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Create a domain event

Abstract

We present in this example the creation and the use of a DomainEvent through a simple Monte-Carlo estimator.

import openturns as ot
import openturns.viewer as otv
from matplotlib import pylab as plt

We consider a standard unit Gaussian bivariate random vector \vect{X} = (X_1,X_2) with independent marginals.

dim = 2
distX = ot.Normal(dim)

We define a model f which maps a vector of \mathbb{R}^2 to another vector of \mathbb{R}^2

f : (x_1, x_2) \mapsto (x_1 + x_2, 2x_1)

f = ot.SymbolicFunction(["x1", "x2"], ["x1+x2", "2*x1"])

We build a RandomVector out of the input distribution and a CompositeRandomVector by using the model.

vecX = ot.RandomVector(distX)
vecY = ot.CompositeRandomVector(f, vecX)

Definition and vizualisation of a domain event

We define for each marginals of the output random vector vecY a domain of interest, say [0,1] \times [0,1]

domain = ot.Interval([0.0, 0.0], [1.0, 1.0])

The DomainEvent is then built from the output random vector vecY and the domain :

event = ot.DomainEvent(vecY, domain)

This domain is

\mathcal{D} = \{ \vect{x}=(x_1, x_2) \in \mathbb{R}^2 \; | \; x_1+x_2 \in [0,1] \; \mathrm{and} \; 2x_1 \in [0,1] \}.

We plot both marginals of the model and the domain of interest for each marginal using contour curves.

We represent the first marginal of vecY.

ot.ResourceMap.SetAsUnsignedInteger("Contour-DefaultLevelsNumber", 7)
graphModel0 = f.draw(0, 1, 0, [0.0, 0.0], [-5.0, -5.0], [5.0, 5.0])
graphModel0.setXTitle(r"$x_1$")
graphModel0.setYTitle(r"$x_2$")
graphModel0.setTitle(r"Isolines of the model : $Y = f(X)$, first marginal")

We represent the second marginal of vecY.

graphModel1 = f.draw(0, 1, 1, [0.0, 0.0], [-5.0, -5.0], [5.0, 5.0])
graphModel1.setXTitle(r"$x_1$")
graphModel1.setYTitle(r"$x_2$")
graphModel1.setTitle(r"Isolines of the model : $Y = f(X)$, second marginal")

We shall now represent the curves delimiting the domain of interest :

nx, ny = 15, 15
xx = ot.Box([nx], ot.Interval([-5.0], [5.0])).generate()
yy = ot.Box([ny], ot.Interval([-5.0], [5.0])).generate()
inputData = ot.Box([nx, ny], ot.Interval([-5.0, -5.0], [5.0, 5.0])).generate()
outputData = f(inputData)

The contour line associated with the 0.0 value for the first marginal.

mycontour0 = ot.Contour(xx, yy, outputData.getMarginal(0))
mycontour0.setLevels([0.0])
mycontour0.setLabels(["0.0"])
mycontour0.setColor("black")
mycontour0.setLineStyle("dashed")
graphModel0.add(mycontour0)

The contour line associated with the 1.0 value for the first marginal.

mycontour1 = ot.Contour(xx, yy, outputData.getMarginal(0))
mycontour1.setLevels([1.0])
mycontour1.setLabels(["1.0"])
mycontour1.setColor("black")
mycontour1.setLineStyle("dashed")
graphModel0.add(mycontour1)
view = otv.View(graphModel0)
Isolines of the model : $Y = f(X)$, first marginal

The contour line associated with the 0.0 value for the second marginal.

mycontour2 = ot.Contour(xx, yy, outputData.getMarginal(1))
mycontour2.setLevels([0.0])
mycontour2.setLabels(["0.0"])
mycontour2.setColor("black")
mycontour2.setLineStyle("dashed")
graphModel1.add(mycontour2)

The contour line associated with the 1.0 value for the second marginal.

mycontour3 = ot.Contour(xx, yy, outputData.getMarginal(1))
mycontour3.setLevels([1.0])
mycontour3.setLabels(["1.0"])
mycontour3.setColor("black")
mycontour3.setLineStyle("dashed")
graphModel1.add(mycontour3)
view = otv.View(graphModel1)
Isolines of the model : $Y = f(X)$, second marginal

For each marginal the domain of interest is the area between the two black dashed curves. The domain event \mathcal{D} is the intersection of these two areas. Here the intersection of both events is a parallelogram with the following vertices :

data = [[0.0, 0.0], [0.5, -0.5], [0.5, 0.5], [0.0, 1.0], [0.0, 0.0]]

We create a polygon from these vertices with the Polygon class : that is our domain event.

myGraph = ot.Graph("Domain event", r"$x_1$", r"$x_2$", True, "", 1.0)
myPolygon = ot.Polygon(data)
myPolygon.setColor("darkgray")
myPolygon.setEdgeColor("darkgray")
myGraph.add(myPolygon)

# Some annotation
texts = [
    r"$\mathcal{D} = \{ \mathbf{x}=(x_1, x_2) \in \mathbb{R}^2 \; | \; x_1+x_2 \in [0,1] \; \mathrm{and} \; 2x_1 \in [0,1] \}$"
]

myText = ot.Text([0.25], [0.0], texts)
myText.setTextSize(1)
myText.setColor("black")
myGraph.add(myText)
view = otv.View(myGraph)
Domain event

An example

Consider the integral

P_f = \int_{\mathcal{D}} \mathbf{1}_{\mathcal{D}}(\vect{x}) f_{X_1,X_2}(\vect{x}) d \vect{x}

where {\mathcal{D}} is the previous domain event, \mathbf{1}_{\mathcal{D}} is the indicator function on the domain and f_{X_1,X_2} is the probability density function of the input variable.

We observe the integration domain {\mathcal{D}} superimposed on the 2D-PDF.

graphPDF = distX.drawPDF([-5.0, -5.0], [5.0, 5.0])
graphPDF.setXTitle(r"$x_1$")
graphPDF.setYTitle(r"$x_2$")
graphPDF.setTitle(r"Isolines of the 2D-PDF")
graphPDF.add(myPolygon)
view = otv.View(graphPDF, contour_kw={"norm": "log"})
Isolines of the 2D-PDF

We shall use a basic Monte Carlo algorithm using the domain event to estimate the probability.

algoMC = ot.ProbabilitySimulationAlgorithm(event)
algoMC.setMaximumOuterSampling(1000)
algoMC.setBlockSize(100)
algoMC.setMaximumCoefficientOfVariation(0.02)
algoMC.run()
print("Pf = %.4f" % algoMC.getResult().getProbabilityEstimate())

Pf = 0.0693

We draw the convergence history :

graphConvergence = algoMC.drawProbabilityConvergence()
view = otv.View(graphConvergence)
ProbabilitySimulationAlgorithm convergence graph at level 0.95

We can use the getSample() method of the event to estimate the probability P_f. This method draws realizations of the underlying random input vector vecX and returns True if the corresponding output random vector is in the domain event. Then the ratio between the number of realizations in the domain and the total of realizations is a rough estimate of the probability P_f which we compare with the previous Monte-Carlo estimator.

N = 30000
samples = event.getSample(N)
print("Basic estimator : %.4f" % (sum(samples)[0] / N))

Basic estimator : 0.0733

Display all figures

plt.show()

Reset default settings

ot.ResourceMap.Reload()