Polynomial chaos over database¶

In this example we are going to create a global approximation of a model response using functional chaos over a design of experiment.

You will need to specify the distribution of the input parameters. If not known, statistical inference can be used to select a possible candidate, and fitting tests can validate such an hypothesis.

```import openturns as ot

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

Create a function R^n –> R^p For example R^4 –> R

```myModel = ot.SymbolicFunction(["x1", "x2", "x3", "x4"], ["1+x1*x2 + 2*x3^2+x4^4"])

# Create a distribution of dimension n
# for example n=3 with independent components
distribution = ot.ComposedDistribution(
[ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)]
)
```

Prepare the input/output samples

```sampleSize = 250
X = distribution.getSample(sampleSize)
Y = myModel(X)
dimension = X.getDimension()
```

build the orthogonal basis

```coll = [
ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i))
for i in range(dimension)
]
enumerateFunction = ot.LinearEnumerateFunction(dimension)
productBasis = ot.OrthogonalProductPolynomialFactory(coll, enumerateFunction)
```

create the algorithm

```degree = 6
productBasis, enumerateFunction.getStrataCumulatedCardinal(degree)
)
projectionStrategy = ot.LeastSquaresStrategy()
algo = ot.FunctionalChaosAlgorithm(
)
algo.run()
```

get the metamodel function

```result = algo.getResult()
metamodel = result.getMetaModel()
```

Print residuals

```result.getResiduals()
```

[2.64115e-15]