# GaussianNonLinearCalibration¶

class `GaussianNonLinearCalibration`(*args)

Gaussian non linear calibration algorithm (3DVAR).

Parameters
model`Function`

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.

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

 Accessor to the bootstrap size used to sample the posterior distribution. Accessor to the parameter candidate. Accessor to the object’s name. Accessor to the observation error covariance. Accessor to the flag for a global observation error covariance. Accessor to the object’s id. Accessor to the input data to be fitted. Accessor to the model to be fitted. Accessor to the object’s name. Accessor to the optimization algorithm used for the computation. Accessor to the output data to be fitted. Accessor to the parameter prior covariance. Accessor to the parameter prior distribution. Get the result structure. Accessor to the object’s shadowed id. Accessor to the object’s visibility state. Test if the object is named. 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. Accessor to the object’s shadowed id. `setVisibility`(visible) Accessor to the object’s visibility state.
`__init__`(*args)

Initialize self. See help(type(self)) for accurate signature.

`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
candidate`Point`

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
error`CovarianceMatrix`

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
data`Sample`

The input data to be fitted.

`getModel`()

Accessor to the model to be fitted.

Returns
data`Function`

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
algo`OptimizationAlgorithm`

Optimization algorithm used for the computation.

`getOutputObservations`()

Accessor to the output data to be fitted.

Returns
data`Sample`

The output data to be fitted.

`getParameterCovariance`()

Accessor to the parameter prior covariance.

Returns
prior`CovarianceMatrix`

Parameter prior covariance.

`getParameterPrior`()

Accessor to the parameter prior distribution.

Returns
prior`Distribution`

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
namestr

The name of the object.

`setOptimizationAlgorithm`(algorithm)

Accessor to the optimization algorithm used for the computation.

Parameters
algo`OptimizationAlgorithm`

Optimization algorithm to use for the computation.

`setResult`(result)

Accessor to optimization result.

Parameters
result`CalibrationResult`

Result class.

`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.