00pcme-package {pcme}R Documentation

Maximum entropy for periodically correlated processes.

Description

This is an implementation of the method of Lambert-Lacroix and Boshnakov for maximum entropy completion of partially specified autocovariance functions of periodically correlated processes.

Details

Package: pcme
Type: Package
Version: 0.3
Date: 2009-03-18
License: GPL (>= 2)
LazyLoad: yes

A process (X_t) is said to be periodically correlated of period T>1 (or simply T-periodically correlated) if for all k and l

Cov(X_{T+k},X_{T+k-l})=Cov(X_{k},X_{k-l}).

Therefore the autocovariance function of (X_t) can be denoted by R_k(l)=Cov(X_k,X_{k-l}), where k=1,...,T and l=0,1,2,.... We refer to (k,l) as a season-lag pair, where k is the season and l is the lag.

If a set of autocovariances is given and l_{max} is the maximal lag of these autocovariances, then we can arrange them in a T x (lmax+1) matrix R such that R_k(l) occupies the (k,l+1)th entry of R:
R_1(0) R_1(1) R_1(2) ... R_1(lmax)
R_2(0) R_2(1) R_2(2) ... R_2(lmax)
vdots vdots vdots vdots vdots
R_T(0) R_T(1) R_T(2) ... R_T(lmax)

where some of the entries may be missing if the autocovariances are not given for every season-lag pair on 1:T x 0:(lmax+1).

The maximum entropy (ME) problem is to find a model whose entropy is maximal among all models whose autocovariances coincide with the given ones.

If the autocovariances are given over a set of lags which is contiguous and satisfies an additional technical condition, then the solution of the maximum entropy (ME) problem can be computed with the periodic Levinson-Durbin algorithm, otherwise the problem is non-linear. In any case the solution is a periodic autoregression model of order, say, p_1,...,p_T. A season-lag pair is called a gap if R_k(l) is missing and l <= p_k. We single out these lags since the maximum entropy problem can be solved by filling the gaps with values that maximise the entropy over the PAR(p_1,...,p_T) models, see Boshnakov and Lacroix (2009?) for details.

The main function in this package is pcme. It solves the ME problem for arbitrary patterns of lags. It implements the method of Boshnakov and Lambert-Lacroix (2009?). See the examples below and also the help page of pcme for examples of its use.

Besides pcme, of independent interest to users may be the functions implementing the periodic Levinson-Durbin algorithm, in particular LD and pldinverse (but see also package pear by McLeod and Mehmet (2008) for this kind of functionality).

LD and pldinverse work with periodic partial autocorrelations, B_k(l). They are arranged in a matrix in a way analogous to the arrangement of periodic autocovariances described above:

β_1(0) β_1(1) β_1(2) ... β_1(lmax)
β_2(0) β_2(1) β_2(2) ... β_2(lmax)
vdots vdots vdots vdots vdots
β_T(0) β_T(1) β_T(2) ... β_T(lmax)

Also, we use the convention (see Lacroix 2005) that B_t(l)=R_k(0) for k=1,...,T. In this way the partial correlations completely determine the covariance structure of the process.

Functions pcme.test1 and pcme.testcombn solve a "truckload" of problems by treating subsets of a given autocovariance sequence as unknown.

Example autocovariances are given in pcme.paperex and pcme.someex.

The autocovariances needed for pcme may be theoretical or sample ones. This package does not provide functions for computing sample autocovariances, use package pear (see McLeod and Mehmet (2008)) and partsm (see L'opez-de-Lacalle (2005)).

Author(s)

Sophie Lambert-Lacroix and Georgi Boshnakov

Maintainer: Georgi Boshnakov <Georgi.Boshnakov@manchester.ac.uk>

References

Boshnakov, Georgi and Lambert-Lacroix, Sophie (2009?) Maximum entropy for periodically correlated processes from nonconsecutive autocovariance coefficients. J. Time Series Anal. (to appear)

Lambert-Lacroix, Sophie (2005) Extension of autocovariance coefficients sequence for periodically correlated processes. Journal of Time Series Analysis, 26, No. 6, 423-435.

Lopez-de-Lacalle, Javier (2005). partsm: Periodic Autoregressive Time Series Models. R package version 1.0.

McLeod, A. I. and Balcilar, Mehmet. (2008). pear: Package for Periodic Autoregression Analysis. R package version 1.0. http://www.r-project.org

See Also

pcme


[Package pcme version 0.51 Index]