Calibration of the deflection of a tube

We consider a calibration of the deflection of a tube as described here.

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)

Create a calibration problem

We load the model from the use case module :

from openturns.usecases import deflection_tube as deflection_tube
dt = deflection_tube.DeflectionTube()

We create a sample out of our input distribution :

sampleSize = 100
inputSample = dt.inputDistribution.getSample(sampleSize)
inputSample[0:5]

	Force	Length	Location	External diameter	Internal diameter	Young Modulus
0	0.8890136	1.5	1	0.8	0.1	199323.1
1	1.056105	1.5	1	0.8	0.1	198446.3
2	1.052138	1.5	1	0.8	0.1	202572.1
3	1.058734	1.5	1	0.8	0.1	199465
4	0.9325915	1.5	1	0.8	0.1	199179.1

We take the image of our input sample by the model :

outputDeflection = dt.model(inputSample)
outputDeflection[0:5]

	Deflection	Left angle	Right angle
0	-6.163458e-06	-1.232692e-05	1.540864e-05
1	-7.354243e-06	-1.470849e-05	1.838561e-05
2	-7.177393e-06	-1.435479e-05	1.794348e-05
3	-7.334891e-06	-1.466978e-05	1.833723e-05
4	-6.470256e-06	-1.294051e-05	1.617564e-05

observationNoiseSigma = [0.1e-6,0.05e-5,0.05e-5]
observationNoiseCovariance = ot.CovarianceMatrix(3)
for i in range(3):
    observationNoiseCovariance[i,i] = observationNoiseSigma[i]**2

noiseSigma = ot.Normal([0.,0.,0.],observationNoiseCovariance)
sampleObservationNoise = noiseSigma.getSample(sampleSize)
observedOutput = outputDeflection + sampleObservationNoise
observedOutput[0:5]

	Deflection	Left angle	Right angle
0	-6.137027e-06	-1.244258e-05	1.518425e-05
1	-7.415107e-06	-1.441052e-05	1.856411e-05
2	-7.305498e-06	-1.315868e-05	1.841629e-05
3	-7.491503e-06	-1.433714e-05	1.829689e-05
4	-6.654401e-06	-1.285798e-05	1.61612e-05

observedInput = ot.Sample(sampleSize,2)
observedInput[:,0] = inputSample[:,0] # F
observedInput[:,1] = inputSample[:,5] # E
observedInput.setDescription(["Force","Young Modulus"])
observedInput[0:5]

	Force	Young Modulus
0	0.8890136	199323.1
1	1.056105	198446.3
2	1.052138	202572.1
3	1.058734	199465
4	0.9325915	199179.1

fullSample = ot.Sample(sampleSize,5)
fullSample[:,0:2] = observedInput
fullSample[:,2:5] = observedOutput
fullSample.setDescription(["Force","Young","Deflection","Left Angle","Right Angle"])
fullSample[0:5]

	Force	Young	Deflection	Left Angle	Right Angle
0	0.8890136	199323.1	-6.137027e-06	-1.244258e-05	1.518425e-05
1	1.056105	198446.3	-7.415107e-06	-1.441052e-05	1.856411e-05
2	1.052138	202572.1	-7.305498e-06	-1.315868e-05	1.841629e-05
3	1.058734	199465	-7.491503e-06	-1.433714e-05	1.829689e-05
4	0.9325915	199179.1	-6.654401e-06	-1.285798e-05	1.61612e-05

ot.Pairs(fullSample)

Out:

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color=blue fillStyle=solid lineStyle=solid pointStyle=fsquare lineWidth=1]]

Setting up the calibration

XL = 1.4 # Exact : 1.5
Xa = 1.2 # Exact : 1.0
XD = 0.7 # Exact : 0.8
Xd = 0.2 # Exact : 0.1
thetaPrior = ot.Point([XL,Xa,XD,Xd])

sigmaXL = 0.1 * XL
sigmaXa = 0.1 * Xa
sigmaXD = 0.1 * XD
sigmaXd = 0.1 * Xd
parameterCovariance = ot.CovarianceMatrix(4)
parameterCovariance[0,0] = sigmaXL**2
parameterCovariance[1,1] = sigmaXa**2
parameterCovariance[2,2] = sigmaXD**2
parameterCovariance[3,3] = sigmaXd**2
parameterCovariance

[[ 0.0196 0 0 0 ]
[ 0 0.0144 0 0 ]
[ 0 0 0.0049 0 ]
[ 0 0 0 0.0004 ]]

calibratedIndices = [1,2,3,4]
calibrationFunction = ot.ParametricFunction(dt.model, calibratedIndices, thetaPrior)

sigmaObservation = [0.2e-6,0.03e-5,0.03e-5] # Exact : 0.1e-6

errorCovariance = ot.CovarianceMatrix(3)
errorCovariance[0,0] = sigmaObservation[0]**2
errorCovariance[1,1] = sigmaObservation[1]**2
errorCovariance[2,2] = sigmaObservation[2]**2

calibrationFunction.setParameter(thetaPrior)
predictedOutput = calibrationFunction(observedInput)
predictedOutput[0:5]

	Deflection	Left angle	Right angle
0	-2.612376e-06	-8.707919e-06	1.415037e-05
1	-3.117089e-06	-1.03903e-05	1.688423e-05
2	-3.042131e-06	-1.014044e-05	1.647821e-05
3	-3.108887e-06	-1.036296e-05	1.68398e-05
4	-2.742412e-06	-9.141372e-06	1.485473e-05

Calibration with gaussian non linear least squares

algo = ot.GaussianNonLinearCalibration(calibrationFunction, observedInput, observedOutput, thetaPrior, parameterCovariance, errorCovariance)

algo.run()

calibrationResult = algo.getResult()

Analysis of the results

thetaMAP = calibrationResult.getParameterMAP()
thetaMAP

[1.49677,0.99466,0.800456,0.199881]

Compute a 95% confidence interval for each marginal.

thetaPosterior = calibrationResult.getParameterPosterior()
alpha = 0.95
dim = thetaPosterior.getDimension()
for i in range(dim):
    print(thetaPosterior.getMarginal(i).computeBilateralConfidenceInterval(alpha))

Out:

[1.47886, 1.51642]
[0.972844, 1.0201]
[0.796505, 0.803986]
[0.199879, 0.199926]

graph = calibrationResult.drawObservationsVsInputs()
view = viewer.View(graph)

graph = calibrationResult.drawObservationsVsPredictions()
view = viewer.View(graph)

graph = calibrationResult.drawResiduals()
view = viewer.View(graph)

graph = calibrationResult.drawParameterDistributions()
view = viewer.View(graph)

plt.show()