pcts (version 0.15.8)

pcalg1: Periodic Levinson-Durbin algorithm

Description

Calculate partial periodic autocorrelations, forward and backward prediction coefficients and error variances using the periodic Levinson-Durbin algorithm.

Usage

alg1(r, p)

Value

A list with the following elements.

orders

autoregression orders

be

partial autocorrelations, a matrix with d rows

fv

forward innovation variances, a matrix with d rows

bv

backward innovation variances, a matrix with d rows

af

forward autoregression parameters, a list with one element for the parameters for each order.

ab

backward autoregression parameters, a list with one element for the parameters for each order.

Arguments

r

periodic autocovariances, a matrix, see `Details'.

p

autoregressive orders, numeric vector.

Author

Georgi N. Boshnakov

Details

alg1(r,p) calculates the partial periodic correlations from autocovariances r and autoregression orders p. The matrix r has the same format as that of the r slot of pcAcvf objects. The periodicity, d, is set equal to the number of rows in r. If the length of p is not equal to the periodicity, all autoregressive orders are set to the first element of p. This last feature is really meant to be used only with a scalar p.

The convention for the signs of the coefficients is the one from Boshnakov(1996) and is consistent with other R time series functions.

pmax below stands for the maximal element of p, i.e. the maximal AR order.

As in the non-periodic case, the periodic Levinson-Durbin algorithm fits recursively models of order 0, 1, ..., pmax. Namely, at step i the AR orders for all seasons are set to i. This is done in a way that correctly handles the case when not all elements of p are equal, see the references.

The essential quantities calculated by the periodic Levinson-Durbin algorithm are returned as matrices, whose \(i\)th rows contain values for season \(i\). The complete details depend on the quantities, as described below.

The partial autocorrelations, the forward innovation variances and the backward innovation variances are returned as matrices with d rows and 1+pmax columns, whose j-th columns contain the quantities for order j-1 (partial autocorrelations, forward innovation variances and backward innovation variances, respectively). Note that the lag-0 partial autocorrelations are the autocovariances for lag 0, see the references for details.

The forward autoregression parameters are returned as a list whose \(j\)th element is a matrix containing the coefficients for order \(j\). Similarly for the backward autoregression parameters.

One often is interested in the model of order p only. Its coefficients are given by af[[pmax]], while the innovation variances are in the last column of fv.

References

boshnakov1996pcarmapcts

Sophie2000parpcts

Sophie2005pacfpcts

See Also

pdSafeParOrder

Examples

r1 <- rbind(c(1,0.81,0.729),c(1,0.90,0.900))
alg1(r1,2)

## pc2 <- pcAcvf(init=r1)
## pc2a <- pcAcvf(init=r1,seasonnames=c("am","pm"), periodunit="day")

# example of Lambert-Lacroix
data(ex1f)
pc3 <- slMatrix(period=2,maxlag=5,f=ex1f,type="tt")
res0p2 <- alg1(pc3[],c(0,2))
res1p2 <- alg1(pc3[],c(1,2))
res3p3 <- alg1(pc3[],c(3,3))

paramsys1 <- pcarma_param_system(pc3, NULL, NULL, 2, 0, 2)
t1 <- solve(paramsys1$A,paramsys1$b)

# this is from tests.r but I have lost t1
# set it to pc3 below
# note: t1 is not the t1 computed above and in other examples!

t1 <- pc3
t1
t1[]
alg1(t1[],c(1,1))
alg1(t1[],c(1,0))
alg1(t1[],c(0,1))
alg1(t1[],c(5,5))
alg1(t1[],c(2,2))
alg1(t1[],c(2,3))
alg1(t1[],c(3,3))
alg1(t1[],c(4,4))
alg1(t1[],c(5,5))