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"""
Estimate a conditional quantile
===============================
"""
# sphinx_gallery_thumbnail_number = 8
# %%
# From a multivariate data sample, we estimate a distribution with kernel smoothing.
# Here we present a bivariate distribution :math:`X= (X_1, X_2)`.
# We use the `computeConditionalQuantile` method to estimate the 90% quantile :math:`Q_1` of the conditional variable :math:`X_2|X_1` :
#
# .. math::
# Q_2 : x_1 \mapsto q_{0.9}(X_2|X_1=x_1)
#
# We then draw the curve :math:`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
ot.Log.Show(ot.Log.NONE)
# %%
# Set the random generator seed
ot.RandomGenerator.SetSeed(0)
# %%
# 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)
# %%
# We draw the isolines of the PDF of :math:`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)
# %%
# We estimate the density with kernel smoothing :
kernel = ot.KernelSmoothing()
estimated = kernel.build(sample)
# %%
# We draw the isolines of the estimated PDF of :math:`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)
# %%
# We can compute the conditional quantile of :math:`X_2 | X_1` with the `computeConditionalQuantile` method and draw it after.
#
# %%
# We first create :math:`\sampleSize` observation points in :math:`[-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 :math:`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)
# %%
# In this case the :math:`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)
# %%
# We draw the isolines of the PDF of :math:`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)
# %%
# We estimate the density with kernel smoothing :
kernel = ot.KernelSmoothing()
estimated = kernel.build(sample)
# %%
# We draw the isolines of the estimated PDF of :math:`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)
# %%
# We can compute the conditional quantile of :math:`X_2 | X_1=x1` with the `computeConditionalQuantile` method and draw it after.
#
# %%
# We first create :math:`\sampleSize` observation points in :math:`[-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 :math:`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)
# %%
# 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()
|
