Fit a distribution from an input sample

In this example we show how to use the BuildDistribution() function to fit a distribution to an input sample. This function is used by the FunctionalChaosAlgorithm class when we want to create a polynomial chaos metamodel and we have a design of experiments which have been computed beforehand. In this case, we have to identify the distributions which best fit to the input sample in order to define the input probabilistic model. This is, in turn, used by in the polynomial chaos to create the orthogonal basis.

import openturns as ot

ot.Log.Show(ot.Log.NONE)

We first create the function model.

ot.RandomGenerator.SetSeed(0)
dimension = 2
input_names = ["x1", "x2"]
formulas = ["cos(x1 + x2)", "(x2 + 1) * exp(x1)"]
model = ot.SymbolicFunction(input_names, formulas)

Then we create a sample x and compute the corresponding output sample y.

distribution = ot.Normal(dimension)
samplesize = 1000
inputSample = distribution.getSample(samplesize)
outputSample = model(inputSample)

Create a functional chaos model. The algorithm used by BuildDistribution() fits a distribution on the input sample. This is done with the Lilliefors test. Please read The Kolmogorov-Smirnov goodness of fit test for continuous data for more details on this topic.

The Lilliefors test is based on sampling the distribution of the Kolmogorov-Smirnov statistics. The sample size corresponding to this algorithm is configured by the “FittingTest-LillieforsMaximumSamplingSize” ResourceMap key. In order to get satisfactory results, the default value of this key is relatively large.

ot.ResourceMap.GetAsUnsignedInteger("FittingTest-LillieforsMaximumSamplingSize")

100000

In order to speed this example up, let us reduce this value.

ot.ResourceMap.SetAsUnsignedInteger("FittingTest-LillieforsMaximumSamplingSize", 100)

Then we fit the distribution.

distribution = ot.FunctionalChaosAlgorithm.BuildDistribution(inputSample)

Let us explore the distribution with its fitted parameters.

distribution

JointDistribution

• name=JointDistribution
• dimension: 2
• description=[X0,X1]
• copula: IndependentCopula(dimension = 2)

Index	Variable	Distribution
0	X0	Normal(mu = -0.0268588, sigma = 0.985838)
1	X1	Normal(mu = 0.00246667, sigma = 1.00015)

We can also analyze its properties in more details.

for i in range(dimension):
    marginal = distribution.getMarginal(i)
    marginalname = marginal.getImplementation().getClassName()
    print("Marginal #", i, ":", marginalname)
distribution.getCopula()

Marginal # 0 : Normal
Marginal # 1 : Normal

IndependentCopula

• name=IndependentCopula
• dimension=2
• weight=1
• range=[0, 1] [0, 1]
• description=[X0,X1]
• isParallel=true
• isCopula=true

The previous call to BuildDistribution() is what is done internally by the following constructor of FunctionalChaosAlgorithm.

algo = ot.FunctionalChaosAlgorithm(inputSample, outputSample)

The previous constructor is the main topic of the example Create a polynomial chaos metamodel from a data set.

ot.ResourceMap.Reload()  # reset default settings