DLLoglikelihood function

Durbin-Levinsion Loglikelihood

The Durbin-Levinsion algorithm is used for the computation of the exact loglikelihood function.

DLLoglikelihood(r, z, useC = TRUE)

Arguments

• r: autocovariance or autocorrelation at lags 0,...,n-1, where n is length(z)
• z: time series data
• useC: TRUE, use compiled C, otherwise R

Details

The concentrated loglikelihood function may be written Lm(beta) = -(n/2)log(S/n)-0.5g, where beta is the parameter vector, n is the length of the time series, S=z'M z, z is the mean-corrected time series, M is the inverse of the covariance matrix setting the innovation variance to one and g=-log(det(M)). This method was given in Li (1981) for evaluating the loglikelihood function in the case of the fractionally differenced white noise.

Returns

The loglikelihood concentrated over the parameter for the innovation variance is returned.

References

W.K. Li (1981). Topics in Time Series Analysis. Ph.D. Thesis, University of Western Ontario.

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

Note

The purpose of this function is to provide a check on the TrenchLoglikelihod function. Completely different algorithms are used in each case but the numerical values should agree.

Author(s)

A.I. McLeod

See Also

TrenchLoglikelihood

Examples

#Example 1
#compute loglikelihood for white noise
z<-rnorm(100)
DLLoglikelihood(c(1,rep(0,length(z)-1)), z)

#Example 2
#simulate a time series and compute the concentrated loglikelihood using DLLoglikelihood and
#compare this with the value given by TrenchLoglikelihood.
phi<-0.8
n<-200
r<-phi^(0:(n-1))
z<-arima.sim(model=list(ar=phi), n=n)
LD<-DLLoglikelihood(r,z)
LT<-TrenchLoglikelihood(r,z)
ans<-c(LD,LT)
names(ans)<-c("DLLoglikelihood","TrenchLoglikelihood")

#Example 3
## Not run:

#Compare direct evaluation of AR(1) loglikelihood with DL method
#First define the exact concentrated loglikelihood function for AR(1)
AR1Loglikelihood <-function(phi,z){
n<-length(z)
S<-(z[1]^2)*(1-phi^2) + sum((z[-1]-phi*z[-n])^2)
0.5*log(1-phi^2)-(n/2)*log(S/n)
}
#Next run script to compare numerically the loglikelihoods.
#They should be identical.
phi<-0.8
n<-200
z<-arima.sim(list(ar=phi), n=n)
phis<-seq(0.1,0.95,0.05)
ansAR1<-ansDL<-numeric(length(phis))
for (i in 1:length(phis)) {
    ansAR1[i] <- AR1Loglikelihood(phis[i],z)
    r<-(1/(1-phis[i]^2))*phis[i]^(0:(n-1))
    ansDL[i] <- DLLoglikelihood(r,z,useC=FALSE)
}
ans<-matrix(c(ansDL,ansAR1),ncol=2)
dimnames(ans)<-list(phis, c("DL-method","AR1-method"))
## End(Not run)

#Example 4
## Not run:

#compare timings. See (McLeod, Yu, Krougly, Table 8).
 n<-5000
 ds<-c(-0.45, -0.25, -0.05, 0.05, 0.25, 0.45)
 tim<-matrix(numeric(3*length(ds)),ncol=3)
 for (i in 1:length(ds)){
    d<-ds[i]
    alpha <- 1-2*d #equivalent hyperbolic autocorrelation
    r <- (1/(1:n))^alpha
    z<-DLSimulate(n,r)
    tim1a<-system.time(LL1<-DLLoglikelihood(r,z))[1]
    tim1b<-system.time(LL1<-DLLoglikelihood(r,z,useC=FALSE))[1]
    tim2<-system.time(LL2<-TrenchLoglikelihood(r,z))[1]
    tim[i,]<-c(tim1a,tim1b, tim2)
    }
 dimnames(tim)<-list(ds, c("DL-C","DL-R","Trench"))
 tim
## End(Not run)