rExamples1D {pdSpecEst}

Several example curves of HPD matrices

Description

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 (Chau 2018).

Usage

rExamples1D(n, d = 3, example = c("bumps", "two-cats", "heaviSine",
  "gaussian", "mix-gaussian", "arma", "peaks", "blocks"), user.f = NULL,
  return.ts = FALSE, replicates = 1, noise = "riem-gaussian",
  noise.level = 1, df.wishart = NULL, nblocks = 10)

Arguments

n

number of sampled matrices to be generated.

d

row- (resp. column-)dimension of the generated matrices. Defaults to d = 3.

example

the example target HPD matrix curve, one of 'bumps', 'two-cats', 'heaviSine', 'gaussian', 'mix-gaussian', 'arma', 'peaks' or 'blocks'.

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 return.ts = TRUE the function also returns time series observations generated via the Cramer representation based on the transfer function of the example HPD spectral matrix and complex normal random variates. Defaults to return.ts = FALSE.

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 replicates = 1

noise

noise distribution for the generated noisy curves of HPD matrices, one of 'riem-gaussian', 'log-gaussian', 'wishart', 'log-wishart' or 'periodogram', defaults to 'riem-gaussian'. Additional details are given below.

noise.level

parameter to tune the signal-to-noise ratio for the generated noisy HPD matrix observations, only used if noise != 'periodogram'. 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.

df.wishart

optional parameter to specify the degrees of freedom in the case of a Wishart noise distribution (noise = 'wishart' or noise = 'log-wishart'); or the number of DPSS tapers in the case of generated periodogram matrices if noise = 'periodogram'. By default df.wishart is equal to the dimension d to guarantee positive definiteness of the generated noisy matrices.

nblocks

optional parameter to specify the number of constant segments in the 'blocks' HPD matrix curve. Only used if example = 'blocks'.

Details

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 (Chau 2018). 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'.

Value

Depending on the input arguments returns a list with two or three components:

f

a (d,d,n)-dimensional array, corresponding to the length n example target curve of (d,d)-dimensional HPD matrices.

P

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 f. If replicates > 1, P is a (d,d,n,length(replicates))-dimensional array, corresponding to a collection of replicated length n curves of noisy (d,d)-dimensional HPD matrices centered around the smooth target HPD matrix curve f.

ts

generated d-dimensional time series observations, only available if return.ts = TRUE.

Note

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., (Brillinger 1981).

References

Brillinger D (1981). Time Series: Data Analysis and Theory. Holden-Day, San Francisco.

Chau J (2018). Advances in Spectral Analysis for Multivariate, Nonstationary and Replicated Time Series. phdthesis, Universite catholique de Louvain.

See Also

rExamples2D, pdPgram, rARMA

Examples

example <- rExamples1D(100, example = "bumps", return.ts = TRUE)
plot.ts(Re(example$ts), main = "3-d time series") # plot generated time series