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

Description

The function ltzb 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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ltzb(r,x)

Arguments

r

autocorelation vector

x

time series data

Value

Vector with values t(en) * solve(R) * x and t(en) * solve(R) * en. 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 <- ltzb(r,Nile)
# Comparison of the algorithm with typical approaches

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

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