Estimate moments from Taylor expansions

In this example we are going to estimate mean and standard deviation of an output variable of interest thanks to the Taylor variance decomposition method of order 1 or 2.

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)

Create a composite random vector

ot.RandomGenerator.SetSeed(0)
input_names = ['x1', 'x2', 'x3', 'x4']
myFunc = ot.SymbolicFunction(input_names,
    ['cos(x2*x2+x4)/(x1*x1+1+x3^4)'])
R = ot.CorrelationMatrix(4)
for i in range(4):
    R[i, i - 1] = 0.25
distribution = ot.Normal([0.2]*4, [0.1, 0.2, 0.3, 0.4], R)
distribution.setDescription(input_names)
# We create a distribution-based RandomVector
X = ot.RandomVector(distribution)
# We create a composite RandomVector Y from X and myFunc
Y = ot.CompositeRandomVector(myFunc, X)

We create a Taylor expansion method to approximate moments

taylor = ot.TaylorExpansionMoments(Y)

get 1st order mean

print(taylor.getMeanFirstOrder())

Out:

[0.932544]

get 2nd order mean

print(taylor.getMeanSecondOrder())

Out:

[0.820295]

get covariance

print(taylor.getCovariance())

Out:

[[ 0.0124546 ]]

draw importance factors

taylor.getImportanceFactors()

[x1 : 0.181718, x2 : 0.0430356, x3 : 0.0248297, x4 : 0.750417]

draw importance factors

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

Get the value of the output at the mean point

taylor.getValueAtMean()

[0.932544]

Get the gradient value of the output at the mean point

taylor.getGradientAtMean()

[[ -0.35812 ]
[ -0.0912837 ]
[ -0.0286496 ]
[ -0.228209 ]]

Get the hessian value of the output at the mean point

taylor.getHessianAtMean()
plt.show()