NonLinearLeastSquaresCalibration

class NonLinearLeastSquaresCalibration(*args)

Non-linear least-squares calibration algorithm.

Parameters
modelFunction

The parametric function to be calibrated.

inputObservations2-d sequence of float

The sample of input observations.

outputObservations2-d sequence of float

The sample of output observations.

candidatesequence of float

The reference value of the parameter.

See also

GaussianLinearCalibration, LinearLeastSquaresCalibration, GaussianNonLinearCalibration

Notes

NonLinearLeastSquaresCalibration is the minimum variance estimator of the parameter of a given model with no assumption on the dependence of the model wrt the parameter.

The prior distribution of the parameter is a Dirac.

The posterior distribution of the parameter is Normal and reflects the variability of the optimum parameter depending on the observation sample. By default, the posterior distribution is evaluated based on a linear approximation of the model at the optimum. This corresponds to using the LinearLeastSquaresCalibration at the optimum, and is named Laplace approximation in the bayesian context. However, if the key NonLinearLeastSquaresCalibration-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 distribution of the output error is a Normal and is computed from the residuals.

If least squares optimization algorithms are enabled, then the algorithm used is the first found by Build of OptimizationAlgorithm. Otherwise, the algorithm TNC is used combined with a multistart algorithm which makes use of the NonLinearLeastSquaresCalibration-MultiStartSize key in the ResourceMap.

Examples

Calibrate a nonlinear model using non-linear least-squares:

>>> 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
>>> algo = ot.NonLinearLeastSquaresCalibration(modelX, x, y, candidate)
>>> algo.run()
>>> print(algo.getResult().getParameterMAP())
[2.773...,1.203...,0.499...]

Methods

BuildResidualFunction(model, …)

Build a residual function given a parametric model, input and output observations.

getBootstrapSize(self)

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

getCandidate(self)

Accessor to the parameter candidate.

getClassName(self)

Accessor to the object’s name.

getId(self)

Accessor to the object’s id.

getName(self)

Accessor to the object’s name.

getOptimizationAlgorithm(self)

Accessor to the optimization algorithm used for the computation.

getOutputObservations(self)

Accessor to the output data to be fitted.

getParameterPrior(self)

Accessor to the parameter prior distribution.

getResult(self)

Get the result structure.

getShadowedId(self)

Accessor to the object’s shadowed id.

getVisibility(self)

Accessor to the object’s visibility state.

hasName(self)

Test if the object is named.

hasVisibleName(self)

Test if the object has a distinguishable name.

run(self, \*args)

Launch the algorithm.

setBootstrapSize(self, bootstrapSize)

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

setName(self, name)

Accessor to the object’s name.

setOptimizationAlgorithm(self, algorithm)

Accessor to the optimization algorithm used for the computation.

setResult(self, result)

Accessor to optimization result.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

setVisibility(self, visible)

Accessor to the object’s visibility state.
__init__(self, \*args)

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

static BuildResidualFunction(model, inputObservations, outputObservations)

Build a residual function given a parametric model, input and output observations.

Parameters
modelFunction

Parametric model.

inputObservations2-d sequence of float

Input observations associated to the output observations.

outputObservationsFunction

Output observations.

Returns
residualFunction
Residual function.

Notes

Given a parametric model F_{\theta}:\Rset^n\rightarrow\Rset^p with parameter \theta\in\Rset^m, a sample of input points (x_i)_{i=1,\dots,N} and the associated output (y_i)_{i=1,\dots,N}, the residual function f is defined by:

\forall \theta\in\Rset^m, f(\theta)=\left(\begin{array}{c} F_{\theta}(x_1)-y_1 \\ \vdots \\ F_{\theta}(x_N)-y_N \end{array} \right)

getBootstrapSize(self)

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(self)

Accessor to the parameter candidate.

Returns
candidatePoint

Parameter candidate.

getClassName(self)

Accessor to the object’s name.

Returns
class_namestr

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

getId(self)

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOptimizationAlgorithm(self)

Accessor to the optimization algorithm used for the computation.

Returns
algoOptimizationAlgorithm

Optimization algorithm used for the computation.

getOutputObservations(self)

Accessor to the output data to be fitted.

Returns
dataSample

The output data to be fitted.

getParameterPrior(self)

Accessor to the parameter prior distribution.

Returns
priorDistribution

The parameter prior distribution.

getResult(self)

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(self)

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getVisibility(self)

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName(self)

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName(self)

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

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

run(self, \*args)

Launch the algorithm.

Notes

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

setBootstrapSize(self, 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(self, name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setOptimizationAlgorithm(self, algorithm)

Accessor to the optimization algorithm used for the computation.

Parameters
algoOptimizationAlgorithm

Optimization algorithm to use for the computation.

setResult(self, result)

Accessor to optimization result.

Parameters
resultCalibrationResult

Result class.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setVisibility(self, visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.