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Estimate a conditional quantile

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From a multivariate data sample, we estimate a distribution with kernel smoothing. Here we present a bivariate distribution X= (X_1, X_2). We use the computeConditionalQuantile method to estimate the 90% quantile Q_1 of the conditional variable X_2|X_1 :

Q_2 : x_1 \mapsto q_{0.9}(X_2|X_1=x_1)

We then draw the curve Q_2 : x_1 \mapsto Q_2(x_1). We first start with independent Normals then we consider dependent marginals with a Clayton copula.

import openturns as ot
import openturns.viewer as viewer
import numpy as np

Defining the marginals

We consider two independent Normal marginals :

X1 = ot.Normal(0.0, 1.0)
X2 = ot.Normal(0.0, 3.0)

Independent marginals

distX = ot.JointDistribution([X1, X2])
sample = distX.getSample(1000)

Let’s see the data

graph = ot.Graph("2D-Normal sample", "x1", "x2", True, "")
cloud = ot.Cloud(sample, "blue", "fsquare", "My Cloud")
graph.add(cloud)
graph.setXTitle("$X_1$")
graph.setYTitle("$X_2$")
graph.setTitle("A sample from $X=(X_1, X_2)$")
view = viewer.View(graph)
A sample from $X=(X_1, X_2)$

We draw the isolines of the PDF of X :

graph = distX.drawPDF()
graph.setXTitle("$X_1$")
graph.setYTitle("$X_2$")
graph.setTitle("iso-PDF of $X=(X_1, X_2)$")
view = viewer.View(graph)
iso-PDF of $X=(X_1, X_2)$

We estimate the density with kernel smoothing :

kernel = ot.KernelSmoothing()
estimated = kernel.build(sample)

We draw the isolines of the estimated PDF of X :

graph = estimated.drawPDF()
graph.setXTitle("$X_1$")
graph.setYTitle("$X_2$")
graph.setTitle("iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing")
view = viewer.View(graph)
iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing

We can compute the conditional quantile of X_2 | X_1 with the computeConditionalQuantile method and draw it after.

We first create \sampleSize observation points in [-3.0, 3.0] :

N = 301
xobs = np.linspace(-3.0, 3.0, N)
sampleObs = ot.Sample([[xi] for xi in xobs])

We create curves of the exact and approximated quantile Q_1

x = [xi for xi in xobs]
yapp = [estimated.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]
yex = [distX.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]

cxy_app = ot.Curve(x, yapp)
cxy_ex = ot.Curve(x, yex)
graph = ot.Graph("90% quantile of $X_2 | X_1=x_1$", "$x_1$", "$Q_2(x_1)$", True, "")
graph.add(cxy_app)
graph.add(cxy_ex)
graph.setLegends(["$Q_2$ kernel smoothing", "$Q_2$ exact"])
graph.setLegendPosition("lower right")
view = viewer.View(graph)
90% quantile of $X_2 | X_1=x_1$

In this case the Q_2 quantile is constant because of the independence of the marginals.

Dependence through a Clayton copula

We now define a Clayton copula to model the dependence between our marginals. The Clayton copula is a bivariate asymmmetric Archimedean copula, exhibiting greater dependence in the negative tail than in the positive.

copula = ot.ClaytonCopula(2.5)
distX = ot.JointDistribution([X1, X2], copula)

We generate a sample from the distribution :

sample = distX.getSample(1000)

Let’s see the data

graph = ot.Graph("2D-Normal sample", "x1", "x2", True, "")
cloud = ot.Cloud(sample, "blue", "fsquare", "My Cloud")
graph.add(cloud)
graph.setXTitle("$X_1$")
graph.setYTitle("$X_2$")
graph.setTitle("A sample from $X=(X_1, X_2)$")
view = viewer.View(graph)
A sample from $X=(X_1, X_2)$

We draw the isolines of the PDF of X :

graph = distX.drawPDF()
graph.setXTitle("$X_1$")
graph.setYTitle("$X_2$")
graph.setTitle("iso-PDF of $X=(X_1, X_2)$")
view = viewer.View(graph)
iso-PDF of $X=(X_1, X_2)$

We estimate the density with kernel smoothing :

kernel = ot.KernelSmoothing()
estimated = kernel.build(sample)

We draw the isolines of the estimated PDF of X :

graph = estimated.drawPDF()
graph.setXTitle("$X_1$")
graph.setYTitle("$X_2$")
graph.setTitle("iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing")
view = viewer.View(graph)
iso-PDF of $X=(X_1, X_2)$ estimated by kernel smoothing

We can compute the conditional quantile of X_2 | X_1=x1 with the computeConditionalQuantile method and draw it after.

We first create \sampleSize observation points in [-3.0, 3.0] :

N = 301
xobs = np.linspace(-3.0, 3.0, N)
sampleObs = ot.Sample([[xi] for xi in xobs])

We create curves of the exact and approximated quantile Q_1

x = [xi for xi in xobs]
yapp = [estimated.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]
yex = [distX.computeConditionalQuantile(0.9, sampleObs[i]) for i in range(N)]

cxy_app = ot.Curve(x, yapp)
cxy_ex = ot.Curve(x, yex)
graph = ot.Graph("90% quantile of $X_2 | X_1=x_1$", "$x_1$", "$Q_2(x_1)$", True, "")
graph.add(cxy_app)
graph.add(cxy_ex)
graph.setLegends(["$Q_2$ kernel smoothing", "$Q_2$ exact"])
graph.setLegendPosition("lower right")
view = viewer.View(graph)
90% quantile of $X_2 | X_1=x_1$

Our estimated conditional quantile is a good approximate and should be better the more data we have. We can observe it by increasing the number of samples.

Display the graphs

view.ShowAll()