SphericalModel¶

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class SphericalModel(*args)¶

Spherical covariance function.

Available constructors:

SphericalModel(spatialDim=1)

SphericalModel(scale, amplitude)

SphericalModel(scale, amplitude, radius)

Parameters

spatialDimint: Spatial dimension $n$ . By default, equal to 1.
scalesequence of positive floats: Scale coefficient $\vect{\theta}\in \Rset^n$ . The size of $\vect{\theta}$ is the input dimension.
amplitudesequence of positive floats: Amplitude of the process $\vect{\sigma} \in \Rset^d$ . Must be of size equal to 1. By default, equal to $[1]$ .
radiusfloat, $a > 0$: Radius of the sphere on which the covariance model is not zero. By default, equal to 1.

See also

CovarianceModel

Notes

The spherical function is a stationary covariance function with dimension $d=1$ .

We consider the scalar stochastic process $X: \Omega \times\cD \mapsto \Rset$ , where $\omega \in \Omega$ is an event, $\cD$ is a domain of $\Rset^n$ .

The spherical function is defined on the sphere which ray is $a$ :

$C(\vect{s}, \vect{t}) = \sigma^2 \left[1 - \frac{1}{2a} \left\|\dfrac{\vect{s}-\vect{t}}{\vect{\theta}}\right\|_2 \left(3 - \frac{1}{a^2}\left\|\dfrac{\vect{s}-\vect{t}}{\vect{\theta}}\right\|_2^2\right) \right], \quad \forall (\vect{s}, \vect{t}), \left\|\dfrac{\vect{s}-\vect{t}}{\vect{\theta}}\right\|_2 \leq a$

The function is equal to zero outside the sphere.

The correlation function $\rho$ writes:

$\rho(\vect{s}, \vect{t}) = 1 - \frac{1}{2a} \left\|\vect{s} - \vect{t}\right\|_2 \left(3 - \frac{1}{a^2}\left\| \vect{s} - \vect{t} \right\|_2^2\right), \quad \forall (\vect{s}, \vect{t}), \left\| \vect{s} - \vect{t} \right\|_2 \leq a$

and is equal to zero outside the sphere.

Examples

Create a standard spherical covariance function:

>>> import openturns as ot
>>> covModel = ot.SphericalModel(2)
>>> t = [0.1, 0.3]
>>> s = [0.2, 0.4]
>>> print(covModel(s, t))
[[ 0.789282 ]]
>>> tau = [0.1, 0.3]
>>> print(covModel(tau))
[[ 0.54147 ]]

Create a spherical covariance function specifying the scale, amplitude vectors:

>>> covarianceModel = ot.SphericalModel([0.2, 0.3], [2.5])

Create a squared exponential covariance function specifying the scale vector, the amplitude and radius:

>>> covModel3 = ot.SphericalModel([0.2, 0.3], [2.5], 2.3)

Methods

`__call__`(*args)	Evaluate the covariance function.
`computeAsScalar`(*args)	Compute the covariance function for scalar model.
`computeCrossCovariance`(*args)	computeCrossCovariance the covariance function on a given mesh.
`discretize`(*args)	Discretize the covariance function on a given mesh.
`discretizeAndFactorize`(*args)	Discretize and factorize the covariance function on a given mesh.
`discretizeAndFactorizeHMatrix`(*args)	Discretize and factorize the covariance function on a given mesh.
`discretizeHMatrix`(*args)	Discretize the covariance function on a given mesh using HMatrix result.
`discretizeRow`(vertices, p)	(TODO)
`draw`(*args)	Draw a specific component of the covariance model with input dimension 1.
`getActiveParameter`()	Accessor to the active parameter set.
`getAmplitude`()	Get the amplitude parameter $\vect{\sigma}$ of the covariance function.
`getClassName`()	Accessor to the object's name.
`getFullParameter`()	Get the full parameters of the covariance function.
`getFullParameterDescription`()	Get the description full parameters of the covariance function.
`getId`()	Accessor to the object's id.
`getInputDimension`()	Get the input dimension $n$ of the covariance function.
`getMarginal`(*args)	Get the ith marginal of the model.
`getName`()	Accessor to the object's name.
`getNuggetFactor`()	Accessor to the nugget factor.
`getOutputCorrelation`()	Get the spatial correlation matrix $\mat{R}$ of the covariance function.
`getOutputDimension`()	Get the dimension $d$ of the covariance function.
`getParameter`()	Get the parameters of the covariance function.
`getParameterDescription`()	Get the description of the covariance function parameters.
`getRadius`()	Accessor to the radius.
`getScale`()	Get the scale parameter $\vect{\theta}$ of the covariance function.
`getShadowedId`()	Accessor to the object's shadowed id.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`isDiagonal`()	Test whether the model is diagonal or not.
`isStationary`()	Test whether the model is stationary or not.
`parameterGradient`(s, t)	Compute the gradient according to the parameters.
`partialGradient`(s, t)	Compute the gradient of the covariance function.
`setActiveParameter`(active)	Accessor to the active parameter set.
`setAmplitude`(amplitude)	Set the amplitude parameter $\vect{\sigma}$ of the covariance function.
`setFullParameter`(parameter)	Set the full parameters of the covariance function.
`setName`(name)	Accessor to the object's name.
`setNuggetFactor`(nuggetFactor)	Set the nugget factor for the variance of the observation error.
`setOutputCorrelation`(correlation)	Set the spatial correlation matrix $\mat{R}$ of the covariance function.
`setParameter`(parameter)	Set the parameters of the covariance function.
`setRadius`(radius)	Accessor to the radius.
`setScale`(scale)	Set the scale parameter $\vect{\theta}$ of the covariance function.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

computeAsScalar(*args)¶

Compute the covariance function for scalar model.

Available usages:

computeAsScalar(s, t)

computeAsScalar(tau)

Parameters

s, tfloats (if $n=1$ ) or sequences of floats (any $n$ ): Multivariate index $(\vect{s}, \vect{t}) \in \cD \times \cD$
taufloat (if $n=1$ ) or sequence of floats (any $n$ ): Multivariate index $\vect{\tau} \in \cD$

Returns

covariancefloat: Covariance.

Notes

The method makes sense only if the dimension of the process is $d=1$ . It evaluates $C(\vect{s}, \vect{t})$ .

In the second usage, the covariance model must be stationary. Then we note $C^{stat}(\vect{\tau})$ for $C(\vect{s}, \vect{s}+\vect{\tau})$ as this quantity does not depend on $\vect{s}$ .

computeCrossCovariance(*args)¶

computeCrossCovariance the covariance function on a given mesh.

Parameters

firstVerticesSample or Point: Container of the first discretization vertices
secondVerticesSample or Point: Container of the second discretization vertices

Returns

MatrixMatrix: Container of the cross covariance

Notes

This method computes a cross-covariance matrix. The cross-covariance is the evaluation of the covariance model on both firstVertices and secondVertices.

If $firstVertices$ contains $n_1$ points and $secondVertices$ contains $n_2$ points, the method returns an $n_1 d \times n_2 d$ matrix ( $d$ being the output dimension).

To make things easier, let us focus on the $d=1$ case. Let $\vect{x}_0, \dots, \vect{x}_{n_1-1}$ be the points of firstVertices and let $\vect{y}_0, \dots, \vect{y}_{n_2-1}$ be the points of secondVertices. The result is the $n_1 \times n_2$ matrix $\mat{M}$ such that for any nonnegative integers $i < n_1$ and $j < n_2$ , $\mat{M}_{i,j} = \mathcal{C}(\vect{x}_i, \vect{y}_j)$ .

discretize(*args)¶

Discretize the covariance function on a given mesh.

Parameters

whereMesh or RegularGrid or Sample: Container of the discretization vertices

Returns

covarianceMatrixCovarianceMatrix: Covariance matrix $\in \cS_{nd}^+(\Rset)$ (if the process is of dimension $d$ )

Notes

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

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

discretizeAndFactorize(*args)¶

Discretize and factorize the covariance function on a given mesh.

Parameters

whereMesh or RegularGrid or Sample: Container of the discretization vertices

Returns

CholeskyMatrixTriangularMatrix: Cholesky factor of the covariance matrix $\in \cM_{nd\times nd}(\Rset)$ (if the process is of dimension $d$ )

Notes

This method makes a discretization of the model on the given Mesh, RegularGrid or Sample composed of the vertices $(\vect{t}_1, \dots, \vect{t}_{N-1})$ thanks to the discretize() method and returns its Cholesky factor.

discretizeAndFactorizeHMatrix(*args)¶

Discretize and factorize the covariance function on a given mesh.

This uses HMatrix.

Parameters

whereMesh or RegularGrid or Sample: Container of the discretization vertices
hmatParamHMatrixParameters: Parameter values for the HMatrix

Returns

HMatrixHMatrix: Cholesk matrix $\in \cS_{nd}^+(\Rset)$ (if the process is of dimension $d$ ), stored in hierarchical format (H-Matrix)

Notes

This method is similar to the discretizeAndFactorize() method. This method requires that requires that OpenTURNS has been compiled with the hmat library. The method is helpful for very large parameters (Mesh, grid, Sample) because it compresses data.

discretizeHMatrix(*args)¶

Discretize the covariance function on a given mesh using HMatrix result.

Parameters

whereMesh or RegularGrid or Sample: Container of the discretization vertices
hmatParamHMatrixParameters: Parameter values for the HMatrix

Returns

HMatrixHMatrix: Covariance matrix $\in\cS_{nd}^+(\Rset)$ (if the process is of dimension $d$ ), stored in hierarchical format (H-Matrix)

Notes

This method is similar to the discretize() method. This method requires that OpenTURNS has been compiled with the hmat library. The method is helpful for very large parameters (Mesh, grid, Sample) because it compresses data.

discretizeRow(vertices, p)¶: (TODO)

draw(*args)¶

Draw a specific component of the covariance model with input dimension 1.

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.
tMinfloat: The lower bound of the range over which the model is plotted. Default value is CovarianceModel-DefaultTMin in ResourceMap.
tMaxfloat: The upper bound of the range over which the model is plotted. Default value is CovarianceModel-DefaultTMax in ResourceMap.
pointNumberint, $pointNumber \geq 2$: The discretization of the range $[tMin,tMax]$ over which the model is plotted. Default value is CovarianceModel-DefaultPointNumber in class:~openturns.ResourceMap.
asStationarybool: Flag to tell if the model has to be plotted as a stationary model, ie as a function of the lag $\tau=t-s$ if equals to True, or as a non-stationary model, ie as a function of $(s,t)$ if equals to False. Default value is True.
correlationFlagbool: Flag to tell if the model has to be plotted as a correlation function if equals to True or as a covariance function if equals to False. Default value is False.

Returns

graphGraph: A graph containing a unique curve if asStationary=True and if the model is actually a stationary model, or containing the iso-values of the model if asStationary=False or if the model is nonstationary.

getActiveParameter()¶

Accessor to the active parameter set.

Returns

activeIndices: Indices of the active parameters.

getAmplitude()¶

Get the amplitude parameter $\vect{\sigma}$ of the covariance function.

Returns

amplitudePoint: The amplitude parameter $\vect{\sigma} \in \Rset^d$ of the covariance function.

getClassName()¶

Accessor to the object’s name.

Returns

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

getFullParameter()¶

Get the full parameters of the covariance function.

Returns

parameterPoint: List the full parameter of the covariance function i.e. scale parameter $\vect{\theta} \in \Rset^n$ , the the amplitude parameter $\vect{\sigma} \in \Rset^d$ , the Spatial correlation parameter $\mat{R} \in \cS_d^+([-1,1])$ ; and potential other parameter depending on the model;

getFullParameterDescription()¶

Get the description full parameters of the covariance function.

Returns

descriptionDescription: Description of the full parameter of the covariance function.

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getInputDimension()¶

Get the input dimension $n$ of the covariance function.

Returns

inputDimensionint: Spatial dimension $n$ of the covariance function.

getMarginal(*args)¶

Get the ith marginal of the model.

Returns

marginalint or sequence of int: index of marginal of the model.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getNuggetFactor()¶

Accessor to the nugget factor.

This parameter allows smooth predictions from noisy data. The nugget is added to the diagonal of the assumed training covariance (thanks to discretize) and acts as a Tikhonov regularization in the problem.

Returns

nuggetFactorfloat: Nugget factor used to model the observation error variance.

getOutputCorrelation()¶

Get the spatial correlation matrix $\mat{R}$ of the covariance function.

Returns

spatialCorrelationCorrelationMatrix: Correlation matrix $\mat{R} \in \cS_d^+(\Rset)$ .

getOutputDimension()¶

Get the dimension $d$ of the covariance function.

Returns

dint: Dimension $d$ such that $C : \cD \times \cD \mapsto \cS_d^+(\Rset).$ This is the dimension of the process $X$ .

getParameter()¶

Get the parameters of the covariance function.

Returns

parametersPoint: List of the scale parameter $\vect{\theta} \in \Rset^n$ and the amplitude parameter $\vect{\sigma} \in \Rset^d$ of the covariance function.

The other specific parameters are not included.

getParameterDescription()¶

Get the description of the covariance function parameters.

Returns

descriptionParamDescription: Description of the components of the parameters obtained with the getParameter method..

getRadius()¶

Accessor to the radius.

Returns

radiusfloat, $a > 0$: Radius of the sphere on which the covariance model is not zero.

getScale()¶

Get the scale parameter $\vect{\theta}$ of the covariance function.

Returns

scalePoint: The scale parameter $\vect{\theta} \in \Rset^n$ used in the covariance function.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

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

isDiagonal()¶

Test whether the model is diagonal or not.

Returns

isDiagonalbool: True if the model is diagonal.

isStationary()¶

Test whether the model is stationary or not.

Returns

isStationarybool: True if the model is stationary.

Notes

The covariance function $C$ is stationary when it is invariant by translation:

$\forall(\vect{s},\vect{t},\vect{h}) \in \cD \times \cD, & \, \quad C(\vect{s}, \vect{s}+\vect{h}) = C(\vect{t}, \vect{t}+\vect{h})$

We note $C^{stat}(\vect{\tau})$ for $C(\vect{s}, \vect{s}+\vect{\tau})$ .

parameterGradient(s, t)¶

Compute the gradient according to the parameters.

Parameters

s, tsequences of float: Multivariate index $(\vect{s}, \vect{t}) \in \cD \times \cD$ .

Returns

gradientMatrix: Gradient of the function according to the parameters.

partialGradient(s, t)¶

Compute the gradient of the covariance function.

Parameters

s, tfloats or sequences of float: Multivariate index $(\vect{s}, \vect{t}) \in \cD \times \cD$ .

Returns

gradientMatrix: Gradient of the covariance function.

setActiveParameter(active)¶

Accessor to the active parameter set.

Parameters

activesequence of int: Indices of the active parameters.

setAmplitude(amplitude)¶

Set the amplitude parameter $\vect{\sigma}$ of the covariance function.

Parameters

amplitudePoint: The amplitude parameter $\vect{\sigma} \in \Rset^d$ to be used in the covariance function. Its size must be equal to the dimension of the covariance function.

setFullParameter(parameter)¶

Set the full parameters of the covariance function.

Parameters

parameterPoint

List the full parameter of the covariance function i.e. scale parameter $\vect{\theta} \in \Rset^n$ , the the amplitude parameter $\vect{\sigma} \in \Rset^d$ , the Spatial correlation parameter $\mat{R} \in \cS_d^+([-1,1])$ ; and potential other parameter depending on the model;

Must be at least of dimension $n+\frac{d(d+1)}{2}$ .

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setNuggetFactor(nuggetFactor)¶

Set the nugget factor for the variance of the observation error.

Acts on the discretized covariance matrix.

Parameters

nuggetFactorfloat: nugget factor to be used to model the variance of the observation error.

setOutputCorrelation(correlation)¶

Set the spatial correlation matrix $\mat{R}$ of the covariance function.

Parameters

spatialCorrelationCorrelationMatrix: Correlation matrix $\mat{R} \in \cS_d^+([-1,1])$ .

setParameter(parameter)¶

Set the parameters of the covariance function.

Parameters

parametersPoint

List of the scale parameter $\vect{\theta} \in \Rset^n$ and the amplitude parameter $\vect{\sigma} \in \Rset^d$ of the covariance function.

Must be of dimension $n+d$ .

setRadius(radius)¶

Accessor to the radius.

Parameters

radiusfloat, $a > 0$: Radius of the sphere on which the covariance model is not zero.

setScale(scale)¶

Set the scale parameter $\vect{\theta}$ of the covariance function.

Parameters

scalePoint: The scale parameter $\vect{\theta} \in \Rset^n$ to be used in the covariance function. Its size must be equal to the input dimension of the covariance function.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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