def test_K():
"""Test matrix _K(m) against relations in Wiktorsson2001 equation (4.3)"""
for q in range(2, 10):
P0 = _P(q)
K0 = _K(q)
M = q*(q-1)/2
Iqs = np.eye(q**2)
IM = np.eye(M)
assert(np.allclose(np.dot(K0, K0.T), IM))
d = []
for k in range(1, q+1):
d.extend([0]*k + [1]*(q-k))
assert(np.allclose(np.dot(K0.T, K0), np.diag(d)))
assert(np.allclose(K0.dot(P0).dot(K0.T), np.zeros((M, M))))
assert(np.allclose(K0.dot(Iqs).dot(K0.T), IM))
assert(np.allclose((Iqs - P0).dot(K0.T).dot(K0).dot(Iqs - P0), Iqs - P0))
def test_Iwik_Jwik_identities():
dW = deltaW(N, m, h).reshape((N, m, 1))
Atilde, I = Iwik(dW, h)
M = m * (m - 1) // 2
assert (Atilde.shape == (N, M, 1) and I.shape == (N, m, m))
Im = broadcast_to(np.eye(m), (N, m, m))
assert (np.allclose(I + _t(I), _dot(dW, _t(dW)) - h * Im))
# can get A from Atilde: (Wiktorsson2001 equation between (4.3) and (4.4))
Ims = broadcast_to(np.eye(m * m), (N, m * m, m * m))
Pm = broadcast_to(_P(m), (N, m * m, m * m))
Km = broadcast_to(_K(m), (N, M, m * m))
A = _unvec(_dot(_dot((Ims - Pm), _t(Km)), Atilde))
# now can test this A against the identities of Wiktorsson eqn (2.1)
assert (np.allclose(A, -_t(A)))
assert (np.allclose(2.0 * (I - A), _dot(dW, _t(dW)) - h * Im))
# and tests for Stratonovich case
Atilde, J = Jwik(dW, h)
assert (Atilde.shape == (N, M, 1) and J.shape == (N, m, m))
assert (np.allclose(J + _t(J), _dot(dW, _t(dW))))
A = _unvec(_dot(_dot((Ims - Pm), _t(Km)), Atilde))
assert (np.allclose(2.0 * (J - A), _dot(dW, _t(dW))))
def test_Iwik_Jwik_identities():
dW = deltaW(N, m, h).reshape((N, m, 1))
Atilde, I = Iwik(dW, h)
M = m*(m-1)/2
assert(Atilde.shape == (N, M, 1) and I.shape == (N, m, m))
Im = broadcast_to(np.eye(m), (N, m, m))
assert(np.allclose(I + _t(I), _dot(dW, _t(dW)) - h*Im))
# can get A from Atilde: (Wiktorsson2001 equation between (4.3) and (4.4))
Ims = broadcast_to(np.eye(m*m), (N, m*m, m*m))
Pm = broadcast_to(_P(m), (N, m*m, m*m))
Km = broadcast_to(_K(m), (N, M, m*m))
A = _unvec(_dot(_dot((Ims - Pm), _t(Km)), Atilde))
# now can test this A against the identities of Wiktorsson eqn (2.1)
assert(np.allclose(A, -_t(A)))
assert(np.allclose(2.0*(I - A), _dot(dW, _t(dW)) - h*Im))
# and tests for Stratonovich case
Atilde, J = Jwik(dW, h)
assert(Atilde.shape == (N, M, 1) and J.shape == (N, m, m))
assert(np.allclose(J + _t(J), _dot(dW, _t(dW))))
A = _unvec(_dot(_dot((Ims - Pm), _t(Km)), Atilde))
assert(np.allclose(2.0*(J - A), _dot(dW, _t(dW))))