Note

Go to the end to download the full example code.

Subset Sampling¶

The objective is to evaluate a probability from the Subset sampling technique.

We consider the function $g : \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by:

$\begin{align*} g(X)= 20-(x_1-x_2)^2-8(x_1+x_2-4)^3 \end{align*}$

and the input random vector $X = (X_1, X_2)$ which follows a Normal distribution with independent components, and identical marginals with 0.25 mean and unit variance:

$\begin{align*} X \sim \mathcal{N}(\mu = [0.25, 0.25], \sigma = [1,1], cor = I_2) \end{align*}$

We want to evaluate the probability:

$\begin{align*} p = \mathbb{P} \{ g(X) \leq 0 \} \end{align*}$

First, import the python modules:

import openturns as ot
import openturns.viewer as otv

Create the probabilistic model $Y = g(X)$ ¶

Create the input random vector $X$ :

X = ot.RandomVector(ot.Normal([0.25] * 2, [1] * 2, ot.IdentityMatrix(2)))

Create the function $g$ :

g = ot.SymbolicFunction(["x1", "x2"], ["20-(x1-x2)^2-8*(x1+x2-4)^3"])
print("function g: ", g)

function g:  [x1,x2]->[20-(x1-x2)^2-8*(x1+x2-4)^3]

Create the output random vector $Y = g(X)$ :

Y = ot.CompositeRandomVector(g, X)

Create the event $\{ Y = g(X) \leq 0 \}$ ¶

event = ot.ThresholdEvent(Y, ot.Less(), 0.0)

Evaluate the probability with the subset sampling technique¶

algo = ot.SubsetSampling(event)

In order to get all the inputs and outputs that realize the event, you have to mention it now:

algo.setKeepSample(True)

Now you can run the algorithm!

algo.run()

result = algo.getResult()
proba = result.getProbabilityEstimate()
print("Proba Subset = ", proba)
print("Current coefficient of variation = ", result.getCoefficientOfVariation())

Proba Subset =  0.0003695999999999999
Current coefficient of variation =  0.08967382696106657

The length of the confidence interval of level $95\%$ is:

length95 = result.getConfidenceLength()
print("Confidence length (0.95) = ", result.getConfidenceLength())

Confidence length (0.95) =  0.00012991992271072015

which enables to build the confidence interval:

print(
    "Confidence interval (0.95) = [",
    proba - length95 / 2,
    ", ",
    proba + length95 / 2,
    "]",
)

Confidence interval (0.95) = [ 0.0003046400386446398 ,  0.00043455996135535995 ]

You can also get the successive thresholds used by the algorithm:

levels = algo.getThresholdPerStep()
print("Levels of g = ", levels)

Levels of g =  [57.4335,18.5265,8.17273,0]

Draw the subset samples used by the algorithm¶

You can get the number $N_s$ of steps with:

Ns = algo.getStepsNumber()
print("Number of steps= ", Ns)

Number of steps=  4

Get all the inputs where $g$ was evaluated at each step

list_subSamples = list()
for step in range(Ns):
    list_subSamples.append(algo.getInputSample(step))

The following graph draws each subset sample and the frontier $g(x_1, x_2) = l_i$ where $l_i$ is the threshold at the step $i$ :

graph = ot.Graph()
graph.setAxes(True)
graph.setGrid(True)
graph.setTitle("Subset sampling: samples")
graph.setXTitle(r"$x_1$")
graph.setYTitle(r"$x_2$")
graph.setLegendPosition("lower left")

Add all the subset samples:

for i in range(Ns):
    cloud = ot.Cloud(list_subSamples[i])
    cloud.setPointStyle("dot")
    graph.add(cloud)

Add the frontiers $g(x_1, x_2) = l_i$ where $l_i$ is the threshold at the step $i$ :

gIsoLines = g.draw([-3] * 2, [5] * 2, [128] * 2)
dr = gIsoLines.getDrawable(0)
dr.setColor("black")
for i in range(levels.getSize()):
    dr.setLevels([levels[i]])
    dr.setLegend(r"$g(X) = $" + str(round(levels[i], 2)))
    graph.add(dr)

_ = otv.View(graph)

Draw the frontiers only¶

The following graph enables to understand the progression of the algorithm:

graph = ot.Graph()
graph.setAxes(True)
graph.setGrid(True)
dr = gIsoLines.getDrawable(0)
colors = ot.Drawable().BuildDefaultPalette(len(levels))
for i in range(levels.getSize()):
    dr.setLevels([levels[i]])
    dr.setLegend(r"$g(X) = $" + str(round(levels[i], 2)))
    dr.setColor(colors[i])
    graph.add(dr)

graph.setLegendPosition("lower left")
graph.setTitle("Subset sampling: thresholds")
graph.setXTitle(r"$x_1$")
graph.setYTitle(r"$x_2$")

_ = otv.View(graph)

Get all the input and output points that realized the event¶

The following lines are possible only if you have mentioned that you wanted to keep samples with the method algo.setKeepSample(True)

select = ot.SubsetSampling.EVENT1  # points that realize the event
step = Ns - 1  # get the working sample from last iteration
inputEventSample = algo.getInputSample(step, select)
outputEventSample = algo.getOutputSample(step, select)
print("Number of event realizations = ", inputEventSample.getSize())

Number of event realizations =  3696

Draw them! They are all in the event space.

graph = ot.Graph()
graph.setAxes(True)
graph.setGrid(True)
cloud = ot.Cloud(inputEventSample)
cloud.setPointStyle("dot")
graph.add(cloud)
gIsoLines = g.draw([-3] * 2, [5] * 2, [1000] * 2)
dr = gIsoLines.getDrawable(0)
dr.setLevels([0.0])
dr.setColor("red")
graph.add(dr)
_ = otv.View(graph)

otv.View.ShowAll()

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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Subset Sampling¶

Create the probabilistic model $Y = g(X)$ ¶

Create the event $\{ Y = g(X) \leq 0 \}$ ¶

Evaluate the probability with the subset sampling technique¶

Draw the subset samples used by the algorithm¶

Draw the frontiers only¶

Get all the input and output points that realized the event¶

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

Subset Sampling¶

Create the probabilistic model ¶

Create the event ¶

Evaluate the probability with the subset sampling technique¶

Draw the subset samples used by the algorithm¶

Draw the frontiers only¶

Get all the input and output points that realized the event¶

Create the probabilistic model $Y = g(X)$ ¶

Create the event $\{ Y = g(X) \leq 0 \}$ ¶