## Contents

## CHOLICM

Calculates a Cholesky-factor of the inverse covariance matrix (ICM) of a multivariable autoregressive process.

## Syntax

`C = cholicm(Af,Ab,Sf,Sb,N)`

## Description

The Inverse Covariance Matrix *S* of a multivariable autoregressive noise process according to [1] is calculated. A Cholesky factor *C* is returned such that *C'C* = *S*

The noise model contains a causal and an auticausal part, both of which describe the actual noise *v(k)*. If *e(k)* is a Gaussian white innovation, the model is given by:

Right arrow denote the causal (Forward) components while left arrows denote the anti-causal (Backward) ones.

The multivariable noise model can be obtained using the `destmar` function

## Inputs

`Af` is an *l* x *ld* matrix containing the causal part of the noise process. `Af = [Af1,Af2,...,Afd]`

`Ab` is an *l* x *ld* matrix containing the anti-causal part of the noise process. `Ab = [Ab1,Ab2,...,Abd]`

`Sf` is an *l* x *l* matrix describing the covariance `E[ef ef']`

`Sb` is an *l* x *l* matrix describing the covariance `E[eb eb']`

`N` is the number of samples

## Outputs

C is a Cholesky factor of the ICM. This matrix is stored in LAPACK/BLAS band-storage; its size is *(d* + *1)l* x *N*, and the bottom row contains the diagonal of *C*. The row above contains a zero, and then the first superdiagonal of *C*. The row above contains two zeros, and then the second superdiagonal, etc. The top row contains *((d* + *1)l* - *1)* zeros, and then the *((d* + *1)l* - *1)* th superdiagonal.

## Limitations

A covariance matrix of a stationary process is always positive definite. However, it is very well possible to specify filter coefficients `Af`, `Ab` and covariances `Sf` and `Sb` such that the theoretical ICM calculated per [1] is not positive definite. In such cases, no Cholesky factor can be calculated, and an identity matrix will be returned along with a warning message. The filter should be checked and adjusted in these cases.

## Algorithm

The upper-triangular block-band of a sparse banded inverse covariance matrix according to [1] is filled. A direct sparse Cholesky factorization is subsequently performed using MATLAB's internal `chol` function.

## Used By

## See Also

## References

[1] B. Davis, *Parameter Estimation in Nonlinear Dynamical Systems with Correlated Noise.* PhD thesis, Universite Catholique de Louvain-La-Neuve, Belgium, 2001.