libopenturns-dev 1.9-5 / usr / include / openturns / swig / KrigingResult_doc.i

%feature("docstring") OT::KrigingResult
"Kriging result.

Available constructors:
    KrigingResult(*inputSample, outputSample, metaModel, residuals, relativeErrors, basis, trendCoefficients, covarianceModel, covarianceCoefficients*)

    KrigingResult(*inputSample, outputSample, metaModel, residuals, relativeErrors, basis, trendCoefficients, covarianceModel, covarianceCoefficients, covarianceCholeskyFactor, covarianceHMatrix*)


Parameters
----------
inputSample, outputSample : 2-d sequence of float
    The samples :math:`(\\\\vect{x}_k)_{1 \\\\leq k \\\\leq N} \\\\in \\\\Rset^d` and :math:`(\\\\vect{y}_k)_{1 \\\\leq k \\\\leq N}\\\\in \\\\Rset^p`.
metaModel : :class:`~openturns.Function`
    The meta model: :math:`\\\\tilde{\\\\cM}: \\\\Rset^d \\\\rightarrow \\\\Rset^p`, defined in :eq:`metaModelKrigFinal`.
residuals : :class:`~openturns.Point`
    The residual errors.
relativeErrors : :class:`~openturns.Point`
    The relative errors.
basis : collection of :class:`~openturns.Basis`
    Collection of the  :math:`p` functional basis: :math:`(\\\\varphi_j^l)_{1 \\\\leq j \\\\leq n_l}` for each :math:`l \\\\in [1, p]` with :math:`\\\\varphi_j^l: \\\\Rset^d \\\\rightarrow \\\\Rset`.
    Its size must be equal to zero if the trend is not estimated.
trendCoefficients : collection of :class:`~openturns.Point`
   The trend coeffient vectors :math:`(\\\\vect{\\\\alpha}^1, \\\\dots, \\\\vect{\\\\alpha}^p)`.
covarianceModel : :class:`~openturns.CovarianceModel`
    Covariance function of the normal process.
covarianceCoefficients : 2-d sequence of float
    The :math:`\\\\vect{\\\\gamma}` defined in :eq:`gammaEq`.
covarianceCholeskyFactor : :class:`~openturns.TriangularMatrix`
    The Cholesky factor :math:`\\\\mat{L}` of :math:`\\\\mat{C}`.
covarianceHMatrix :  :class:`~openturns.HMatrix`
    The *hmat* implementation of :math:`\\\\mat{L}`.


Notes
-----
The Kriging meta model :math:`\\\\tilde{\\\\cM}` is defined by:

.. math::
    :label: metaModelKrig
    
    \\\\tilde{\\\\cM}(\\\\vect{x}) =  \\\\vect{\\\\mu}(\\\\vect{x}) + \\\\Expect{\\\\vect{Y}(\\\\omega, \\\\vect{x})\\\\,| \\\\,\\\\cC}

where :math:`\\\\cC` is the condition :math:`\\\\vect{Y}(\\\\omega, \\\\vect{x}_k) = \\\\vect{y}_k` for each :math:`k \\\\in [1, N]`.
    
Equation :eq:`metaModelKrig` writes:

.. math::

    \\\\tilde{\\\\cM}(\\\\vect{x}) = \\\\vect{\\\\mu}(\\\\vect{x}) + \\\\Cov{\\\\vect{Y}(\\\\omega, \\\\vect{x}), (\\\\vect{Y}(\\\\omega,\\\\vect{x}_1),\\\\dots,\\\\vect{Y}(\\\\omega, \\\\vect{x}_N))}\\\\vect{\\\\gamma}

where 

.. math::

    \\\\Cov{\\\\vect{Y}(\\\\omega, \\\\vect{x}), (\\\\vect{Y}(\\\\omega, \\\\vect{x}_1),\\\\dots,\\\\vect{Y}(\\\\omega, \\\\vect{x}_N))} = \\\\left(\\\\mat{C}(\\\\vect{x},\\\\vect{x}_1)|\\\\dots|\\\\mat{C}(\\\\vect{x},\\\\vect{x}_N)\\\\right)\\\\in \\\\cM_{p,NP}(\\\\Rset)

and 

.. math::
    :label: gammaEq

    \\\\vect{\\\\gamma} = \\\\mat{C}^{-1}(\\\\vect{y}-\\\\vect{m})

At the end, the meta model writes:

.. math::
    :label: metaModelKrigFinal

    \\\\tilde{\\\\cM}(\\\\vect{x}) = \\\\vect{\\\\mu}(\\\\vect{x}) + \\\\sum_{i=1}^N \\\\gamma_i  \\\\mat{C}(\\\\vect{x},\\\\vect{x}_i)




Examples
--------
Create the model :math:`\\\\cM: \\\\Rset \\\\mapsto \\\\Rset` and the samples:

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x'],  ['x * sin(x)'])
>>> sampleX = [[1.0], [2.0], [3.0], [4.0], [5.0], [6.0]]
>>> sampleY = f(sampleX)

Create the algorithm:

>>> basis = ot.Basis([ot.SymbolicFunction(['x'], ['x']), ot.SymbolicFunction(['x'], ['x^2'])])
>>> covarianceModel = ot.GeneralizedExponential([2.0], 2.0)
>>> algoKriging = ot.KrigingAlgorithm(sampleX, sampleY, covarianceModel, basis)
>>> algoKriging.run()

Get the result:

>>> resKriging = algoKriging.getResult()

Get the meta model:

>>> metaModel = resKriging.getMetaModel()
"

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%feature("docstring") OT::KrigingResult::getCovarianceCoefficients
"Accessor to the covariance coefficients.

Returns
-------
covCoeff : :class:`~openturns.Sample`
    The :math:`\\\\vect{\\\\gamma}` defined in :eq:`gammaEq`.
"

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%feature("docstring") OT::KrigingResult::getTrendCoefficients
"Accessor to the trend coefficients.

Returns
-------
trendCoef : collection of :class:`~openturns.Point`
    The trend coefficients vectors :math:`(\\\\vect{\\\\alpha}^1, \\\\dots, \\\\vect{\\\\alpha}^p)`
"

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%feature("docstring") OT::KrigingResult::getCovarianceModel
"Accessor to the covariance model.

Returns
-------
covModel : :class:`~openturns.CovarianceModel`
    The covariance model of the Normal process *W* with its optimized parameters.
"

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%feature("docstring") OT::KrigingResult::getBasisCollection
"Accessor to the collection of basis.

Returns
-------
basisCollection : collection of :class:`~openturns.Basis`
    Collection of the :math:`p` function basis: :math:`(\\\\varphi_j^l)_{1 \\\\leq j \\\\leq n_l}` for each :math:`l \\\\in [1, p]` with :math:`\\\\varphi_j^l: \\\\Rset^d \\\\rightarrow \\\\Rset`.

Notes
-----
If the trend is not estimated, the collection is empty. 
"


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%feature("docstring") OT::KrigingResult::getConditionalMean
"Compute the expected mean of the Gaussian process on a point or a sample of points.

Available usages:
    getConditionalMean(x)

    getConditionalMean(sampleX)

Parameters
----------
x : sequence of float
    The point :math:`\\\\vect{x}` where the conditional mean of the output has to be evaluated.
sampleX : 2-d sequence of float
     The sample :math:`(\\\\vect{\\\\xi}_1, \\\\dots, \\\\vect{\\\\xi}_M)` where the conditional mean of the output has to be evaluated (*M* can be equal to 1).

Returns
-------
condMean : :class:`~openturns.Point`
    The conditional mean :math:`\\\\Expect{\\\\vect{Y}(\\\\omega, \\\\vect{x})\\\\, | \\\\,  \\\\cC}` at point :math:`\\\\vect{x}`.
   
    Or the conditional mean matrix at the sample :math:`(\\\\vect{\\\\xi}_1, \\\\dots, \\\\vect{\\\\xi}_M)`:

.. math::

    \\\\left(
      \\\\begin{array}{l}
         \\\\Expect{\\\\vect{Y}(\\\\omega, \\\\vect{\\\\xi}_1)\\\\, | \\\\,  \\\\cC}\\\\\\\\
        \\\\dots  \\\\\\\\
        \\\\Expect{\\\\vect{Y}(\\\\omega, \\\\vect{\\\\xi}_M)\\\\, | \\\\,  \\\\cC}
       \\\\end{array}
     \\\\right)

"


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%feature("docstring") OT::KrigingResult::getConditionalCovariance
"Compute the expected covariance of the Gaussian process on a point (or several points).

Available usages:
    getConditionalCovariance(x)

    getConditionalCovariance(sampleX)

Parameters
----------
x : sequence of float
    The point :math:`\\\\vect{x}` where the conditional mean of the output has to be evaluated.
sampleX : 2-d sequence of float
     The sample :math:`(\\\\vect{\\\\xi}_1, \\\\dots, \\\\vect{\\\\xi}_M)` where the conditional mean of the output has to be evaluated (*M* can be equal to 1).

Returns
-------
condCov : :class:`~openturns.CovarianceMatrix`
    The conditional covariance :math:`\\\\Cov{\\\\vect{Y}(\\\\omega, \\\\vect{x})\\\\, | \\\\,  \\\\cC}` at point :math:`\\\\vect{x}`.
   
    Or the conditional covariance matrix at the sample :math:`(\\\\vect{\\\\xi}_1, \\\\dots, \\\\vect{\\\\xi}_M)`:

.. math::

    \\\\left(
      \\\\begin{array}{lcl}
         \\\\Sigma_{11} & \\\\dots & \\\\Sigma_{1M} \\\\\\\\
        \\\\dots  \\\\\\\\
        \\\\Sigma_{M1} & \\\\dots & \\\\Sigma_{MM}
       \\\\end{array}
     \\\\right)

where :math:`\\\\Sigma_{ij} = \\\\Cov{\\\\vect{Y}(\\\\omega, \\\\vect{\\\\xi}_i), \\\\vect{Y}(\\\\omega, \\\\vect{\\\\xi}_j)\\\\, | \\\\,  \\\\cC}`.
"

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%feature("docstring") OT::KrigingResult::getMetaModel
"Accessor to the metamodel.

Returns
-------
metaModel : :class:`~openturns.Function`
    The meta model :math:`\\\\tilde{\\\\cM}: \\\\Rset^d \\\\rightarrow \\\\Rset^p`, defined in :eq:`metaModelKrigFinal`.
"

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%feature("docstring") OT::KrigingResult::setTransformation
"Accessor to the normalizing transformation.

Parameters
----------
transformation : :class:`~openturns.Function`
    The transformation *T* that normalizes the input sample."

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%feature("docstring") OT::KrigingResult::getTransformation
"Accessor to the normalizing transformation.

Returns
-------
transformation : :class:`~openturns.Function`
    The transformation *T* that normalizes the input sample."

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%feature("docstring") OT::KrigingResult::getInputSample
"Accessor to the input sample.

Returns
-------
inputSample : :class:`~openturns.Sample`
    The input sample."

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%feature("docstring") OT::KrigingResult::getOutputSample
"Accessor to the output sample.

Returns
-------
outputSample : :class:`~openturns.Sample`
    The output sample."