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

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

Pacin Al
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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Re: 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.
>
>
>
>
>  
>
> --
> View this message in context: http://r.789695.n4.nabble.com/Ensuring-a-matrix-to-be-positive-definite-case-involving-three-matrices-tp4083376p4083376.html
> Sent from the R help mailing list archive at Nabble.com.
>
> ______________________________________________
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> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

--

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

______________________________________________
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