def test_eigenvalues(self):
P = self.bdc.transition_matrix()
ev = eigvals(P)
"""Sort with decreasing magnitude"""
ev = ev[np.argsort(np.abs(ev))[::-1]]
"""k=None"""
evn = eigenvalues(P)
assert_allclose(ev, evn)
"""k is not None"""
evn = eigenvalues(P, k=self.k)
assert_allclose(ev[0:self.k], evn)
def test_eigenvalues_reversible(self):
P = self.bdc.transition_matrix()
ev = eigvals(P)
"""Sort with decreasing magnitude"""
ev = ev[np.argsort(np.abs(ev))[::-1]]
"""reversible without given mu"""
evn = eigenvalues(P, reversible=True)
assert_allclose(ev, evn)
"""reversible with given mu"""
evn = eigenvalues(P,
reversible=True,
mu=self.bdc.stationary_distribution())
assert_allclose(ev, evn)
def test_eigenvalues(self):
P = self.bdc.transition_matrix_sparse()
P_dense = self.bdc.transition_matrix()
ev = eigvals(P_dense)
"""Sort with decreasing magnitude"""
ev = ev[np.argsort(np.abs(ev))[::-1]]
"""k=None"""
with self.assertRaises(ValueError):
evn = eigenvalues(P)
"""k is not None"""
evn = eigenvalues(P, k=self.k)
assert_allclose(ev[0:self.k], evn)
"""k is not None and ncv is not None"""
evn = eigenvalues(P, k=self.k, ncv=self.ncv)
assert_allclose(ev[0:self.k], evn)
def eigenvalues(self):
"""Get the eigenvalues of the transition matrix"""
if isinstance(self._eigenvalues, np.ndarray):
return self._eigenvalues
else:
self._eigenvalues = mana.eigenvalues(self.transition_matrix)
return self._eigenvalues
def test_eigenvalues_rev(self):
P = self.bdc.transition_matrix_sparse()
P_dense = self.bdc.transition_matrix()
ev = eigvals(P_dense)
"""Sort with decreasing magnitude"""
ev = ev[np.argsort(np.abs(ev))[::-1]]
"""k=None"""
with self.assertRaises(ValueError):
evn = eigenvalues(P, reversible=True)
"""k is not None"""
evn = eigenvalues(P, k=self.k, reversible=True)
assert_allclose(ev[0:self.k], evn)
"""k is not None and ncv is not None"""
evn = eigenvalues(P, k=self.k, ncv=self.ncv, reversible=True)
assert_allclose(ev[0:self.k], evn)
"""mu is not None"""
mu = self.bdc.stationary_distribution()
"""k=None"""
with self.assertRaises(ValueError):
evn = eigenvalues(P, reversible=True, mu=mu)
"""k is not None"""
evn = eigenvalues(P, k=self.k, reversible=True, mu=mu)
assert_allclose(ev[0:self.k], evn)
"""k is not None and ncv is not None"""
evn = eigenvalues(P, k=self.k, ncv=self.ncv, reversible=True, mu=mu)
assert_allclose(ev[0:self.k], evn)
def pcca(P, m):
"""
PCCA+ spectral clustering method with optimized memberships [1]_
Clusters the first m eigenvectors of a transition matrix in order to cluster the states.
This function does not assume that the transition matrix is fully connected. Disconnected sets
will automatically define the first metastable states, with perfect membership assignments.
Parameters
----------
P : ndarray (n,n)
Transition matrix.
m : int
Number of clusters to group to.
Returns
-------
chi by default, or (chi,rot) if return_rot = True
chi : ndarray (n x m)
A matrix containing the probability or membership of each state to be assigned to each cluster.
The rows sum to 1.
References
----------
[1] S. Roeblitz and M. Weber, Fuzzy spectral clustering by PCCA+:
application to Markov state models and data classification.
Adv Data Anal Classif 7, 147-179 (2013).
[2] F. Noe, multiset PCCA and HMMs, in preparation.
"""
# imports
from msmtools.estimation import connected_sets
from msmtools.analysis import eigenvalues, is_transition_matrix, hitting_probability
# validate input
n = np.shape(P)[0]
if (m > n):
raise ValueError(
"Number of metastable states m = " + str(m) +
" exceeds number of states of transition matrix n = " + str(n))
if not is_transition_matrix(P):
raise ValueError("Input matrix is not a transition matrix.")
# prepare output
chi = np.zeros((n, m))
# test connectivity
components = connected_sets(P)
# print "all labels ",labels
n_components = len(
components
) # (n_components, labels) = connected_components(P, connection='strong')
# print 'n_components'
# store components as closed (with positive equilibrium distribution)
# or as transition states (with vanishing equilibrium distribution)
closed_components = []
transition_states = []
for i in range(n_components):
component = components[i] # np.argwhere(labels==i).flatten()
rest = list(set(range(n)) - set(component))
# is component closed?
if (np.sum(P[component, :][:, rest]) == 0):
closed_components.append(component)
else:
transition_states.append(component)
n_closed_components = len(closed_components)
closed_states = np.concatenate(closed_components)
if len(transition_states) == 0:
transition_states = np.array([], dtype=int)
else:
transition_states = np.concatenate(transition_states)
# check if we have enough clusters to support the disconnected sets
if (m < len(closed_components)):
raise ValueError("Number of metastable states m = " + str(m) +
" is too small. Transition matrix has " +
str(len(closed_components)) +
" disconnected components")
# We collect eigenvalues in order to decide which
closed_components_Psub = []
closed_components_ev = []
closed_components_enum = []
for i in range(n_closed_components):
component = closed_components[i]
# print "component ",i," ",component
# compute eigenvalues in submatrix
Psub = P[component, :][:, component]
closed_components_Psub.append(Psub)
closed_components_ev.append(eigenvalues(Psub))
closed_components_enum.append(i * np.ones((component.size), dtype=int))
# flatten
closed_components_ev_flat = np.array(closed_components_ev).flatten()
closed_components_enum_flat = np.array(closed_components_enum).flatten()
# which components should be clustered?
component_indexes = closed_components_enum_flat[np.argsort(
closed_components_ev_flat)][0:m]
# cluster each component
ipcca = 0
for i in range(n_closed_components):
component = closed_components[i]
# how many PCCA states in this component?
m_by_component = np.shape(np.argwhere(component_indexes == i))[0]
# if 1, then the result is trivial
if (m_by_component == 1):
chi[component, ipcca] = 1.0
ipcca += 1
elif (m_by_component > 1):
#print "submatrix: ",closed_components_Psub[i]
chi[component, ipcca:ipcca + m_by_component] = _pcca_connected(
closed_components_Psub[i], m_by_component)
ipcca += m_by_component
else:
raise RuntimeError("Component " + str(i) + " spuriously has " +
str(m_by_component) + " pcca sets")
# finally assign all transition states
# print "chi\n", chi
# print "transition states: ",transition_states
# print "closed states: ", closed_states
if (transition_states.size > 0):
# make all closed states absorbing, so we can see which closed state we hit first
Pabs = P.copy()
Pabs[closed_states, :] = 0.0
Pabs[closed_states, closed_states] = 1.0
for i in range(closed_states.size):
# hitting probability to each closed state
h = hitting_probability(Pabs, closed_states[i])
for j in range(transition_states.size):
# transition states belong to closed states with the hitting probability, and inherit their chi
chi[transition_states[j]] += h[transition_states[j]] * chi[
closed_states[i]]
# check if we have m metastable sets. If less than m, we must raise
nmeta = np.count_nonzero(chi.sum(axis=0))
assert m <= nmeta, str(m) + " metastable states requested, but transition matrix only has " + str(nmeta) \
+ ". Consider using a prior or request less metastable states. "
# print "chi\n", chi
return chi
def est_ts_k ( T, tau, k ):
ts_est = eigenvalues(T,k=k+1)
ts_est = np.real(ts_est[1:k+1])
ts_est = -tau / np.ma.log(ts_est)
return np.array(ts_est)