00pcme-package {pcme} | R Documentation |
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.
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)).
Sophie Lambert-Lacroix and Georgi Boshnakov
Maintainer: Georgi Boshnakov <Georgi.Boshnakov@manchester.ac.uk>
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