Create a covariance model with and without a nugget effect

This example shows the impact of a nugget effect on the covariance function of a process:

• we create two covariance functions: the second one adds a nugget effect to the first one,

• we draw the covariance functions.

The nugget effect allows us to model a noise observed in the output values of a process. This noise may be, for example, a measurement noise from a sensor with finite precision.

The noise is characterized by the scalar which is added to the correlation function evaluated on each where . See Covariance models to get more details on covariance models and the introduction of a nugget factor, and in particular see equation (5).

import openturns as ot
import openturns.viewer as otv

# sphinx_gallery_thumbnail_number = 1

First we create the covariance function of a process defined on and with scalar outputs. We use the SquaredExponential with unit scale, unit amplitude. By default, the nugget factor is equal to (defined through the CovarianceModel-DefaultNuggetFactor key in the class ResourceMap) which allows the regularization of the covariance matrix. We set it to zero.

cov_model = ot.SquaredExponential()
cov_model.setNuggetFactor(0.0)

Then, we create a second covariance function which adds a nugget factor to the first one with. We take the nugget factor .

cov_model_with_nugget = ot.SquaredExponential()
epsilon_nugget = 1
cov_model_with_nugget.setNuggetFactor(epsilon_nugget)

We draw the covariance function with and without the nugget effect.

g = cov_model.draw()
g.setBoundingBox(ot.Interval([-5, 0.0], [5.0, 2.0]))
g.setTitle("Without nugget effect")
g_nugget = cov_model_with_nugget.draw()
g_nugget.setBoundingBox(ot.Interval([-5, 0.0], [5.0, 2.0]))
g_nugget.setTitle("With nugget effect")

grid = ot.GridLayout(1, 2)
grid.setGraph(0, 0, g)
grid.setGraph(0, 1, g_nugget)
view = otv.View(grid)
otv.View.ShowAll()

Display all figures.

otv.View.ShowAll()