def test_acm_1d(self):
"""Test autocorrelation matrix for 1D input"""
v = np.array([1, 2, 0, 0, 1, 2, 0, 0])
acm = lambda l: datatools.acm(v, l)
self.assertEqual(np.mean(v**2), acm(0))
for l in range(1, 6):
self.assertEqual(np.correlate(v[l:], v[:-l]) / (len(v) - l),
acm(l))
def test_acm_1d(self):
"""Test autocorrelation matrix for 1D input"""
v = np.array([1, 2, 0, 0, 1, 2, 0, 0])
acm = lambda l: datatools.acm(v, l)
self.assertEqual(np.mean(v**2), acm(0))
for l in range(1, 6):
self.assertEqual(
np.correlate(v[l:], v[:-l]) / (len(v) - l), acm(l))
def from_yw(self, acms):
"""Determine VAR model from autocorrelation matrices by solving the
Yule-Walker equations.
Parameters
----------
acms : array, shape (n_lags, n_channels, n_channels)
acms[l] contains the autocorrelation matrix at lag l. The highest
lag must equal the model order.
Returns
-------
self : :class:`VAR`
The :class:`VAR` object to facilitate method chaining (see usage
example).
"""
if len(acms) != self.p + 1:
raise ValueError("Number of autocorrelation matrices ({}) does not"
" match model order ({}) + 1.".format(
len(acms), self.p))
n_channels = acms[0].shape[0]
acm = lambda l: acms[l] if l >= 0 else acms[-l].T
r = np.concatenate(acms[1:], 0)
rr = np.array([[acm(m - k) for k in range(self.p)]
for m in range(self.p)])
rr = np.concatenate(np.concatenate(rr, -2), -1)
c = sp.linalg.solve(rr, r)
# calculate residual covariance
r = acm(0)
for k in range(self.p):
bs = k * n_channels
r -= np.dot(c[bs:bs + n_channels, :].T, acm(k + 1))
self.coef = np.concatenate(
[c[m::n_channels, :] for m in range(n_channels)]).T
self.rescov = r
return self
def from_yw(self, acms):
"""Determine VAR model from autocorrelation matrices by solving the
Yule-Walker equations.
Parameters
----------
acms : array, shape (n_lags, n_channels, n_channels)
acms[l] contains the autocorrelation matrix at lag l. The highest
lag must equal the model order.
Returns
-------
self : :class:`VAR`
The :class:`VAR` object to facilitate method chaining (see usage
example).
"""
if len(acms) != self.p + 1:
raise ValueError("Number of autocorrelation matrices ({}) does not"
" match model order ({}) + 1.".format(len(acms),
self.p))
n_channels = acms[0].shape[0]
acm = lambda l: acms[l] if l >= 0 else acms[-l].T
r = np.concatenate(acms[1:], 0)
rr = np.array([[acm(m-k) for k in range(self.p)]
for m in range(self.p)])
rr = np.concatenate(np.concatenate(rr, -2), -1)
c = sp.linalg.solve(rr, r)
# calculate residual covariance
r = acm(0)
for k in range(self.p):
bs = k * n_channels
r -= np.dot(c[bs:bs + n_channels, :].T, acm(k + 1))
self.coef = np.concatenate([c[m::n_channels, :]
for m in range(n_channels)]).T
self.rescov = r
return self
def _calc_q_statistic(x, h, nt):
"""Calculate Portmanteau statistics up to a lag of h.
"""
t, m, n = x.shape
# covariance matrix of x
c0 = acm(x, 0)
# LU factorization of covariance matrix
c0f = sp.linalg.lu_factor(c0, overwrite_a=False, check_finite=True)
q = np.zeros((3, h + 1))
for l in range(1, h + 1):
cl = acm(x, l)
# calculate tr(cl' * c0^-1 * cl * c0^-1)
a = sp.linalg.lu_solve(c0f, cl)
b = sp.linalg.lu_solve(c0f, cl.T)
tmp = a.dot(b).trace()
# Box-Pierce
q[0, l] = tmp
# Ljung-Box
q[1, l] = tmp / (nt - l)
# Li-McLeod
q[2, l] = tmp
q *= nt
q[1, :] *= (nt + 2)
q = np.cumsum(q, axis=1)
for l in range(1, h + 1):
q[2, l] = q[0, l] + m * m * l * (l + 1) / (2 * nt)
return q
def _calc_q_statistic(x, h, nt):
"""Calculate Portmanteau statistics up to a lag of h.
"""
t, m, n = x.shape
# covariance matrix of x
c0 = acm(x, 0)
# LU factorization of covariance matrix
c0f = sp.linalg.lu_factor(c0, overwrite_a=False, check_finite=True)
q = np.zeros((3, h + 1))
for l in range(1, h + 1):
cl = acm(x, l)
# calculate tr(cl' * c0^-1 * cl * c0^-1)
a = sp.linalg.lu_solve(c0f, cl)
b = sp.linalg.lu_solve(c0f, cl.T)
tmp = a.dot(b).trace()
# Box-Pierce
q[0, l] = tmp
# Ljung-Box
q[1, l] = tmp / (nt - l)
# Li-McLeod
q[2, l] = tmp
q *= nt
q[1, :] *= (nt + 2)
q = np.cumsum(q, axis=1)
for l in range(1, h+1):
q[2, l] = q[0, l] + m * m * l * (l + 1) / (2 * nt)
return q
def test_yulewalker(self):
np.random.seed(7353)
x, var0 = self.generate_data([[1, 2], [3, 4]])
acms = [acm(x, l) for l in range(var0.p+1)]
var = VAR(var0.p)
var.from_yw(acms)
assert_allclose(var0.coef, var.coef, rtol=1e-2, atol=1e-2)
# that limit is rather generous, but we don't want tests to fail due to random variation
self.assertTrue(np.all(np.abs(var0.coef - var.coef) < 0.02))
self.assertTrue(np.all(np.abs(var0.rescov - var.rescov) < 0.02))