GaussianProcessConditionalCovariance¶

class GaussianProcessConditionalCovariance(*args)¶

Conditional covariance post processing of a Gaussian Process Regression result.

Warning

This class is experimental and likely to be modified in future releases. To use it, import the openturns.experimental submodule.

Parameters:

gprResultGaussianProcessRegressionResult: The result class built by GaussianProcessRegression.

Methods

`getClassName`()	Accessor to the object's name.
`getConditionalCovariance`(*args)	Compute the conditional covariance of the Gaussian process on a point (or several points).
`getConditionalMarginalVariance`(*args)	Compute the conditional variance of the Gaussian process on a point (or several points).
`getConditionalMean`(*args)	Compute the conditional mean of the Gaussian process on a point or a sample of points.
`getDiagonalCovariance`(xi)	Compute the diagonal conditional covariance of the Gaussian process on a point.
`getDiagonalCovarianceCollection`(xi)	Compute the conditional covariance of the Gaussian process on a sample.
`getName`()	Accessor to the object's name.
`hasName`()	Test if the object is named.
`setName`(name)	Accessor to the object's name.

Notes

Refer to Gaussian process regression (step 3) to get all the notations and the theoretical aspects. We only detail here the notions related to the class.

We suppose we have a sample $(\vect{x}_k, \vect{y}_k)_{1 \leq k \leq \sampleSize}$ where $\vect{y}_k = \model(\vect{x}_k)$ for all k, with $\model:\Rset^{\inputDim} \mapsto \Rset^{\outputDim}$ the model. The Gaussian process approximation $\vect{Y}$ is defined by:

$\vect{Y}(\omega, \vect{x}) = \vect{\mu}(\vect{x}) + \vect{W}(\omega, \vect{x})$

where $\vect{\mu} : \Rset^\inputDim \rightarrow \Rset^outputDim$ is the trend function and $\vect{W}$ is a Gaussian process of dimension $\outputDim$ with zero mean and a specified covariance function. The Gaussian process regression denoted by $\vect{Z}$ is defined by:

$\vect{Z}(\omega, \vect{x}) = \vect{Y}(\omega, \vect{x})\, | \, \cC$

where $\cC$ is the condition $\vect{Y}(\omega, \vect{x}_k) = \vect{y}_k$ for $1 \leq k \leq \sampleSize$ .

The class provides services related to the conditional covariance of the Gaussian process regression $\vect{Z}$ .

Examples

Create the model $g: \Rset \mapsto \Rset$ and the samples:

>>> import openturns as ot
>>> from openturns.experimental import GaussianProcessRegression
>>> from openturns.experimental import GaussianProcessConditionalCovariance
>>> trend = ot.SymbolicFunction(['x'],  ['1'])
>>> sampleX = [[1.0], [2.0], [3.0], [4.0], [5.0], [6.0]]
>>> sampleY = trend(sampleX)

Create the algorithm:

>>> covarianceModel = ot.SquaredExponential([1.0])
>>> covarianceModel.setActiveParameter([])

>>> algo = GaussianProcessRegression(sampleX, sampleY, covarianceModel, trend)
>>> algo.run()
>>> result = algo.getResult()
>>> condCov = GaussianProcessConditionalCovariance(result)
>>> c = condCov([1.1])

__init__(*args)¶

getClassName()¶

Accessor to the object’s name.

Returns:

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

getConditionalCovariance(*args)¶

Compute the conditional covariance of the Gaussian process on a point (or several points).

Parameters:

xsequence of float: The point $\vect{x}$ where the conditional covariance of the output has to be evaluated.
sampleX2-d sequence of float: The sample $(\vect{\xi}_1, \dots, \vect{\xi}_N)$ where the conditional covariance of the output has to be evaluated (N can be equal to 1).

Returns:

condCovCovarianceMatrix: The conditional covariance of the Gaussian process regression $\vect{Z}$ defined in (2) at point $\vect{x}$ is defined in (5). When computed on the sample $(\vect{\xi}_1, \dots, \vect{\xi}_N)$ , the covariance matrix is defined in (6).

getConditionalMarginalVariance(*args)¶

Compute the conditional variance of the Gaussian process on a point (or several points).

Parameters:

xsequence of float

The point $\vect{x}$ where the conditional variance of the output has to be evaluated.

sampleX2-d sequence of float

The sample $(\vect{\xi}_1, \dots, \vect{\xi}_N)$ where the conditional variance of the output has to be evaluated (N can be equal to 1).

marginalIndexint

Marginal of interest (for multiple outputs).

Default value is 0 (first component).

marginalIndicessequence of int

Marginals of interest (for multiple outputs).

Returns:

varfloat: The variance of the specified marginal of the Gaussian process regression $\vect{Z}$ defined in (2) at the specified point.
varPointsequence of float: The marginal variances of each marginal of interest computed at each given point.

Notes

If only one marginal $k$ of interest and one point $\vect{x}$ have been specified, the method returns $\Var{Z_k(\omega, \vect{x})}$ where $\vect{Z}$ is the Gaussian process regression defined in (2).

If several marginal of interest $(k_1, \dots, k_M)$ or several points $(\vect{\xi}_1, \dots, \vect{\xi}_N)$ have been specified, the method returns the concatenation of sequence of variances $(\Var{Z_{k_1}(\omega, \vect{\xi}_j)}, \dots, \Var{Z_{k_M}(\omega, \vect{\xi}_j)})$ for each $1 \leq j \leq N$ .

getConditionalMean(*args)¶

Compute the conditional mean of the Gaussian process on a point or a sample of points.

Parameters:

xsequence of float: The point $\vect{x}$ where the conditional mean of the output has to be evaluated.
sampleX2-d sequence of float: The sample $(\vect{\xi}_1, \dots, \vect{\xi}_N)$ where the conditional mean of the output has to be evaluated (N can be equal to 1).

Returns:

condMeanPoint

The conditional mean the Gaussian process regression $\vect{Z}$ defined in (2) at point $\vect{x}$ or on the sample $(\vect{\xi}_1, \dots, \vect{\xi}_N)$ :

$\left( \begin{array}{l} \Expect{\vect{Z}(\omega, \vect{\xi}_1)}\\ \dots \\ \Expect{\vect{Z}(\omega, \vect{\xi}_N)} \end{array} \right)$

This vector is in $\Rset^{N \times \outputDim}$ .

getDiagonalCovariance(xi)¶

Compute the diagonal conditional covariance of the Gaussian process on a point.

Parameters:

xsequence of float: The point $\vect{x}$ where the conditional marginal covariance of the output has to be evaluated.

Returns:

condCovCovarianceMatrix: The conditional covariance $\Cov{\vect{Y}(\omega, \vect{x})\, | \, \cC}$ at point $\vect{x}$ .

getDiagonalCovarianceCollection(xi)¶

Compute the conditional covariance of the Gaussian process on a sample.

Parameters:

sampleX2-d sequence of float: The sample $(\vect{\xi}_1, \dots, \vect{\xi}_N)$ where the conditional marginal covariance of the output has to be evaluated (N can be equal to 1).

Returns:

condCovCovarianceMatrixCollection: The collection of conditional covariance matrices $\Cov{\vect{Z}(\omega, \vect{\xi}_i)} = \mat{\Sigma_{ii}}$ for $1 \leq i \leq N$ defined in (6).

Notes

Each element of the collection corresponds to the conditional covariance with respect to the input learning set (e.g. a pointwise evaluation of the getDiagonalCovariance). The returned collection is of size $N$ and contains matrices in $\cM_{\outputDim, \outputDim}(\Rset)$ .

getName()¶

Accessor to the object’s name.

Returns: