Autocorrelations to AR parameters
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.
DLAcfToAR(r, useC = TRUE, PDSequenceTestQ = FALSE)
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 |
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).
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.
A.I. McLeod
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.
#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))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.