WavTransf1D computes a forward intrinsic average-interpolating (AI) wavelet transform for a
curve in the manifold of HPD matrices equipped with a metric specified by the user, such as the
affine-invariant Riemannian metric, as described in CvS17pdSpecEst and Chapter 3 of
C18pdSpecEst.
WavTransf1D(P, order = 5, jmax, periodic = FALSE,
metric = "Riemannian", ...)a (\(d,d,m\))-dimensional array of HPD matrices, corresponding to a sequence of \((d,d)\)-dimensional HPD matrices of length \(m\), with \(m = 2^J\) for some \(J > 0\).
an odd integer larger or equal to 1 corresponding to the order of the intrinsic AI refinement scheme,
defaults to order = 5. Note that if order > 9, the computational cost
significantly increases as the wavelet transform no longer uses a fast wavelet refinement scheme based
on pre-determined weights.
the maximum scale up to which the wavelet coefficients are computed. If jmax is not
specified, it is set equal to the maximum possible scale jmax = J-1, where J = log2(m).
a logical value determining whether the curve of HPD matrices can be reflected at the boundary for
improved wavelet refinement schemes near the boundaries of the domain. This is useful for spectral matrix estimation,
in which case the spectral matrix is a symmetric and periodic curve in the frequency domain. Defaults to periodic = FALSE.
the metric that the space of HPD matrices is equipped with. The default choice is "Riemannian",
but this can also be one of: "logEuclidean", "Cholesky", "rootEuclidean",
"Euclidean" or "Riemannian-Rahman". See also the Details section below.
additional arguments for internal use.
The function returns a list with three components:
the pyramid of wavelet coefficients. This is a list of arrays, where each array contains the
(\(d,d\))-dimensional Hermitian wavelet coefficients from the coarsest wavelet scale j = 0 up to
the finest wavelet scale j = jmax
the pyramid of whitened wavelet coefficients. The structure of D.white is the same as
D, but with the wavelet coefficients replaced by their whitened counterparts as explained in
CvS17pdSpecEst.
a numeric array containing the midpoint(s) at the coarsest scale j = 0 in the midpoint pyramid.
The input array P corresponds to a discretized curve of \((d,d)\)-dimensional HPD matrices of
dyadic length. WavTransf1D then computes the intrinsic AI wavelet transform of P based on
the given refinement order and the chosen metric. If the refinement order is an odd integer smaller or
equal to 9, the function computes the wavelet transform using a fast wavelet refinement scheme based on weighted
intrinsic averages with pre-determined weights as explained in CvS17pdSpecEst and Chapter 3 of
C18pdSpecEst. If the refinement order is an odd integer larger than 9, the wavelet refinement
scheme uses intrinsic polynomial prediction based on Neville's algorithm in the Riemannian manifold (via pdNeville).
The function computes the intrinsic AI wavelet transform in the space of HPD matrices equipped with
one of the following metrics: (i) the affine-invariant Riemannian metric (default) as detailed in e.g., B09pdSpecEst[Chapter 6]
or PFA05pdSpecEst; (ii) the log-Euclidean metric, the Euclidean inner product between matrix logarithms;
(iii) the Cholesky metric, the Euclidean inner product between Cholesky decompositions; (iv) the Euclidean metric; or
(v) the root-Euclidean metric. The default choice of metric (affine-invariant Riemannian) satisfies several useful properties
not shared by the other metrics, see CvS17pdSpecEst or C18pdSpecEst for more details. Note that this comes
at the cost of increased computation time in comparison to one of the other metrics.
# NOT RUN {
P <- rExamples1D(2^8, example = "bumps")
P.wt <- WavTransf1D(P$f, periodic = FALSE)
# }
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