WavTransf2D {pdSpecEst}

Forward AI wavelet transform for surface of HPD matrices

Description

WavTransf2D computes a forward intrinsic average-interpolation (AI) wavelet transform for a rectangular surface in the manifold of HPD matrices equipped with a metric specified by the user, such as the affine-invariant Riemannian metric, as described in Chapter 5 of (Chau 2018).

Usage

WavTransf2D(P, order = c(3, 3), jmax, metric = "Riemannian", ...)

Arguments

P

a (d,d,n1,n2)-dimensional array of HPD matrices corresponding to a rectangular surface of (d,d)-dimensional HPD matrices of size n_1 \times n_2, with n_1 = 2^{J_1} and n_2 = 2^{J_2} for some J_1, J_2 > 0.

order

a 2-dimensional numeric vector (1,1) \le order \le (9,9) corresponding to the marginal orders of the intrinsic 2D AI refinement scheme, defaults to order = c(3, 3).

jmax

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 = max(J1, J2) - 1.

metric

the metric that the space of HPD matrices is equipped with. The default choice is "Riemannian", but this can be one of: "Riemannian", "logEuclidean", "Cholesky", "rootEuclidean" or "Euclidean". See also the Details section below.

...

additional arguments for internal use.

Details

The 4-dimensional array P corresponds to a discretized rectangular surface of (d,d)-dimensional HPD matrices. The rectangular surface is of size n_1 by n_2, where both n_1 and n_2 are supposed to be dyadic numbers. WavTransf2D then computes the intrinsic AI wavelet transform of P based on the given refinement orders and the chosen metric. The marginal refinement orders should be smaller or equal to 9, and the function computes the wavelet transform using a fast wavelet refinement scheme based on weighted intrinsic averages with pre-determined weights as explained in Chapter 5 of (Chau 2018). By default WavTransf2D computes the intrinsic 2D AI wavelet transform equipping the space of HPD matrices with (i) the affine-invariant Riemannian metric as detailed in e.g., (Bhatia 2009)[Chapter 6] or (Pennec et al. 2006). Instead, the space of HPD matrices can also be equipped with one of the following metrics; (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 and (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 2018) for more details. Note that this comes at the cost of increased computation time in comparison to one of the other metrics.

Value

The function returns a list with three components:

D

the 2D pyramid of wavelet coefficients. This is a list of arrays, where each 4-dimensional array contains the (d,d)-dimensional wavelet coefficients in a 2D grid of locations from the coarsest wavelet scale j = 0 up to the finest wavelet scale j = jmax.

D.white

the 2D 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 Chapter 5 of (Chau 2018).

M0

a numeric array containing the midpoint(s) at the coarsest scale j = 0 in the 2D midpoint pyramid.

Note

The function does not check for positive definiteness of the input matrices, and (depending on the specified metric) may fail if matrices are close to being singular.

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.

Pennec X, Fillard P, Ayache N (2006). “A Riemannian framework for tensor computing.” International Journal of Computer Vision, 66(1), 41–66.

See Also

InvWavTransf2D, pdSpecEst2D, pdNeville

Examples

P <- rExamples2D(c(2^4, 2^4), 2, example = "tvar")
P.wt <- WavTransf2D(P$f)