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

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

A copy of the underlying implementation object.

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.

Examples using the class

Estimate a spectral density function

Estimate a spectral density function