Create a compound distribution

In this example we are going to build the distribution of the random vector defined by the conditional distribution of:

where is the output of the random variable through the link function :

This example creates a CompoundDistribution which offers all the methods attached to the distributions.

We consider the case where is of dimension 1 and follows a uniform distribution defined by:

Variable	Distribution	Parameter
	Uniform ()
	Uniform ()

import openturns as ot
import openturns.viewer as otv

Create the distribution.

YDist = ot.Uniform(-1.0, 1.0)

Create the link function .

f = ot.SymbolicFunction(["y"], ["y", "1+y^2"])

Create the conditional distribution of : as the parameters have no importance, we use the default distribution.

XgivenThetaDist = ot.Uniform()

In this example, the range of depends on its parameters which are random. Thus, the model is not regular. We advice to change the discretization method that performs the integration. By default, this method is the quadrature method GaussProduct. We prefer to use the QMC. We use the ResourceMap.

ot.ResourceMap.SetAsString('CompoundDistribution-ContinuousDiscretizationMethod', 'QMC')

Create the compound distribution of .

XDist = ot.CompoundDistribution(XgivenThetaDist, YDist, f)
XDist.setDescription([r"$X|\mathbf{\boldsymbol{\Theta}} = f(Y)$"])
XDist

CompoundDistribution

• name=CompoundDistribution
• dimension=1
• weight=1
• range=[-0.999969, 1.99997]
• description=[$X|\mathbf{\boldsymbol{\Theta}} = f(Y)$]
• isParallel=true
• isCopula=false

Get a sample:

XDist.getSample(5)

	$X|\mathbf{\boldsymbol{\Theta}} = f(Y)$
0	0.9728113
1	-0.655941
2	1.050775
3	1.277257
4	-0.101663

Draw the PDF.

graph = XDist.drawPDF()
view = otv.View(graph)

view.ShowAll()