def get_mi_asymptotics(M, R):
"""
Return a multiline string with some asymptotics.
@param M: pure mutation rate matrix
@param R: mutation-selection balance rate matrix
@return: multiline string
"""
out = StringIO()
# get the stationary distributions
M_v = mrate.R_to_distn(M)
R_v = mrate.R_to_distn(R)
# The shannon entropy of the stationary distribution of the process
# determines the mutual information at small times.
M_shannon_entropy = -np.dot(np.log(M_v), M_v)
R_shannon_entropy = -np.dot(np.log(R_v), R_v)
if not np.allclose(M_shannon_entropy, R_shannon_entropy):
print >> out, 'At small enough times'
if R_shannon_entropy < M_shannon_entropy:
print >> out, '* pure mutation',
else:
print >> out, '* mutation-selection balance',
print >> out, 'will be more informative'
print >> out, 'because its stationary distribution has greater',
print >> out, 'Shannon entropy.'
else:
print >> out, 'There is not enough difference between the'
print >> out, 'Shannon entropies of the stationary distributions'
print >> out, 'to determine which process'
print >> out, 'is more informative at times near zero'
print >> out
# The spectral gap of the process
# determines the mutual information at large times.
M_spectral_gap = sorted(abs(w) for w in scipy.linalg.eigvals(M))[1]
R_spectral_gap = sorted(abs(w) for w in scipy.linalg.eigvals(R))[1]
M_cheeg_low, M_cheeg_mid, M_cheeg_high = cheeger.get_cheeger_bounds(M, M_v)
R_cheeg_low, R_cheeg_mid, R_cheeg_high = cheeger.get_cheeger_bounds(R, R_v)
if not np.allclose(M_spectral_gap, R_spectral_gap):
print >> out, 'At large enough times'
if R_spectral_gap < M_spectral_gap:
print >> out, '* mutation-selection balance',
else:
print >> out, '* pure mutation',
print >> out, 'will be more informative'
print >> out, 'because it has a smaller spectral gap.'
if (R_cheeg_high < M_cheeg_low) or (M_cheeg_high < R_cheeg_low):
print >> out, 'And also because of the isoperimetric bounds.'
else:
print >> out, 'There is not enough difference between the'
print >> out, 'spectral gaps to determine which process'
print >> out, 'is more informative at times near infinity'
print >> out
# return the text
return out.getvalue().strip()
def get_mi_asymptotics(M, R):
"""
Return a multiline string with some asymptotics.
@param M: pure mutation rate matrix
@param R: mutation-selection balance rate matrix
@return: multiline string
"""
out = StringIO()
# get the stationary distributions
M_v = mrate.R_to_distn(M)
R_v = mrate.R_to_distn(R)
# The shannon entropy of the stationary distribution of the process
# determines the mutual information at small times.
M_shannon_entropy = -np.dot(np.log(M_v), M_v)
R_shannon_entropy = -np.dot(np.log(R_v), R_v)
if not np.allclose(M_shannon_entropy, R_shannon_entropy):
print >> out, 'At small enough times'
if R_shannon_entropy < M_shannon_entropy:
print >> out, '* pure mutation',
else:
print >> out, '* mutation-selection balance',
print >> out, 'will be more informative'
print >> out, 'because its stationary distribution has greater',
print >> out, 'Shannon entropy.'
else:
print >> out, 'There is not enough difference between the'
print >> out, 'Shannon entropies of the stationary distributions'
print >> out, 'to determine which process'
print >> out, 'is more informative at times near zero'
print >> out
# The spectral gap of the process
# determines the mutual information at large times.
M_spectral_gap = sorted(abs(w) for w in scipy.linalg.eigvals(M))[1]
R_spectral_gap = sorted(abs(w) for w in scipy.linalg.eigvals(R))[1]
M_cheeg_low, M_cheeg_mid, M_cheeg_high = cheeger.get_cheeger_bounds(M, M_v)
R_cheeg_low, R_cheeg_mid, R_cheeg_high = cheeger.get_cheeger_bounds(R, R_v)
if not np.allclose(M_spectral_gap, R_spectral_gap):
print >> out, 'At large enough times'
if R_spectral_gap < M_spectral_gap:
print >> out, '* mutation-selection balance',
else:
print >> out, '* pure mutation',
print >> out, 'will be more informative'
print >> out, 'because it has a smaller spectral gap.'
if (R_cheeg_high < M_cheeg_low) or (M_cheeg_high < R_cheeg_low):
print >> out, 'And also because of the isoperimetric bounds.'
else:
print >> out, 'There is not enough difference between the'
print >> out, 'spectral gaps to determine which process'
print >> out, 'is more informative at times near infinity'
print >> out
# return the text
return out.getvalue().strip()
def __init__(self, Q):
"""
@param Q: rate matrix
"""
# define intermediate variables
v = mrate.R_to_distn(Q)
n = len(v)
psi = np.sqrt(v)
c_low, c_mid, c_high = cheeger.get_cheeger_bounds(Q, v)
# define member variables to summarize the rate matrix
self.rate_matrix = Q
self.exch_matrix = Q / v
if not np.allclose(self.exch_matrix, self.exch_matrix.T):
print self.exch_matrix
raise ValueError('expected symmetry')
self.sim_sym_matrix = np.outer(psi, 1 / psi) * Q
if not np.allclose(self.sim_sym_matrix, self.sim_sym_matrix.T):
print self.sim_sym_matrix
raise ValueError('expected symmetry')
self.distn = v
self.distn_shannon_entropy = -ndot(np.log(v), v)
self.distn_logical_entropy = ndot(v, 1 - v)
self.expected_rate = -ndot(np.diag(Q), v)
self.spectrum = scipy.linalg.eigvalsh(self.sim_sym_matrix)
self.spectral_gap = -self.spectrum[-2]
self.isoperimetric_low = c_low
self.isoperimetric_constant = c_mid
self.isoperimetric_high = c_high
self.trace_bound_high = -sum(np.diag(Q)) / (n - 1)
def __init__(self, Q):
"""
@param Q: rate matrix
"""
# define intermediate variables
v = mrate.R_to_distn(Q)
n = len(v)
psi = np.sqrt(v)
c_low, c_mid, c_high = cheeger.get_cheeger_bounds(Q, v)
# define member variables to summarize the rate matrix
self.rate_matrix = Q
self.exch_matrix = Q / v
if not np.allclose(self.exch_matrix, self.exch_matrix.T):
print self.exch_matrix
raise ValueError('expected symmetry')
self.sim_sym_matrix = np.outer(psi, 1/psi) * Q
if not np.allclose(self.sim_sym_matrix, self.sim_sym_matrix.T):
print self.sim_sym_matrix
raise ValueError('expected symmetry')
self.distn = v
self.distn_shannon_entropy = -ndot(np.log(v), v)
self.distn_logical_entropy = ndot(v, 1-v)
self.expected_rate = -ndot(np.diag(Q), v)
self.spectrum = scipy.linalg.eigvalsh(self.sim_sym_matrix)
self.spectral_gap = -self.spectrum[-2]
self.isoperimetric_low = c_low
self.isoperimetric_constant = c_mid
self.isoperimetric_high = c_high
self.trace_bound_high = -sum(np.diag(Q)) / (n-1)