# Linear Time Series Analysis Linear time series modelling. Methods are given for loglikelihood computation, forecasting and simulation. package ## Details ||| |:--|:--| |Package:|ltsa| |Type:|Package| |Version:|1.4.5| |Date:|2015-08-22| |License:|GPL (>= 2)| ||| |:--|:--| |FUNCTION|SUMMARY| |DHSimulate|Davies and Harte algorithm for time series simulation| |DLAcfToAR|from Acf to AR using Durbin-Levinson recursion| |DLLoglikelihood|exact loglikelihood using Durbin-Levinson algorithm| |DLResiduals|exact one-step residuals, Durbin-Levision algorithm| |DLSimulate|exact simulation of Gaussian time series using DL| |is.toeplitz|test for Toeplitz matrix| |PredictionVariance|two methods provided| |tacvfARMA|theoretical autocovariances| |ToeplitzInverseUpdate|update inverse| |TrenchForecast|general algorithm for forecasting| |TrenchInverse|efficient algorithm for inverse of Toeplitz matrix| |TrenchLogLikelihood|exact loglikelihood| |TrenchMean|exact MLE for mean| ## Author(s) A. I. McLeod, Hao Yu and Zinovi Krougly. Maintainer: aimcleod@uwo.ca ## References Hipel, K.W. and McLeod, A.I., (2005). Time Series Modelling of Water Resources and Environmental Systems. Electronic reprint of our book orginally published in 1994. [http://www.stats.uwo.ca/faculty/aim/1994Book/](http://www.stats.uwo.ca/faculty/aim/1994Book/). McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software. ## See Also `DHSimulate`, `DLAcfToAR`, `DLLoglikelihood`, `DLResiduals`, `DLSimulate`, `exactLoglikelihood`, `is.toeplitz`, `PredictionVariance`, `tacvfARMA`, `ToeplitzInverseUpdate`, `TrenchForecast`, `TrenchInverse`, `TrenchLoglikelihood`, `TrenchMean`, ## Examples ```r #Example 1: DHSimulate #First define acf for fractionally-differenced white noise and then simulate using DHSimulate `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<-1000 rZ<-tacvfFdwn(0.25, n-1) #length 1000 Z<-DHSimulate(n, rZ) acf(Z) #Example 2: DLAcfToAR # n<-10 d<-0.4 r<-tacvfFdwn(d, n) r<-(r/r[1])[-1] HoskingPacf<-d/(-d+(1:n)) cbind(DLAcfToAR(r),HoskingPacf) #Example 3: DLLoglikelihood #Using Z and rZ in Example 1. DLLoglikelihood(rZ, Z) #Example 4: DLResiduals #Using Z and rZ in Example 1. DLResiduals(rZ, Z) #Example 5: DLSimulate #Using Z in Example 1. z<-DLSimulate(n, rZ) plot.ts(z) #Example 6: is.toeplitz is.toeplitz(toeplitz(1:5)) #Example 7: PredictionVariance #Compare with predict.Arima #general script, just change z, p, q, ML z<-sqrt(sunspot.year) n<-length(z) p<-9 q<-0 ML<-10 #for different data/model just reset above out<-arima(z, order=c(p,0,q)) sda<-as.vector(predict(out, n.ahead=ML)$se) # phi<-theta<-numeric(0) if (p>0) phi<-coef(out)[1:p] if (q>0) theta<-coef(out)[(p+1):(p+q)] zm<-coef(out)[p+q+1] sigma2<-out$sigma2 r<-sigma2*tacvfARMA(phi, theta, maxLag=n+ML-1) sdb<-sqrt(PredictionVariance(r, maxLead=ML)) cbind(sda,sdb) #Example 8: tacfARMA #There are two methods: tacvfARMA and ARMAacf. #tacvfARMA is more general since it computes the autocovariances function # given the ARMA parameters and the innovation variance whereas ARMAacf # only computes the autocorrelations. Sometimes tacvfARMA is more suitable # for what is needed and provides a better result than ARMAacf as in the # the following example. # #general script, just change z, p, q, ML z<-sqrt(sunspot.year) n<-length(z) p<-9 q<-0 ML<-5 #for different data/model just reset above out<-arima(z, order=c(p,0,q)) phi<-theta<-numeric(0) if (p>0) phi<-coef(out)[1:p] if (q>0) theta<-coef(out)[(p+1):(p+q)] zm<-coef(out)[p+q+1] sigma2<-out$sigma2 rA<-tacvfARMA(phi, theta, maxLag=n+ML-1, sigma2=sigma2) rB<-var(z)*ARMAacf(ar=phi, ma=theta, lag.max=n+ML-1) #rA and rB are slighly different cbind(rA[1:5],rB[1:5]) #Example 9: ToeplitzInverseUpdate #In this example we compute the update inverse directly and using ToeplitzInverseUpdate and #compare the result. phi<-0.8 sde<-30 n<-30 r<-arima.sim(n=30,list(ar=phi),sd=sde) r<-phi^(0:(n-1))/(1-phi^2)*sde^2 n1<-25 G<-toeplitz(r[1:n1]) GI<-solve(G) #could also use TrenchInverse GIupdate<-ToeplitzInverseUpdate(GI,r[1:n1],r[n1+1]) GIdirect<-solve(toeplitz(r[1:(n1+1)])) ERR<-sum(abs(GIupdate-GIdirect)) ERR #Example 10: TrenchForecast #Compare TrenchForecast and predict.Arima #general script, just change z, p, q, ML z<-sqrt(sunspot.year) n<-length(z) p<-9 q<-0 ML<-10 #for different data/model just reset above out<-arima(z, order=c(p,0,q)) Fp<-predict(out, n.ahead=ML) phi<-theta<-numeric(0) if (p>0) phi<-coef(out)[1:p] if (q>0) theta<-coef(out)[(p+1):(p+q)] zm<-coef(out)[p+q+1] sigma2<-out$sigma2 #r<-var(z)*ARMAacf(ar=phi, ma=theta, lag.max=n+ML-1) #When r is computed as above, it is not identical to below r<-sigma2*tacvfARMA(phi, theta, maxLag=n+ML-1) F<-TrenchForecast(z, r, zm, n, maxLead=ML) #the forecasts are identical using tacvfARMA # #Example 11: TrenchInverse #invert a matrix of order n and compute the maximum absolute error # in the product of this inverse with the original matrix n<-5 r<-0.8^(0:(n-1)) G<-toeplitz(r) Gi<-TrenchInverse(G) GGi<-crossprod(t(G),Gi) id<-matrix(0, nrow=n, ncol=n) diag(id)<-1 err<-max(abs(id-GGi)) err #Example 12: TrenchLoglikelihood #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 13: TrenchMean phi<- -0.9 a<-rnorm(100) z<-numeric(length(a)) phi<- -0.9 n<-100 a<-rnorm(n) z<-numeric(n) mu<-100 sig<-10 z[1]<-a[1]*sig/sqrt(1-phi^2) for (i in 2:n) z[i]<-phi*z[i-1]+a[i]*sig z<-z+mu r<-phi^(0:(n-1)) meanMLE<-TrenchMean(r,z) meanBLUE<-mean(z) ans<-c(meanMLE, meanBLUE) names(ans)<-c("BLUE", "MLE") ans ```

ltsa-package