Description Usage Arguments Details Value Note References See Also Examples
rExamples1D()
generates several example (locally) smooth target curves 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 3 in
\insertCiteC18pdSpecEst.
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n |
number of sampled matrices to be generated. |
d |
row- (resp. column-)dimension of the generated matrices. Defaults to |
example |
the example target HPD matrix curve, one of |
user.f |
user-specified target HPD matrix curve, should be a (d,d,n)-dimensional array, corresponding to a length n curve of (d,d)-dimensional HPD matrices. |
return.ts |
a logical value, if |
replicates |
a positive integer specifying the number of replications of noisy HPD matrix curves to be generated based on the
target curve of HPD matrices. Defaults to |
noise |
noise distribution for the generated noisy curves of HPD matrices, one of |
noise.level |
parameter to tune the signal-to-noise ratio for the generated noisy HPD matrix observations, only used if |
df.wishart |
optional parameter to specify the degrees of freedom in the case of a Wishart noise distribution ( |
nblocks |
optional parameter to specify the number of constant segments in the |
The examples include: (i) a (3,3)-dimensional 'bumps'
HPD matrix curve containing peaks and bumps of various smoothness degrees;
(ii) a (3,3)-dimensional 'two-cats'
HPD matrix curve visualizing the contour of two side-by-side cats, with inhomogeneous
smoothness across the domain; (iii) a (3,3)-dimensional 'heaviSine'
HPD matrix curve consisting of smooth sinosoids with a break;
(iv) a (2,2)-dimensional 'gaussian'
HPD matrix curve consisting of smooth Gaussian functions; (v) a (d,d)-dimensional
'mix-gaussian'
HPD matrix curve consisting of a weighted linear combination of smooth Gaussian functions; (vi) a (2,2)-dimensional
'arma'
HPD matrix curve generated from the smooth spectral matrix of a 2-dimensional stationary ARMA(1,1)-process; (vii) a (d, d)-
dimensional 'peaks'
HPD matrix curve containing several sharp peaks across the domain; and (viii) a (d, d)-'blocks'
HPD matrix
curve generated from locally constant segments of HPD matrices.
In addition to the smooth target curve of HPD matrices, the function also returns a noisy version of the target curve 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 curve of HPD matrices. Additional details can be found in Chapters 2, 3,
and 5 of \insertCiteC18pdSpecEst. 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'
); and (v) noisy periodogram observations obtained with pdPgram
from a stationary time series
generated via the Cramer representation based on the transfer function of the target HPD spectral matrix curve and complex normal random variates
(noise = 'periodogram'
). If return.ts = TRUE
, the function also returns the generated time series observations, which are not generated
by default if noise != 'periodogram'
.
Depending on the input arguments returns a list with two or three components:
|
a (d,d,n)-dimensional array, corresponding to the length n example target curve of (d,d)-dimensional HPD matrices. |
|
a (d,d,n)-dimensional array, corresponding to a length n curve of noisy (d,d)-dimensional
HPD matrices centered around the smooth target HPD matrix curve |
|
generated d-dimensional time series observations, only available if |
If noise = 'wishart'
, the generated noisy HPD matrix observations are independent complex Wishart matrices, which can be
interpreted informally as pseudo-periodogram matrix observations, as the periodogram matrices based on strictly stationary time series
observations obtained with noise = 'periodogram'
are asymptotically independent and asymptotically complex Wishart distributed,
see e.g., \insertCiteB81pdSpecEst.
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