def fit_model(x1,
x2,
order=None,
max_order=10,
criterion=utils.bayesian_information_criterion):
"""
Fit the auto-regressive model used in calculation of Granger 'causality'.
Parameters
----------
x1,x2: float arrays (n)
x1,x2 bivariate combination.
order: int (optional)
If known, the order of the autoregressive process
max_order: int (optional)
If the order is not known, this will be the maximal order to fit.
criterion: callable
A function which defines an information criterion, used to determine the
order of the model.
"""
c_old = np.inf
n_process = 2
Ntotal = n_process * x1.shape[-1]
# If model order was provided as an input:
if order is not None:
lag = order + 1
Rxx = utils.autocov_vector(np.vstack([x1, x2]), nlags=lag)
coef, ecov = alg.lwr_recursion(np.array(Rxx).transpose(2, 0, 1))
# If the model order is not known and provided as input:
else:
for lag in xrange(1, max_order):
Rxx_new = utils.autocov_vector(np.vstack([x1, x2]), nlags=lag)
coef_new, ecov_new = alg.lwr_recursion(
np.array(Rxx_new).transpose(2, 0, 1))
order_new = coef_new.shape[0]
c_new = criterion(ecov_new, n_process, order_new, Ntotal)
if c_new > c_old:
# Keep the values you got in the last round and break out:
break
else:
# Replace the output values with the new calculated values and
# move on to the next order:
c_old = c_new
order = order_new
Rxx = Rxx_new
coef = coef_new
ecov = ecov_new
else:
e_s = (
"Model estimation order did not converge at max_order = %s" %
max_order)
raise ValueError(e_s)
return order, Rxx, coef, ecov
def fit_model(x1, x2, order=None, max_order=10,
criterion=utils.bayesian_information_criterion):
"""
Fit the auto-regressive model used in calculation of Granger 'causality'.
Parameters
----------
x1,x2: float arrays (n)
x1,x2 bivariate combination.
order: int (optional)
If known, the order of the autoregressive process
max_order: int (optional)
If the order is not known, this will be the maximal order to fit.
criterion: callable
A function which defines an information criterion, used to determine the
order of the model.
"""
c_old = np.inf
n_process = 2
Ntotal = n_process * x1.shape[-1]
# If model order was provided as an input:
if order is not None:
lag = order + 1
Rxx = utils.autocov_vector(np.vstack([x1, x2]), nlags=lag)
coef, ecov = alg.lwr_recursion(np.array(Rxx).transpose(2, 0, 1))
# If the model order is not known and provided as input:
else:
for lag in range(1, max_order):
Rxx_new = utils.autocov_vector(np.vstack([x1, x2]), nlags=lag)
coef_new, ecov_new = alg.lwr_recursion(
np.array(Rxx_new).transpose(2, 0, 1))
order_new = coef_new.shape[0]
c_new = criterion(ecov_new, n_process, order_new, Ntotal)
if c_new > c_old:
# Keep the values you got in the last round and break out:
break
else:
# Replace the output values with the new calculated values and
# move on to the next order:
c_old = c_new
order = order_new
Rxx = Rxx_new
coef = coef_new
ecov = ecov_new
else:
e_s = ("Model estimation order did not converge at max_order = %s"
% max_order)
raise ValueError(e_s)
return order, Rxx, coef, ecov
def test_lwr():
"test solution of lwr recursion"
for trial in xrange(3):
nc = np.random.randint(2, high=10)
P = np.random.randint(2, high=6)
# nc is channels, P is lags (order)
r = np.random.randn(P + 1, nc, nc)
r[0] = np.dot(r[0], r[0].T) # force r0 to be symmetric
a, Va = tsa.lwr_recursion(r)
# Verify the "orthogonality" principle of the mAR system
# Set up a system in blocks to compute, for each k
# sum_{i=1}^{P} A(i)R(k-i) = -R(k) k > 0
# = sum_{i=1}^{P} R(k-i)^T A(i)^T = -R(k)^T
# = sum_{i=1}^{P} R(i-k)A(i)^T = -R(k)^T
rmat = np.zeros((nc * P, nc * P))
for k in xrange(1, P + 1):
for i in xrange(1, P + 1):
im = i - k
if im < 0:
r1 = r[-im].T
else:
r1 = r[im]
rmat[(k - 1) * nc:k * nc, (i - 1) * nc:i * nc] = r1
rvec = np.zeros((nc * P, nc))
avec = np.zeros((nc * P, nc))
for m in xrange(P):
rvec[m * nc:(m + 1) * nc] = -r[m + 1].T
avec[m * nc:(m + 1) * nc] = a[m].T
l2_d = np.dot(rmat, avec) - rvec
l2_d = (l2_d ** 2).sum() ** 0.5
l2_r = (rvec ** 2).sum() ** 0.5
# compute |Ax-b| / |b| metric
npt.assert_almost_equal(l2_d / l2_r, 0, decimal=5)
def test_lwr():
"test solution of lwr recursion"
for trial in range(3):
nc = np.random.randint(2, high=10)
P = np.random.randint(2, high=6)
# nc is channels, P is lags (order)
r = np.random.randn(P + 1, nc, nc)
r[0] = np.dot(r[0], r[0].T) # force r0 to be symmetric
a, Va = tsa.lwr_recursion(r)
# Verify the "orthogonality" principle of the mAR system
# Set up a system in blocks to compute, for each k
# sum_{i=1}^{P} A(i)R(k-i) = -R(k) k > 0
# = sum_{i=1}^{P} R(k-i)^T A(i)^T = -R(k)^T
# = sum_{i=1}^{P} R(i-k)A(i)^T = -R(k)^T
rmat = np.zeros((nc * P, nc * P))
for k in range(1, P + 1):
for i in range(1, P + 1):
im = i - k
if im < 0:
r1 = r[-im].T
else:
r1 = r[im]
rmat[(k - 1) * nc:k * nc, (i - 1) * nc:i * nc] = r1
rvec = np.zeros((nc * P, nc))
avec = np.zeros((nc * P, nc))
for m in range(P):
rvec[m * nc:(m + 1) * nc] = -r[m + 1].T
avec[m * nc:(m + 1) * nc] = a[m].T
l2_d = np.dot(rmat, avec) - rvec
l2_d = (l2_d**2).sum()**0.5
l2_r = (rvec**2).sum()**0.5
# compute |Ax-b| / |b| metric
npt.assert_almost_equal(l2_d / l2_r, 0, decimal=5)
def test_information_criteria():
"""
Test the implementation of information criteria:
"""
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_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)
AIC = []
BIC = []
AICc = []
# The total number data points available for estimation:
Ntotal = L * n_process
for n_lags in range(1, 10):
Rxx = np.empty((N, n_process, n_process, n_lags))
for i in xrange(N):
Rxx[i] = utils.autocov_vector(z[i], nlags=n_lags)
Rxx = Rxx.mean(axis=0)
Rxx = Rxx.transpose(2, 0, 1)
a, ecov = alg.lwr_recursion(Rxx)
IC = utils.akaike_information_criterion(ecov, n_process, n_lags,
Ntotal)
AIC.append(IC)
IC = utils.akaike_information_criterion(ecov,
n_process,
n_lags,
Ntotal,
corrected=True)
AICc.append(IC)
IC = utils.bayesian_information_criterion(ecov, n_process, n_lags,
Ntotal)
BIC.append(IC)
# The model has order 2, so this should minimize on 2:
# We do not test this for AIC/AICc, because these sometimes do not minimize
# (see Ding and Bressler)
# nt.assert_equal(np.argmin(AIC), 2)
# nt.assert_equal(np.argmin(AICc), 2)
nt.assert_equal(np.argmin(BIC), 2)
def test_information_criteria():
"""
Test the implementation of information criteria:
"""
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_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)
AIC = []
BIC = []
AICc = []
# The total number data points available for estimation:
Ntotal = L * n_process
for n_lags in range(1, 10):
Rxx = np.empty((N, n_process, n_process, n_lags))
for i in xrange(N):
Rxx[i] = utils.autocov_vector(z[i], nlags=n_lags)
Rxx = Rxx.mean(axis=0)
Rxx = Rxx.transpose(2, 0, 1)
a, ecov = alg.lwr_recursion(Rxx)
IC = utils.akaike_information_criterion(ecov, n_process, n_lags, Ntotal)
AIC.append(IC)
IC = utils.akaike_information_criterion(ecov, n_process, n_lags, Ntotal, corrected=True)
AICc.append(IC)
IC = utils.bayesian_information_criterion(ecov, n_process, n_lags, Ntotal)
BIC.append(IC)
# The model has order 2, so this should minimize on 2:
# We do not test this for AIC/AICc, because these sometimes do not minimize
# (see Ding and Bressler)
# nt.assert_equal(np.argmin(AIC), 2)
# nt.assert_equal(np.argmin(AICc), 2)
nt.assert_equal(np.argmin(BIC), 2)
Rxx = Rxx.mean(axis=0)
R0 = Rxx[..., 0]
Rm = Rxx[..., 1:]
Rxx = Rxx.transpose(2, 0, 1)
"""
We use the Levinson-Whittle(-Wiggins) and Robinson algorithm, as described in [Morf1978]_
, in order to estimate the MAR coefficients and the covariance matrix:
"""
a, ecov = alg.lwr_recursion(Rxx)
"""
Next, we use the calculated coefficients and covariance matrix, in order to
calculate Granger 'causality':
"""
w, f_x2y, f_y2x, f_xy, Sw = alg.granger_causality_xy(a,
ecov,
n_freqs=n_freqs)
"""
xzRa = extract_ij(0, 2, Raxx) yzRa = extract_ij(1, 2, Raxx) Rbxx = Rbxx.mean(axis=0) xyRb = extract_ij(0, 1, Rbxx) xzRb = extract_ij(0, 2, Rbxx) yzRb = extract_ij(1, 2, Rbxx) """ Now estimate mAR coefficients and covariance from the full and pairwise relationships: """ Raxx = Raxx.transpose(2, 0, 1) a_est, cov_est1 = alg.lwr_recursion(Raxx) a_xy_est, cov_xy_est1 = alg.lwr_recursion(xyRa.transpose(2, 0, 1)) a_xz_est, cov_xz_est1 = alg.lwr_recursion(xzRa.transpose(2, 0, 1)) a_yz_est, cov_yz_est1 = alg.lwr_recursion(yzRa.transpose(2, 0, 1)) Rbxx = Rbxx.transpose(2, 0, 1) b_est, cov_est2 = alg.lwr_recursion(Rbxx) b_xy_est, cov_xy_est2 = alg.lwr_recursion(xyRb.transpose(2, 0, 1)) b_xz_est, cov_xz_est2 = alg.lwr_recursion(xzRb.transpose(2, 0, 1)) b_yz_est, cov_yz_est2 = alg.lwr_recursion(yzRb.transpose(2, 0, 1)) """ We proceed to visualize these relationships:
xyRa = extract_ij(0, 1, Raxx) xzRa = extract_ij(0, 2, Raxx) yzRa = extract_ij(1, 2, Raxx) Rbxx = Rbxx.mean(axis=0) xyRb = extract_ij(0, 1, Rbxx) xzRb = extract_ij(0, 2, Rbxx) yzRb = extract_ij(1, 2, Rbxx) """ Now estimate mAR coefficients and covariance from the full and pairwise relationships: """ Raxx = Raxx.transpose(2, 0, 1) a_est, cov_est1 = alg.lwr_recursion(Raxx) a_xy_est, cov_xy_est1 = alg.lwr_recursion(xyRa.transpose(2, 0, 1)) a_xz_est, cov_xz_est1 = alg.lwr_recursion(xzRa.transpose(2, 0, 1)) a_yz_est, cov_yz_est1 = alg.lwr_recursion(yzRa.transpose(2, 0, 1)) Rbxx = Rbxx.transpose(2, 0, 1) b_est, cov_est2 = alg.lwr_recursion(Rbxx) b_xy_est, cov_xy_est2 = alg.lwr_recursion(xyRb.transpose(2, 0, 1)) b_xz_est, cov_xz_est2 = alg.lwr_recursion(xzRb.transpose(2, 0, 1)) b_yz_est, cov_yz_est2 = alg.lwr_recursion(yzRb.transpose(2, 0, 1)) """ We proceed to visualize these relationships: """
