UnivariateLinearModelAnalysis¶

class UnivariateLinearModelAnalysis(*args)¶

Linear regression analysis with residuals hypothesis tests.

Available constructors:

UnivariateLinearModelAnalysis(inputSample, outputSample)

UnivariateLinearModelAnalysis(inputSample, outputSample, noiseThres, saturationThres, resDistFact, boxCox)

Parameters:

inputSample2-d sequence of float: Vector of the defect sizes, of dimension 1.
outputSample2-d sequence of float: Vector of the signals, of dimension 1.
noiseThresfloat: Value for low censored data. Default is None.
saturationThresfloat: Value for high censored data. Default is None.
resDistFactopenturns.DistributionFactory: Distribution hypothesis followed by the residuals. Default is openturns.NormalFactory.
boxCoxbool 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.

Methods

`drawBoxCoxLikelihood`([name])	Draw the loglikelihood versus the Box Cox parameter.
`drawLinearModel`([model, name])	Draw the linear regression prediction versus the true data.
`drawResiduals`([model, name])	Draw the residuals versus the defect values.
`drawResidualsDistribution`([model, name])	Draw the residuals histogram with the fitted distribution.
`drawResidualsQQplot`([model, name])	Draw the residuals QQ plot with the fitted distribution.
`getAndersonDarlingPValue`()	Accessor to the Anderson Darling test p-value.
`getBoxCoxParameter`()	Accessor to the Box Cox parameter.
`getBreuschPaganPValue`()	Accessor to the Breusch Pagan test p-value.
`getCramerVonMisesPValue`()	Accessor to the Cramer Von Mises test p-value.
`getDurbinWatsonPValue`()	Accessor to the Durbin Watson test p-value.
`getHarrisonMcCabePValue`()	Accessor to the Harrison McCabe test p-value.
`getInputSample`()	Accessor to the input sample.
`getIntercept`()	Accessor to the intercept of the linear regression model.
`getKolmogorovPValue`()	Accessor to the Kolmogorov test p-value.
`getNoiseThreshold`()	Accessor to the noise threshold.
`getOutputSample`()	Accessor to the output sample.
`getR2`()	Accessor to the R2 value.
`getResiduals`()	Accessor to the residuals.
`getResidualsDistribution`()	Accessor to the residuals distribution.
`getResults`()	Print results of the linear analysis.
`getSaturationThreshold`()	Accessor to the saturation threshold.
`getSlope`()	Accessor to the slope of the linear regression model.
`getStandardError`()	Accessor to the standard error of the estimate.
`getZeroMeanPValue`()	Accessor to the Zero Mean test p-value.
`saveResults`(name)	Save all analysis test results in a file.

Notes

This method automatically :

computes the Box Cox parameter if boxCox is True,
computes the transformed signals if boxCox is True or a float,
builds the univariate linear regression model on the data,
computes the linear regression parameters for censored data if needed,
computes the residuals,
runs all hypothesis tests.

Examples

Generate data :

>>> import openturns as ot
>>> import otpod
>>> N = 100
>>> ot.RandomGenerator.SetSeed(0)
>>> defectDist = ot.Uniform(0.1, 0.6)
>>> epsilon = ot.Normal(0, 1.9)
>>> defects = defectDist.getSample(N)
>>> signalsInvBoxCox = defects * 43. + epsilon.getSample(N) + 2.5
>>> invBoxCox = ot.InverseBoxCoxTransform(0.3)
>>> signals = invBoxCox(signalsInvBoxCox)

Run analysis with gaussian hypothesis on the residuals :

>>> analysis = otpod.UnivariateLinearModelAnalysis(defects, signals, boxCox=True)
>>> print analysis.getIntercept() # get intercept value
[Intercept for uncensored case : 2.51037]
>>> print analysis.getKolmogorovPValue()
[Kolmogorov p-value for uncensored case : 0.835529]

Run analysis with noise and saturation threshold :

>>> analysis = otpod.UnivariateLinearModelAnalysis(defects, signals, 60., 1700., boxCox=True)
>>> print analysis.getIntercept() # get intercept value for uncensored and censored case
[Intercept for uncensored case : 4.28758, Intercept for censored case : 3.11243]
>>> print analysis.getKolmogorovPValue()
[Kolmogorov p-value for uncensored case : 0.346827, Kolmogorov p-value for censored case : 0.885006]

Run analysis with a Weibull distribution hypothesis on the residuals

>>> analysis = otpod.UnivariateLinearModelAnalysis(defects, signals, 60., 1700., ot.WeibullMinFactory(), boxCox=True)
>>> print analysis.getIntercept() # get intercept value for uncensored and censored case
[Intercept for uncensored case : 4.28758, Intercept for censored case : 3.11243]
>>> print analysis.getKolmogorovPValue()
[Kolmogorov p-value for uncensored case : 0.476036, Kolmogorov p-value for censored case : 0.71764]

drawBoxCoxLikelihood(name=None)¶

Draw the loglikelihood versus the Box Cox parameter.

Parameters:

namestring: 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.

Returns:

figmatplotlib.figure: Matplotlib figure object.
axmatplotlib.axes: Matplotlib axes object.

Notes

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

drawLinearModel(model='uncensored', name=None)¶

Draw the linear regression prediction versus the true data.

Parameters:

modelstring: The linear regression model to be used, either uncensored or censored if censored threshold were given. Default is uncensored.
namestring: 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.

Returns:

figmatplotlib.figure: Matplotlib figure object.
axmatplotlib.axes: Matplotlib axes object.

drawResiduals(model='uncensored', name=None)¶

Draw the residuals versus the defect values.

Parameters:

modelstring: The residuals to be used, either uncensored or censored if censored threshold were given. Default is uncensored.
namestring: 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.

Returns:

figmatplotlib.figure: Matplotlib figure object.
axmatplotlib.axes: Matplotlib axes object.

drawResidualsDistribution(model='uncensored', name=None)¶

Draw the residuals histogram with the fitted distribution.

Parameters:

modelstring: The residuals to be used, either uncensored or censored if censored threshold were given. Default is uncensored.
namestring: 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.

Returns:

figmatplotlib.figure: Matplotlib figure object.
axmatplotlib.axes: Matplotlib axes object.

drawResidualsQQplot(model='uncensored', name=None)¶

Draw the residuals QQ plot with the fitted distribution.

Parameters:

modelstring: The residuals to be used, either uncensored or censored if censored threshold were given. Default is uncensored.
namestring: 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.

Returns:

figmatplotlib.figure: Matplotlib figure object.
axmatplotlib.axes: Matplotlib axes object.

getAndersonDarlingPValue()¶

Accessor to the Anderson Darling test p-value.

Returns:

pValueopenturns.Point: Either the p-value for the uncensored case or for both cases.

getBoxCoxParameter()¶

Accessor to the Box Cox parameter.

Returns:

lambdaBoxCoxfloat: The Box Cox parameter used to transform the data. If the transformation is not enabled None is returned.

getBreuschPaganPValue()¶

Accessor to the Breusch Pagan test p-value.

Returns:

pValueopenturns.Point: Either the p-value for the uncensored case or for both cases.

getCramerVonMisesPValue()¶

Accessor to the Cramer Von Mises test p-value.

Returns:

pValueopenturns.Point: Either the p-value for the uncensored case or for both cases.

getDurbinWatsonPValue()¶

Accessor to the Durbin Watson test p-value.

Returns:

pValueopenturns.Point: Either the p-value for the uncensored case or for both cases.

getHarrisonMcCabePValue()¶

Accessor to the Harrison McCabe test p-value.

Returns:

pValueopenturns.Point: Either the p-value for the uncensored case or for both cases.

getInputSample()¶

Accessor to the input sample.

Returns:

defectsopenturns.Sample: The input sample which is the defect values.

getIntercept()¶

Accessor to the intercept of the linear regression model.

Returns:

interceptopenturns.Point: The intercept parameter for the uncensored and censored (if so) linear regression model.

getKolmogorovPValue()¶

Accessor to the Kolmogorov test p-value.

Returns:

pValueopenturns.Point: Either the p-value for the uncensored case or for both cases.

getNoiseThreshold()¶

Accessor to the noise threshold.

Returns:

noiseThresfloat: The noise threhold if it exists, if not it returns None.

getOutputSample()¶

Accessor to the output sample.

Returns:

signalsopenturns.Sample: The input sample which is the signal values.

getR2()¶

Accessor to the R2 value.

Returns:

R2openturns.Point: Either the R2 for the uncensored case or for both cases.

getResiduals()¶

Accessor to the residuals.

Returns:

residualsopenturns.Sample: The residuals computed from the uncensored and censored linear regression model. The first column corresponds with the uncensored case.

getResidualsDistribution()¶

Accessor to the residuals distribution.

Returns:

distributionlist of openturns.Distribution: The fitted distribution on the residuals, computed in the uncensored and censored (if so) case.

getResults()¶: Print results of the linear analysis.

getSaturationThreshold()¶

Accessor to the saturation threshold.

Returns:

saturationThresfloat: The saturation threhold if it exists, if not it returns None.

getSlope()¶

Accessor to the slope of the linear regression model.

Returns:

slopeopenturns.Point: The slope parameter for the uncensored and censored (if so) linear regression model.

getStandardError()¶

Accessor to the standard error of the estimate.

Returns:

stderropenturns.Point: The standard error of the estimate for the uncensored and censored (if so) linear regression model.

getZeroMeanPValue()¶

Accessor to the Zero Mean test p-value.

Returns:

pValueopenturns.Point: Either the p-value for the uncensored case or for both cases.

saveResults(name)¶

Save all analysis test results in a file.

Parameters:

namestring: Name of the file or full path name.

Notes

The file can be saved as a csv file. Separations are made with tabulations.

If name is the file name, then it is saved in the current working directory.

otpod

A module to build Probability of Detection for Non Destructive Testing.

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