Create a functional basis processΒΆ

The objective of this example is to define X: \Omega \times \mathcal{D} \rightarrow \mathbb{R}^d a multivariate stochastic process of dimension d where \mathcal{D} \in \mathbb{R}^n, as a linear combination of K deterministic functions (\phi_i)_{i=1,\dots,K}: \mathbb{R}^n \rightarrow \mathbb{R}^d:

\begin{aligned} X(\omega,\underline{t})=\sum_{i=1}^KA_i(\omega)\phi_i(\underline{t})\end{aligned}

where \underline{A}=(A_1,\dots, A_K) is a random vector of dimension K.

We suppose that \mathcal{M} is discretized on the mesh \mathcal{M} wich has N vertices.

A realization of X on \mathcal{M} consists in generating a realization \underline{\alpha} of the random vector \underline{A} and in evaluating the functions (\phi_i)_{i=1,\dots,K} on the mesh \mathcal{M}.

If we note (\underline{x}_0, \dots, \underline{x}_{N-1}) the realization of X, where X(\omega, \underline{t}_k) = \underline{x}_k, we have:

\begin{aligned} \forall k \in [0, N-1], \quad \underline{x}_k = \sum_{i=1}^K\alpha_i\phi_i(\underline{t}_k)\end{aligned}

[40]:

from __future__ import print_function
import openturns as ot
import math as m

[41]:

# Define the coefficients distribution
mu = [2.0]*2
sigma = [5.0]*2
R = ot.CorrelationMatrix(2)
coefDist = ot.Normal(mu, sigma, R)

[42]:

# Create a basis of functions
phi_1 = ot.SymbolicFunction(['t'], ['sin(t)'])
phi_2 = ot.SymbolicFunction(['t'], ['cos(t)^2'])
myBasis = ot.Basis([phi_1, phi_2])

[43]:

# Create the mesh
myMesh = ot.RegularGrid(0.0, 0.1, 100)

[44]:

# Create the process
process = ot.FunctionalBasisProcess(coefDist, myBasis, myMesh)

[45]:

# Draw a sample
N = 6
sample = process.getSample(N)
graph = sample.drawMarginal(0)
graph.setTitle(str(N)+' realizations of functional basis process')
graph

[45]:
../../_images/examples_probabilistic_modeling_functional_basis_process_7_0.png