[R] Ensuring a matrix to be positive definite, case involving three matrices

[Giovanni Petris]

You may want to look at the following paper.

Best,
Giovanni

Reference Type: Journal Article
Author: Pinheiro, José C.
Author: Bates, Douglas M.
Primary Title: Unconstrained parametrizations for variance-covariance
matrices
Journal Name: Statistics and Computing
Cover Date: 1996-09-01
Publisher: Springer Netherlands
Issn: 0960-3174
Subject: Computer Science
Start Page: 289
End Page: 296
Volume: 6
Issue: 3
Url: http://dx.doi.org/10.1007/BF00140873
Doi: 10.1007/BF00140873


On Fri, 2011-11-18 at 06:07 -0800, Pacin Al wrote:
> Hi,
> 
> I would like to know what should I garantee about P and GGt in order to have
> 
> F = Z %*% P %*% t(Z) + GGt always as a positive definite matrix.
> 
> Being more precise:
> 
> I am trying to find minimum likelihood parameters by using the function
> 'optim' to find the lowest value generated by $LogLik from the function
> 'fkf' (http://127.0.0.1:27262/library/FKF/html/fkf.html).
> 
> The variable Kt within the algorithm used to generate the Kalman Filter
> equations needs in each iteration the inverse of the variable Ft, ("Kt[,, i]
> = Pt[,, i] %*% t(Zt[,, i]) %*% solve(Ft[,, i])") which is updated by "Ft[,,
> i] = Zt[,, i] %*% Pt[,, i] %*% t(Zt[,, i]) + GGt[,, i]".
> 
> Zt is a constant 2x4 matrix and can't be changed. Gt (2x2) and P0 (4x4) are
> inputs for 'fkf'. Pt is updated in each iteration, starting with P0. GGt is
> constant and one of the parameters tested by 'optim' to minimize the LogLik
> (by the way, GGt is always positive definite). Except for the first
> parameters that I give to 'optim', I can't control its tested parameters,
> which will be used as the inputs of 'fkf' (except, as I sad, for Gt, because
> I ask 'optim' to give GLt, the lower triangular matrix of Gt, giving as
> input to 'fkf' GLt %*% t(GLt) ).
> 
> Since the process stops every time a non positive matrix Ft appears, I would
> like to know if are there any transformations that could be applied to GGt
> and P0, given by 'optim', to be sure that Ft will be always positive
> definite.
> 
> 
> 
> 
>  
> 
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-- 

Giovanni Petris  <GPetris at uark.edu>
Associate Professor
Department of Mathematical Sciences
University of Arkansas - Fayetteville, AR 72701
Ph: (479) 575-6324, 575-8630 (fax)
http://definetti.uark.edu/~gpetris/