AutocorrIP function

Wavelet Autocorrelation Inner Product Functions

Inner product of cross-level wavelet autocorrelation functions. methods

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.

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:

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 $\lambda$:

Returns

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.

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

## 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))
    }
  }
}
## End(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)