\name{DLAcfToAR}
\alias{DLAcfToAR}
\title{ 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)
}
\arguments{
  \item{r}{ autocorrelations starting at lag 1}
  \item{useC}{ TRUE, C-interface function used. Otherwise if FALSE
    calculations are done in R }
  \item{PDSequenceTestQ}{ FALSE, an error message is given if the 
  autocorrelation sequence in not pd otherwise test for pd }
}
\details{
This function is more general than the built-in \code{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). 
}
\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.
}
\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.
}

\author{A.I. McLeod  }
\seealso{ 
\code{\link{acf2AR}}, 
\code{\link{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))
}
\keyword{ ts }