1d:T775,# Autocovariance function of ARTFIMA Theoretical autocovariance function of ARTFIMA model ```r artfimaTACVF(d = numeric(0), lambda = numeric(0), phi = numeric(0), theta = numeric(0), maxlag, sigma2 = 1, obj = NULL) ``` ## Arguments - `d`: ARTFIMA difference parameter, any real value. When d=0, reduces to ARMA and lambda is ignored. - `lambda`: ARTFIMA tempered decay parameter. When lambda=0, reduces to ARFIMA - `phi`: AR coefficients - `theta`: MA coefficients, Box-Jenkins definition - `maxlag`: maxlag+1 lags computed corresponding to 0,1,...,maxlag - `sigma2`: innovation variance - `obj`: output from artfima function ## Returns vector of length maxlag+1 of the specified autocovariances ## Author(s) A. I. McLeod, aimcleod@uwo.ca ## See Also `ARMAacf`, `artfimaSDF`, `artsim`, `artfima` ## Examples ```r #ARTFIMA - area under SDF equals theoretical Var(z[t]) #and sample variance = mean of periodogram # lambda <- 0.045 d <- 5/6 TVarZ <- artfimaTACVF(d=d, lambda=lambda, maxlag=3)[1] TVarZ n <- 256 set.seed(4337751) z <- artsim(n, lambda=lambda, d=d) VarZ <- mean((z-mean(z))^2) Ip <- Periodogram(z) mean(Ip) length(Ip) x <- (1/n)*(1:length(Ip)) plot(x, Ip, xlab="frequency", ylab="Spectral density & Periodogram", main=paste("lambda, d =", lambda, d), type="l", col=rgb(0,0,1,0.5)) n <- 5000 y <- artfimaSDF(n, lambda=lambda, d=d) x <- (1/n)*(1:length(y)) lines(x, y, type="l", lwd=1.25) h <- x[2]-x[1] #step length SimpsonsRule <- function(h, y) { n <- length(y) h/3*sum(y * c(1, rep(c(4,2), n-1), 1)) } AreaApprox <- SimpsonsRule(h, y) text(0.2, 230, labels=paste("Area under SDF using Simpson's Rule =", round(AreaApprox,4))) text(0.2, 200, labels=paste("Theoretical ARTFIMA Variance =", round(TVarZ,4))) text(0.2, 170, labels=paste("mean(Ip) =", round(mean(Ip),4))) text(0.2, 140, labels=paste("sample variance =", round(VarZ,4))) ```

artfimaTACVF function