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Create a white noise processΒΆ

This example details how to create and manipulate a white noise. A second order white noise \varepsilon: \Omega \times \mathbb{D} \rightarrow \mathbb{R}^d is a stochastic process of dimension d such that the covariance function C(\underline{s},\underline{t})=\delta(\underline{t}-\underline{s})C(\underline{s},\underline{s}) where C(\underline{s},\underline{s}) is the covariance matrix of the process at vertex \underline{s} and \delta the Kroenecker function.

A process \varepsilon is a white noise if all finite family of locations (\underline{t}_i)_{i=1, \dots, n} \in \mathbb{D}, (\varepsilon_{\underline{t}_i})_{i=1, \dots, n} is independent and identically distributed.

The library proposes to model it through the object WhiteNoise defined on a mesh and a distribution with zero mean and finite standard deviation.

If the distribution has a mean different from zero, The library writes message to prevent the User and does not allow the creation of such a white noise.

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)

Define the distribution

sigma = 1.0
dist = ot.Normal(0.0, sigma)

Define the mesh

tgrid = ot.RegularGrid(0.0, 1.0, 100)

Create the process

process = ot.WhiteNoise(dist, tgrid)
process
class=WhiteNoise distribution=class=Normal name=Normal dimension=1 mean=class=Point name=Unnamed dimension=1 values=[0] sigma=class=Point name=Unnamed dimension=1 values=[1] correlationMatrix=class=CorrelationMatrix dimension=1 implementation=class=MatrixImplementation name=Unnamed rows=1 columns=1 values=[1]


Draw a realization

realization = process.getRealization()
graph = realization.drawMarginal(0)
graph.setTitle("Realization of a white noise with distribution N(0,1)")
view = viewer.View(graph)
Realization of a white noise with distribution N(0,1)

Draw a sample

sample = process.getSample(5)
graph = sample.drawMarginal(0)
graph.setTitle(
    str(sample.getSize()) + " realizations of a white noise with distribution N(0,1)"
)
for k in range(sample.getSize()):
    drawable = graph.getDrawable(k)
    drawable.setLegend("realization " + str(k + 1))
    graph.setDrawable(drawable, k)
view = viewer.View(graph)
plt.show()
5 realizations of a white noise with distribution N(0,1)