Transform a distribution

In this example we are going to use distribution algebra and distribution transformation via functions.

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)

We define some (classical) distribution :

distribution1 = ot.Uniform(0.0, 1.0)
distribution2 = ot.Uniform(0.0, 2.0)
distribution3 = ot.WeibullMin(1.5, 2.0)

Sum & difference of distributions

It is easy to compute the sum of distributions. For example:

distribution = distribution1 + distribution2
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)

Out:

Trapezoidal(a = 0, b = 1, c = 2, d = 3)

We might also use substraction even with scalar values:

distribution = 3.0 - distribution3
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)

Out:

RandomMixture(3 - WeibullMin(beta = 1.5, alpha = 2, gamma = 0))

Product & inverse

We might also compute the product of two (or more) distributions. For example:

distribution = distribution1 * distribution2
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)

Out:

ProductDistribution(Uniform(a = 0, b = 1) * Uniform(a = 0, b = 2))

We could also inverse a distribution :

distribution = 1 / distribution1
print(distribution)
graph = distribution.drawPDF()
view = viewer.View(graph)

Out:

CompositeDistribution=f(Uniform(a = 0, b = 1)) with f=[x]->[1.0 / x]

Or compute a ratio distrobution:

ratio = distribution2 / distribution1
print(ratio)
graph = ratio.drawPDF()
view = viewer.View(graph)

Out:

ProductDistribution(Uniform(a = 0, b = 2) * CompositeDistribution=f(Uniform(a = 0, b = 1)) with f=[x]->[1.0 / x])

Transformation using functions

The library provides methods to get the full distributions of f(x) where f can be equal to :

• sin,

• asin,

• cos,

• acos,

• tan,

• atan,

• sinh,

• asinh,

• cosh,

• acosh,

• tanh,

• atanh,

• sqr (for square),

• inverse,

• sqrt,

• exp,

• log/ln,

• abs,

• cbrt.

If one wants a specific method, user might rely on CompositeDistribution.

For example for the usual log transformation:

graph =distribution1.log().drawPDF()
view = viewer.View(graph)

And for the log2 function:

f = ot.SymbolicFunction(['x'], ['log2(x)'])
f.setDescription(["X","ln(X)"])
graph = ot.CompositeDistribution(f, distribution1).drawPDF()
view = viewer.View(graph)
plt.show()