mvLSW (version 1.2.5)

AutoCorrIP: Wavelet Autocorrelation Inner Product Functions

Description

Inner product of cross-level wavelet autocorrelation functions.

Usage

AutoCorrIP(J, filter.number = 1, family = "DaubExPhase",
    crop = TRUE)

Arguments

J

Number of levels.

filter.number

Number of vanishing moments of the wavelet function.

family

Wavelet family, either "DaubExPhase" or "DaubLeAsymm". The Haar wavelet is defined as default.

crop

Logical, should the output of AutoCorrIP be cropped such that the first dimension of the returned array relate to the offset range -\(2^J\):\(2^J\).This is set at TRUE by default.

Value

A 4D array (invisibly returned) of order LxJxJxJ where L depends on the specified wavelet function. If crop=TRUE then L=\(2^{J+1}\)+1. The first dimension defines the offset \(\lambda\), whilst the second to fourth dimensions identify the levels indexed by j, l & h respectively.

Details

Let \(\psi(x)\) denote the mother wavelet and the wavelet defined for level j as \(\psi_{j,k}(x) = 2^{j/2}\psi(2^{j}x-k)\). The wavelet autocorrelation function between levels j & l is therefore:

$$\Psi_{j,l}(\tau) = \sum_\tau \psi_{j,k}(0)\psi_{l,k-\tau}(0)$$

Here, integer \(\tau\) defines the offset of the latter wavelet function relative to the first.

The inner product of this wavelet autocorrelation function is defined as follows for level indices j, l & h and offset \(\lambda\):

$$A^{\lambda}_{j,l,h} = \sum_{\tau} \Psi_{j,l}(\lambda - \tau) \Psi_{h,h}(\tau)$$

References

Taylor, S.A.C., Park, T.A. and Eckley, I. (2019) Multivariate locally stationary wavelet analysis with the mvLSW R package. Journal of statistical software 90(11) pp. 1--16, doi: 10.18637/jss.v090.i11.

Fryzlewicz, P. and Nason, G. (2006) HaarFisz estimation of evolutionary wavelet spectra. Journal of the Royal Statistical Society. Series B, 68(4) pp. 611-634.

See Also

ipndacw.

Examples

# NOT RUN {
## Plot Haar autocorrelation wavelet functions inner product
AInnProd <- AutoCorrIP(J = 8, filter.number = 1, family = "DaubExPhase")
# }
# NOT RUN {
MaxOffset <- 2^8
for(h in 6:8){
  x11()
  par(mfrow = c(3, 3))
  for(l in 6:8){
    for(j in 6:8){
      plot(-MaxOffset:MaxOffset, AInnProd[, j, l, h], type = "l", 
        xlab = "lambda", ylab = "Autocorr Inner Prod", 
        main = paste("j :", j, "- l :", l, "- h :", h))
    }
  }
}
# }
# NOT RUN {
## Special case relating to ipndacw function from wavethresh package
Amat <- matrix(NA, ncol = 8, nrow = 8)
for(j in 1:8) Amat[, j] <- AInnProd[2^8 + 1, j, j, ]
round(Amat, 5)
round(ipndacw(J = -8, filter.number = 1, family = "DaubExPhase"), 5)
# }