Estimate a multivariate ARMA process

The objective of the Use Case is to estimate a multivariate ARMA model from a stationary time series using the maximum likelihood estimator and a centered normal white noise. The data can be a unique time series or several time series collected in a process sample.

We estimate thanks to the ARMALikelihoodFactory object and its method build, acting on a time series or on a sample of time series. It produces a result of type ARMA. Note that no evaluation of selection criteria such as AIC and BIC is done.

The synthetic data is generated from the 2-d ARMA model:

with E the white noise:

from __future__ import print_function
import openturns as ot
ot.Log.Show(ot.Log.NONE)

Create a 2-d ARMA process

p = 2
q = 1
dim = 2

# Tmin , Tmax and N points for TimeGrid
dt = 1.0
size = 400
timeGrid = ot.RegularGrid(0.0, dt, size)

# white noise
cov = ot.CovarianceMatrix([[0.1, 0.0], [0.0, 0.2]])
whiteNoise = ot.WhiteNoise(ot.Normal([0.0] * dim, cov), timeGrid)

# AR/MA coefficients
ar = ot.ARMACoefficients(p, dim)
ar[0] = ot.SquareMatrix([[-0.5, -0.1], [-0.4, -0.5]])
ar[1] = ot.SquareMatrix([[0.0, 0.0], [-0.25, 0.0]])

ma = ot.ARMACoefficients(q, dim)
ma[0] = ot.SquareMatrix([[-0.4, 0.0], [0.0, -0.4]])

# ARMA model creation
arma = ot.ARMA(ar, ma, whiteNoise)
arma

ARMA(X_{0,t} - 0.5 X_{0,t-1} - 0.1 X_{1,t-1} = E_{0,t} - 0.4 E_{0,t-1}
X_{1,t} - 0.4 X_{0,t-1} - 0.5 X_{1,t-1} - 0.25 X_{0,t-2} = E_{1,t} - 0.4 E_{1,t-1}, E_t ~ Normal(mu = [0,0], sigma = [0.316228,0.447214], R = [[ 1 0 ]
[ 0 1 ]]))

Create a realization

timeSeries = ot.TimeSeries(arma.getRealization())

Estimate the process from the previous realization

factory = ot.ARMALikelihoodFactory(p, q, dim)
factory.setInitialConditions(ar, ma, cov)

arma_est = ot.ARMA(factory.build(timeSeries))
print('Estimated ARMA= ', arma_est)

Out:

Estimated ARMA=  ARMA(X_{0,t} - 0.556835 X_{0,t-1} - 0.172264 X_{1,t-1} + 0.060772 X_{0,t-2} + 0.0199867 X_{1,t-2} = E_{0,t} - 0.56517 E_{0,t-1} - 0.0885182 E_{1,t-1}
X_{1,t} - 0.312348 X_{0,t-1} - 0.323698 X_{1,t-1} - 0.345647 X_{0,t-2} - 0.106516 X_{1,t-2} = E_{1,t} + 0.13192 E_{0,t-1} - 0.233798 E_{1,t-1}, E_t ~ Normal(mu = [0,0], sigma = [0.299585,0.455287], R = [[ 1        0.015397 ]
 [ 0.015397 1        ]]))