mvLSW (version 1.1)

Autocorrelation: Wavelet Autocorrelation Functions

Description

Wavelet autocorrelation functions between two wavelet functions associated to level j & l for all pairs j, l = 1,…,J and associated inner product.

Usage

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

Arguments

J

Number of levels.

filter.number

Number of vanishing moments for wavelet.

family

The family of wavelet, 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 is limited to -\(2^J\):\(2^J\). 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

This function returns the wavelet autocorrelation function between wavelet functions at level pair j & l.

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

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)
# }