Note

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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], cov = 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
from openturns.viewer import View

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]

In order to be able to get the subset samples used in the algorithm, it is necessary to transform the SymbolicFunction into a MemoizeFunction:

g = ot.MemoizeFunction(g)

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

Y = ot.CompositeRandomVector(g, X)

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

myEvent = ot.ThresholdEvent(Y, ot.LessOrEqual(), 0.0)

Evaluate the probability with the subset sampling technique

algo = ot.SubsetSampling(myEvent)

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

algo.setKeepEventSample(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 =  0.00040940000000000025
Current coefficient of variation =  0.08584233339124359

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

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

Confidence length (0.95) =  0.00013776136561433357

which enables to build the confidence interval:

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

Confidence interval (0.95) = [ 0.0003405193171928335 ,  0.000478280682807167 ]

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

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

Levels of g =  [54.0207,18.0184,7.60975,0]

Draw the subset samples used by the algorithm

The following manipulations are possible onfly if you have created a MemoizeFunction that enables to store all the inputs and output of the function g.

Get all the inputs where g were evaluated:

inputSampleSubset = g.getInputHistory()
nTotal = inputSampleSubset.getSize()
print("Number of evaluations of g = ", nTotal)

Number of evaluations of g =  40000

Within each step of the algorithm, a sample of size N is created, where:

N = algo.getMaximumOuterSampling() * algo.getBlockSize()
print("Size of each subset = ", N)

Size of each subset =  10000

You can get the number N_s of steps with:

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

Number of steps=  4

and you can verify that N_s is equal to \frac{nTotal}{N}:

print("nTotal / N = ", int(nTotal / N))

nTotal / N =  4

Now, we can split the initial sample into subset samples of size N_s:

list_subSamples = list()
for i in range(Ns):
    list_subSamples.append(inputSampleSubset[i * N: i * N + N])

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("bottomleft")

Add all the subset samples:

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

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)
for i in range(levels.getSize()):
    dr.setLevels([levels[i]])
    dr.setLineStyle("solid")
    dr.setLegend(r"$g(X) = $" + str(round(levels[i], 2)))
    dr.setLineWidth(3)
    dr.setColor(col[i])
    graph.add(dr)

_ = View(graph)
Subset sampling: samples

Draw the frontiers only

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

graph = ot.Graph()
graph.setAxes(True)
graph.setGrid(True)
dr = gIsoLines.getDrawable(0)
for i in range(levels.getSize()):
    dr.setLevels([levels[i]])
    dr.setLineStyle("solid")
    dr.setLegend(r"$g(X) = $" + str(round(levels[i], 2)))
    dr.setLineWidth(3)
    graph.add(dr)

graph.setColors(col)
graph.setLegendPosition("bottomleft")
graph.setTitle("Subset sampling: thresholds")
graph.setXTitle(r"$x_1$")
graph.setYTitle(r"$x_2$")

_ = View(graph)
Subset sampling: thresholds

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 the points that realize the event with the method algo.setKeepEventSample(True)

inputEventSample = algo.getEventInputSample()
outputEventSample = algo.getEventOutputSample()
print("Number of event realizations = ", inputEventSample.getSize())

Number of event realizations =  4094

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)
_ = View(graph)
plot subset sampling