Analyse the central tendency of a cantilever beam

In this example we perform a central tendency analysis of a random variable Y using the various methods available. We consider the cantilever beam example and show how to use the TaylorExpansionMoments and ExpectationSimulationAlgorithm classes.

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

We first load the data class from the usecases module :

cb = cantilever_beam.CantileverBeam()

We want to create the random variable of interest Y=g(X) where is the physical model and is the input vectors. For this example we consider independent marginals.

We set a mean vector and a unitary standard deviation :

dim = cb.dim
mean = [50.0, 1.0, 10.0, 5.0]
sigma = [1.0] * dim
R = ot.IdentityMatrix(dim)

We create the input parameters distribution and make a random vector :

distribution = ot.Normal(mean, sigma, R)
X = ot.RandomVector(distribution)
X.setDescription(['E', 'F', 'L', 'I'])

f is the cantilever beam model :

f = cb.model

The random variable of interest Y is then

Y = ot.CompositeRandomVector(f, X)
Y.setDescription('Y')

Taylor expansion

Perform Taylor approximation to get the expected value of Y and the importance factors.

taylor = ot.TaylorExpansionMoments(Y)
taylor_mean_fo = taylor.getMeanFirstOrder()
taylor_mean_so = taylor.getMeanSecondOrder()
taylor_cov = taylor.getCovariance()
taylor_if = taylor.getImportanceFactors()
print('model evaluation calls number=', f.getGradientCallsNumber())
print('model gradient calls number=', f.getGradientCallsNumber())
print('model hessian calls number=', f.getHessianCallsNumber())
print('taylor mean first order=', taylor_mean_fo)
print('taylor variance=', taylor_cov)
print('taylor importance factors=', taylor_if)

Out:

model evaluation calls number= 1
model gradient calls number= 1
model hessian calls number= 1
taylor mean first order= [1.33333]
taylor variance= [[ 2.0096 ]]
taylor importance factors= [E : 0.000353857, F : 0.884642, L : 0.079618, I : 0.0353857]

graph = taylor.drawImportanceFactors()
view = viewer.View(graph)

We see that, at first order, the variable explains 88.5% of the variance of the output . On the other hand, the variable is not significant in the variance of the output: at first order, the random variable could be replaced by a constant with no change to the output variance.

Monte-Carlo simulation

Perform a Monte Carlo simulation of Y to estimate its mean.

algo = ot.ExpectationSimulationAlgorithm(Y)
algo.setMaximumOuterSampling(1000)
algo.setCoefficientOfVariationCriterionType('NONE')
algo.run()
print('model evaluation calls number=', f.getEvaluationCallsNumber())
expectation_result = algo.getResult()
expectation_mean = expectation_result.getExpectationEstimate()
print('monte carlo mean=', expectation_mean, 'var=',
      expectation_result.getVarianceEstimate())

Out:

model evaluation calls number= 1001
monte carlo mean= [1.45846] var= [0.00299836]

Central dispersion analysis based on a sample

Directly compute statistical moments based on a sample of Y. Sometimes the probabilistic model is not available and the study needs to start from the data.

Y_s = Y.getSample(1000)
y_mean = Y_s.computeMean()
y_stddev = Y_s.computeStandardDeviation()
y_quantile_95p = Y_s.computeQuantilePerComponent(0.95)
print('mean=', y_mean, 'stddev=', y_stddev, 'quantile@95%', y_quantile_95p)

Out:

mean= [1.40943] stddev= [1.63795] quantile@95% [4.36899]

graph = ot.KernelSmoothing().build(Y_s).drawPDF()
graph.setTitle("Kernel smoothing approximation of the output distribution")
view = viewer.View(graph)

plt.show()