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# 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:

```
from openturns import *
from openturns.viewer import View
from math import *
```

## 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 = RegularGrid(t0, step, n)
```

Create the squared exeponential covariance model:

where the scale parameter is and the amplitude .

```
l = 10/sqrt(2)
myCovKernel = SquaredExponential([l])
print('cov model = ', myCovKernel)
```

Out:

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

Create the gaussian process :

```
S_proc = GaussianProcess(myCovKernel, myMesh)
```

## 2. Create the process ¶

First, create the random variable , with and :

```
muR = 5
sigR = 0.3
R = Normal(muR, sigR)
```

The create the Dirac random variable :

```
B = Dirac(b)
```

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

with independent.

```
const_func = SymbolicFunction(['t'], ['1'])
linear_func = SymbolicFunction(['t'], ['-t'])
myBasis = Basis([const_func, linear_func])
coef = ComposedDistribution([R, B])
R_proc = FunctionalBasisProcess(coef, myBasis, myMesh)
```

## 3. Create the process ¶

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

```
myRS_proc = 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 = SymbolicFunction(['x1', 'x2'], ['x1-x2'])
gDyn = ValueFunction(g, myMesh)
```

Now you have to create the final process thanks to :

```
Z_proc = 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 $Z(\omega, t)$')
Show(graph)
```

## 5. Evaluate the probability that ¶

We define the domaine and the event :

```
domain = Interval([2], [4])
print('D = ', domain)
event = ProcessEvent(Z_proc, domain)
```

Out:

```
D = [2, 4]
```

We use the Monte Carlo sampling to evaluate the probability:

```
MC_algo = 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, ']')
```

Out:

```
Probability = 0.7551515151515152
Variance Estimate = 5.659164649247292e-05
IC (90%) = [ 0.7427777057653888 , 0.7675253245376417 ]
```

**Total running time of the script:** ( 0 minutes 0.129 seconds)