ltsa (version 1.4.6.1)

DLAcfToAR: Autocorrelations to AR parameters

Description

Given autocorrelations at lags 1,...,n the AR parameters corresponding to the AR coefficients, partial autocorrelations (pacf) and standarized minimum-mean-square predictor variance (sigsqk) are computed. Can also be used as a test for valid acf sequence.

Usage

DLAcfToAR(r, useC = TRUE, PDSequenceTestQ = FALSE)

Value

a matrix with 3 columns and length(r) rows is returned corresponding to the ar coefficients, pacf and sigsqk when PDSequenceTestQ = FALSE. Otherwise when PDSequenceTestQ = TRUE, the result is TRUE or FALSE according as the autocorrelation is a valid positive-definite sequence.

Arguments

r

autocorrelations starting at lag 1

useC

TRUE, C-interface function used. Otherwise if FALSE calculations are done in R

PDSequenceTestQ

FALSE, an error message is given if the autocorrelation sequence in not pd otherwise test for pd

Author

A.I. McLeod

Details

This function is more general than the built-in acf2AR since it provides the pacf and standardized minimum-mean-square error predictors. The standardized minimum-mean-square error predictor variances are defined as the minimum-mean-square error predictor variance for an AR process with unit variance. So for a sufficiently high-order, an approximation to the innovation variance is obtained.

The pacf may be used as an alternative parameterization for the linear time series model (McLeod and Zhang, 2006).

References

McLeod, A.I. and Zhang, Y. (2006). Partial autocorrelation parameterization for subset autoregression. Journal of Time Series Analysis, 27, 599-612.

McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software.

See Also

acf2AR, ar

Examples

#Example 1:  Yule-Walker estimates
z<-log(lynx)
p<-11
r<-(acf(z, lag.max=p, plot=FALSE)$acf)[-1]
ans<-DLAcfToAR(r)
#compare with built-in ar
phiAR<-ar(z,aic=FALSE, order.max=p, method="yw")$ar
#yet another way is to use acf2AR
phi2<-(acf2AR(c(1,r)))[p,]
cbind(ans,phiAR,phi2)
#   
#Example 2:  AR(1) illustration
#For AR(1) case compare useC = T and F
r<-0.9^(1:3)
DLAcfToAR(r, useC=TRUE)
DLAcfToAR(r, useC=FALSE)
DLAcfToAR(r, useC=TRUE, PDSequenceTestQ=TRUE)
DLAcfToAR(r, useC=FALSE, PDSequenceTestQ=TRUE)
#
#Example 3: test for valid tacf
r<-c(0.8, rep(0,99))
DLAcfToAR(r, PDSequenceTestQ=TRUE)
#   
#Example 4: Fractional-difference example
#Hosking (1981), pacf, zeta[k]=d/(k-d)
#we compare this numerically with our procedure
`tacvfFdwn` <-
function(d, maxlag)
{
    x <- numeric(maxlag + 1)
    x[1] <- gamma(1 - 2 * d)/gamma(1 - d)^2
    for(i in 1:maxlag) 
        x[i + 1] <- ((i - 1 + d)/(i - d)) * x[i]
    x
}
n<-10
d<-0.4
r<-tacvfFdwn(d, n)
r<-(r/r[1])[-1]
HoskingPacf<-d/(-d+(1:n))
cbind(DLAcfToAR(r),HoskingPacf)
#
# Example 5: Determining a suitable MA approximation
#Find MA approximation to hyperbolic decay series
N<-10^4  #pick N so large that mmse forecast error converged
r<-1/sqrt(1:N)
out<-DLAcfToAR(r[-1])
InnovationVariance<-out[nrow(out),3]
phi<-out[,1]
psi<-ARMAtoMA(ar=phi, lag.max=N)
Error<-r[1]-InnovationVariance*(1+sum(psi^2))