def __call__(self):
"""
Look for a counterexample.
"""
# Sample a rate matrix.
# Use a trick by Robert Kern to left and right multiply by diagonals.
# http://mail.scipy.org/pipermail/numpy-discussion/2007-March/
# 026809.html
S = MatrixUtil.sample_pos_sym_matrix(self.nstates)
v = mrate.sample_distn(self.nstates)
R = (v**-0.5)[:,np.newaxis] * S * (v**0.5)
R -= np.diag(np.sum(R, axis=1))
# Construct a parent-independent process
# with the same max rate and stationary distribution
# as the sampled process.
#max_rate = max(-np.diag(R))
#expected_rate = np.dot(v, -np.diag(R))
#logical_entropy = np.dot(v, 1-v)
#randomization_rate = expected_rate / logical_entropy
Q = self.simplification(R)
#Q = np.outer(np.ones(self.nstates), v)
#Q -= np.diag(np.sum(Q, axis=1))
#Q *= max(np.diag(R) / np.diag(Q))
# sample a random time
t = random.expovariate(1)
# Check that the mutual information of the
# parent independent process is smaller.
mi_R = ctmcmi.get_expected_ll_ratio(R, t)
mi_Q = ctmcmi.get_expected_ll_ratio(Q, t)
if np.allclose(mi_R, mi_Q):
self.n_too_close += 1
return False
if mi_R < mi_Q:
out = StringIO()
print >> out, 'found a counterexample'
print >> out
print >> out, 'sampled symmetric matrix S:'
print >> out, S
print >> out
print >> out, 'sampled stationary distribution v:'
print >> out, v
print >> out
print >> out, 'implied rate matrix R:'
print >> out, R
print >> out
print >> out, 'parent independent process Q:'
print >> out, Q
print >> out
print >> out, 'sampled time t:', t
print >> out
print >> out, 'mutual information of sampled process:', mi_R
print >> out, 'mutual information of p.i. process:', mi_Q
print >> out
self.counterexample = out.getvalue().rstrip()
return True
def __call__(self):
"""
Look for a counterexample.
"""
# Sample a rate matrix.
# Use a trick by Robert Kern to left and right multiply by diagonals.
# http://mail.scipy.org/pipermail/numpy-discussion/2007-March/
# 026809.html
S = MatrixUtil.sample_pos_sym_matrix(self.nstates)
v = mrate.sample_distn(self.nstates)
R = (v**-0.5)[:, np.newaxis] * S * (v**0.5)
R -= np.diag(np.sum(R, axis=1))
# Construct a parent-independent process
# with the same max rate and stationary distribution
# as the sampled process.
#max_rate = max(-np.diag(R))
#expected_rate = np.dot(v, -np.diag(R))
#logical_entropy = np.dot(v, 1-v)
#randomization_rate = expected_rate / logical_entropy
Q = self.simplification(R)
#Q = np.outer(np.ones(self.nstates), v)
#Q -= np.diag(np.sum(Q, axis=1))
#Q *= max(np.diag(R) / np.diag(Q))
# sample a random time
t = random.expovariate(1)
# Check that the mutual information of the
# parent independent process is smaller.
mi_R = ctmcmi.get_expected_ll_ratio(R, t)
mi_Q = ctmcmi.get_expected_ll_ratio(Q, t)
if np.allclose(mi_R, mi_Q):
self.n_too_close += 1
return False
if mi_R < mi_Q:
out = StringIO()
print >> out, 'found a counterexample'
print >> out
print >> out, 'sampled symmetric matrix S:'
print >> out, S
print >> out
print >> out, 'sampled stationary distribution v:'
print >> out, v
print >> out
print >> out, 'implied rate matrix R:'
print >> out, R
print >> out
print >> out, 'parent independent process Q:'
print >> out, Q
print >> out
print >> out, 'sampled time t:', t
print >> out
print >> out, 'mutual information of sampled process:', mi_R
print >> out, 'mutual information of p.i. process:', mi_Q
print >> out
self.counterexample = out.getvalue().rstrip()
return True
def __call__(self):
"""
Look for a counterexample.
"""
# Sample a rate matrix.
# Use a trick by Robert Kern to left and right multiply by diagonals.
# http://mail.scipy.org/pipermail/numpy-discussion/2007-March/
# 026809.html
S = MatrixUtil.sample_pos_sym_matrix(self.nstates)
v = mrate.sample_distn(self.nstates)
R = (v**-0.5)[:,np.newaxis] * S * (v**0.5)
R -= np.diag(np.sum(R, axis=1))
# sample a random time
rate = 1.0 / self.etime
t = random.expovariate(rate)
# sample one side of the bipartition and get the mutual information
k = random.randrange(1, self.nstates)
A = random.sample(range(self.nstates), k)
mi_non_markov = get_mutual_information(R, A, t)
# get summary statistics of the non-markov process
Q = msimpl.get_fast_two_state(R, A)
mi_markov = ctmcmi.get_expected_ll_ratio(Q, t)
# check if the mutual informations are indistinguishable
if np.allclose(mi_non_markov, mi_markov):
self.n_too_close += 1
return False
if mi_non_markov < mi_markov:
out = StringIO()
print >> out, 'found a counterexample'
print >> out
print >> out, 'sampled symmetric matrix S:'
print >> out, S
print >> out
print >> out, 'sampled stationary distribution v:'
print >> out, v
print >> out
print >> out, 'implied rate matrix R:'
print >> out, R
print >> out
print >> out, 'reduced rate matrix Q'
print >> out, Q
print >> out
print >> out, 'sampled time t:', t
print >> out
print >> out, 'non-markov mutual information:', mi_non_markov
print >> out, 'markov mutual information:', mi_markov
print >> out
self.counterexample = out.getvalue().rstrip()
return True
def __call__(self):
"""
Look for a counterexample.
"""
# Sample a rate matrix.
# Use a trick by Robert Kern to left and right multiply by diagonals.
# http://mail.scipy.org/pipermail/numpy-discussion/2007-March/
# 026809.html
S = MatrixUtil.sample_pos_sym_matrix(self.nstates)
v = mrate.sample_distn(self.nstates)
R = (v**-0.5)[:, np.newaxis] * S * (v**0.5)
R -= np.diag(np.sum(R, axis=1))
# sample a random time
rate = 1.0 / self.etime
t = random.expovariate(rate)
# sample one side of the bipartition and get the mutual information
k = random.randrange(1, self.nstates)
A = random.sample(range(self.nstates), k)
mi_non_markov = get_mutual_information(R, A, t)
# get summary statistics of the non-markov process
Q = msimpl.get_fast_two_state(R, A)
mi_markov = ctmcmi.get_expected_ll_ratio(Q, t)
# check if the mutual informations are indistinguishable
if np.allclose(mi_non_markov, mi_markov):
self.n_too_close += 1
return False
if mi_non_markov < mi_markov:
out = StringIO()
print >> out, 'found a counterexample'
print >> out
print >> out, 'sampled symmetric matrix S:'
print >> out, S
print >> out
print >> out, 'sampled stationary distribution v:'
print >> out, v
print >> out
print >> out, 'implied rate matrix R:'
print >> out, R
print >> out
print >> out, 'reduced rate matrix Q'
print >> out, Q
print >> out
print >> out, 'sampled time t:', t
print >> out
print >> out, 'non-markov mutual information:', mi_non_markov
print >> out, 'markov mutual information:', mi_markov
print >> out
self.counterexample = out.getvalue().rstrip()
return True
def process(fs):
n = fs.nstates
np.set_printoptions(linewidth=200)
out = StringIO()
# Sample a symmetric rate matrix and a stationary distribution,
# then construct the rate matrix R.
S = MatrixUtil.sample_pos_sym_matrix(n)
v = mrate.sample_distn(n)
psi = np.sqrt(v)
R = (S.T / psi).T * psi
R -= np.diag(np.sum(R, axis=1))
R_W, R_V = scipy.linalg.eig(R)
# Add some deletion.
deletion = 0.1
Q = np.zeros((n + 1, n + 1))
for i in range(n):
for j in range(n):
Q[i, j] = R[i, j]
for i in range(n):
Q[i, -1] = deletion
Q -= np.diag(np.sum(Q, axis=1))
Q_W, Q_V = scipy.linalg.eig(Q)
print >> out, 'deletion rate:'
print >> out, deletion
print >> out
print >> out, 'sampled rate matrix R:'
print >> out, R
print >> out
print >> out, 'spectrum of R:'
print >> out, R_W
print >> out
print >> out, 'rate matrix with deletion Q:'
print >> out, Q
print >> out
print >> out, 'spectrum of Q:'
print >> out, Q_W
print >> out
return out.getvalue().rstrip()
def process(fs):
n = fs.nstates
np.set_printoptions(linewidth=200)
out = StringIO()
# Sample a symmetric rate matrix and a stationary distribution,
# then construct the rate matrix R.
S = MatrixUtil.sample_pos_sym_matrix(n)
v = mrate.sample_distn(n)
psi = np.sqrt(v)
R = (S.T / psi).T * psi
R -= np.diag(np.sum(R, axis=1))
R_W, R_V = scipy.linalg.eig(R)
# Add some deletion.
deletion = 0.1
Q = np.zeros((n+1, n+1))
for i in range(n):
for j in range(n):
Q[i, j] = R[i, j]
for i in range(n):
Q[i, -1] = deletion
Q -= np.diag(np.sum(Q, axis=1))
Q_W, Q_V = scipy.linalg.eig(Q)
print >> out, 'deletion rate:'
print >> out, deletion
print >> out
print >> out, 'sampled rate matrix R:'
print >> out, R
print >> out
print >> out, 'spectrum of R:'
print >> out, R_W
print >> out
print >> out, 'rate matrix with deletion Q:'
print >> out, Q
print >> out
print >> out, 'spectrum of Q:'
print >> out, Q_W
print >> out
return out.getvalue().rstrip()
def __call__(self):
"""
Look for a counterexample.
"""
# Sample a rate matrix.
# Use a trick by Robert Kern to left and right multiply by diagonals.
# http://mail.scipy.org/pipermail/numpy-discussion/2007-March/
# 026809.html
S = MatrixUtil.sample_pos_sym_matrix(self.nstates)
v = mrate.sample_distn(self.nstates)
R = (v**-0.5)[:, np.newaxis] * S * (v**0.5)
R -= np.diag(np.sum(R, axis=1))
# Sample a short time and a longer time.
# For each row of the transition matrix,
# look at the K-L divergence to the stationary distribution,
# and check that it is smaller at the larger time.
ta, tb = sorted((random.expovariate(1), random.expovariate(1)))
Pa = scipy.linalg.expm(R * ta)
Pb = scipy.linalg.expm(R * tb)
for rowa, rowb in zip(Pa, Pb):
self.nrows_total += 1
if np.allclose(rowa, rowb):
self.nrows_allclose += 1
continue
kla = sum(x * math.log(x / y) for x, y in zip(rowa, v))
klb = sum(x * math.log(x / y) for x, y in zip(rowb, v))
if kla < klb:
out = StringIO()
print >> out, 'found a counterexample'
print >> out
print >> out, 'sampled symmetric matrix S:'
print >> out, S
print >> out
print >> out, 'sampled stationary distribution v:'
print >> out, v
print >> out
print >> out, 'implied rate matrix R:'
print >> out, R
print >> out
print >> out, 'sampled time ta:', ta
print >> out, 'sampled time tb:', tb
print >> out
print >> out, 'transition matrix Pa:'
print >> out, Pa
print >> out
print >> out, 'transition matrix Pb:'
print >> out, Pb
print >> out
print >> out, 'relevant row of Pa:'
print >> out, rowa
print >> out
print >> out, 'relevant row of Pb:'
print >> out, rowb
print >> out
print >> out, 'K-L divergence of row of Pa from v:'
print >> out, kla
print >> out
print >> out, 'K-L divergence of row of Pb from v:'
print >> out, klb
print >> out
self.counterexample = out.getvalue().rstrip()
return True
def process(fs):
n = fs.nstates
np.set_printoptions(linewidth=200)
out = StringIO()
# Sample a symmetric rate matrix and a stationary distribution,
# then construct the rate matrix R.
S = MatrixUtil.sample_pos_sym_matrix(n)
v = mrate.sample_distn(n)
psi = np.sqrt(v)
R = (S.T / psi).T * psi
R -= np.diag(np.sum(R, axis=1))
R_W, R_V = scipy.linalg.eig(R)
# construct the symmetric matrix that is similar to R
R_sim = (R.T * psi).T / psi
if not np.allclose(R_sim, R_sim.T):
raise ValueError('the similar symmetric matrix is not symmetric...')
R_sim_W, R_sim_V = scipy.linalg.eigh(R_sim)
R_gap = -R_sim_W[-2]
v2 = R_sim_V.T[-2]**2
# reconstruct the eigenvectors of R
R_V_rebuilt = (R_sim_V.T / psi).T
# Try to make the commute time matrix.
# R_sim is a lot like a Laplacian matrix, so lets pseudoinvert it.
R_sim_pinv = scipy.linalg.pinv(R_sim)
myouter = np.outer(np.ones(n), np.diag(R_sim_pinv))
D = -(myouter + myouter.T - 2 * R_sim_pinv)
D_commute = mrate.get_commute_distance_matrix(R, v)
if not np.allclose(D, D_commute):
raise ValueError('error computing commute distances')
HDH = MatrixUtil.double_centered(D)
HDH_W, HDH_V = scipy.linalg.eigh(HDH)
# compute squared pairwise distances brutely
X = R_sim_V.T[:-1].T / np.sqrt(-R_sim_W[:-1])
D_brute = np.array([[np.dot(b - a, b - a) for a in X] for b in X])
print >> out, 'reconstructed EDM:'
print >> out, D
print >> out
D = (D.T / psi).T / psi
print >> out, 'divide by square roots of stationary probabilities:'
print >> out, D
print >> out
print >> out, 'eigh of centered EDM:'
print >> out, 'eigenvalues:'
print >> out, HDH_W
print >> out, 'reciprocal nonzero eigenvalues:'
print >> out, 1 / HDH_W
print >> out, 'eigenvectors:'
print >> out, HDH_V
print >> out
print >> out, 'squared distances computed brutely:'
print >> out, D_brute
print >> out
print >> out, '1 / (h * max(D)):', 1 / (np.dot(v, 1 - v) * np.max(D))
print >> out, '1 / max(D):', 1 / np.max(D)
print >> out
# report some more standard stuff
print >> out, 'sampled rate matrix R:'
print >> out, R
print >> out, 'stationary distn:', v
print >> out, '1/R01 + 1/R10:', 1 / R[0, 1] + 1 / R[1, 0]
print >> out
print >> out, 'scipy.linagl.eig(R):'
print >> out, R_W
print >> out, R_V
print >> out
print >> out, 'symmetric matrix similar to R:'
print >> out, R_sim
print >> out
print >> out, 'eigh of the symmetric similar matrix to R:'
print >> out, R_sim_W
print >> out, R_sim_V
print >> out, 'spectral gap:', R_gap
print >> out, 'entrywise squares of eigenvectors:'
print >> out, R_sim_V**2
print >> out, 'a bilinear form involving a fiedler-like eigenvector:'
print >> out, ndot(R_sim_V.T[-2], R_sim, R_sim_V.T[-2])
print >> out, 'expected rate:', -np.dot(v, np.diag(R))
print >> out, 'second order expected rate:', -np.dot(v2, np.diag(R))
print >> out
print >> out, 'eigenvectors of R from eigenvectors of the similar matrix:'
print >> out, R_sim_W
print >> out, R_V_rebuilt
print >> out
return out.getvalue().rstrip()
def process(fs):
n = fs.nstates
np.set_printoptions(linewidth=200)
out = StringIO()
# Sample a symmetric rate matrix and a stationary distribution,
# then construct the rate matrix R.
S = MatrixUtil.sample_pos_sym_matrix(n)
v = mrate.sample_distn(n)
psi = np.sqrt(v)
R = (S.T / psi).T * psi
R -= np.diag(np.sum(R, axis=1))
R_W, R_V = scipy.linalg.eig(R)
# construct the symmetric matrix that is similar to R
R_sim = (R.T * psi).T / psi
if not np.allclose(R_sim, R_sim.T):
raise ValueError('the similar symmetric matrix is not symmetric...')
R_sim_W, R_sim_V = scipy.linalg.eigh(R_sim)
R_gap = -R_sim_W[-2]
v2 = R_sim_V.T[-2]**2
# reconstruct the eigenvectors of R
R_V_rebuilt = (R_sim_V.T / psi).T
# Try to make the commute time matrix.
# R_sim is a lot like a Laplacian matrix, so lets pseudoinvert it.
R_sim_pinv = scipy.linalg.pinv(R_sim)
myouter = np.outer(np.ones(n), np.diag(R_sim_pinv))
D = -(myouter + myouter.T - 2*R_sim_pinv)
D_commute = mrate.get_commute_distance_matrix(R, v)
if not np.allclose(D, D_commute):
raise ValueError('error computing commute distances')
HDH = MatrixUtil.double_centered(D)
HDH_W, HDH_V = scipy.linalg.eigh(HDH)
# compute squared pairwise distances brutely
X = R_sim_V.T[:-1].T / np.sqrt(-R_sim_W[:-1])
D_brute = np.array([[np.dot(b - a, b - a) for a in X] for b in X])
print >> out, 'reconstructed EDM:'
print >> out, D
print >> out
D = (D.T / psi).T / psi
print >> out, 'divide by square roots of stationary probabilities:'
print >> out, D
print >> out
print >> out, 'eigh of centered EDM:'
print >> out, 'eigenvalues:'
print >> out, HDH_W
print >> out, 'reciprocal nonzero eigenvalues:'
print >> out, 1 / HDH_W
print >> out, 'eigenvectors:'
print >> out, HDH_V
print >> out
print >> out, 'squared distances computed brutely:'
print >> out, D_brute
print >> out
print >> out, '1 / (h * max(D)):', 1 / (np.dot(v, 1-v) * np.max(D))
print >> out, '1 / max(D):', 1 / np.max(D)
print >> out
# report some more standard stuff
print >> out, 'sampled rate matrix R:'
print >> out, R
print >> out, 'stationary distn:', v
print >> out, '1/R01 + 1/R10:', 1/R[0,1] + 1/R[1,0]
print >> out
print >> out, 'scipy.linagl.eig(R):'
print >> out, R_W
print >> out, R_V
print >> out
print >> out, 'symmetric matrix similar to R:'
print >> out, R_sim
print >> out
print >> out, 'eigh of the symmetric similar matrix to R:'
print >> out, R_sim_W
print >> out, R_sim_V
print >> out, 'spectral gap:', R_gap
print >> out, 'entrywise squares of eigenvectors:'
print >> out, R_sim_V ** 2
print >> out, 'a bilinear form involving a fiedler-like eigenvector:'
print >> out, ndot(R_sim_V.T[-2], R_sim, R_sim_V.T[-2])
print >> out, 'expected rate:', -np.dot(v, np.diag(R))
print >> out, 'second order expected rate:', -np.dot(v2, np.diag(R))
print >> out
print >> out, 'eigenvectors of R from eigenvectors of the similar matrix:'
print >> out, R_sim_W
print >> out, R_V_rebuilt
print >> out
return out.getvalue().rstrip()
def __call__(self):
"""
Look for a counterexample.
"""
# Sample a rate matrix.
# Use a trick by Robert Kern to left and right multiply by diagonals.
# http://mail.scipy.org/pipermail/numpy-discussion/2007-March/
# 026809.html
S = MatrixUtil.sample_pos_sym_matrix(self.nstates)
v = mrate.sample_distn(self.nstates)
R = (v**-0.5)[:,np.newaxis] * S * (v**0.5)
R -= np.diag(np.sum(R, axis=1))
# Sample a short time and a longer time.
# For each row of the transition matrix,
# look at the K-L divergence to the stationary distribution,
# and check that it is smaller at the larger time.
ta, tb = sorted((random.expovariate(1), random.expovariate(1)))
Pa = scipy.linalg.expm(R * ta)
Pb = scipy.linalg.expm(R * tb)
for rowa, rowb in zip(Pa, Pb):
self.nrows_total += 1
if np.allclose(rowa, rowb):
self.nrows_allclose += 1
continue
kla = sum(x*math.log(x/y) for x, y in zip(rowa, v))
klb = sum(x*math.log(x/y) for x, y in zip(rowb, v))
if kla < klb:
out = StringIO()
print >> out, 'found a counterexample'
print >> out
print >> out, 'sampled symmetric matrix S:'
print >> out, S
print >> out
print >> out, 'sampled stationary distribution v:'
print >> out, v
print >> out
print >> out, 'implied rate matrix R:'
print >> out, R
print >> out
print >> out, 'sampled time ta:', ta
print >> out, 'sampled time tb:', tb
print >> out
print >> out, 'transition matrix Pa:'
print >> out, Pa
print >> out
print >> out, 'transition matrix Pb:'
print >> out, Pb
print >> out
print >> out, 'relevant row of Pa:'
print >> out, rowa
print >> out
print >> out, 'relevant row of Pb:'
print >> out, rowb
print >> out
print >> out, 'K-L divergence of row of Pa from v:'
print >> out, kla
print >> out
print >> out, 'K-L divergence of row of Pb from v:'
print >> out, klb
print >> out
self.counterexample = out.getvalue().rstrip()
return True
def process(fs):
n = fs.nstates
np.set_printoptions(linewidth=200)
out = StringIO()
# Sample a symmetric rate matrix and a stationary distribution,
# then construct the rate matrix R.
S = MatrixUtil.sample_pos_sym_matrix(n)
v = mrate.sample_distn(n)
psi = np.sqrt(v)
R = (S.T / psi).T * psi
R -= np.diag(np.sum(R, axis=1))
R_W, R_V = scipy.linalg.eig(R)
# construct the symmetric matrix that is similar to R
R_sim = (R.T * psi).T / psi
if not np.allclose(R_sim, R_sim.T):
raise ValueError('the similar symmetric matrix is not symmetric...')
R_sim_W, R_sim_V = scipy.linalg.eigh(R_sim)
R_gap = -R_sim_W[-2]
v2 = R_sim_V.T[-2]**2
# reconstruct the eigenvectors of R
R_V_rebuilt = (R_sim_V.T / psi).T
# Sample some numbers then subtract mean then normalize.
dv = np.random.exponential(1, n)
dv -= np.mean(dv)
dv *= fs.eps / np.dot(dv, dv)
qv = v + dv
if any(qv < 0) or any(1 < qv):
raise ValueError(
'the stationary distribution change was too large '
'for the randomly sampled process')
qpsi = np.sqrt(qv)
# define the rate matrix
if fs.knudsen:
Q = (S.T / qpsi).T * qpsi
elif fs.sella:
Q = R.copy()
for a in range(n):
for b in range(n):
if a != b:
tau = (qv[b] / v[b]) / (qv[a] / v[a])
Q[a, b] *= math.log(tau) / (1 - 1/tau)
Q -= np.diag(np.sum(Q, axis=1))
# construct the symmetric matrix that is similar to Q
Q_sim = (Q.T * qpsi).T / qpsi
Q_sim_W, Q_sim_V = scipy.linalg.eigh(Q_sim)
Q_gap = -Q_sim_W[-2]
# report some stuff
print >> out, 'sampled rate matrix R:'
print >> out, R
print >> out, 'stationary distn:', v
print >> out
print >> out, 'scipy.linagl.eig(R):'
print >> out, R_W
print >> out, R_V
print >> out
print >> out, 'symmetric matrix similar to R:'
print >> out, R_sim
print >> out
print >> out, 'eigh of the symmetric similar matrix to R:'
print >> out, R_sim_W
print >> out, R_sim_V
print >> out, 'spectral gap:', R_gap
print >> out, 'entrywise squares of eigenvectors:'
print >> out, R_sim_V ** 2
print >> out, 'a bilinear form involving a fiedler-like eigenvector:'
print >> out, ndot(R_sim_V.T[-2], R_sim, R_sim_V.T[-2])
print >> out, 'expected rate:', -np.dot(v, np.diag(R))
print >> out, 'second order expected rate:', -np.dot(v2, np.diag(R))
print >> out
print >> out, 'eigenvectors of R from eigenvectors of the similar matrix:'
print >> out, R_sim_W
print >> out, R_V_rebuilt
print >> out
print >> out, 'mutation-selection balance matrix Q:'
print >> out, Q
print >> out, 'stationary distn:', qv
print >> out, 'spectral gap:', Q_gap
print >> out
print >> out, 'symmetric matrix similar to Q:'
print >> out, Q_sim
print >> out
print >> out, 'pi(Q) - pi(R):', dv
print >> out, 'gap(Q) - gap(R):', Q_gap - R_gap
print >> out, 'diag(Q) - diag(R):', np.diag(Q) - np.diag(R)
print >> out, 'trace(Q) - trace(R):', np.trace(Q) - np.trace(R)
print >> out
print >> out, 'rate away estimate of spectral gap change:'
print >> out, np.dot(np.diag(Q) - np.diag(R), R_sim_V.T[-2]**2)
print >> out
return out.getvalue().rstrip()
def process(fs):
n = fs.nstates
np.set_printoptions(linewidth=200)
out = StringIO()
# Sample a symmetric rate matrix and a stationary distribution,
# then construct the rate matrix R.
S = MatrixUtil.sample_pos_sym_matrix(n)
v = mrate.sample_distn(n)
psi = np.sqrt(v)
R = (S.T / psi).T * psi
R -= np.diag(np.sum(R, axis=1))
R_W, R_V = scipy.linalg.eig(R)
# construct the symmetric matrix that is similar to R
R_sim = (R.T * psi).T / psi
if not np.allclose(R_sim, R_sim.T):
raise ValueError('the similar symmetric matrix is not symmetric...')
R_sim_W, R_sim_V = scipy.linalg.eigh(R_sim)
R_gap = -R_sim_W[-2]
v2 = R_sim_V.T[-2]**2
# reconstruct the eigenvectors of R
R_V_rebuilt = (R_sim_V.T / psi).T
# Sample some numbers then subtract mean then normalize.
dv = np.random.exponential(1, n)
dv -= np.mean(dv)
dv *= fs.eps / np.dot(dv, dv)
qv = v + dv
if any(qv < 0) or any(1 < qv):
raise ValueError('the stationary distribution change was too large '
'for the randomly sampled process')
qpsi = np.sqrt(qv)
# define the rate matrix
if fs.knudsen:
Q = (S.T / qpsi).T * qpsi
elif fs.sella:
Q = R.copy()
for a in range(n):
for b in range(n):
if a != b:
tau = (qv[b] / v[b]) / (qv[a] / v[a])
Q[a, b] *= math.log(tau) / (1 - 1 / tau)
Q -= np.diag(np.sum(Q, axis=1))
# construct the symmetric matrix that is similar to Q
Q_sim = (Q.T * qpsi).T / qpsi
Q_sim_W, Q_sim_V = scipy.linalg.eigh(Q_sim)
Q_gap = -Q_sim_W[-2]
# report some stuff
print >> out, 'sampled rate matrix R:'
print >> out, R
print >> out, 'stationary distn:', v
print >> out
print >> out, 'scipy.linagl.eig(R):'
print >> out, R_W
print >> out, R_V
print >> out
print >> out, 'symmetric matrix similar to R:'
print >> out, R_sim
print >> out
print >> out, 'eigh of the symmetric similar matrix to R:'
print >> out, R_sim_W
print >> out, R_sim_V
print >> out, 'spectral gap:', R_gap
print >> out, 'entrywise squares of eigenvectors:'
print >> out, R_sim_V**2
print >> out, 'a bilinear form involving a fiedler-like eigenvector:'
print >> out, ndot(R_sim_V.T[-2], R_sim, R_sim_V.T[-2])
print >> out, 'expected rate:', -np.dot(v, np.diag(R))
print >> out, 'second order expected rate:', -np.dot(v2, np.diag(R))
print >> out
print >> out, 'eigenvectors of R from eigenvectors of the similar matrix:'
print >> out, R_sim_W
print >> out, R_V_rebuilt
print >> out
print >> out, 'mutation-selection balance matrix Q:'
print >> out, Q
print >> out, 'stationary distn:', qv
print >> out, 'spectral gap:', Q_gap
print >> out
print >> out, 'symmetric matrix similar to Q:'
print >> out, Q_sim
print >> out
print >> out, 'pi(Q) - pi(R):', dv
print >> out, 'gap(Q) - gap(R):', Q_gap - R_gap
print >> out, 'diag(Q) - diag(R):', np.diag(Q) - np.diag(R)
print >> out, 'trace(Q) - trace(R):', np.trace(Q) - np.trace(R)
print >> out
print >> out, 'rate away estimate of spectral gap change:'
print >> out, np.dot(np.diag(Q) - np.diag(R), R_sim_V.T[-2]**2)
print >> out
return out.getvalue().rstrip()
def process(fs):
nstates = fs.nstates
np.set_printoptions(linewidth=200)
t = fs.t
### sample a random time
##time_mu = 0.01
##t = random.expovariate(1 / time_mu)
# Sample a rate matrix.
# Use a trick by Robert Kern to left and right multiply by diagonals.
# http://mail.scipy.org/pipermail/numpy-discussion/2007-March/
# 026809.html
S = MatrixUtil.sample_pos_sym_matrix(nstates)
v = mrate.sample_distn(nstates)
R = (v**-0.5)[:,np.newaxis] * S * (v**0.5)
R -= np.diag(np.sum(R, axis=1))
# Construct a parent-independent process
# with the same max rate and stationary distribution
# as the sampled process.
if fs.parent_indep:
Q = np.outer(np.ones(nstates), v)
Q -= np.diag(np.sum(Q, axis=1))
pi_rescaling_factor = max(np.diag(R) / np.diag(Q))
Q *= pi_rescaling_factor
Z = msimpl.get_fast_meta_f81_autobarrier(Q)
# Construct a child-independent process
# with the same expected rate
# as the sampled process
if fs.child_indep:
C = np.outer(1/v, np.ones(nstates))
C -= np.diag(np.sum(C, axis=1))
ci_rescaling_factor = np.max(R / C)
#expected_rate = -ndot(np.diag(R), v)
#ci_rescaling_factor = expected_rate / (nstates*(nstates-1))
#ci_rescaling_factor = expected_rate / (nstates*nstates)
C *= ci_rescaling_factor
Q = C
if fs.bipartitioned:
Q = msimpl.get_fast_meta_f81_autobarrier(R)
# Check that the mutual information of the
# parent independent process is smaller.
out = StringIO()
print >> out, 'sampled symmetric part of the rate matrix S:'
print >> out, S
print >> out
print >> out, 'sampled stationary distribution v:'
print >> out, v
print >> out
print >> out, 'shannon entropy of stationary distribution v:'
print >> out, -np.dot(np.log(v), v)
print >> out
print >> out, 'sqrt stationary distribution:'
print >> out, np.sqrt(v)
print >> out
print >> out, 'implied rate matrix R:'
print >> out, R
print >> out
print >> out, 'eigenvalues of R:', scipy.linalg.eigvals(R)
print >> out
print >> out, 'relaxation rate of R:',
print >> out, sorted(np.abs(scipy.linalg.eigvals(R)))[1]
print >> out
print >> out, 'expected rate of R:', mrate.Q_to_expected_rate(R)
print >> out
print >> out, 'cheeger bounds of R:', get_cheeger_bounds(R, v)
print >> out
print >> out, 'randomization rate of R:', get_randomization_rate(R, v)
print >> out
candidates = [get_randomization_candidate(R, v, i) for i in range(nstates)]
if np.allclose(get_randomization_rate(R, v), candidates):
print >> out, 'all candidates are equal to this rate'
else:
print >> out, 'not all candidates are equal to this rate'
print >> out
print >> out, 'simplified rate matrix Q:'
print >> out, Q
print >> out
qv = mrate.R_to_distn(Q)
print >> out, 'stationary distribution of Q:'
print >> out, qv
print >> out
print >> out, 'ratio qv/v:'
print >> out, qv / v
print >> out
print >> out, 'shannon entropy of stationary distribution of Q:'
print >> out, -np.dot(np.log(qv), qv)
print >> out
if fs.parent_indep:
print >> out, 'parent independent rescaling factor:'
print >> out, pi_rescaling_factor
print >> out
if fs.child_indep:
print >> out, 'child independent rescaling factor:'
print >> out, ci_rescaling_factor
print >> out
print >> out, 'eigenvalues of Q:', scipy.linalg.eigvals(Q)
print >> out
print >> out, 'relaxation rate of Q:',
print >> out, sorted(np.abs(scipy.linalg.eigvals(Q)))[1]
print >> out
print >> out, 'expected rate of Q:', mrate.Q_to_expected_rate(Q)
print >> out
print >> out, 'cheeger bounds of Q:', get_cheeger_bounds(Q, v)
print >> out
print >> out, 'randomization rate of Q:', get_randomization_rate(Q, v)
print >> out
candidates = [get_randomization_candidate(Q, v, i) for i in range(nstates)]
if np.allclose(get_randomization_rate(Q, v), candidates):
print >> out, 'all candidates are equal to this rate'
else:
print >> out, 'warning: not all candidates are equal to this rate'
print >> out
print >> out, 'E(rate) of Q divided by logical entropy:',
print >> out, mrate.Q_to_expected_rate(Q) / ndot(v, 1-v)
print >> out
print >> out, 'symmetric matrix similar to Q:'
S = ndot(np.diag(np.sqrt(v)), Q, np.diag(1/np.sqrt(v)))
print >> out, S
print >> out
print >> out, 'eigendecomposition of the similar matrix:'
W, V = scipy.linalg.eigh(S)
print >> out, V
print >> out, np.diag(W)
print >> out, V.T
print >> out
#
print >> out, 'time:', t
print >> out
print >> out, 'stationary distn logical entropy:', ndot(v, 1-v)
print >> out
#
P_by_hand = get_pi_transition_matrix(Q, v, t)
print >> out, 'simplified-process transition matrix computed by hand:'
print >> out, P_by_hand
print >> out
print >> out, 'simplified-process transition matrix computed by expm:'
print >> out, scipy.linalg.expm(Q*t)
print >> out
#
print >> out, 'simplified-process m.i. by hand:'
print >> out, get_pi_mi(Q, v, t)
print >> out
print >> out, 'simplified-process m.i. by expm:'
print >> out, ctmcmi.get_expected_ll_ratio(Q, t)
print >> out
#
print >> out, 'original process m.i. by expm:'
print >> out, ctmcmi.get_expected_ll_ratio(R, t)
print >> out
#
print >> out, 'stationary distn Shannon entropy:'
print >> out, -ndot(v, np.log(v))
print >> out
#
if fs.parent_indep:
print >> out, 'approximate simplified process m.i. 2nd order approx:'
print >> out, get_pi_mi_t2_approx(Q, v, t)
print >> out
print >> out, 'approximate simplified process m.i. "better" approx:'
print >> out, get_pi_mi_t2_diag_approx(Q, v, t)
print >> out
print >> out, '"f81-ization plus barrier" of pure f81-ization:'
print >> out, Z
print >> out
#
return out.getvalue().rstrip()
