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 
userspecified 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 signaltonoise 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 'twocats'
HPD matrix curve visualizing the contour of two sidebyside 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
'mixgaussian'
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 2dimensional 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 userspecified noise distribution. By default, the noisy HPD matrix observations follow an intrinsic signal plus i.i.d. noise model with
respect to the affineinvariant Riemannian metric, with a matrix logGaussian noise distribution (noise = 'riemgaussian'
), 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 signalnoise models include: (ii) a LogEuclidean signal plus i.i.d. noise model, with
a matrix logGaussian noise distribution (noise = 'loggaussian'
); (iii) a Riemannian signal plus i.i.d. noise model, with a complex
Wishart noise distribution (noise = 'wishart'
); (iv) a LogEuclidean signal plus i.i.d. noise model, with a complex Wishart noise
distribution (noise = 'logwishart'
); 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 ddimensional 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 pseudoperiodogram 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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