PacfDL: Partial Autocorrelations via Durbin-Levinson
In FitAR: Subset AR Model Fitting

Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

Given autocovariances, the partial autocorrelations and/or autoregressive coefficients in an AR may be determined using the Durbin-Levinson algorithm. If the autocovariances are sample autocovariances, this is equivalent to using the Yule-Walker equations. But as noted below our function is more general than the built-in R functions.

1	PacfDL(c, LinearPredictor = FALSE)

`c`	autocovariances at lags 0,1,...,p = length(c)-1
`LinearPredictor`	if TRUE, AR coefficients are also determined using the Yule-Walker method

The Durbin-Levinson algorithm is described in many books on time series and numerical methods, for example Percival and Walden (1993, eqn 403).

If LinearPredictor = FALSE, vector of length p = length(c)-1 containing the partial autocorrelations at lags 1,...,p. Otherwise a list with components:

`Pacf`	vector of partial autocorrelations
`ARCoefficients`	vector of AR coefficients
`ResidualVariance`	residual variance for AR(p)

Stationarity is not tested.

Sample partial autocorrelations can also be computed with the acf function and Yule-Walker estimates can be computed with the ar function. Our function PacfDL provides more flexibility since then input c may be any valid autocovariances not just the usual sample autocovariances. For example, we can determine the minimum mean square error one-step ahead linear predictor of order p for theoretical autocovariances from a fractional arma or other linear process.

A.I. McLeod and Y. Zhang

Percival, D.B. and Walden, A.T. (1993). Spectral Analysis For Physical Applications, Cambridge University Press.

acf, ar

#first define a function to compute the Sample Autocovariances
 sacvf<-function(z, lag.max){
 c(acf(z, plot=FALSE, lag.max=lag.max)$acf)*(length(z)-1)/length(z)
 }
#now compute PACF and also fit AR(7) to SeriesA
 ck<-sacvf(SeriesA, 7)
 PacfDL(ck)
 PacfDL(ck, LinearPredictor = TRUE)
#compare with built-in functions
 pacf(SeriesA, lag.max=7, plot=FALSE)
 ar(SeriesA, lag.max=7, method="yw")
#fit an optimal linear predictor of order 10 to MA(1)
 g<-TacvfMA(0.8,5)
 PacfDL(g, LinearPredictor=TRUE)
#
#Compute the theoretical pacf for MA(1) and plot it
 ck<-c(1,-0.4,rep(0,18))
 AcfPlot(PacfDL(ck)$Pacf)
 title(main="Pacf of MA(1), r(1)=-0.4")