# Fitting an MAR model: analyzer interface¶

In this example, we will use the Analyzer interface to fit a multivariate auto-regressive model with two time-series influencing each other.

We start by importing 3rd party modules:

import numpy as np
import matplotlib.pyplot as plt


And then by importing Granger analysis sub-module, which we will use for fitting the MAR model:

import nitime.analysis.granger as gc


The utils sub-module includes a function to generate auto-regressive processes based on provided coefficients:

import nitime.utils as utils


Generate some MAR processes (according to Ding and Bressler [Ding2006]),

a1 = np.array([[0.9, 0],
[0.16, 0.8]])

a2 = np.array([[-0.5, 0],
[-0.2, -0.5]])

am = np.array([-a1, -a2])

x_var = 1
y_var = 0.7
xy_cov = 0.4
cov = np.array([[x_var, xy_cov],
[xy_cov, y_var]])


Number of realizations of the process:

N = 500


Length of each realization:

L = 1024

order = am.shape[0]
n_lags = order + 1

n_process = am.shape[-1]

z = np.empty((N, n_process, L))
nz = np.empty((N, n_process, L))

np.random.seed(1981)
for i in range(N):
z[i], nz[i] = utils.generate_mar(am, cov, L)


We start by estimating the order of the model from the data:

est_order = []
for i in range(N):
this_order, this_Rxx, this_coef, this_ecov = gc.fit_model(z[i][0], z[i][1])
est_order.append(this_order)

order = int(np.round(np.mean(est_order)))


Once we have estimated the order, we go ahead and fit each realization of the MAR model, constraining the model order accordingly (by setting the order keyword argument) to be always equal to the model order estimated above.

Rxx = np.empty((N, n_process, n_process, n_lags))
coef = np.empty((N, n_process, n_process, order))
ecov = np.empty((N, n_process, n_process))

for i in range(N):
this_order, this_Rxx, this_coef, this_ecov = gc.fit_model(z[i][0], z[i][1], order=order)
Rxx[i] = this_Rxx
coef[i] = this_coef
ecov[i] = this_ecov


We generate a time-series from the recovered coefficients, using the same randomization seed as the first mar. These should look pretty similar to each other:

np.random.seed(1981)
est_ts, _ = utils.generate_mar(np.mean(coef, axis=0), np.mean(ecov, axis=0), L)

fig01 = plt.figure()

ax.plot(est_ts[0][0:100])
ax.plot(z[0][0][0:100], 'g--')

plt.show()

 [Ding2006] M. Ding, Y. Chen and S.L. Bressler (2006) Granger causality: basic theory and application to neuroscience. In Handbook of Time Series Analysis, ed. B. Schelter, M. Winterhalder, and J. Timmer, Wiley-VCH Verlage, 2006: 451-474

Example source code

You can download the full source code of this example. This same script is also included in the Nitime source distribution under the doc/examples/ directory.