Note

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Create a Point Conditional Distribution¶

In this example, we create distributions conditionned by a particular value of their marginals.

We consider the following cases:

Case 1: Bivariate distribution with continuous marginals,
Case 2: Bivariate distribution with continuous and discrete marginals,
Case 3: Trivariate distribution with continuous and discrete marginals,
Case 4: Extreme values conditioning.

We illustrate the fact that the range of the conditioned distribution is updated.

import openturns as ot
import openturns.viewer as otv
import openturns.experimental as otexp

ot.ResourceMap.SetAsString("Contour-DefaultColorMapNorm", "rank")

Case 1: Bivariate distribution with continuous marginals¶

Here, we consider the bivariate distribution of $(X_0, X_1)$ defined by:

Variable	Distribution	Parameter
$X_0$	`Gamma` ( $k, \lambda$ )	$(k, \lambda) = (5, 0.5)$
$X_1$	`Student` ( $\nu$ )	$\nu = 5$
Copula	`ClaytonCopula` ( $\theta$ )	$\theta = 2.5$

We condition the marginal $X_1$ to be equal to its quantiles of order 0.05, 0.5 and then 0.95. Then, we draw the resulting conditioned distribution of:

X_0 | X_1 = x_1

We define initial distribution of $(X_0, X_1)$ .

coll_marg = [ot.Gamma(5.0, 0.5), ot.Student(5.0)]
cop = ot.ClaytonCopula(2.5)
dist_X = ot.JointDistribution(coll_marg, cop)

We draw the PDF of $(X_0, X_1)$ .

g_X = dist_X.drawPDF()
g_X.setTitle(r"$(X_0, X_1)$: iso-lines PDF")
g_X.setXTitle(r"$x_0$")
g_X.setYTitle(r"$x_1$")

We define the conditioned marginal, the list of the conditioning values and we create all the conditioned distributions, whose PDF we draw on the same graph.

We also print the updated range of the distribution of $X_0$ .

cond_indices = [1]
q_list = [0.05, 0.5, 0.95]
cond_value_list = [coll_marg[1].computeQuantile(q)[0] for q in q_list]

g_cond = coll_marg[0].drawPDF()
g_cond.setLegends([r"dist of $X_0$"])
for index, cond_value in enumerate(cond_value_list):
    cond_value = cond_value_list[index]
    cond_dist = otexp.PointConditionalDistribution(dist_X, cond_indices, [cond_value])
    draw = cond_dist.drawPDF().getDrawable(0)
    draw.setLegend(r"$x_1 = q($" + str(q_list[index]) + ")")
    g_cond.add(draw)
    print(
        "Quantile of the cond value = ", q_list[index], "Range : ", cond_dist.getRange()
    )

g_cond.setTitle(r"PDF of $X_0|X_1 = x_1$")
g_cond.setColors(ot.Drawable.BuildDefaultPalette(len(q_list) + 1))

Quantile of the cond value =  0.05 Range :  [0.0612939, (35.8447) +inf[
Quantile of the cond value =  0.5 Range :  [0.263597, (49.7477) +inf[
Quantile of the cond value =  0.95 Range :  [0.867931, (66.3894) +inf[

We gather both graphs into one grid.

grid = ot.GridLayout(1, 2)
grid.setGraph(0, 0, g_X)
grid.setGraph(0, 1, g_cond)
# sphinx_gallery_thumbnail_number =  1
view = otv.View(grid)

, $(X_0, X_1)$: iso-lines PDF, PDF of $X_0|X_1 = x_1$

Case 2: Bivariate distribution with continuous and discrete marginals¶

Here, we consider the bivariate distribution of $(X_0, X_1)$ defined by:

Variable	Distribution	Parameter
$X_0$	`Binomial` ( $n,p$ )	$(n,p) = (25, 0.2)$
$X_1$	`Poisson` ( $\lambda$ )	$\lambda = 1$
Copula	`GumbelCopula` ( $\theta$ )	$\theta = 2$

We condition the marginal $X_1$ to be equal to its quantiles of order 0.05, 0.5 and then 0.95. Then, we draw the resulting conditioned distribution of:

X_0 | X_1 = x_1

We go through the same steps as before.

coll_marg = [ot.Binomial(25, 0.2), ot.Poisson(1.0)]
cop = ot.GumbelCopula(2)
dist_X = ot.JointDistribution(coll_marg, cop)

We draw the PDF of $(X_0, X_1)$ .

g_X = dist_X.drawPDF()
g_X.setTitle(r"$(X_0, X_1)$: iso-lines PDF")
g_X.setXTitle(r"$x_0$")
g_X.setYTitle(r"$x_1$")

We define the conditioned marginal, the list of the conditioning values and we create all the conditioned distributions, whose PDF we draw on the same graph. To make them all visible, we artificially shifted the discrete distributions by a factor 0.1, 0.2 and 0.3.

We also print the updated range of the distribution of $X_0$ .

cond_value_list = [coll_marg[1].computeQuantile(q)[0] for q in q_list]

g_cond = coll_marg[0].drawPDF()
g_cond.setLegends([r"dist of $X_0$"])
for index, cond_value in enumerate(cond_value_list):
    cond_value = cond_value_list[index]
    cond_dist = otexp.PointConditionalDistribution(dist_X, cond_indices, [cond_value])
    # Here, we shift the distribution to make it visible!
    draw = ((index / 10.0) + cond_dist).drawPDF().getDrawable(0)
    draw.setLegend(r"$x_1 = q($" + str(q_list[index]) + ")")
    g_cond.add(draw)
    print(
        "Quantile of the cond value = ", q_list[index], "Range : ", cond_dist.getRange()
    )

g_cond.setTitle(r"PDF of $X_0|X_1 = x_1$")
g_cond.setLegendPosition("topleft")
g_cond.setColors(ot.Drawable.BuildDefaultPalette(len(q_list) + 1))

Quantile of the cond value =  0.05 Range :  [0, 16]
Quantile of the cond value =  0.5 Range :  [0, 18]
Quantile of the cond value =  0.95 Range :  [0, 20]

We gather both graphs into one grid.

grid = ot.GridLayout(1, 2)
grid.setGraph(0, 0, g_X)
grid.setGraph(0, 1, g_cond)
view = otv.View(grid)

Case 3: Trivariate distribution with continuous and discrete marginals¶

Here, we consider the trivariate distribution of $(X_0, X_1, X_2)$ defined by:

Variable	Distribution	Parameter
$X_0$	`Normal` ( $\mu, \sigma$ )	$(\mu, \sigma) = (0,1)$
$X_1$	`Poisson` ( $\lambda$ )	$\lambda = 1$
$X_2$	`Uniform` ( $a,b$ )	$(a,b) = (-1, 1)$
Copula	`NormalCopula` ( $\mat{R}$ )	see below

where the correlation matrix $\mat{R}$ of the normal copula is defined by:

$R = \left( \begin{array}{ccc} 1 & 0.2 & 0.7 \\ 0.2 & 1 & -0.5 \\ 0.7 & -0.5 & 1 \end{array} \right)$

We condition the marginal $X_1$ to be equal to its quantiles of order 0.05, 0.5 and then 0.95. Then, we draw the resulting bivariate conditioned distribution of:

(X_0,X_2) | X_1 = x_1

We go through the same steps as before.

coll_marg = [ot.Normal(), ot.Poisson(1.0), ot.Uniform()]
R = ot.CorrelationMatrix(3)
R[0, 1] = 0.2
R[0, 2] = 0.7
R[1, 2] = -0.5
cop = ot.NormalCopula(R)
dist_X = ot.JointDistribution(coll_marg, cop)

We draw the distribution of $(X_0, X_2)$ .

g_X02 = dist_X.getMarginal([0, 2]).drawPDF([-4.0, -1.0], [4.0, 1.0], [256] * 2)
contour = g_X02.getDrawable(0).getImplementation()
contour.setIsFilled(True)
contour.buildDefaultLevels(50)
g_X02.setDrawable(contour, 0)
g_X02.setTitle(r"$(X_0, X_2)$: iso-lines PDF")

We draw all the conditioned distributions $(X_0, X_2)|X_1 = x_1$ .

cond_value_list = [coll_marg[1].computeQuantile(q)[0] for q in q_list]
graph_list = list()
graph_list.append(g_X02)

for index, cond_value in enumerate(cond_value_list):
    cond_value = cond_value_list[index]
    cond_dist = otexp.PointConditionalDistribution(dist_X, cond_indices, [cond_value])
    g_cond = cond_dist.drawPDF([-4.0, -1.0], [4.0, 1.0], [256] * 2)
    contour = g_cond.getDrawable(0).getImplementation()
    contour.setIsFilled(True)
    contour.buildDefaultLevels(50)
    g_cond.setDrawable(contour, 0)
    g_cond.setTitle(r"$(X_0, X_2)|X_1 = q($" + str(q_list[index]) + "): iso-lines PDF")
    g_cond.setXTitle(r"$x_0$")
    g_cond.setYTitle(r"$x_2$")
    graph_list.append(g_cond)
    print("Quantile of the cond value = ", q_list[index])
    print("Range :\n", cond_dist.getRange())

Quantile of the cond value =  0.05
Range :
 ]-inf (-7.56355), (7.42856) +inf[
[-1, 1]
Quantile of the cond value =  0.5
Range :
 ]-inf (-7.36999), (7.62212) +inf[
[-1, 1]
Quantile of the cond value =  0.95
Range :
 ]-inf (-7.08103), (7.91108) +inf[
[-1, 1]

We gather both graphs into one grid.

grid = ot.GridLayout(2, 2)
grid.setGraph(0, 0, graph_list[0])
grid.setGraph(0, 1, graph_list[1])
grid.setGraph(1, 0, graph_list[2])
grid.setGraph(1, 1, graph_list[3])
view = otv.View(grid)

, $(X_0, X_2)$: iso-lines PDF, $(X_0, X_2)|X_1 = q($0.05): iso-lines PDF, $(X_0, X_2)|X_1 = q($0.5): iso-lines PDF, $(X_0, X_2)|X_1 = q($0.95): iso-lines PDF

Case 4: Extreme values conditioning¶

Here, we show that it is possible to condition one marginal to a value which is outside its numerical range. Be careful that the conditioning value still must be in the theoretical range of the distribution!

We consider the bivariate distribution of $(X_0, X_1)$ defined by:

Variable	Distribution	Parameter
$X_0$	`Normal` ( $\mu, \sigma$ )	$(\mu, \sigma) = (0,1)$
$X_1$	`Normal` ( $\mu, \sigma$ )	$(\mu, \sigma) = (0,1)$
Copula	`ClaytonCopula` ( $\theta$ )	$\theta = 2$

We condition the marginal $X_1$ to be equal to the value $x_1 = -9$ , which is outside the numerical range of its distribution: $[-7.65063, +7.65063]$ , but inside the theoretical range which is $\Rset$ .

We draw the distribution of:

X_0 | X_1 = -9.0

dist_X = ot.JointDistribution([ot.Normal(), ot.Normal()], ot.ClaytonCopula(2.0))
cond_dist = otexp.PointConditionalDistribution(dist_X, [1], [-9.0])

g_cond = cond_dist.drawPDF(-10.0, -8.0, 256)
g_cond.setTitle(r"$X_0|X_1 = -9$: iso-lines PDF")
g_cond.setXTitle(r"$x_0$")
g_cond.setLegendPosition("")
view = otv.View(g_cond)

We print the updated numerical range of $X_0| X_1 = -9.0$ . We note that it has moved into $[-11.76, -0.82]$ which is not included in the numerical range of $X_0$ .

print("Range of X0|X1 = -9.0 :\n", cond_dist.getRange())

Range of X0|X1 = -9.0 :
 ]-inf (-11.7626), (-0.825411) +inf[

view.ShowAll()

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Create a Point Conditional Distribution¶

Case 1: Bivariate distribution with continuous marginals¶

Case 2: Bivariate distribution with continuous and discrete marginals¶

Case 3: Trivariate distribution with continuous and discrete marginals¶

Case 4: Extreme values conditioning¶