pdSpecClust1D performs clustering of HPD spectral matrices corrupted by noise (e.g. HPD periodograms)
by combining wavelet thresholding and fuzzy clustering in the intrinsic wavelet coefficient domain according to
the following steps:
Transform a collection of noisy HPD spectral matrices to the intrinsic wavelet domain and denoise the
HPD matrix curves by (tree-structured) thresholding of wavelet coefficients with
Apply an intrinsic fuzzy c-means algorithm to the coarsest midpoints at scale
j = 0 across subjects.
Taking into account the fuzzy cluster assignments in the previous step, apply a weighted fuzzy c-means
algorithm to the nonzero thresholded wavelet coefficients across subjects from scale
j = 1 up to
j = jmax.
More details can be found in Chapter 3 of \insertCiteC18pdSpecEst and the accompanying vignettes.
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a (d,d,n,S)-dimensional array of HPD matrices, corresponding to a collection of sequences of (d,d)-dimensional HPD matrices of length n, with n = 2^J for some J > 0, for S different subjects.
the number of clusters, a positive integer larger than 1.
an upper bound on the maximum wavelet scale to be considered in the clustering procedure. If
the metric that the space of HPD matrices is equipped with. The default choice is
the fuzziness parameter for both fuzzy c-means algorithms.
a proportion that is used to determine the maximum wavelet scale to be considered in the clustering
procedure. A larger value
an optional vector with two components determining the stopping criterion. The first step in the cluster procedure
terminates if the (integrated) intrinsic distance between cluster centers is smaller than
an optional argument tuning the weight given to the cluster assignments obtained in the first step of
the clustering algorithm. If
an optional argument tuning the maximum number of iterations in both the first and second step of the
clustering algorithm, defaults to
should the cluster centers transformed back the space of HPD matrices also be returned?
additional arguments passed on to
The input array
P corresponds to a collection of initial noisy HPD spectral estimates of the (d,d)-dimensional
spectral matrix at
n different frequencies, with n = 2^J for some J > 0, for S different subjects.
These can be e.g., multitaper HPD periodograms given as output by the function
First, for each subject s = 1,…,S, thresholded wavelet coefficients in the intrinsic wavelet domain are calculated by
pdSpecEst1D, see the function documentation for additional details on the wavelet thresholding
The maximum wavelet scale taken into account in the clustering procedure is determined by the arguments
d.jmax. The maximum scale is set to the minimum of
jmax and the wavelet
scale j for which the proportion of nonzero thresholded wavelet coefficients (averaged
across subjects) is smaller than
The S subjects are assigned to K different clusters in a probabilistic fashion according to a two-step procedure:
In the first step, an intrinsic fuzzy c-means algorithm, with fuzziness parameter m is applied to the
S coarsest midpoints at scale
j = 0 in the subject-specific midpoint pyramids. Note that the distance
function in the intrinsic c-means algorithm relies on the chosen metric on the space of HPD matrices.
In the second step, a weighted fuzzy c-means algorithm based on the Euclidean
distance function, also with fuzziness parameter m, is applied to the nonzero thresholded wavelet
coefficients of the S different subjects. The tuning parameter
tau controls the weight given
to the cluster assignments obtained in the first step of the clustering algorithm.
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., \insertCiteB09pdSpecEst[Chapter 6]
or \insertCitePFA05pdSpecEst; (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 \insertCiteCvS17pdSpecEst or \insertCiteC18pdSpecEst for more details. Note that this comes
at the cost of increased computation time in comparison to one of the other metrics.
return.centers = TRUE, the function also returns the
K HPD spectral matrix curves corresponding to
the cluster centers based on the given metric by applying the intrinsic inverse AI wavelet transform (
InvWavTransf1D) to the cluster centers in the wavelet domain.
Depending on the input the function returns a list with five or six components:
an (S,K)-dimensional matrix, where the value at position (s,k) in the matrix corresponds to the probabilistic cluster membership assignment of subject s with respect to cluster k.
a list of
K wavelet coefficient pyramids, where each pyramid of wavelet
coefficients is associated to a cluster center.
a list of
K arrays of coarse-scale midpoints at scale
j = 0, where each
array is associated to a cluster center.
only available if
return.centers = TRUE, returning a list of
K (d,d,n)-dimensional arrays,
where each array corresponds to a length n curve of (d,d)-dimensional HPD matrices associated to a cluster center.
the maximum wavelet scale taken into account in the clustering procedure determined by
the input arguments
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## ARMA(1,1) process: Example 11.4.1 in (Brockwell and Davis, 1991) Phi1 <- array(c(0.5, 0, 0, 0.6, rep(0, 4)), dim = c(2, 2, 2)) Phi2 <- array(c(0.7, 0, 0, 0.4, rep(0, 4)), dim = c(2, 2, 2)) Theta <- array(c(0.5, -0.7, 0.6, 0.8, rep(0, 4)), dim = c(2, 2, 2)) Sigma <- matrix(c(1, 0.71, 0.71, 2), nrow = 2) ## Generate periodogram data for 10 subjects in 2 groups pgram <- function(Phi) pdPgram(rARMA(2^9, 2, Phi, Theta, Sigma)$X)$P P <- array(c(replicate(5, pgram(Phi1)), replicate(5, pgram(Phi2))), dim=c(2,2,2^8,10)) cl <- pdSpecClust1D(P, K = 2, metric = "logEuclidean")
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