# Create a process from random vectors and processes¶

The objective is to create a process defined from a random vector and a process.

We consider the following limit state function, defined as the difference between a degrading resistance and a time-varying load :

We propose the following probabilistic model:

• is the initial resistance, and ;

• is the deterioration rate of the resistance; it is deterministic;

• is the time-varying stress, which is modeled by a stationary Gaussian process of mean value , standard deviation and a squared exponential covariance model;

• is the time, varying in .

First, import the python modules:

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


## 1. Create the gaussian process ¶

Create the mesh which is a regular grid on , with , by step =1:

b = 0.01
t0 = 0.0
step = 1
tfin = 50
n = round((tfin - t0) / step)
myMesh = ot.RegularGrid(t0, step, n)


Create the squared exeponential covariance model:

where the scale parameter is and the amplitude .

ll = 10 / m.sqrt(2)
myCovKernel = ot.SquaredExponential([ll])
print("cov model = ", myCovKernel)

cov model =  SquaredExponential(scale=[7.07107], amplitude=[1])


Create the gaussian process :

S_proc = ot.GaussianProcess(myCovKernel, myMesh)


## 2. Create the process ¶

First, create the random variable , with and :

muR = 5
sigR = 0.3
R = ot.Normal(muR, sigR)


The create the Dirac random variable :

B = ot.Dirac(b)


Then create the process using the class and the functional basis and :

with independent.

const_func = ot.SymbolicFunction(["t"], ["1"])
linear_func = ot.SymbolicFunction(["t"], ["-t"])
myBasis = ot.Basis([const_func, linear_func])

coef = ot.JointDistribution([R, B])

R_proc = ot.FunctionalBasisProcess(coef, myBasis, myMesh)


## 3. Create the process ¶

First, aggregate both processes into one process of dimension 2:

myRS_proc = ot.AggregatedProcess([R_proc, S_proc])


Then create the spatial field function that acts only on the values of the process, keeping the mesh unchanged, using the ValueFunction class. We define the function on by:

in order to define the spatial field function that acts on fields, defined by:

g = ot.SymbolicFunction(["x1", "x2"], ["x1-x2"])
gDyn = ot.ValueFunction(g, myMesh)


Now you have to create the final process thanks to :

Z_proc = ot.CompositeProcess(gDyn, myRS_proc)


## 4. Draw some realizations of the process¶

N = 10
sampleZ_proc = Z_proc.getSample(N)
graph = sampleZ_proc.drawMarginal(0)
graph.setTitle(r"Some realizations of $Z(\omega, t)$")
view = View(graph)


## 5. Evaluate the probability that ¶

We define the domaine and the event :

domain = ot.Interval([2], [4])
print("D = ", domain)
event = ot.ProcessEvent(Z_proc, domain)

D =  [2, 4]


We use the Monte Carlo sampling to evaluate the probability:

MC_algo = ot.ProbabilitySimulationAlgorithm(event)
MC_algo.setMaximumOuterSampling(1000000)
MC_algo.setBlockSize(100)
MC_algo.setMaximumCoefficientOfVariation(0.01)
MC_algo.run()

result = MC_algo.getResult()

proba = result.getProbabilityEstimate()
print("Probability = ", proba)
variance = result.getVarianceEstimate()
print("Variance Estimate = ", variance)
IC90_low = proba - result.getConfidenceLength(0.90) / 2
IC90_upp = proba + result.getConfidenceLength(0.90) / 2
print("IC (90%) = [", IC90_low, ", ", IC90_upp, "]")
view.ShowAll()

Probability =  0.7572727272727273
Variance Estimate =  5.625950969752621e-05
IC (90%) = [ 0.7449352823106309 ,  0.7696101722348238 ]