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 \rho writes:

\rho(\underline{s}, \underline{t} ) = e^{-\left\| \underline{s}- \underline{t} \right\|_2} \quad \forall (\underline{s}, \underline{t}) \in \mathcal{D}

It is possible to define the exponential model from the spatial covariance matrix \underline{\underline{C}}^{spat} rather than the correlation matrix \underline{\underline{R}}:

\forall \underline{t} \in \mathcal{D},\quad \underline{\underline{C}}^{spat} = \mathbb{E} \left[ X_{\underline{t}} X^t_{\underline{t}} \right] = \underline{\underline{A}}\,\underline{\underline{R}}\, \underline{\underline{A}}

with:

\underline{\underline{A}} = \mbox{Diag}(a_1, \dots, a_d)

We call \underline{a} the amplitude vector and \underline{\lambda} the scale vector.

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

  • the scale and amplitude vectors (\underline{\lambda}, \underline{a}): in that case, by default \underline{\underline{R}} = \underline{\underline{I}};

  • the scale and amplitude vectors and the spatial correlation matrix (\underline{\lambda}, \underline{a},\underline{\underline{R}});

  • the scale and amplitude vectors and the spatial covariance matrix (\underline{\lambda}, \underline{a},\underline{\underline{C}}); Then \underline{\underline{C}} is mapped into the associated correlation matrix \underline{\underline{R}} and the previous constructor is used.

[6]:

from __future__ import print_function
import openturns as ot
import math as m

[7]:

# 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

[8]:

# Create the covariance model from the amplitude and scale, no spatial correlation
ot.ExponentialModel(scale, amplitude)

[8]:

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

[9]:

# or from the amplitude, scale and spatialCovariance
ot.ExponentialModel(scale, amplitude, spatialCorrelation)

[9]:

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

[10]:

# or from the scale and spatialCovariance
ot.ExponentialModel(scale, spatialCovariance)

[10]:

ExponentialModel(scale=[4], amplitude=[2,2.23607,2.44949], no spatial correlation)