y_var = 0.7
xy_cov = 0.4
cov = np.array([[x_var, xy_cov],
[xy_cov, y_var]])
"""
We can calculate the spectral matrix analytically, based on the known
coefficients, for 1024 frequency bins:
"""
n_freqs = 1024
w, Hw = alg.transfer_function_xy(am, n_freqs=n_freqs)
Sw_true = alg.spectral_matrix_xy(Hw, cov)
"""
Next, we will generate 500 example sets of 100 points of these processes, to analyze:
"""
#Number of realizations of the process
N = 500
#Length of each realization:
L = 1024
order = am.shape[0]
def test_MAR_est_LWR():
"""
Test the LWR MAR estimator against the power of the signal
This also tests the functions: transfer_function_xy, spectral_matrix_xy,
coherence_from_spectral and granger_causality_xy
"""
# This is the same processes as those in doc/examples/ar_est_2vars.py:
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] ])
n_freqs = 1024
w, Hw = tsa.transfer_function_xy(am, n_freqs=n_freqs)
Sw = tsa.spectral_matrix_xy(Hw, cov)
# This many realizations of the process:
N = 500
# Each one this long
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))
for i in xrange(N):
z[i], nz[i] = utils.generate_mar(am, cov, L)
a_est = []
cov_est = []
# This loop runs MAR_est_LWR:
for i in xrange(N):
Rxx = (tsa.MAR_est_LWR(z[i],order=n_lags))
a_est.append(Rxx[0])
cov_est.append(Rxx[1])
a_est = np.mean(a_est,0)
cov_est = np.mean(cov_est,0)
# This tests transfer_function_xy and spectral_matrix_xy:
w, Hw_est = tsa.transfer_function_xy(a_est, n_freqs=n_freqs)
Sw_est = tsa.spectral_matrix_xy(Hw_est, cov_est)
# coherence_from_spectral:
c = tsa.coherence_from_spectral(Sw)
c_est = tsa.coherence_from_spectral(Sw_est)
# granger_causality_xy:
w, f_x2y, f_y2x, f_xy, Sw = tsa.granger_causality_xy(am,
cov,
n_freqs=n_freqs)
w, f_x2y_est, f_y2x_est, f_xy_est, Sw_est = tsa.granger_causality_xy(a_est,
cov_est,
n_freqs=n_freqs)
# interdependence_xy
i_xy = tsa.interdependence_xy(Sw)
i_xy_est = tsa.interdependence_xy(Sw_est)
# This is all very approximate:
npt.assert_almost_equal(Hw,Hw_est,decimal=1)
npt.assert_almost_equal(Sw,Sw_est,decimal=1)
npt.assert_almost_equal(c,c_est,1)
npt.assert_almost_equal(f_xy,f_xy_est,1)
npt.assert_almost_equal(f_x2y,f_x2y_est,1)
npt.assert_almost_equal(f_y2x,f_y2x_est,1)
npt.assert_almost_equal(i_xy,i_xy_est,1)
a2[0, 1] = -0.2
b = np.array([a1.copy(), a2.copy()])
def extract_ij(i, j, m):
m_ij_rows = m[[i, j]]
return m_ij_rows[:, [i, j]]
"""
We calculate the transfer function based on the coefficients:
"""
w, Haw = alg.transfer_function_xy(a)
w, Hbw = alg.transfer_function_xy(b)
"""
Generate 500 sets of 100 points
"""
N = 500
L = 100
"""
def test_MAR_est_LWR():
"""
Test the LWR MAR estimator against the power of the signal
This also tests the functions: transfer_function_xy, spectral_matrix_xy,
coherence_from_spectral and granger_causality_xy
"""
# This is the same processes as those in doc/examples/ar_est_2vars.py:
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]])
n_freqs = 1024
w, Hw = tsa.transfer_function_xy(am, n_freqs=n_freqs)
Sw = tsa.spectral_matrix_xy(Hw, cov)
# This many realizations of the process:
N = 500
# Each one this long
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))
for i in range(N):
z[i], nz[i] = utils.generate_mar(am, cov, L)
a_est = []
cov_est = []
# This loop runs MAR_est_LWR:
for i in range(N):
Rxx = (tsa.MAR_est_LWR(z[i], order=n_lags))
a_est.append(Rxx[0])
cov_est.append(Rxx[1])
a_est = np.mean(a_est, 0)
cov_est = np.mean(cov_est, 0)
# This tests transfer_function_xy and spectral_matrix_xy:
w, Hw_est = tsa.transfer_function_xy(a_est, n_freqs=n_freqs)
Sw_est = tsa.spectral_matrix_xy(Hw_est, cov_est)
# coherence_from_spectral:
c = tsa.coherence_from_spectral(Sw)
c_est = tsa.coherence_from_spectral(Sw_est)
# granger_causality_xy:
w, f_x2y, f_y2x, f_xy, Sw = tsa.granger_causality_xy(am,
cov,
n_freqs=n_freqs)
w, f_x2y_est, f_y2x_est, f_xy_est, Sw_est = tsa.granger_causality_xy(
a_est, cov_est, n_freqs=n_freqs)
# interdependence_xy
i_xy = tsa.interdependence_xy(Sw)
i_xy_est = tsa.interdependence_xy(Sw_est)
# This is all very approximate:
npt.assert_almost_equal(Hw, Hw_est, decimal=1)
npt.assert_almost_equal(Sw, Sw_est, decimal=1)
npt.assert_almost_equal(c, c_est, 1)
npt.assert_almost_equal(f_xy, f_xy_est, 1)
npt.assert_almost_equal(f_x2y, f_x2y_est, 1)
npt.assert_almost_equal(f_y2x, f_y2x_est, 1)
npt.assert_almost_equal(i_xy, i_xy_est, 1)
b = np.array([a1.copy(), a2.copy()])
def extract_ij(i, j, m):
m_ij_rows = m[[i, j]]
return m_ij_rows[:, [i, j]]
"""
We calculate the transfer function based on the coefficients:
"""
w, Haw = alg.transfer_function_xy(a)
w, Hbw = alg.transfer_function_xy(b)
"""
Generate 500 sets of 100 points
"""
N = 500
L = 100
"""
Generate the instances of the time-series based on the coefficients:
"""
xy_cov = 0.4
cov = np.array([ [x_var, xy_cov],
[xy_cov, y_var] ])
"""
Calculate the spectral matrix analytically ( z-transforms evaluated at
z=exp(j*omega) from omega in [0,pi] )
"""
Nfreqs=1024
w, Hw = alg.transfer_function_xy(am, Nfreqs=Nfreqs)
Sw_true = alg.spectral_matrix_xy(Hw, cov)
"""
generate 500 sets of 100 points
"""
#Number of realizations of the process
N = 500
#Length of each realization:
L = 1024
order = am.shape[0]
n_lags = order + 1