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

Its is associated to the Exponential covariance model. The Cauchy spectral model is defined by:

(1)S_{ij}(f) = \displaystyle \frac{4R_{ij}a_ia_j(\lambda_i+ \lambda_j)}{(\lambda_i+ \lambda_j)^2 + (4\pi f)^2}, \quad \forall (i,j) \leq d

where \mat{R}, \vect{a} and \vect{\lambda} are the parameters of the Exponential covariance model defined in section [ParamStationaryCovarianceFunction]. The relation (1) can be explained with the spatial covariance function \mat{C}^{spat}(\tau) defined in (6).