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)

Out:

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())

Out:

Proba Subset =  0.000405100000000001
Current coefficient of variation =  0.08824365916251675

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

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

Out:

Confidence length (0.95) =  0.00014012764987503907

which enables to build the confidence interval:

print('Confidence intervalle (0.95) = [', proba - length95/2, ', ', proba + length95/2, ']')

Out:

Confidence intervalle (0.95) = [ 0.0003350361750624815 ,  0.00047516382493752055 ]

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

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

Out:

Levels of g =  [56.6226,18.2723,7.85413,0]

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)

Out:

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)

Out:

Size of each subset =  10000

You can get the number N_s of steps with:

Ns = algo.getNumberOfSteps()
print('Number of steps= ', Ns)

Out:

Number of steps=  4

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

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

Out:

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

Out:

<openturns.viewer.View object at 0x7fd4cc742880>

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

Out:

<openturns.viewer.View object at 0x7fd4cd14b460>

Get all the input and output points that realized the event

The following lines are possible only if you have mentionned 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())

Out:

Number of event realizations =  4051

Here we have to avoid a bug of the version 1.15 because getEventInputSample() gives the sample in the stadrad space: we have to push it backward to the physical space.

dist = ot.Normal([0.25]*2, [1]*2, ot.IdentityMatrix(2))
transformFunc = dist.getInverseIsoProbabilisticTransformation()
inputEventSample = transformFunc(inputEventSample)

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

Out:

<openturns.viewer.View object at 0x7fd4a7f7e370>

Total running time of the script: ( 0 minutes 1.048 seconds)

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