# 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 analyse 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
```

Total running time of the script: (0 minutes 2.354 seconds)