rExamples2D() generates several example (locally) smooth target surfaces of HPD matrices corrupted by
noise in a manifold of HPD matrices for testing and simulation purposes. For more details, see also Chapter 2 and 5 in
C18pdSpecEst.
rExamples2D(n, d = 2, example = c("smiley", "tvar", "facets", "peak"),
replicates = 1, noise = "riem-gaussian", noise.level = 1,
df.wishart = NULL)integer vector c(n1, n2) specifying the number of sampled matrices to be generated on a rectangular surface.
row- (resp. column-)dimension of the generated matrices. Defaults to d = 2.
the example target HPD matrix surface, one of 'smiley', 'tvar', 'facets' or 'peak'.
a positive integer specifying the number of replications of noisy HPD matrix surfaces to be generated based on the
target surface of HPD matrices. Defaults to replicates = 1
noise distribution for the generated noisy surfaces of HPD matrices, one of 'riem-gaussian',
'log-gaussian', 'wishart', 'log-wishart' or 'periodogram', defaults to 'riem-gaussian'.
Additional details are given below.
parameter to tune the signal-to-noise ratio for the generated noisy HPD matrix observations.
If noise.level = 0, the noise distributions are degenerate and the noisy HPD matrix observations coincide with the target HPD matrices.
Defaults to noise.level = 1.
optional parameter to specify the degrees of freedom in the case of a Wishart noise distribution (noise = 'wishart' or
noise = 'log-wishart'). By default df.wishart is equal to the dimension d to guarantee positive definiteness of the
generated noisy matrices.
Returns a list with two components:
f a (d,d,n[1],n[2])-dimensional array, corresponding to the \((n_1 \times n_2)\)-sized example target surface of
\((d,d)\)-dimensional HPD matrices.
P a (d,d,n[1],n[2])-dimensional array, corresponding to the \((n_1 \times n_2)\)-sized surface of noisy \((d,d)\)-dimensional
HPD matrices centered around the smooth target HPD matrix surface f. If replicates > 1, P is a
(d,d,n[1],n[2],length(replicates))-dimensional array, corresponding to a collection of replicated \((n_1 \times n_2)\)-sized surfaces
of noisy \((d,d)\)-dimensional HPD matrices centered around the smooth target HPD matrix surface f.
The examples include: (i) a \((d,d)\)-dimensional 'smiley' HPD matrix surface consisting of constant surfaces of random HPD matrices in
the shape of a smiley face; (ii) a \((d,d)\)-dimensional 'tvar' HPD matrix surface generated from a time-varying vector-auto-
regressive process of order 1 with random time-varying coefficient matrix (\(\Phi\)); (iii) a \((d,d)\)-dimensional 'facets' HPD matrix
surface consisting of several facets generated from random geodesic surfaces; and (iv) a \((d,d)\)-dimensional 'peak' HPD matrix surface
containing a pronounced peak in the center of its 2-d (e.g., time-frequency) domain.
In addition to the (locally) smooth target surface of HPD matrices, the function also returns a noisy version of the target surface of HPD matrices, corrupted
by a user-specified noise distribution. By default, the noisy HPD matrix observations follow an intrinsic signal plus i.i.d. noise model with
respect to the affine-invariant Riemannian metric, with a matrix log-Gaussian noise distribution (noise = 'riem-gaussian'), such that the
Riemannian Karcher means of the observations coincide with the target surface of HPD matrices. Additional details can be found in Chapters 2, 3,
and 5 of C18pdSpecEst. Other available signal-noise models include: (ii) a Log-Euclidean signal plus i.i.d. noise model, with
a matrix log-Gaussian noise distribution (noise = 'log-gaussian'); (iii) a Riemannian signal plus i.i.d. noise model, with a complex
Wishart noise distribution (noise = 'wishart'); (iv) a Log-Euclidean signal plus i.i.d. noise model, with a complex Wishart noise
distribution (noise = 'log-wishart').
# NOT RUN {
example <- rExamples2D(n = c(32, 32), example = "smiley")
# }
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