ExponentialCauchy

class ExponentialCauchy(*args)

Absolute exponential covariance model.

Available constructors:

ExponentialCauchy(theta, sigma)
Parameters:

theta : sequence of floats

Scale coefficients. The spatial dimension is the size of vector.

sigma : sequence of floats

Amplitude coefficients \sigma. Should be of size 1

Notes

The second order model instanciate both a openturns.AbsoluteExponential as covariance function and openturns.CauchyModel as spectral model.

Examples

>>> import openturns as ot
>>> model = ot.ExponentialCauchy([10.0, 10.0], [1.0])
>>> t = [0.1, 0.3]
>>> s = [0.2, 0.4]
>>> print(model.computeCovariance(s, t))
[[ 0.980199 ]]
>>> tau = [0.1, 0.3]
>>> print(model.computeCovariance(tau))
[[ 0.960789 ]]
>>> f = 0.3
>>> print(model.computeSpectralDensity(f))
[[ (0.00315075,0) ]]
>>> f = 10
>>> print(model.computeSpectralDensity(f))
[[ (2.56648e-09,0) ]]

Methods

computeCovariance(*args) Evaluate the covariance function.
computeSpectralDensity(frequency) Evaluate the spectral density function for a specific frequency.
discretize(timeGrid) Discretize the second order on a given RegularGrid/Mesh model using its covariance function.
getAmplitude() Get the amplitude parameter of the second order model.
getClassName() Accessor to the object’s name.
getCovarianceModel() Return the covariance model.
getDimension() Get the dimension of the SecondOrderModel.
getId() Accessor to the object’s id.
getName() Accessor to the object’s name.
getScale() Get the scale parameter of the second order model.
getShadowedId() Accessor to the object’s shadowed id.
getSpatialDimension() Get the spatial dimension of the spectral density function.
getSpectralModel() Return the spectral model.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
setModels(covarianceModel, spectralModel) Set both the covariance and spectral models of a second order model.
setName(name) Accessor to the object’s name.
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
computeCovariance(*args)

Evaluate the covariance function.

Available usages:

computeCovariance(s, t)

computeCovariance(tau)

Parameters:

s, t : floats or sequence of floats.

Inputs.

tau : float or sequence of floats.

Input.

Returns:

covariance : CovarianceMatrix

The evaluation of the covariance function.

Notes

computeCovariance evaluates the covariance model C : \cD \times \cD \mapsto  \cM_{d \times d}(\Rset) at (s,t)\in \Rset^n:

C(\vect{s}, \vect{t})=\Expect{(X_{\vect{s}}-m(\vect{s}))\Tr{(X_{\vect{t}}-m(\vect{t}))}}

We note that the first usage calls the second as model is stationary. Thus,
C(\vect{s}, \vect{t}) = C^{stat}(\vect{\tau}) with \vect{\tau}=\vect{s}-\vect{t}.
computeSpectralDensity(frequency)

Evaluate the spectral density function for a specific frequency.

Parameters:

f : float

Frequency value.

Returns:

spd : HermitianMatrixs

The evaluation of spectral density function at frequency f.

Notes

computeSpectralDensity evaluates the spectral model S : \Rset^n \mapsto  \cH^+_{d} at f\in \Rset^n:

\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

discretize(timeGrid)

Discretize the second order on a given RegularGrid/Mesh model using its covariance function.

Parameters:

meshOrGrid : Mesh or RegularGrid

Mesh or time grid of size N associated with the process.

Returns:

covarianceMatrix : CovarianceMatrix

Covariance matrix \in\cM_{nd\times nd}(\Rset) (if the process is of dimension d).

Notes

This method makes a discretization of the covariance model on meshOrGrid composed of the vertices (\vect{t}_1, \dots, \vect{t}_{N-1}) and returns the covariance matrix:

\mat{C}_{1,\dots,k} = \left(
    \begin{array}{cccc}
    C(\vect{t}_1, \vect{t}_1) &C(\vect{t}_1, \vect{t}_2) & \dots & C(\vect{t}_1, \vect{t}_{k}) \\
    \dots & C(\vect{t}_2, \vect{t}_2)  & \dots & C(\vect{t}_2, \vect{t}_{k}) \\
    \dots & \dots & \dots & \dots \\
    \dots & \dots & \dots & C(\vect{t}_{k}, \vect{t}_{k})
    \end{array} \right)

getAmplitude()

Get the amplitude parameter of the second order model.

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__).

getCovarianceModel()

Return the covariance model.

Returns:

covarianceModel : CovarianceModel

The covariance model of the second order model.

getDimension()

Get the dimension of the SecondOrderModel.

Returns:

dimension : int

Dimension of the SecondOrderModel.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getScale()

Get the scale parameter of the second order model.

Returns:

scale : NumericalPoint

The used scale parameter.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getSpatialDimension()

Get the spatial dimension of the spectral density function.

Returns:

spatialDimension : int

SpatialDimension of the SecondOrderModel.

getSpectralModel()

Return the spectral model.

Returns:

spectralModel : SpectralModel

The spectral model of the second order model.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

setModels(covarianceModel, spectralModel)

Set both the covariance and spectral models of a second order model.

Parameters:

covarianceModel : CovarianceModel

The covariance model of the second order model.

spectralModel : SpectralModel

The spectral model of the second order model.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.