| InvWavTransf1D {pdSpecEst} | R Documentation |
Inverse AI wavelet transform for curve of HPD matrices
Description
InvWavTransf1D computes an inverse intrinsic average-interpolation (AI) wavelet
transform mapping an array of coarsest-scale HPD midpoints combined with a pyramid of Hermitian
wavelet coefficients to a curve in the manifold of HPD matrices equipped with a metric specified by the user,
as described in (Chau and von
Sachs 2019) and Chapter 3 of (Chau 2018). This is
the inverse operation of the function WavTransf1D.
Usage
InvWavTransf1D(D, M0, order = 5, jmax, periodic = FALSE,
metric = "Riemannian", ...)
Arguments
D |
a list of arrays containing the pyramid of wavelet coefficients, where each array contains the
( |
M0 |
a numeric array containing the midpoint(s) at the coarsest scale |
order |
an odd integer larger or equal to 1 corresponding to the order of the intrinsic AI refinement scheme,
defaults to |
jmax |
the maximum scale (resolution) up to which the HPD midpoints (i.e. scaling coefficients) are reconstructed.
If |
periodic |
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,
where the spectral matrix is a symmetric and periodic curve in the frequency domain. Defaults to |
metric |
the metric that the space of HPD matrices is equipped with. The default choice is |
... |
additional arguments for internal use. |
Details
The input list of arrays D and array M0 correspond to a pyramid of wavelet coefficients and
the coarsest-scale HPD midpoints respectively, both are structured in the same way as in the output of
WavTransf1D. As in the forward AI wavelet transform, if the refinement order is an odd integer smaller or
equal to 9, the function computes the inverse wavelet transform using a fast wavelet refinement scheme based on
weighted intrinsic averages with pre-determined weights as explained in (Chau and von
Sachs 2019) and Chapter 3 of
(Chau 2018). 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 inverse 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., (Bhatia 2009)[Chapter 6]
or (Pennec et al. 2006); (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 (Chau and von
Sachs 2019) or (Chau 2018) for more details. Note that this comes
at the cost of increased computation time in comparison to one of the other metrics.
Value
Returns a (d, d, m)-dimensional array corresponding to a length m curve of
(d,d)-dimensional HPD matrices.
References
Bhatia R (2009).
Positive Definite Matrices.
Princeton University Press, New Jersey.
Chau J (2018).
Advances in Spectral Analysis for Multivariate, Nonstationary and Replicated Time Series.
phdthesis, Universite catholique de Louvain.
Chau J, von
Sachs R (2019).
“Intrinsic wavelet regression for curves of Hermitian positive definite matrices.”
Journal of the American Statistical Association.
doi: 10.1080/01621459.2019.1700129.
Pennec X, Fillard P, Ayache N (2006).
“A Riemannian framework for tensor computing.”
International Journal of Computer Vision, 66(1), 41–66.
See Also
WavTransf1D, pdSpecEst1D, pdNeville
Examples
P <- rExamples1D(2^8, example = "bumps")
P.wt <- WavTransf1D(P$f) ## forward transform
P.f <- InvWavTransf1D(P.wt$D, P.wt$M0) ## backward transform
all.equal(P.f, P$f)