def test_rdl_decomposition(scenario, ncv_values, norm, statdist, reversible):
k, bdc = scenario
dim = bdc.q.shape[0]
P = bdc.transition_matrix
mu = bdc.stationary_distribution
with assert_raises(
ValueError) if bdc.sparse and k is None else nullcontext():
Rn, Dn, Ln = rdl_decomposition(P,
reversible=reversible,
norm=norm,
ncv=ncv_values,
k=k,
mu=mu if statdist else None)
Xn = Ln @ Rn
"""Right-eigenvectors"""
assert_allclose(P @ Rn, Rn @ Dn)
"""Left-eigenvectors"""
assert_allclose(Ln @ P, Dn @ Ln)
"""Orthonormality"""
assert_allclose(Xn, np.eye(dim if k is None else k))
"""Probability vector"""
assert_allclose(np.sum(Ln[0, :]), 1.0)
if norm == 'standard' and reversible:
"""Standard l2-normalization of right eigenvectors except dominant one"""
Yn = Rn.T @ Rn
assert_allclose(np.diag(Yn)[1:], 1.0)
if norm == 'reversible':
"""Reversibility"""
assert_allclose(Ln.transpose(), mu[:, np.newaxis] * Rn)
def test_eigenvalues(scenario, ncv_values):
k, bdc = scenario
P = bdc.transition_matrix
ev = eigvals(P)
"""Sort with decreasing magnitude"""
ev = ev[np.argsort(np.abs(ev))[::-1]]
"""k=None"""
with assert_raises(
ValueError) if bdc.sparse and k is None else nullcontext():
evn = eigenvalues(P, ncv=ncv_values, k=k)
assert_allclose(ev[:k], evn)
def test_timescales(scenario, ncv_values, tau, statdist, reversible):
k, bdc = scenario
P = bdc.transition_matrix
mu = bdc.stationary_distribution
ev = eigvals(P)
"""Sort with decreasing magnitude"""
ev = ev[np.argsort(np.abs(ev))[::-1]]
ts = -1.0 / np.log(np.abs(ev))
with assert_raises(
ValueError) if bdc.sparse and k is None else nullcontext():
tsn = timescales(P,
k=k,
tau=tau,
mu=mu if statdist else None,
reversible=reversible)
assert_allclose(tau * ts[1:k], tsn[1:k])
def test_eigenvalues_reversible(scenario, ncv_values):
k, bdc = scenario
P = bdc.transition_matrix
ev = eigvals(P)
"""Sort with decreasing magnitude"""
ev = ev[np.argsort(np.abs(ev))[::-1]]
with assert_raises(
ValueError) if bdc.sparse and k is None else nullcontext():
"""reversible without given mu"""
evn = eigenvalues(P, reversible=True, k=k, ncv=ncv_values)
assert_allclose(ev[:k], evn)
"""reversible with given mu"""
evn = eigenvalues(P,
reversible=True,
mu=bdc.stationary_distribution,
k=k,
ncv=ncv_values)
assert_allclose(ev[:k], evn)
def test_eigenvectors(scenario, ncv_values):
k, bdc = scenario
P = bdc.transition_matrix
ev = eigvals(P)
ev = ev[np.argsort(np.abs(ev))[::-1]]
Dn = np.diag(ev)
Dnk = Dn[:, :k][:k, :]
with assert_raises(
ValueError) if bdc.sparse and k is None else nullcontext():
# right eigenvectors
Rn = eigenvectors(P, k=k, ncv=ncv_values)
assert_allclose(P @ Rn, Rn @ Dnk)
# left eigenvectors
Ln = eigenvectors(P, right=False, k=k, ncv=ncv_values).T
assert_allclose(Ln.T @ P, Dnk @ Ln.T)
# orthogonality
Xn = Ln.T @ Rn
di = np.diag_indices(Xn.shape[0] if k is None else k)
Xn[di] = 0.0
assert_allclose(Xn, 0)
def test_eigenvectors_reversible(scenario, ncv_values):
k, bdc = scenario
P = bdc.transition_matrix
ev = eigvals(P)
ev = ev[np.argsort(np.abs(ev))[::-1]]
Dn = np.diag(ev)
Dnk = Dn[:, :k][:k, :]
with assert_raises(
ValueError) if bdc.sparse and k is None else nullcontext():
# right eigenvectors
Rn = eigenvectors(P, k=k, reversible=True, ncv=ncv_values)
assert_allclose(P @ Rn, Rn @ Dnk)
# left eigenvectors
Ln = eigenvectors(P, right=False, k=k, reversible=True,
ncv=ncv_values).T
assert_allclose(Ln.T @ P, Dnk @ Ln.T)
# orthogonality
Xn = Ln.T @ Rn
di = np.diag_indices(Xn.shape[0] if k is None else k)
Xn[di] = 0.0
assert_allclose(Xn, 0)
Rn = eigenvectors(P,
k=k,
ncv=ncv_values,
reversible=True,
mu=bdc.stationary_distribution)
assert_allclose(ev[:k][np.newaxis, :] * Rn, P.dot(Rn))
Ln = eigenvectors(P,
right=False,
k=k,
ncv=ncv_values,
reversible=True,
mu=bdc.stationary_distribution).T
assert_allclose(P.transpose().dot(Ln), ev[:k][np.newaxis, :] * Ln)