pdSpecEst1D {pdSpecEst}

Intrinsic wavelet HPD spectral estimation

Description

pdSpecEst1D calculates a (d,d)-dimensional HPD wavelet-denoised spectral matrix estimator by applying the following steps to an initial noisy HPD spectral estimate (obtained with e.g., pdPgram):

1. a forward intrinsic AI wavelet transform, with WavTransf1D,

2. (tree-structured) thresholding of the wavelet coefficients, with pdCART,

3. an inverse intrinsic AI wavelet transform, with InvWavTransf1D.

The complete estimation procedure is described in more detail in (Chau and von Sachs 2019) or Chapter 3 of (Chau 2018).

Usage

pdSpecEst1D(P, order = 5, metric = "Riemannian", alpha = 1,
  return_val = "f", ...)

Arguments

P

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.

order

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.

metric

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.

alpha

an optional tuning parameter in the wavelet thresholding procedure. The penalty (or sparsity) parameter in the tree-structured wavelet thresholding procedure in pdCART is set to alpha times the estimated universal threshold, defaults to alpha = 1.

return_val

an optional argument that specifies whether the denoised spectral estimator is returned or not. See the Details section below.

...

additional arguments for internal use.

Details

The input array P corresponds to an initial noisy HPD spectral estimate of the (d,d)-dimensional spectral matrix at m different frequencies, with m = 2^J for some J > 0. This can be e.g., a multitaper HPD periodogram given as output by the function pdPgram.
P is transformed to the wavelet domain by the function WavTransf1D, which applies an intrinsic 1D AI wavelet transform based on a metric specified by the user. The noise is removed by tree-structured thresholding of the wavelet coefficients based on the trace of the whitened coefficients with pdCART by minimization of a complexity penalized residual sum of squares (CPRESS) criterion via the fast tree-pruning algorithm in (Donoho 1997). The penalty or sparsity parameter in the optimization procedure is set equal to alpha times the universal threshold, where the noise variance of the traces of the whitened wavelet coefficients are determined from the finest wavelet scale. See (Chau and von Sachs 2019) and Chapter 3 of (Chau 2018) for further details.
The function computes the forward and 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.
If return_val = 'f' the thresholded wavelet coefficients are transformed back to the frequency domain by the inverse intrinsic 1D AI wavelet transform via InvWavTransf1D, returning the wavelet-denoised HPD spectral estimate.

Value

The function returns a list with the following five components:

f

a (d,d,m)-dimensional array of HPD matrices, corresponding to the HPD wavelet-denoised estimate of the same resolution as the input array P. If return_val != 'f', the inverse wavelet transform of the thresholded wavelet coefficients is not computed and f is set equal to NULL.

D

the pyramid of threshold wavelet coefficients. This is a list of arrays, where each array contains the (d,d)-dimensional thresholded wavelet coefficients from the coarsest wavelet scale j = 0 up to the finest wavelet scale j = jmax.

M0

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

tree.weights

a list of logical values specifying which coefficients to keep, with each list component corresponding to an individual wavelet scale starting from the coarsest wavelet scale j = 0.

D.raw

the pyramid of non-thresholded wavelet coefficients in the same format as the component $D.

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.

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.

Donoho D (1997). “CART and best-ortho-basis: a connection.” The Annals of Statistics, 25(5), 1870–1911.

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

See Also

pdPgram, WavTransf1D, InvWavTransf1D, pdCART

Examples

P <- rExamples1D(2^8, example = "bumps")$P
f <- pdSpecEst1D(P)