# PolynomialChaosPOD¶

class PolynomialChaosPOD(*args)

Polynomial chaos based POD.

Available constructor:

PolynomialChaosPOD(inputSample, outputSample, detection, noiseThres, saturationThres, boxCox)

Parameters: inputSample : 2-d sequence of float Vector of the input values. The first column must correspond with the defect sizes. outputSample : 2-d sequence of float Vector of the signals, of dimension 1. detection : float Detection value of the signal. noiseThres : float Value for low censored data. Default is None. saturationThres : float Value for high censored data. Default is None boxCox : bool or float Enable or not the Box Cox transformation. If boxCox is a float, the Box Cox transformation is enabled with the given value. Default is False.

Warning

The first column of the input sample must corresponds with the defects sample.

Notes

This class aims at building the POD based on a polynomial chaos model. This method must be used under the assumption that the residuals follows a Normal distribution.

The return POD model corresponds with an interpolate function built with the POD values computed for the given defect sizes. The default values are 20 defect sizes between the minimum and maximum value of the defect sample. The defect sizes can be changed using the method setDefectSizes.

The default polynomial chaos model is built with uniform distributions for each parameters. Coefficients are computed using the LAR algorithm combined with the KFold. The AdaptiveStrategy is chosen fixed with a linear enumerate function of maximum degree 3.

For advanced use, all parameters can be defined thanks to dedicated set methods. Moreover, if the user has already built a polynomial chaos result, it can be given as parameter using the method setPolynomialChaosResult, then the POD is computed based on this polynomial chaos result.

A progress bar is shown if the verbosity is enabled. It can be disabled using the method setVerbose.

Methods

 computeDetectionSize(probabilityLevel[, …]) Compute the detection size for a given probability level. drawBoxCoxLikelihood([name]) Draw the loglikelihood versus the Box Cox parameter. drawPOD([probabilityLevel, confidenceLevel, …]) Draw the POD curve. drawPolynomialChaosModel([name]) Draw the polynomial chaos prediction versus the true data. drawValidationGraph([name]) Draw the validation graph of the metamodel. getAdaptiveStrategy() Accessor to the adaptive strategy. getBoxCoxParameter() Accessor to the Box Cox parameter. getCoefficientDistribution() Accessor to the distribution of the polynomial chaos coefficients. getDefectSizes() Accessor to the defect size where POD is computed. getDegree() Accessor to the polynomial chaos degree. getDistribution() Accessor to the parameters distribution. getPODCLModel([confidenceLevel]) Accessor to the POD model at a given confidence level. getPODModel() Accessor to the POD model. getPolynomialChaosResult() Accessor to the polynomial chaos result. getProjectionStrategy() Accessor to the projection strategy. getQ2() Accessor to the Q2 value. getR2() Accessor to the R2 value. getSamplingSize() Accessor to the Monte Carlo sampling size. getSimulationSize() Accessor to the simulation size. getVerbose() Accessor to the verbosity. run() Build the POD models. setAdaptiveStrategy(strategy) Accessor to the adaptive strategy. setDefectSizes(size) Accessor to the defect size where POD is computed. setDegree(degree) Accessor to the polynomial chaos degree. setDistribution(distribution) Accessor to the parameters distribution. setPolynomialChaosResult(chaosResult) Accessor to the polynomial chaos result. setProjectionStrategy(strategy) Accessor to the projection strategy. setSamplingSize(size) Accessor to the Monte Carlo sampling size. setSimulationSize(size) Accessor to the simulation size. setVerbose(verbose) Accessor to the verbosity.
computeDetectionSize(probabilityLevel, confidenceLevel=None)

Compute the detection size for a given probability level.

Parameters: probabilityLevel : float The probability level for which the defect size is computed. confidenceLevel : float The confidence level associated to the given probability level the defect size is computed. Default is None. result : collection of openturns.PointWithDescription A PointWithDescription containing the detection size computed at the given probability level and confidence level if provided.
drawBoxCoxLikelihood(name=None)

Draw the loglikelihood versus the Box Cox parameter.

Parameters: name : string name of the figure to be saved with transparent option sets to True and bbox_inches=’tight’. It can be only the file name or the full path name. Default is None. fig : matplotlib.figure Matplotlib figure object. ax : matplotlib.axes Matplotlib axes object.

Notes

This method is available only when the parameter boxCox is set to True.

drawPOD(probabilityLevel=None, confidenceLevel=None, defectMin=None, defectMax=None, nbPt=100, name=None)

Draw the POD curve.

Parameters: probabilityLevel : float The probability level for which the defect size is computed. Default is None. confidenceLevel : float The confidence level associated to the given probability level the defect size is computed. Default is None. defectMin, defectMax : float Define the interval where the curve is plotted. Default : min and max values of the input sample. nbPt : int The number of points to draw the curves. Default is 100. name : string name of the figure to be saved with transparent option sets to True and bbox_inches=’tight’. It can be only the file name or the full path name. Default is None. fig : matplotlib.figure Matplotlib figure object. ax : matplotlib.axes Matplotlib axes object.
drawPolynomialChaosModel(name=None)

Draw the polynomial chaos prediction versus the true data.

Parameters: name : string name of the figure to be saved with transparent option sets to True and bbox_inches=’tight’. It can be only the file name or the full path name. Default is None. fig : matplotlib.figure Matplotlib figure object. ax : matplotlib.axes Matplotlib axes object.

Notes

This method only works if the dimension of the input sample is 1.

drawValidationGraph(name=None)

Draw the validation graph of the metamodel.

Parameters: name : string name of the figure to be saved with transparent option sets to True and bbox_inches=’tight’. It can be only the file name or the full path name. Default is None. fig : matplotlib.figure Matplotlib figure object. ax : matplotlib.axes Matplotlib axes object.
getAdaptiveStrategy()

Returns: strategy : openturns.AdaptiveStrategy The adaptive strategy for the polynomial chaos.
getBoxCoxParameter()

Accessor to the Box Cox parameter.

Returns: lambdaBoxCox : float The Box Cox parameter used to transform the data. If the transformation is not enabled None is returned.
getCoefficientDistribution()

Accessor to the distribution of the polynomial chaos coefficients.

Returns: dist : openturns.Distribution The distribution of the coefficients.
getDefectSizes()

Accessor to the defect size where POD is computed.

Returns: defectSize : sequence of float The defect sizes where the Monte Carlo simulation is performed to compute the POD.
getDegree()

Accessor to the polynomial chaos degree.

Returns: degree : int The degree of the polynomial chaos.
getDistribution()

Accessor to the parameters distribution.

Returns: distribution : openturns.ComposedDistribution The input parameters distribution, default is a Uniform distribution for all parameters.
getPODCLModel(confidenceLevel=0.95)

Accessor to the POD model at a given confidence level.

Parameters: confidenceLevel : float The confidence level the POD must be computed. Default is 0.95 PODModelCl : openturns.Function The function which computes the probability of detection for a given defect value at the confidence level given as parameter.
getPODModel()

Accessor to the POD model.

Returns: PODModel : openturns.Function The function which computes the probability of detection for a given defect value.
getPolynomialChaosResult()

Accessor to the polynomial chaos result.

Returns: result : openturns.FunctionalChaosResult The polynomial chaos result.
getProjectionStrategy()

Accessor to the projection strategy.

Returns: strategy : openturns.ProjectionStrategy The projection strategy for the polynomial chaos.
getQ2()

Accessor to the Q2 value.

Returns: Q2 : float The Q2 value computed analytically.
getR2()

Accessor to the R2 value.

Returns: R2 : float The R2 value.
getSamplingSize()

Accessor to the Monte Carlo sampling size.

Returns: size : int The size of the Monte Carlo simulation used to compute the POD for each defect size.
getSimulationSize()

Accessor to the simulation size.

Returns: size : int The size of the simulation used to compute the confidence interval.
getVerbose()

Accessor to the verbosity.

Returns: verbose : bool Enable or disable the verbosity. Default is True.
run()

Build the POD models.

Notes

This method build the polynomial chaos model. First the censored data are filtered if needed. The Box Cox transformation is performed if it is enabled. Then it builds the POD models, the Monte Carlo simulation is performed for each given defect sizes. The confidence interval is computed by simulating new coefficients of the polynomial chaos, then Monte Carlo simulations are performed.

setAdaptiveStrategy(strategy)

Parameters: strategy : openturns.AdaptiveStrategy The adaptive strategy for the polynomial chaos.
setDefectSizes(size)

Accessor to the defect size where POD is computed.

Parameters: defectSize : sequence of float The defect sizes where the Monte Carlo simulation is performed to compute the POD.
setDegree(degree)

Accessor to the polynomial chaos degree.

Parameters: degree : int The degree of the polynomial chaos.
setDistribution(distribution)

Accessor to the parameters distribution.

Parameters: distribution : openturns.ComposedDistribution The input parameters distribution.
setPolynomialChaosResult(chaosResult)

Accessor to the polynomial chaos result.

Parameters: chaosResult : openturns.FunctionalChaosResult The polynomial chaos result.
setProjectionStrategy(strategy)

Accessor to the projection strategy.

Parameters: strategy : openturns.ProjectionStrategy The projection strategy for the polynomial chaos.
setSamplingSize(size)

Accessor to the Monte Carlo sampling size.

Parameters: size : int The size of the Monte Carlo simulation used to compute the POD for each defect size.
setSimulationSize(size)

Accessor to the simulation size.

Parameters: size : int The size of the simulation used to compute the confidence interval.
setVerbose(verbose)

Accessor to the verbosity.

Parameters: verbose : bool Enable or disable the verbosity.