# Central tendency analysis on the cantilever beam example¶

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 __future__ import print_function
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 :

```from openturns.usecases import cantilever_beam as cantilever_beam
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 = ot.Point(dim, 1.0)
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 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 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= [1.39543] var= [0.00271142]
```

## 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.computeStandardDeviationPerComponent()
y_quantile_95p = Y_s.computeQuantilePerComponent(0.95)
print('mean=', y_mean, 'stddev=', y_stddev, 'quantile@95%', y_quantile_95p)
```

Out:

```mean= [1.3887] stddev= [1.61762] quantile@95% [4.21421]
```
```graph = ot.KernelSmoothing().build(Y_s).drawPDF()
graph.setTitle("Kernel smoothing approximation of the output distribution")
view = viewer.View(graph)

plt.show()
```

Total running time of the script: ( 0 minutes 0.133 seconds)

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