Parametric spectral density functions

Let X: \Omega \times \cD \rightarrow \Rset^d be a multivariate stationary normal process of dimension d. We only treat here the case where the domain is of dimension 1: \cD \in \Rset (n=1).
If the process is continuous, then \cD=\Rset. In the discrete case, \cD is a lattice.
X is supposed to be a second order process with zero mean and we suppose that its spectral density function S : \Rset \rightarrow \mathcal{H}^+(d) defined in (8) exists. \mathcal{H}^+(d) \in \mathcal{M}_d(\Cset) is the set of d-dimensional positive definite hermitian matrices.
This use case illustrates how the User can create a density spectral function from parametric models. The library proposes the Cauchy spectral model as a parametric model for the spectral density function S.

The Cauchy spectral model

It is associated to the Kronecker covariance model built upon an exponential covariance model (AbsoluteExponential). The Cauchy spectral model is defined by:

(1)[S(f)]_{ij} = \displaystyle 2 \mat{\Sigma}_{ij}\prod_{k=1}^{n} \frac{\theta_k}{1 + (2\pi \theta_k f)^2}, \quad \forall (i,j) \leq d

where \mat{\Sigma} is the covariance matrix of the Kronecker covariance model and \vect{\theta} = (\theta_1, \dots, \theta_n) is the vector of scale parameters of the AbsoluteExponential covariance model.

API:

Examples: