Value of quadratic forms for the inverse of a symmetric positive definite autocorrelation matrix.

Description

The function ltza is used to calculate the value of quadratic forms for the inverse of a symmetric positive definite autocorrelation matrix, using the Levinson algorithm (Golub and Van Loan 1996, Algorithm 4.7.2).

Usage

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ltza(r,x)

Arguments

r

autocorelation vector

x

time series data

Value

Vector with values t(x) * solve(R) * x, t(en) * solve(R) * x, t(en) * solve(R) * en and the natural logarithm of the determinant of R. t(.) denotes the transpose of a vector, en = (1,1,...,1) and R is the autocorrelation matrix.

Author(s)

Hristos Tyralis

References

Golub G.H., Van Loan C.F. (1996) Matrix Computations, Baltimore: John Hopkins University Press.

Examples

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# Estimate the parameters for the Nile time series.

r <- acfHKp(H = 0.8,maxlag = length(Nile)-1)

examp <- ltza(r,Nile)

# Comparison of the algorithm with typical approaches

examp[1] - as.numeric(t(Nile) %*% solve(toeplitz(r)) %*% Nile)

examp[2] - as.numeric(t(rep(1,length(r))) %*% solve(toeplitz(r)) %*% Nile)

examp[3] - as.numeric(t(rep(1,length(r))) %*% solve(toeplitz(r)) %*%
rep(1,length(r)))

examp[4] - log(det(toeplitz(r)))