# Create a stationary covariance modelΒΆ

This use case illustrates how the User can create a covariance function from parametric models. The library implements the multivariate Exponential model as a parametric model for the covariance function where the spatial covariance function writes:

It is possible to define the exponential model from the spatial covariance matrix rather than the correlation matrix :

with:

We call the amplitude vector and the scale vector.

The library implements the multivariate exponential model thanks to the object ExponentialModel which is created from:

• the scale and amplitude vectors : in that case, by default ;

• the scale and amplitude vectors and the spatial correlation matrix ;

• the scale and amplitude vectors and the spatial covariance matrix ; Then is mapped into the associated correlation matrix and the previous constructor is used.

import openturns as ot

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

Create the amplitude vector (output dimension 3)

amplitude = [1.0, 2.0, 3.0]

# Scale vector (input dimension 1)
scale = [4.0]

# spatialCorrelation
spatialCorrelation = ot.CorrelationMatrix(3)
spatialCorrelation[0, 1] = 0.8
spatialCorrelation[0, 2] = 0.6
spatialCorrelation[1, 2] = 0.1

# spatialCovariance
spatialCovariance = ot.CovarianceMatrix(3)
spatialCovariance[0, 0] = 4.0
spatialCovariance[1, 1] = 5.0
spatialCovariance[2, 2] = 6.0
spatialCovariance[0, 1] = 1.2
spatialCovariance[0, 2] = 0.9
spatialCovariance[1, 2] = -0.2

Create the covariance model from the amplitude and scale, no spatial correlation

ot.ExponentialModel(scale, amplitude)

ExponentialModel(scale=[4], amplitude=[1,2,3], no spatial correlation)

or from the amplitude, scale and spatialCovariance

ot.ExponentialModel(scale, amplitude, spatialCorrelation)

ExponentialModel(scale=[4], amplitude=[1,2,3], spatial correlation=
[[ 1 0.8 0.6 ]
[ 0.8 1 0.1 ]
[ 0.6 0.1 1 ]])

or from the scale and spatialCovariance

ot.ExponentialModel(scale, spatialCovariance)

ExponentialModel(scale=[4], amplitude=[2,2.23607,2.44949], spatial correlation=
[[ 1 0.268328 0.183712 ]
[ 0.268328 1 -0.0365148 ]
[ 0.183712 -0.0365148 1 ]])