Description Usage Arguments Details Value References See Also Examples

Inner product of cross-level wavelet autocorrelation functions.

1 2 | ```
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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
## 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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