AutoCorrIP | R Documentation |
Inner product of cross-level wavelet autocorrelation functions.
AutoCorrIP(J, filter.number = 1, family = "DaubExPhase", crop = TRUE)
J |
Number of levels. |
filter.number |
Number of vanishing moments of the wavelet function. |
family |
Wavelet family, either |
crop |
Logical, should the output of |
Let ψ(x) denote the mother wavelet and the wavelet defined for level j as ψ_{j,k}(x) = 2^{j/2}ψ(2^{j}x-k). The wavelet autocorrelation function between levels j & l is therefore:
Ψ_{j,l}(τ) = ∑_τ ψ_{j,k}(0)ψ_{l,k-τ}(0)
Here, integer τ 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 λ:
A^{λ}_{j,l,h} = ∑_{τ} Ψ_{j,l}(λ - τ) Ψ_{h,h}(τ)
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 λ, whilst the second to
fourth dimensions identify the levels indexed by j, l & h
respectively.
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.
ipndacw
.
## 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)
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