Note

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Non parametric Adaptive Importance Sampling (NAIS)

The objective is to evaluate a probability from the Non parametric Adaptive Importance Sampling (NAIS) technique.

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

\begin{align*} g(\vect{X}) = \min \begin{pmatrix}5+0.1(x_1-x_2)^2-\frac{(x_1+x_2)}{\sqrt{2}}\\ 5+0.1(x_1-x_2)^2+\frac{(x_1+x_2)}{\sqrt{2}}\\ (x_1-x_2)+ \frac{9}{\sqrt{2}}\\ (x_2-x_1)+ \frac{9}{\sqrt{2}} \end{pmatrix} \end{align*}

and the input random vector \vect{X} = (X_1, X_2) which follows the standard 2-dimensional Normal distribution:

\begin{align*} \vect{X} \sim \mathcal{N}(\mu = [0, 0], \sigma = [1,1], corr = \mat{I}_2) \end{align*}

We want to evaluate the probability:

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

First, import the python modules:

import openturns as ot
from openturns.viewer import View
import math

Create the probabilistic model Y = g(\vect{X})

Create the input random vector \vect{X}:

X = ot.RandomVector(ot.Normal(2))

Create the function g from a PythonFunction:

def fourBranch(x):
    x1 = x[0]
    x2 = x[1]

    g1 = 5 + 0.1 * (x1 - x2) ** 2 - (x1 + x2) / math.sqrt(2)
    g2 = 5 + 0.1 * (x1 - x2) ** 2 + (x1 + x2) / math.sqrt(2)
    g3 = (x1 - x2) + 9 / math.sqrt(2)
    g4 = (x2 - x1) + 9 / math.sqrt(2)

    return [min((g1, g2, g3, g4))]


g = ot.PythonFunction(2, 1, fourBranch)

Draw the function g to help to understand the shape of the limit state function:

graph = ot.Graph("Four Branch function", "x1", "x2", True, "upper right")
drawfunction = g.draw([-8] * 2, [8] * 2, [100] * 2)
graph.add(drawfunction)
view = View(graph)
Four Branch function

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

g = ot.MemoizeFunction(g)

Create the output random vector Y = g(\vect{X}):

Y = ot.CompositeRandomVector(g, X)

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

threshold = 0.0
myEvent = ot.ThresholdEvent(Y, ot.Less(), threshold)

Evaluate the probability with the NAIS technique

quantileLevel = 0.1
algo = ot.NAIS(myEvent, quantileLevel)

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 NAIS = ", proba)
print("Current coefficient of variation = ", result.getCoefficientOfVariation())

Proba NAIS =  6.783040767375951e-06
Current coefficient of variation =  0.09906313104733465

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

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

Confidence length (0.95) =  2.6339926841138084e-06

which enables to build the confidence interval:

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

Confidence interval (0.95) = [ 5.466044425319046e-06 ,  8.100037109432855e-06 ]

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

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

Levels of g =  [3.40173,2.03082,0.448448,0]

Draw the NAIS 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 NAIS 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("NAIS sampling: samples")
graph.setXTitle(r"$x_1$")
graph.setYTitle(r"$x_2$")
graph.setLegendPosition("lower left")

Add all the NAIS samples:

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

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

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

_ = View(graph)
NAIS sampling: samples

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

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

_ = View(graph)
NAIS 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 samples with the method algo.setKeepSample(True)

select = ot.NAIS.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 =  441

Draw them! They are all in the event space.

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

View.ShowAll()
plot nais

Total running time of the script: (0 minutes 2.842 seconds)