`ipynb source code `_ Kriging POD =========== .. code:: python # import relevant module import openturns as ot import otpod # enable display figure in notebook %matplotlib inline Generate data ------------- .. code:: python inputSample = ot.NumericalSample( [[4.59626812e+00, 7.46143339e-02, 1.02231538e+00, 8.60042277e+01], [4.14315790e+00, 4.20801346e-02, 1.05874908e+00, 2.65757364e+01], [4.76735111e+00, 3.72414824e-02, 1.05730385e+00, 5.76058433e+01], [4.82811977e+00, 2.49997658e-02, 1.06954641e+00, 2.54461380e+01], [4.48961094e+00, 3.74562922e-02, 1.04943946e+00, 6.19483646e+00], [5.05605334e+00, 4.87599783e-02, 1.06520409e+00, 3.39024904e+00], [5.69679328e+00, 7.74915877e-02, 1.04099514e+00, 6.50990466e+01], [5.10193991e+00, 4.35520544e-02, 1.02502536e+00, 5.51492592e+01], [4.04791970e+00, 2.38565932e-02, 1.01906882e+00, 2.07875350e+01], [4.66238956e+00, 5.49901237e-02, 1.02427200e+00, 1.45661275e+01], [4.86634219e+00, 6.04693570e-02, 1.08199374e+00, 1.05104730e+00], [4.13519347e+00, 4.45225831e-02, 1.01900124e+00, 5.10117047e+01], [4.92541940e+00, 7.87692335e-02, 9.91868726e-01, 8.32302238e+01], [4.70722074e+00, 6.51799251e-02, 1.10608515e+00, 3.30181002e+01], [4.29040932e+00, 1.75426222e-02, 9.75678838e-01, 2.28186756e+01], [4.89291400e+00, 2.34997929e-02, 1.07669835e+00, 5.38926138e+01], [4.44653744e+00, 7.63175936e-02, 1.06979154e+00, 5.19109415e+01], [3.99977452e+00, 5.80430585e-02, 1.01850716e+00, 7.61988190e+01], [3.95491570e+00, 1.09302814e-02, 1.03687664e+00, 6.09981789e+01], [5.16424368e+00, 2.69026464e-02, 1.06673711e+00, 2.88708887e+01], [5.30491620e+00, 4.53802273e-02, 1.06254792e+00, 3.03856837e+01], [4.92809155e+00, 1.20616369e-02, 1.00700410e+00, 7.02512744e+00], [4.68373805e+00, 6.26028935e-02, 1.05152117e+00, 4.81271603e+01], [5.32381954e+00, 4.33013582e-02, 9.90522007e-01, 6.56015973e+01], [4.35455857e+00, 1.23814619e-02, 1.01810539e+00, 1.10769534e+01]]) signals = ot.NumericalSample( [[ 37.305445], [ 35.466919], [ 43.187991], [ 45.305165], [ 40.121222], [ 44.609524], [ 45.14552 ], [ 44.80595 ], [ 35.414039], [ 39.851778], [ 42.046049], [ 34.73469 ], [ 39.339349], [ 40.384559], [ 38.718623], [ 46.189709], [ 36.155737], [ 31.768369], [ 35.384313], [ 47.914584], [ 46.758537], [ 46.564428], [ 39.698493], [ 45.636588], [ 40.643948]]) Build POD with Kriging model ---------------------------- .. code:: python # signal detection threshold detection = 38. # The POD with censored data actually builds a POD only on filtered data. # A warning is diplayed in this case. POD = otpod.KrigingPOD(inputSample, signals, detection, noiseThres=35., saturationThres=45.) User-defined defect sizes ~~~~~~~~~~~~~~~~~~~~~~~~~ The user-defined defect sizes must range between the minimum and maximum of the defect values after filtering. An error is raised if it is not the case. The available range is then returned to the user. .. code:: python # Default defect sizes print('Default defect sizes : ') print(POD.getDefectSizes()) # Wrong range try: POD.setDefectSizes([3.2, 3.6, 4.5, 5.5]) except ValueError as e: print('Range of the defect sizes is too large, it returns a value error : ') print(e) .. parsed-literal:: Default defect sizes : [ 3.9549157 4.0152854 4.07565509 4.13602479 4.19639448 4.25676418 4.31713387 4.37750357 4.43787326 4.49824296 4.55861265 4.61898235 4.67935204 4.73972174 4.80009143 4.86046113 4.92083082 4.98120052 5.04157021 5.10193991] Range of the defect sizes is too large, it returns a value error : Defect sizes must range between 3.9550 and 5.1019. .. code:: python # Good range POD.setDefectSizes([4., 4.3, 4.6, 4.9, 5.1]) print('User-defined defect size : ') print(POD.getDefectSizes()) .. parsed-literal:: User-defined defect size : [ 4. 4.3 4.6 4.9 5.1] Running the Kriging based POD ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The computing time can be reduced by setting the simulation size attribute to another value. However the confidence interval is less accurate. The sampling size is the number of the samples used to compute the POD with the Monte Carlo simulation for each defect sizes. A progress is displayed, which can be disabled with the method *setVerbose*. .. code:: python POD = otpod.KrigingPOD(inputSample, signals, detection) # we can change the number of initial random search for the best starting point # of the TNC algorithm which optimizes the covariance model parameters POD.setInitialStartSize(500) # default is 1000 # we can change the sample size of the Monte Carlo simulation POD.setSamplingSize(2000) # default is 5000 # we can also change the size of the simulation to compute the confidence interval POD.setSimulationSize(500) # default is 1000 %time POD.run() .. parsed-literal:: Start optimizing covariance model parameters... Kriging optimizer completed kriging validation Q2 (>0.9): 1.0000 Computing POD per defect: [==================================================] 100.00% Done CPU times: user 37.7 s, sys: 13.2 s, total: 50.9 s Wall time: 52.3 s Compute detection size ---------------------- .. code:: python # Detection size at probability level 0.9 # and confidence level 0.95 print(POD.computeDetectionSize(0.9, 0.95)) # probability level 0.95 with confidence level 0.99 print(POD.computeDetectionSize(0.95, 0.99)) .. parsed-literal:: [a90 : 4.62318, a90/95 : 4.63983] [a95 : 4.66733, a95/99 : 4.6837] get POD NumericalMathFunction ----------------------------- .. code:: python # get the POD model PODmodel = POD.getPODModel() # get the POD model at the given confidence level PODmodelCl95 = POD.getPODCLModel(0.95) # compute the probability of detection for a given defect value print('POD : {:0.3f}'.format(PODmodel([4.2])[0])) print('POD at level 0.95 : {:0.3f}'.format(PODmodelCl95([4.2])[0])) .. parsed-literal:: POD : 0.148 POD at level 0.95 : 0.126 Compute the Q2 -------------- Enable to check the quality of the model. .. code:: python print('Q2 : {:0.4f}'.format(POD.getQ2())) .. parsed-literal:: Q2 : 1.0000 Draw the validation graph ~~~~~~~~~~~~~~~~~~~~~~~~~ The predictions are the one computed by leave one out. .. code:: python fig, ax = POD.drawValidationGraph() fig.show() .. parsed-literal:: /home/dumas/anaconda2/lib/python2.7/site-packages/matplotlib/figure.py:397: UserWarning: matplotlib is currently using a non-GUI backend, so cannot show the figure "matplotlib is currently using a non-GUI backend, " .. image:: krigingPOD_files/krigingPOD_19_1.png Show POD graphs --------------- Mean POD and POD at confidence level with the detection size for a given probability level ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code:: python fig, ax = POD.drawPOD(probabilityLevel=0.9, confidenceLevel=0.95, name='figure/PODKriging.png') # The figure is saved in PODPolyChaos.png fig.show() .. image:: krigingPOD_files/krigingPOD_21_0.png Advanced user mode ------------------ The user can defined one or both parameters of the kriging algorithm : - the basis - the covariance model The user can also defined the input parameter distribution it is known. The user can set the KrigingResult object if it built from other data. .. code:: python # new POD study PODnew = otpod.KrigingPOD(inputSample, signals, detection) .. code:: python # set the basis constant basis = ot.ConstantBasisFactory(4).build() PODnew.setBasis(basis) .. code:: python # set the covariance Model as an absolute exponential model covColl = ot.CovarianceModelCollection(4) for i in xrange(4): covColl[i] = ot.AbsoluteExponential([1], [1.]) covarianceModel = ot.ProductCovarianceModel(covColl) PODnew.setCovarianceModel(covarianceModel) .. code:: python PODnew.run() .. parsed-literal:: Start optimizing covariance model parameters... Kriging optimizer completed kriging validation Q2 (>0.9): 0.9660 Computing POD per defect: [==================================================] 100.00% Done .. code:: python print(PODnew.computeDetectionSize(0.9, 0.95)) print('Q2 : {:0.4f}'.format(POD.getQ2())) .. parsed-literal:: [a90 : 4.63513, a90/95 : 4.77085] Q2 : 1.0000 .. code:: python fig, ax = PODnew.drawPOD(probabilityLevel=0.9, confidenceLevel=0.95) fig.show() .. image:: krigingPOD_files/krigingPOD_28_0.png .. code:: python fig, ax = PODnew.drawValidationGraph() fig.show() .. image:: krigingPOD_files/krigingPOD_29_0.png