Biomedical Engineering Reference
In-Depth Information
have point mass at zero, non-signicant random eects can be estimated at
zero variance (i.e., removed from the model).
3. Covariance Selection
As demonstrated in the last section, selection of signicant random eects is
closely related to covariance selection. A modied Cholesky decomposition,
slightly dierent from (2.9), is the main device for handling covariance se-
lection problems. There are a number of works on applications of Cholesky
decomposition to longitudinal data analysis and estimation of large covari-
ance matrices
39;40;35;49;22;28
. In this section, we introduce the covariance
selection problem in general terms. The methods described here can be di-
rectly applied to longitudinal data analysis and parsimonious estimation of
large covariance matrices.
Suppose that e
1
;; e
n
are a m-dimensional random sample from a
population with mean zero and covariance matrix . Using the modied
Cholesky decomposition, we have
LL
T
= ;
where L is a lower triangular matrix having ones on its diagonal, and
= diagf
1
;;
d
)
T
is a diagonal matrix. Note that is symmetric
and positive denite. The modied Cholesky decomposition allows us to
use m parameters in and m(m1)=2 parameters in L to model .
Parsimonious estimation of can be done by imposing sparsity on the el-
ements of L. Smith and Kohn
44
assumed that the random samples follow
a m-dimensional normal distribution, and applied a Bayesian variable se-
lection approach for by specifying prior distribution for D and L and
by allowing l
ij
(i > j), the strictly lower diagonal elements of L having a
positive mass at 0. Following a typical Bayesian variable selection for linear
regression models, they obtained the posterior distribution using MCMC.
Since the prior for l
ij
has positive mass at 0, the Bayesian approach may
yield a sparse model for L.
Let
tj
, 1j < tm, be the (t; j)th of L, Denote u
i
= Le
i
=
(u
i1
;; u
im
)
T
. Thus, for 2tm
t1
X
e
it
=
tj
e
ij
+ u
it
(3.1)
j=1
where (e
i1
;; e
im
) = e
i
. That is, the e
it
, t = 2;; m, is an autoregressive
(AR) series, which gives an interpretation for elements of L and D. Huang
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