SpectralModel

class SpectralModel(*args)

Spectral density model.

Notes

We consider X: \Omega \times\cD \mapsto \Rset^d a multivariate stochastic process of dimension d, where \omega \in \Omega is an event, \cD is a domain of \Rset^n, \vect{t}\in \cD is a multivariate index and X(\omega, \vect{t}) \in \Rset^d.

We note X_{\vect{t}}: \Omega \rightarrow \Rset^d the random variable at index \vect{t} \in \cD defined by X_{\vect{t}}(\omega)=X(\omega, \vect{t}) and X(\omega): \cD  \mapsto \Rset^d a realization of the process X, for a given \omega \in \Omega defined by X(\omega)(\vect{t})=X(\omega, \vect{t}).

If the process is a second order process, zero-mean and weakly stationary, we define its bilateral spectral density function S : \Rset^n \mapsto  \cH^+_{d} with:

  • \cH^+(d) \in \cM^+(d)(\Cset) is the set of d-dimensional positive hermitian matrices

Using the stationary covariance function C^{stat} with C^{stat} : \cD \mapsto  \cM_{d \times d}(\Rset) and the Fourier transform, the spectral density writes:

\forall \vect{f} \in \Rset^n, \cS(\vect{f}) = \int_{\Rset^n} \exp\left(-2i \pi <\vect{f}, \vect{\tau}>\right) C^{stat}(\vect{\tau})\di{\vect{\tau}}

A SpectralModel object can be created only through its derived classes: CauchyModel

Methods

__call__(frequency) Evaluate the spectral density function for a specific frequency.
computeStandardRepresentative(frequency) Compute the standard representant of the spectral density function.
draw(*args) Draw a specific component of the spectral density function.
getAmplitude() Get the amplitude parameter of the spectral density function.
getClassName() Accessor to the object’s name.
getDimension() Get the dimension of the SpectralModel.
getId() Accessor to the object’s id.
getImplementation(*args) Accessor to the underlying implementation.
getName() Accessor to the object’s name.
getScale() Get the scale parameter of the spectral density function.
getSpatialCorrelation() Get the spatial correlation matrix of the spectral density function.
getSpatialDimension() Get the spatial dimension of the spectral density function.
setAmplitude(amplitude) Set the amplitude parameter of the spectral density function.
setName(name) Accessor to the object’s name.
setScale(scale) Set the scale parameter of the spectral density function.
__init__(*args)
computeStandardRepresentative(frequency)

Compute the standard representant of the spectral density function.

Parameters:

tau : float

Frequency value.

Returns:

rho : Complex

Standard representant factor of the spectral density function.

Notes

According to definitions in CovarianceModel, as the spectral density function is the Fourier transform of the stationary covariance function and using the expression of the last one, the spectral density function writes as a matrix-complex product where the matrix is the constant spatial covariance structure and the complex represents the standard representative:

Thus,

\forall \vect{f} \in \Rset^n, \cS(\vect{f}) = \prod_{k=1}^{n} \vect{\theta}_k \mat{\Sigma} \rho(\vect{f} \odot \vect{\theta})

where \mat{\Sigma} is a covariance matrix that explains the covariance structure and (\vect{f} \odot \vect{\theta})_k = \vect{f}_k \vect{\theta}_k

draw(*args)

Draw a specific component of the spectral density function.

Parameters:

rowIndex : int

i-th row component to draw. Default value is 0

columnIndex : int

j-th column component to draw . Default value is 0

module : bool

Tells if module has to be drawn(True) or if it is the argument to be drawn (False). Default value is True.

Returns:

graph : Graph

Graphic of the specified component

Notes

The method plots a specific component of the spectral density function over a frequency grid. Thus, this last one should be specified thanks to the setFrequencyGrid method.

getAmplitude()

Get the amplitude parameter of the spectral density function.

Returns:

amplitude : NumericalPoint

The used amplitude parameter.

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

The object class name (object.__class__.__name__).

getDimension()

Get the dimension of the SpectralModel.

Returns:

dimension : int

Dimension of the SpectralModel.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getImplementation(*args)

Accessor to the underlying implementation.

Returns:

impl : Implementation

The implementation class.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getScale()

Get the scale parameter of the spectral density function.

Returns:

scale : NumericalPoint

The used scale parameter.

getSpatialCorrelation()

Get the spatial correlation matrix of the spectral density function.

Returns:

spatialCorrelation : CorrelationMatrix

Correlation matrix \mat{R} \in \mathcal{M}_{dimension \times dimension}([-1, 1]).

getSpatialDimension()

Get the spatial dimension of the spectral density function.

Returns:

spatialDimension : int

SpatialDimension of the SpectralModel.

setAmplitude(amplitude)

Set the amplitude parameter of the spectral density function.

Parameters:

amplitude : NumericalPoint

The amplitude parameter to be used in the spectral density function.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setScale(scale)

Set the scale parameter of the spectral density function.

Parameters:

scale : NumericalPoint

The scale parameter to be used in the spectral density function. It should be of size dimension.