# Simulate General Linear Process Uses the Davies-Harte algorithm to simulate a Gaussian time series with specified autocovariance function. ```r DHSimulate(n, r, ReportTestOnly = FALSE, rand.gen = rnorm, ...) ``` ## Arguments - `n`: length of time series to be generated - `r`: autocovariances at lags 0,1,... - `ReportTestOnly`: FALSE -- Run normally so terminates with an error if Davies-Harte condition does not hold. Othewise if TRUE, then output is TRUE if the Davies-Harte condition holds and FALSE if it does not. - `rand.gen`: random number generator to use. It is assumed to have mean zero and variance one. - ``...``: optional arguments passed to `rand.gen` ## Details The method uses the FFT and so is most efficient if the series length, n, is a power of 2. The method requires that a complicated non-negativity condition be satisfed. Craigmile (2003) discusses this condition in more detail and shows for anti-persistent time series this condition will always be satisfied. Sometimes, as in the case of fractinally differenced white noise with parameter d=0.45 and n=5000, this condition fails and the algorithm doesn't work. In this case, an error message is generated and the function halts. ## Returns Either a vector of length containing the simulated time series if Davies-Harte condition holds and ReportTestOnly = FALSE. If argument ReportTestOnly is set to TRUE, then output is logical variable indicating if Davies-Harte condition holds, TRUE, or if it does not, FALSE. ## Author(s) A.I. McLeod ## References Craigmile, P.F. (2003). Simulating a class of stationary Gaussian processes using the Davies-Harte algorithm, with application to long memory processes. Journal of Time Series Analysis, 24, 505-511. Davies, R. B. and Harte, D. S. (1987). Tests for Hurst Effect. Biometrika 74, 95--101. McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software. ## See Also \ `DLSimulate` , `SimGLP`, `arima.sim` ## Examples ```r #simulate a process with autocovariance function 1/(k+1), k=0,1,... # and plot it n<-2000 r<-1/sqrt(1:n) z<-DHSimulate(n, r) plot.ts(z) #simulate AR(1) and produce a table comparing the theoretical and sample # autocovariances and autocorrelations phi<- -0.8 n<-4096 g0<-1/(1-phi^2) #theoretical autocovariances tacvf<-g0*(phi^(0:(n-1))) z<-DHSimulate(n, tacvf) #autocorrelations sacf<-acf(z, plot=FALSE)$acf #autocovariances sacvf<-acf(z, plot=FALSE,type="covariance")$acf tacf<-tacvf/tacvf[1] tb<-matrix(c(tacvf[1:10],sacvf[1:10],tacf[1:10],sacf[1:10]),ncol=4) dimnames(tb)<-list(0:9, c("Tacvf","Sacvf","Tacf","Sacf")) tb #Show the Davies-Harte condition sometimes hold and sometimes does not # in the case of fractionally differenced white noise # #Define autocovariance function for fractionally differenced white noise `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 } #Build table to show values of d for which condition is TRUE when n=5000 n<-5000 ds<-c(-0.45, -0.25, -0.05, 0.05, 0.25, 0.45) tb<-logical(length(ds)) names(tb)<-ds for (kd in 1:length(ds)){ d<-ds[kd] r<-tacvfFdwn(d, n-1) tb[kd]<-DHSimulate(n, r, ReportTestOnly = TRUE) } tb ```

DHSimulate function