Model a singular multivariate distribution

From time to time we need to model singular distributions (e.g. the joint distribution of KL coefficients for curves resulting from the transport of a low dimensional random vector). A way to do that is to use an EmpiricalBernsteinCopula with a bin number equal to the sample size (also called the empirical beta copula in this case).

import openturns as ot
import openturns.viewer as viewer
import math as m

ot.Log.Show(ot.Log.NONE)
ot.RandomGenerator.SetSeed(0)

routine to draw a distribution cloud and a sample

def draw(dist, Y):
    g = ot.Graph()
    g.setAxes(True)
    g.setGrid(True)
    c = ot.Cloud(dist.getSample(10000))
    c.setColor("red")
    c.setPointStyle("bullet")
    g.add(c)
    c = ot.Cloud(Y)
    c.setColor("black")
    c.setPointStyle("bullet")
    g.add(c)
    g.setBoundingBox(
        ot.Interval(
            Y.getMin() - 0.5 * Y.computeRange(), Y.getMax() + 0.5 * Y.computeRange()
        )
    )
    return g

generate some multivariate data to estimate, with correlation

f = ot.SymbolicFunction(
    ["U", "xi1", "xi2"],
    ["sin(U)/(1+cos(U)^2)+0.05*xi1", "sin(U)*cos(U)/(1+cos(U)^2)+0.05*xi2"],
)
U = ot.Uniform(-0.85 * m.pi, 0.85 * m.pi)
xi = ot.Normal(2)
X = ot.BlockIndependentDistribution([U, xi])
N = 200
Y = f(X.getSample(N))

estimation by multivariate kernel smoothing

multi_ks = ot.KernelSmoothing().build(Y)
view = viewer.View(draw(multi_ks, Y))

estimation by empirical beta copula

beta_copula = ot.EmpiricalBernsteinCopula(Y, len(Y))
marginals = [
    ot.KernelSmoothing().build(Y.getMarginal(j)) for j in range(Y.getDimension())
]
beta_dist = ot.JointDistribution(marginals, beta_copula)
view = viewer.View(draw(beta_dist, Y))

viewer.View.ShowAll()