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.

getId()

Accessor to the object's id.

getImplementation()

Accessor to the underlying implementation.

getInputDimension()

Get the input dimension of the spectral density function.

getName()

Accessor to the object's name.

getOutputCorrelation()

Get the spatial correlation matrix of the spectral density function.

getOutputDimension()

Get the dimension of the SpectralModel.

getScale()

Get the scale parameter 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
taufloat

Frequency value.

Returns
rhoComplex

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
rowIndexint, 0 \leq rowIndex < dimension

The row index of the component to draw. Default value is 0.

columnIndex: int, :math:`0 leq columnIndex < dimension`

The column index of the component to draw. Default value is 0.

minimumFrequencyfloat

The lower bound of the frequency range over which the model is plotted. Default value is SpectralModel-DefaultMinimumFrequency in ResourceMap.

maximumFrequencyfloat

The upper bound of the frequency range over which the model is plotted. Default value is SpectralModel-DefaultMaximumFrequency in ResourceMap.

frequencyNumberint, pointNumber \geq 2

The discretization of the frequency range [minimumFrequency, maximumFrequency] over which the model is plotted. Default value is SpectralModel-DefaultFrequencyNumber in class:~openturns.ResourceMap.

modulebool

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

Returns
graphGraph

Graphic of the specified component

getAmplitude()

Get the amplitude parameter of the spectral density function.

Returns
amplitudePoint

The used amplitude parameter.

getClassName()

Accessor to the object’s name.

Returns
class_namestr

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

getId()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getImplementation()

Accessor to the underlying implementation.

Returns
implImplementation

The implementation class.

getInputDimension()

Get the input dimension of the spectral density function.

Returns
inputDimensionint

SpatialDimension of the SpectralModel.

getName()

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOutputCorrelation()

Get the spatial correlation matrix of the spectral density function.

Returns
spatialCorrelationCorrelationMatrix

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

getOutputDimension()

Get the dimension of the SpectralModel.

Returns
dimensionint

Dimension of the SpectralModel.

getScale()

Get the scale parameter of the spectral density function.

Returns
scalePoint

The used scale parameter.

setAmplitude(amplitude)

Set the amplitude parameter of the spectral density function.

Parameters
amplitudePoint

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

setName(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setScale(scale)

Set the scale parameter of the spectral density function.

Parameters
scalePoint

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