Create a custom covariance model

This example illustrates how the user can define his own covariance model.

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

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

Construct the covariance model

Create the time grid

N = 32
a = 4.0
mesh = ot.IntervalMesher([N]).build(ot.Interval(-a, a))

Create the covariance function depending on (s,t)

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

Create the large covariance matrix

covariance = ot.CovarianceMatrix(mesh.getVerticesNumber())
for k in range(mesh.getVerticesNumber()):
    t = mesh.getVertices()[k]
    for ll in range(k + 1):
        s = mesh.getVertices()[ll]
        covariance[k, ll] = C(s[0], t[0])

Create the covariance model

covmodel = ot.UserDefinedCovarianceModel(mesh, covariance)

Draw the covariance model as a function

Define the function to draw

def f(x):
    return [covmodel([x[0]], [x[1]])[0, 0]]


func = ot.PythonFunction(2, 1, f)
func.setDescription(["$s$", "$t$", "$cov$"])

Draw the function with default options

cov_graph = func.draw([-a] * 2, [a] * 2, [512] * 2)
cov_graph.setLegendPosition("")
view = viewer.View(cov_graph)

Draw the function in a filled contour graph

cov_graph = func.draw(
    0, 1, 0, [0] * 2, [-a] * 2, [a] * 2, [512] * 2, ot.GraphImplementation.NONE, True
)
view = viewer.View(cov_graph)

Draw the covariance model as a matrix

Use raw matshow

plt.matshow(covariance)

<matplotlib.image.AxesImage object at 0x765a44030e30>

Draw the covariance model as a matrix with the correct axes.

To obtain the correct orientation of the y axis we use the origin argument. To obtain the correct graduations we use the extent argument. We also change the colormap used.

pas = 2 * a / (N - 1)
plt.matshow(
    covariance,
    cmap="gray",
    origin="lower",
    extent=(-a - pas / 2, a + pas / 2, -a - pas / 2, a + pas / 2),
)
plt.show()