and genomics respectively, where one of
the dimensions of X is in millions or at least hundreds of thousands. The other
dimension, in both of our examples, is unknown.
It is evident that the problem is ill posed and has many feasible solutions. Our first
objective would be to propose a model that captures the desired higher dimensional
understanding, which we do by using domain knowledge as far as possible. Next is
to obtain estimate of the random components and parameters involved. For which
we use an Estimation-Maximization (EM) type algorithm for one and the Bayesian
hierarchical modelling for the other example
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