Note

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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 . We use the computeConditionalQuantile method to estimate the 90% quantile of the conditional variable :

We then draw the curve . We first start with independent normals then we consider dependent marginals with a Clayton copula.

```
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
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.ComposedDistribution([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 :

```
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 :

```
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 with the computeConditionalQuantile method and draw it after.

We first create N observation points in :

```
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

```
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("bottomright")
graph.setColors(["red", "blue"])
view = viewer.View(graph)
```

In this case the 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.ComposedDistribution([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 :

```
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 :

```
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 with the computeConditionalQuantile method and draw it after.

We first create N observation points in :

```
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

```
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("bottomright")
graph.setColors(["red", "blue"])
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.

```
plt.show()
```