arma_cov: Theoretical Autocovariances for ARMA(p,q) model, given by...
In hlennon/copulaIVTS: Dependence modelling of a digitised Gaussian ARMA(p,q) copula model for Integer-Valued Time Series

Description Usage Arguments Details Value Note Author(s) See Also Examples

Computes the theoretical Autocovariances for parameters a,b of length n where a = ar parameters b = ma parameters Note: This is Dr Yuan's Matlab Code written in R

1	arma_cov(a, b, n)

`a`	AR coefficients in a (0 if MA)
`b`	MA coefficients in b (0 if AR)
`n`	n number of autocovariances required including variance

Requires: a,b,n, (could use p and q)

model: x(t)-a(1)x(t-1)-...-a(p)x(t-p)=e(t)+b(1)e(t-1)+...+b(q)e(t-q)

r is a vector of theoretical autocovariances of length n, output: variance and autocovariances of x(t) up to lag n-1 assuming variance of e(t) = 1

Note: This is Dr Yuan's Matlab Code written in R

The methods used follow Brockwell & Davis (1991, section 3.3). Their equations (3.3.8) are solved for the autocovariances at lags 0, …, max(p, q+1), and the remaining autocorrelations are given by a recursive filter. Note 1: arma.cov() assumes sigma^2 = 1.... it doesnot therefre arma.cov outputs sigma^2 (R(0), R(1), .. R(n-1)) so we must divide by sigma^2 divide by R(0) to get the theoretical autocovariances R(0), R(1), etc r <- arma.cov(0.5, 0.1, 5) r <- r/r[1]

Note 2: ARMAacf is much faster and gives the same result r <- arma.cov(0.7, -0.5, 10) ; r <- r/r[1] r <- as.numeric(ARMAacf(0.7, -0.5, lag=10)) aram.cov (Yuans modified) is 2.5 times faster than arma.covmatlab but hard to see the difference for n<1000

Hannah Lennon and Jingsong Yuan

ARMAacf gives a vector of autocorrelations, named by the lags.

arma_cov(0.7, -0.5, 5)
r <- arma_cov(0.7, -0.5, 5)
r <- r/r[1]


ARMAacf(ar=0.7, ma=-0.5, lag.max=5, pacf=FALSE)
## This gives the theoretical autoCORRELATION funcition (e.g. divide by Var(Y_t)=r[1])


## The function is currently defined as
function (a, b, n) 
{
    p <- length(a)
    q <- length(b)
    c <- ARMAtoMA(a, b, q)
    aa <- c(1, -a)
    A <- matrix(0, p + 1, p + 1)
    for (i in 0:p) {
        for (j in 0:p) {
            k <- abs(i - j)
            A[i + 1, k + 1] <- A[i + 1, k + 1] + aa[j + 1]
        }
    }
    b <- c(1, b)
    c <- c(1, c)
    d <- NULL
    for (i in 0:p) {
        d <- rbind(d, sum(b * c))
        b <- c(b[-1], 0)
    }
    r <- solve(A, d)
    for (j in (p + 1):n) {
        dum <- 0
        for (i in 1:p) {
            dum <- dum + a[i] * r[j - i + 1]
        }
        r <- c(r, dum + sum(b * c))
        b <- c(b[-1], 0)
    }
    return(r[1:(n + 1)])
  }