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Create the distribution of the maximum of distributions

Context

In this example, we define how to build a distribution defined as the maximum of some other ones.

We detail some interesting cases:

  • Case 1: The maximum of correlated scalar distributions,

  • Case 2: The maximum of independent scalar distributions,

  • Case 3: The maximum of independent and identically distributed scalar distributions.

import openturns as ot
import openturns.viewer as otv

Case 1: The maximum correlated scalar distributions

We consider d=3 scalar distributions X_i that form a random vector (X_1, X_2, X_3) which copula is denoted by C.

We assume that:

X_1 \sim \cU(2.5, 3.5)\\ X_2 \sim \mbox{Log}\cU(1.0, 1.2)\\ X_3 \sim \cT(2.0, 3.0, 4.0)\\ C \sim \mbox{NormalCopula}(\mat{M})

where \mat{M} is a correlation matix.

We want to define the distribution of:

Y_1 = \max(X_1, X_2, X_3).

We create the distribution of (X_1, X_2, X_3):

dist_X1 = ot.Uniform(2.5, 3.5)
dist_X2 = ot.LogUniform(1.0, 1.2)
dist_X3 = ot.Triangular(2.0, 3.0, 4.0)
coll = [dist_X1, dist_X2, dist_X3]
cor_mat = ot.CorrelationMatrix([[1.0, -0.9, 0.0], [-0.9, 1.0, 0.3], [0.0, 0.3, 1.0]])
cop = ot.NormalCopula(cor_mat)
dist_X = ot.JointDistribution(coll, cop)

We create the distribution of Y_1:

dist_Y_1 = ot.MaximumDistribution(dist_X)
print(dist_Y_1)

MaximumDistribution(JointDistribution(Uniform(a = 2.5, b = 3.5), LogUniform(aLog = 1, bLog = 1.2), Triangular(a = 2, m = 3, b = 4), NormalCopula(R = [[  1   -0.9  0   ]
 [ -0.9  1    0.3 ]
 [  0    0.3  1   ]])))

We draw its PDF:

graph = dist_Y_1.drawPDF()
graph.setTitle(r" Distribution of $Y_1 = max(X_1, X_2, X_3)$")
graph.setXTitle(r"$Y_1$")
graph.setLegendPosition("")
view = otv.View(graph)
Distribution of $Y_1 = max(X_1, X_2, X_3)$

Case 2: The maximum of independent scalar distributions

Here, we define (\tilde{X}_1,\tilde{X}_2, \tilde{X}_3) the random vector where the \tilde{X}_i are distributed as X_i and are independent.

We want to define the distribution of:

Y_2 = \max(\tilde{X}_1,\tilde{X}_2, \tilde{X}_3).

We create the distribution of Y_2:

dist_Y_2 = ot.MaximumDistribution(coll)

We could have written:

dist_X_tilde = ot.JointDistribution(coll)
dist_Y_2 = ot.MaximumDistribution(dist_X_tilde)

We draw its PDF:

graph = dist_Y_2.drawPDF()
graph.setTitle(r" Distribution of $Y_2 = max(\tilde{X}_1,\tilde{X}_2, \tilde{X}_3)$")
graph.setXTitle(r"$Y_2$")
graph.setLegendPosition("")
view = otv.View(graph)
Distribution of $Y_2 = max(\tilde{X}_1,\tilde{X}_2, \tilde{X}_3)$

Case 3: The maximum of a independent and identically distributed scalar distributions

Here we suppose that X_4 and X_5 are independent and identically distributed as X_1.

We want to define the distribution of:

Y_3 = \max(X_1, X_4, X_5).

We create the distribution of Y_3:

dist_Y_3 = ot.MaximumDistribution(dist_X1, 3)

We draw its PDF:

graph = dist_Y_3.drawPDF()
graph.setTitle(r" Distribution of $Y_3 = max(X_1, X_4, X_5)$")
graph.setXTitle(r"$Y_3$")
graph.setLegendPosition("")
view = otv.View(graph)
Distribution of $Y_3 = max(X_1, X_4, X_5)$
otv.View.ShowAll()