Note

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# Fitting a distribution from data based on OpenTURNS¶

```
from __future__ import annotations
from gemseo import configure_logger
from gemseo.uncertainty.distributions.openturns.fitting import OTDistributionFitter
from numpy.random import randn
from numpy.random import seed
configure_logger()
```

```
<RootLogger root (INFO)>
```

In this example,
we will see how to fit a distribution from data.
For a purely pedagogical reason,
we consider a synthetic dataset made of 100 realizations of *‘X’*,
a random variable distributed according to the standard normal distribution.
These samples are generated from the NumPy library.

```
seed(1)
data = randn(100)
variable_name = "X"
```

## Create a distribution fitter¶

Then,
we create an `OTDistributionFitter`

from these data and this variable name:

```
fitter = OTDistributionFitter(variable_name, data)
```

## Fit a distribution¶

From this distribution fitter, we can easily fit any distribution available in the OpenTURNS library:

```
print(fitter.available_distributions)
```

```
['Arcsine', 'Beta', 'Burr', 'Chi', 'ChiSquare', 'Dirichlet', 'Exponential', 'FisherSnedecor', 'Frechet', 'Gamma', 'GeneralizedPareto', 'Gumbel', 'Histogram', 'InverseNormal', 'Laplace', 'LogNormal', 'LogUniform', 'Logistic', 'MeixnerDistribution', 'Normal', 'Pareto', 'Rayleigh', 'Rice', 'Student', 'Trapezoidal', 'Triangular', 'TruncatedNormal', 'Uniform', 'VonMises', 'WeibullMax', 'WeibullMin']
```

For example, we can fit a normal distribution:

```
norm_dist = fitter.fit("Normal")
print(norm_dist)
```

```
Normal([0.0605829,0.889615])
```

or an exponential one:

```
exp_dist = fitter.fit("Exponential")
print(exp_dist)
```

```
Exponential([0.419342,-2.3241])
```

The returned object is an `OTDistribution`

that we can represent graphically
in terms of probability and cumulative density functions:

```
norm_dist.plot()
```

```
<Figure size 640x320 with 2 Axes>
```

## Measure the goodness-of-fit¶

We can also measure the goodness-of-fit of a distribution by means of a fitting criterion. Some fitting criteria are based on significance tests made of a test statistics, a p-value and a significance level. We can access the names of the available fitting criteria:

```
print(fitter.available_criteria)
print(fitter.available_significance_tests)
```

```
['BIC', 'ChiSquared', 'Kolmogorov']
[<SignificanceTest.ChiSquared: 'ChiSquared'>, <SignificanceTest.Kolmogorov: 'Kolmogorov'>]
```

For example, we can measure the goodness-of-fit of the previous distributions by considering the Bayesian information criterion (BIC):

```
quality_measure = fitter.compute_measure(norm_dist, "BIC")
print("Normal: ", quality_measure)
quality_measure = fitter.compute_measure(exp_dist, "BIC")
print("Exponential: ", quality_measure)
```

```
Normal: 2.5939451287694295
Exponential: 3.7381346286469945
```

Here, the fitted normal distribution is better than the fitted exponential one in terms of BIC. We can also the Kolmogorov fitting criterion which is based on the Kolmogorov significance test:

```
acceptable, details = fitter.compute_measure(norm_dist, "Kolmogorov")
print("Normal: ", acceptable, details)
acceptable, details = fitter.compute_measure(exp_dist, "Kolmogorov")
print("Exponential: ", acceptable, details)
```

```
Normal: True {'p-value': 0.9879299613543082, 'statistics': 0.04330972976650932, 'level': 0.05}
Exponential: False {'p-value': 5.628454180958696e-11, 'statistics': 0.34248997332293696, 'level': 0.05}
```

In this case,
the `OTDistributionFitter.measure()`

method returns a tuple with two values:

a boolean indicating if the measured distribution is acceptable to model the data,

a dictionary containing the test statistics, the p-value and the significance level.

Note

We can also change the significance level for significance tests
whose default value is 0.05.
For that, use the `level`

argument.

## Select an optimal distribution¶

Lastly,
we can also select an optimal `OTDistribution`

based on a collection of distributions names,
a fitting criterion,
a significance level
and a selection criterion:

‘best’: select the distribution minimizing (or maximizing, depending on the criterion) the criterion,

‘first’: select the first distribution for which the criterion is greater (or lower, depending on the criterion) than the level.

By default,
the `OTDistributionFitter.select()`

method uses a significance level equal to 0.5
and ‘best’ selection criterion.

```
selected_distribution = fitter.select(["Exponential", "Normal"], "Kolmogorov")
print(selected_distribution)
```

```
Normal([0.0605829,0.889615])
```

**Total running time of the script:** ( 0 minutes 0.177 seconds)