GaussianNonLinearCalibration¶

class GaussianNonLinearCalibration(*args)¶

Gaussian non linear calibration algorithm (3DVAR).

Parameters:

modelFunction: The parametric function to be calibrated.
inputObservations2-d sequence of float: The sample of input observations. Can have dimension 0 to specify no observations.
outputObservations2-d sequence of float: The sample of output observations.
candidatesequence of float: The mean of the gaussian prior distribution of the parameter.
parameterCovariance2-d sequence of float: The covariance matrix of the gaussian prior distribution of the parameter.
errorCovariance2-d sequence of float: The covariance matrix of the gaussian distribution of the observations error.

See also

GaussianLinearCalibration, LinearLeastSquaresCalibration, NonLinearLeastSquaresCalibration

Notes

GaussianNonLinearCalibration is the minimum variance estimator of the parameter of a given model under the assumption that this parameter is dispersed according to a known covariance matrix. This algorithm is also known as 3DVAR.

The given observation error covariance can be either local, ie the same matrix applies for each observation and is of dimension the output dimension of the model, or global, ie the matrix describes the full set of observation errors, in which case its dimension is the product of the output dimension of the model and the number of observations.

The prior distribution of the parameter is a Normal.

The posterior distribution of the parameter is Normal. By default, the posterior distribution is evaluated based on a linear approximation of the model at the optimum. This corresponds to using the GaussianLinearCalibration at the optimum, and is named Laplace approximation in the bayesian context. However, if the key GaussianNonLinearCalibration-BootstrapSize in the ResourceMap is set to a nonzero positive integer, then a bootstrap resampling of the observations is performed and the posterior distribution is based on a KernelSmoothing of the sample of boostrap optimum parameters.

The resulting error distribution is Normal with a zero mean and with a covariance matrix equal to the errorCovariance input argument.

Examples

Calibrate a nonlinear model using 3DVAR:

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> m = 10
>>> x = [[0.5 + i] for i in range(m)]
>>> inVars = ['a', 'b', 'c', 'x']
>>> formulas = ['a + b * exp(c * x)']
>>> model = ot.SymbolicFunction(inVars, formulas)
>>> p_ref = [2.8, 1.2, 0.5]
>>> params = [0, 1, 2]
>>> modelX = ot.ParametricFunction(model, params, p_ref)
>>> y = modelX(x)
>>> y += ot.Normal(0.0, 0.05).getSample(m)
>>> candidate = [1.0]*3
>>> priorCovariance = ot.CovarianceMatrix(3)
>>> errorCovariance = ot.CovarianceMatrix(1, [0.1])
>>> algo = ot.GaussianNonLinearCalibration(modelX, x, y, candidate, priorCovariance, errorCovariance)
>>> algo.run()
>>> print(algo.getResult().getParameterMAP())
[2.705...,1.212...,0.499...]

Methods

`getBootstrapSize`()	Accessor to the bootstrap size used to sample the posterior distribution.
`getCandidate`()	Accessor to the parameter candidate.
`getClassName`()	Accessor to the object's name.
`getErrorCovariance`()	Accessor to the observation error covariance.
`getGlobalErrorCovariance`()	Accessor to the flag for a global observation error covariance.
`getId`()	Accessor to the object's id.
`getInputObservations`()	Accessor to the input data to be fitted.
`getModel`()	Accessor to the model to be fitted.
`getName`()	Accessor to the object's name.
`getOptimizationAlgorithm`()	Accessor to the optimization algorithm used for the computation.
`getOutputObservations`()	Accessor to the output data to be fitted.
`getParameterCovariance`()	Accessor to the parameter prior covariance.
`getParameterPrior`()	Accessor to the parameter prior distribution.
`getResult`()	Get the result structure.
`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.
`run`(*args)	Launch the algorithm.
`setBootstrapSize`(bootstrapSize)	Accessor to the bootstrap size used to sample the posterior distribution.
`setName`(name)	Accessor to the object's name.
`setOptimizationAlgorithm`(algorithm)	Accessor to the optimization algorithm used for the computation.
`setResult`(result)	Accessor to optimization result.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

getBootstrapSize()¶

Accessor to the bootstrap size used to sample the posterior distribution.

Returns:

sizeint: Bootstrap size used to sample the posterior distribution. A value of 0 means that no bootstrap has been done but a linear approximation has been used to get the posterior distribution, using the GaussianLinearCalibration algorithm at the maximum a posteriori estimate.

getCandidate()¶

Accessor to the parameter candidate.

Returns:

candidatePoint: Parameter candidate.

getClassName()¶

Accessor to the object’s name.

Returns:

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

getErrorCovariance()¶

Accessor to the observation error covariance.

Returns:

errorCovarianceMatrix: Observation error covariance.

getGlobalErrorCovariance()¶

Accessor to the flag for a global observation error covariance.

Returns:

flagbool: Flag telling if the given observation error covariance is global or not.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getInputObservations()¶

Accessor to the input data to be fitted.

Returns:

dataSample: The input data to be fitted.

getModel()¶

Accessor to the model to be fitted.

Returns:

dataFunction: The model to be fitted.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOptimizationAlgorithm()¶

Accessor to the optimization algorithm used for the computation.

Returns:

algoOptimizationAlgorithm: Optimization algorithm used for the computation.

getOutputObservations()¶

Accessor to the output data to be fitted.

Returns:

dataSample: The output data to be fitted.

getParameterCovariance()¶

Accessor to the parameter prior covariance.

Returns:

priorCovarianceMatrix: Parameter prior covariance.

getParameterPrior()¶

Accessor to the parameter prior distribution.

Returns:

priorDistribution: The parameter prior distribution.

getResult()¶

Get the result structure.

Returns:

resCalibration: CalibrationResult: The structure containing all the results of the calibration problem.

Notes

The structure contains all the results of the calibration problem.

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.

run(*args)¶

Launch the algorithm.

Notes

It launches the algorithm and creates a CalibrationResult, structure containing all the results.

setBootstrapSize(bootstrapSize)¶

Accessor to the bootstrap size used to sample the posterior distribution.

Parameters:

sizeint: Bootstrap size used to sample the posterior distribution. A value of 0 means that no bootstrap has to be done but a linear approximation has been used to get the posterior distribution, using the GaussianLinearCalibration algorithm at the maximum a posteriori estimate.

setName(name)¶

Accessor to the object’s name.

Parameters: