Note

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Estimate a non stationary covariance function

The objective of this use case is to estimate C from several fields generated by the process X. We suppose that the process is not stationary.

In the following example, we illustrate the estimation of the non stationary covariance model

C : \mathcal{D} \times [-4, 4] \rightarrow \mathbb{R} defined by:

\begin{aligned} \displaystyle C(s,t) = \exp\left(-\dfrac{4|s-t|}{1+s^2+t^2}\right) \end{aligned}

The domain \mathcal{D} is discretized on a mesh \mathcal{M} which is a time grid with 64 points. We build a normal process X: \Omega \times [-4, 4] \rightarrow \mathbb{R} with zero mean and C as covariance function. We discretize the covariance model C using C(t_k, t_\ell) for each (t_k, t_\ell)\in \mathcal{M} \times \mathcal{M}. We get a N=10^3 fields from the process X from which we estimate the covariance model C.

We use the object NonStationaryCovarianceModelFactory which creates a UserDefinedCovarianceModel.

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

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

Create the time grid

t0 = -4.0
tmax = 4.0
N = 64
dt = (tmax - t0) / N
tgrid = ot.RegularGrid(t0, dt, N)

Create the covariance function at (s,t)

def C(s, t):
    return m.exp(-4.0 * abs(s - t) / (1 + (s * s + t * t)))

Draw…

def f(X):
    s, t = X
    return [C(s, t)]


func = ot.PythonFunction(2, 1, f)
func.setDescription([":math:`s`", ":math:`t`", ":math:`cov`"])
graph = func.draw([t0] * 2, [tmax] * 2)
graph.setTitle("Original covariance model")
graph.setLegendPosition("")
view = viewer.View(graph)
Original covariance model

Create data from a non stationary normal process Omega * [0,T]–> R

# Create the collection of HermitianMatrix
covariance = ot.CovarianceMatrix(N)
for k in range(N):
    s = tgrid.getValue(k)
    for ll in range(k + 1):
        t = tgrid.getValue(ll)
        covariance[k, ll] = C(s, t)

covmodel = ot.UserDefinedCovarianceModel(tgrid, covariance)

Create the normal process with that covariance model based on the mesh tgrid

process = ot.GaussianProcess(covmodel, tgrid)

# Get a sample of fields from the process
N = 1000
sample = process.getSample(N)

The covariance model factory

factory = ot.NonStationaryCovarianceModelFactory()

# Estimation on a sample
estimatedModel = factory.build(sample)

graph = estimatedModel.draw(0, 0, t0, tmax, 256, False)
graph.setTitle("Estimated covariance model")
graph.setLegendPosition("")
view = viewer.View(graph)
plt.show()
Estimated covariance model