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

ltzb(r,x)

Arguments

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

# 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)))

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

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Type 'demo()' for some demos, 'help()' for on-line help, or
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> library(HKprocess)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/HKprocess/ltzb.Rd_%03d_medium.png", width=480, height=480)
> ### Name: ltzb
> ### Title: Value of quadratic forms for the inverse of a symmetric positive
> ###   definite autocorrelation matrix.
> ### Aliases: ltzb
> ### Keywords: array
> 
> ### ** Examples
> 
> # 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)
t(en) * solve(R) * x 
        9.094947e-13 
> 
> examp[2] - as.numeric(t(rep(1,length(r))) %*% solve(toeplitz(r)) %*%
+ rep(1,length(r)))
t(en) * solve(R) * en 
        -8.881784e-16 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>