Objects and Methods for Multi-Companion Matrices

Provides a class for multi-companion matrices with methods for arithmetic and factorization. A method for generation of multi-companion matrices with prespecified spectral properties is provided, as well as some utilities for periodically correlated and multivariate time series models. See Boshnakov (2002) <doi:10.1016/S0024-3795(01)00475-X> and Boshnakov & Iqelan (2009) <doi:10.1111/j.1467-9892.2009.00617.x>.

Details

Index of the main exported objects, classes and methods:

Classes and generators

MultiCompanion-class    Class "MultiCompanion"
MultiFilter-class       Class "MultiFilter"
VAR2pcfilter            PAR representations of VAR models
mCompanion              Create objects from class MultiCompanion
mcSpec                  Generate objects of class mcSpec
mcSpec-class            A class for spectral specifications of
                        multi-companion matrices
mf_VSform               Extract properties of multi-filters

Utilities for multi-companion matrices

mc_eigen                The eigen decomposition of a multi-companion
                        matrix
mc_factorize            Factorise multi-companion matrices
mc_factors              Factors of multi-companion matrices
mc_from_factors         Multi-companion matrix from factors

Simulation

sim_mc                  Simulate a multi-companion matrix
sim_pcfilter            Generate periodic filters

Generic matrix utilities

Jordan_matrix           Utilities for Jordan matrices
mcStable                Check if an object is stable
rblockmult              Right-multiply a matrix by a block

Spectral description of mc-matrices

spec_core               Parameterise Jordan chains of multi-companion
                        matrices
spec_root0              Give the spectral parameters for zero
                        eigenvalues of mc-matrices
spec_root1              Give the spectral parameters for eigenvalues of
                        mc-matrices equal to one
spec_seeds1             Generate seed parameters for unit
                        mc-eigenvectors

Low-level functions

mc_chain_extend         Extend multi-companion eigenvectors

Overview of the package

Package "mcompanion" implements multi-companion matrices as discussed by Boshnakov (2002) and Boshnakov and Iqelan (2009). The main feature is the provided parsimonious parameterisation of such matrices based on their eigenvalues and the seeds for their eigenvectors. This can be used for specification and parameterisation of models for time series and dynamical systems in terms of spectral characteristics, such as the poles of the associated filters or transition matrices.

A multi-companion matrix of order k is a square \(n\times n\) matrix with arbitrary k rows put on top of an identity \((n-k)\times(n-k)\) matrix and a zero \((n-k)\times k\) matrix. The number \(k\) is the multi-companion order of the matrix. It may happen that the top \(k \times n\) block, say T, of an mc-matrix has columns of zeroes at its end. In this documentation we say that an \(n\times n\) matrix has dimension \(n\) and size \(n\times n\).

Multi-companion matrices can be created by the functions new and mCompanion, the latter being more versatile. Some of the other functions in this package return such objects, as well.

sim_mc generates a multi-companion matrix with partially or fully specified spectral properties. If the specification is incomplete, it completes it with simulated values.

sim_pcfilter is a convenience function (it uses sim_mc) for generation of filters for periodically correlated models. These can be converted to various multivariate models, such as VAR, most conveniently using class MultiFilter, see below.

Class "MultiFilter" is a formal representation of periodic filters with methods for conversion between periodic and (non-periodic) multivariate filters. Several forms of VAR models are provided, see mf_VSform, VAR2pcfilter, MultiFilter, and the examples there.

Author

Georgi N. Boshnakov [aut, cre]

Maintainer: Georgi N. Boshnakov <georgi.boshnakov@manchester.ac.uk>

References

Boshnakov GN (2002). “Multi-companion matrices.” Linear Algebra Appl., 354, 53--83. ISSN 0024-3795, doi:10.1016/S0024-3795(01)00475-X .

Boshnakov GN (2007). “Singular value decomposition of multi-companion matrices.” Linear Algebra Appl., 424(2-3), 393--404. ISSN 0024-3795, doi:10.1016/j.laa.2007.02.010 .

Boshnakov GN, Iqelan BM (2009). “Generation of time series models with given spectral properties.” J. Time Series Anal., 30(3), 349--368. ISSN 0143-9782, doi:10.1111/j.1467-9892.2009.00617.x .

Examples

## see the examples in the links in section 'See Also' above.