Parametric spectral density functions

Let X: \Omega \times \cD \rightarrow \Rset^{\inputDim} be a multivariate stationary normal process of dimension \inputDim. We only treat here the case where the domain is of dimension 1: \cD \in \Rset. 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}^+(\inputDim) defined in (8) exists. \mathcal{H}^+(\inputDim) \in \mathcal{M}_{\inputDim}(\Cset) is the set of \inputDim-dimensional positive definite Hermitian matrices. This page illustrates how to 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.

Example: 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.