def get_response_content(fs):
out = StringIO()
np.set_printoptions(linewidth=200)
# get the user defined variables
n = fs.nstates
# sample a random reversible CTMC rate matrix
v = divtime.sample_distribution(n)
S = divtime.sample_symmetric_rate_matrix(n)
R = mrate.to_gtr_halpern_bruno(S, v)
distn = mrate.R_to_distn(R)
spectrum = scipy.linalg.eigvalsh(mrate.symmetrized(R))
print >> out, "random reversible CTMC rate matrix:"
print >> out, R
print >> out
print >> out, "stationary distribution:"
print >> out, distn
print >> out
print >> out, "spectrum:"
print >> out, spectrum
print >> out
Q = aggregate(R)
distn = mrate.R_to_distn(Q)
spectrum = scipy.linalg.eigvalsh(mrate.symmetrized(Q))
print >> out, "aggregated rate matrix:"
print >> out, Q
print >> out
print >> out, "stationary distribution:"
print >> out, distn
print >> out
print >> out, "spectrum:"
print >> out, spectrum
print >> out
return out.getvalue()
def get_mutual_information_diff_b(R, t):
"""
This is a more symmetrized version.
Note that two of the three terms are probably structurally zero.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R * t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
G = np.zeros_like(R)
for i in range(n):
for j in range(n):
G[i, j] = 0
for k in range(n):
G[i, j] += U[i, k] * U[j, k] * math.exp(t * w[k])
G_diff = np.zeros_like(R)
for i in range(n):
for j in range(n):
G_diff[i, j] = 0
for k in range(n):
G_diff[i, j] += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
B = np.outer(U.T[-1], U.T[-1])
term_a = np.sum(B * G_diff)
term_b = np.sum(B * G_diff * np.log(G))
term_c = -np.sum(B * G_diff * np.log(B))
#print term_a
#print term_b
#print term_c
return term_b
def get_mutual_information_diff_c(R, t):
"""
This is a more symmetrized version.
Some structurally zero terms have been removed.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R * t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
B = np.outer(U.T[-1], U.T[-1])
G = np.zeros_like(R)
for i in range(n):
for j in range(n):
G[i, j] = 0
for k in range(n):
G[i, j] += U[i, k] * U[j, k] * math.exp(t * w[k])
G_diff = np.zeros_like(R)
for i in range(n):
for j in range(n):
G_diff[i, j] = 0
for k in range(n):
G_diff[i, j] += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
return np.sum(B * G_diff * np.log(G))
def get_asymptotic_variance_b(R, t):
"""
Break up the sum into two parts and investigate each separately.
The second part with only the second derivative is zero.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R*t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance
accum_a = 0
for i in range(n):
for j in range(n):
# define f
f = p[i] * P[i, j]
# define the first derivative of f
f_dt = 0
for k in range(n):
f_dt += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
f_dt *= (p[i] * p[j])**.5
accum_a -= (f_dt * f_dt) / f
accum_b = 0
for i in range(n):
for j in range(n):
# define the second derivative of f
f_dtt = 0
for k in range(n):
f_dtt += U[i, k] * U[j, k] * w[k] * w[k] * math.exp(t * w[k])
f_dtt *= (p[i] * p[j])**.5
# accumulate the contribution of this entry to the expectation
accum_b += f_dtt
return - 1 / (accum_a + accum_b)
def get_mutual_information_diff_zero(R):
"""
Derivative of mutual information at time zero.
Haha apparently this does not exist.
"""
# get non-spectral summaries
n = len(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
B = np.outer(U.T[-1], U.T[-1])
G = np.zeros_like(R)
for i in range(n):
for j in range(n):
G[i, j] = 0
for k in range(n):
G[i, j] += U[i, k] * U[j, k]
G_diff = np.zeros_like(R)
for i in range(n):
for j in range(n):
G_diff[i, j] = 0
for k in range(n):
G_diff[i, j] += U[i, k] * U[j, k] * w[k]
print G
print G_diff
print B
return np.sum(B * G_diff * np.log(G))
def get_mutual_information_small_approx_d(R, t):
"""
This is an approximation for small times.
This uses all of the off-diagonal entries of the mutual information
and also uses an approximation of the off-diagonal entries.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum_diag_a = 0
accum_diag_b = 0
accum_diag_c = 0
accum_diag_d = 0
for i in range(n):
a = 0
b = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
for k in range(n - 1):
prefix = U[i, k] * U[i, k]
b += prefix * math.exp(t * w[k])
x1 = v[i] * v[i]
x2 = v[i] * b
y1 = math.log(a)
y2 = -math.log(v[i])
accum_diag_a += x1 * y1
accum_diag_b += x1 * y2
accum_diag_c += x2 * y1
accum_diag_d += x2 * y2
accum_a = 0
accum_b = 0
accum_c = 0
accum_d = 0
for i in range(n):
for j in range(n):
if i != j:
prefix = (v[i] * v[j])**.5
a = 0
for k in range(n):
a += U[i, k] * U[j, k] * math.exp(t * w[k])
b = 0
for k in range(n - 1):
b += U[i, k] * U[j, k] * math.exp(t * w[k])
x1 = v[i] * v[j]
x2 = prefix * b
y1 = math.log(a)
y2 = -math.log(prefix)
accum_a += x1 * y1
accum_b += x1 * y2
accum_c += x2 * y1
accum_d += x2 * y2
terms = [
accum_diag_a, accum_diag_b, accum_diag_c, accum_diag_d, accum_a,
accum_b, accum_c, accum_d
]
for term in terms:
print term
return sum(terms)
def get_mutual_information_stable(R, t):
"""
This is a more stable function.
@return: unscaled_result, log_of_scaling_factor
"""
#FIXME under construction
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
P = np.zeros_like(R)
accum = 0
for i in range(n):
for j in range(n):
for k in range(n):
a = (v[j] / v[i])**0.5
b = U[i, k] * U[j, k]
c = math.exp(t * w[k])
P[i, j] += a * b * c
# compute the unscaled part of log(X(i,j)/(X(i)*X(j)))
for i in range(n):
for j in range(n):
if v[i] and P[i, j]:
coeff = v[i] * P[i, j]
numerator = P[i, j]
denominator = v[j]
# the problem is that the following log is nearly zero
value = coeff * math.log(numerator / denominator)
accum += np.real(value)
return accum
def get_asymptotic_variance_b(R, t):
"""
Break up the sum into two parts and investigate each separately.
The second part with only the second derivative is zero.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R * t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance
accum_a = 0
for i in range(n):
for j in range(n):
# define f
f = p[i] * P[i, j]
# define the first derivative of f
f_dt = 0
for k in range(n):
f_dt += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
f_dt *= (p[i] * p[j])**.5
accum_a -= (f_dt * f_dt) / f
accum_b = 0
for i in range(n):
for j in range(n):
# define the second derivative of f
f_dtt = 0
for k in range(n):
f_dtt += U[i, k] * U[j, k] * w[k] * w[k] * math.exp(t * w[k])
f_dtt *= (p[i] * p[j])**.5
# accumulate the contribution of this entry to the expectation
accum_b += f_dtt
return -1 / (accum_a + accum_b)
def get_mutual_information_small_approx_b(R, t):
"""
This is an approximation for small times.
Check a decomposition.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum_a = 0
accum_b = 0
accum_c = 0
accum_d = 0
for i in range(n):
a = 0
b = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
for k in range(n - 1):
prefix = U[i, k] * U[i, k]
b += prefix * math.exp(t * w[k])
x1 = v[i] * v[i]
x2 = v[i] * b
y1 = math.log(a)
y2 = -math.log(v[i])
accum_a += x1 * y1
accum_b += x1 * y2
accum_c += x2 * y1
accum_d += x2 * y2
return accum_a + accum_b + accum_c + accum_d
def get_asymptotic_variance_e(R, t):
"""
Try to mitigate the damage of the aggressive approximation.
The next step is to try to simplify this complicated correction.
But I have not been able to do this.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R * t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance approximation
accum = 0
for k in range(n - 1):
accum += w[k] * w[k] * math.exp(2 * t * w[k])
accum_b = 0
G_a = np.zeros_like(R)
G_b = np.zeros_like(R)
for i in range(n):
for j in range(n):
prefix = (p[i] * p[j])**-.5
a = 0
for k in range(n - 1):
a += U[i, k] * U[j, k] * math.exp(t * w[k])
b = 0
for k in range(n - 1):
b += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
suffix = a * b * b
value = prefix * suffix
accum_b += value
return 1 / (accum - accum_b)
def get_mutual_information_diff_b(R, t):
"""
This is a more symmetrized version.
Note that two of the three terms are probably structurally zero.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R*t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
G = np.zeros_like(R)
for i in range(n):
for j in range(n):
G[i, j] = 0
for k in range(n):
G[i, j] += U[i, k] * U[j, k] * math.exp(t * w[k])
G_diff = np.zeros_like(R)
for i in range(n):
for j in range(n):
G_diff[i, j] = 0
for k in range(n):
G_diff[i, j] += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
B = np.outer(U.T[-1], U.T[-1])
term_a = np.sum(B * G_diff)
term_b = np.sum(B * G_diff * np.log(G))
term_c = -np.sum(B * G_diff * np.log(B))
#print term_a
#print term_b
#print term_c
return term_b
def get_mutual_information_diff_c(R, t):
"""
This is a more symmetrized version.
Some structurally zero terms have been removed.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R*t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
B = np.outer(U.T[-1], U.T[-1])
G = np.zeros_like(R)
for i in range(n):
for j in range(n):
G[i, j] = 0
for k in range(n):
G[i, j] += U[i, k] * U[j, k] * math.exp(t * w[k])
G_diff = np.zeros_like(R)
for i in range(n):
for j in range(n):
G_diff[i, j] = 0
for k in range(n):
G_diff[i, j] += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
return np.sum(B * G_diff * np.log(G))
def get_mutual_information_small_approx_b(R, t):
"""
This is an approximation for small times.
Check a decomposition.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum_a = 0
accum_b = 0
accum_c = 0
accum_d = 0
for i in range(n):
a = 0
b = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
for k in range(n-1):
prefix = U[i, k] * U[i, k]
b += prefix * math.exp(t * w[k])
x1 = v[i] * v[i]
x2 = v[i] * b
y1 = math.log(a)
y2 = -math.log(v[i])
accum_a += x1 * y1
accum_b += x1 * y2
accum_c += x2 * y1
accum_d += x2 * y2
return accum_a + accum_b + accum_c + accum_d
def get_mutual_information_stable(R, t):
"""
This is a more stable function.
@return: unscaled_result, log_of_scaling_factor
"""
#FIXME under construction
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
P = np.zeros_like(R)
accum = 0
for i in range(n):
for j in range(n):
for k in range(n):
a = (v[j] / v[i])**0.5
b = U[i, k] * U[j, k]
c = math.exp(t * w[k])
P[i, j] += a * b * c
# compute the unscaled part of log(X(i,j)/(X(i)*X(j)))
for i in range(n):
for j in range(n):
if v[i] and P[i, j]:
coeff = v[i] * P[i, j]
numerator = P[i, j]
denominator = v[j]
# the problem is that the following log is nearly zero
value = coeff * math.log(numerator / denominator)
accum += np.real(value)
return accum
def get_asymptotic_variance_e(R, t):
"""
Try to mitigate the damage of the aggressive approximation.
The next step is to try to simplify this complicated correction.
But I have not been able to do this.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R*t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance approximation
accum = 0
for k in range(n-1):
accum += w[k] * w[k] * math.exp(2 * t * w[k])
accum_b = 0
G_a = np.zeros_like(R)
G_b = np.zeros_like(R)
for i in range(n):
for j in range(n):
prefix = (p[i] * p[j]) ** -.5
a = 0
for k in range(n-1):
a += U[i, k] * U[j, k] * math.exp(t * w[k])
b = 0
for k in range(n-1):
b += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
suffix = a * b * b
value = prefix * suffix
accum_b += value
return 1 / (accum - accum_b)
def get_mutual_information_diff_zero(R):
"""
Derivative of mutual information at time zero.
Haha apparently this does not exist.
"""
# get non-spectral summaries
n = len(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
B = np.outer(U.T[-1], U.T[-1])
G = np.zeros_like(R)
for i in range(n):
for j in range(n):
G[i, j] = 0
for k in range(n):
G[i, j] += U[i, k] * U[j, k]
G_diff = np.zeros_like(R)
for i in range(n):
for j in range(n):
G_diff[i, j] = 0
for k in range(n):
G_diff[i, j] += U[i, k] * U[j, k] * w[k]
print G
print G_diff
print B
return np.sum(B * G_diff * np.log(G))
def get_mutual_information_small_approx_d(R, t):
"""
This is an approximation for small times.
This uses all of the off-diagonal entries of the mutual information
and also uses an approximation of the off-diagonal entries.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum_diag_a = 0
accum_diag_b = 0
accum_diag_c = 0
accum_diag_d = 0
for i in range(n):
a = 0
b = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
for k in range(n-1):
prefix = U[i, k] * U[i, k]
b += prefix * math.exp(t * w[k])
x1 = v[i] * v[i]
x2 = v[i] * b
y1 = math.log(a)
y2 = -math.log(v[i])
accum_diag_a += x1 * y1
accum_diag_b += x1 * y2
accum_diag_c += x2 * y1
accum_diag_d += x2 * y2
accum_a = 0
accum_b = 0
accum_c = 0
accum_d = 0
for i in range(n):
for j in range(n):
if i != j:
prefix = (v[i] * v[j]) ** .5
a = 0
for k in range(n):
a += U[i, k] * U[j, k] * math.exp(t * w[k])
b = 0
for k in range(n-1):
b += U[i, k] * U[j, k] * math.exp(t * w[k])
x1 = v[i] * v[j]
x2 = prefix * b
y1 = math.log(a)
y2 = -math.log(prefix)
accum_a += x1 * y1
accum_b += x1 * y2
accum_c += x2 * y1
accum_d += x2 * y2
terms = [
accum_diag_a, accum_diag_b, accum_diag_c, accum_diag_d,
accum_a, accum_b, accum_c, accum_d]
for term in terms:
print term
return sum(terms)
def get_mutual_information_b(R, t):
"""
This uses some cancellation.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum_diag_a = 0
accum_diag_b = 0
accum_diag_c = 0
for i in range(n):
a = 0
b = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
for k in range(n - 1):
prefix = U[i, k] * U[i, k]
b += prefix * math.exp(t * w[k])
x1 = v[i] * v[i]
x2 = v[i] * b
y1 = math.log(a)
y2 = -math.log(v[i])
accum_diag_a += x1 * y1
accum_diag_b += x1 * y2
accum_diag_c += x2 * y1
accum_a = 0
accum_b = 0
accum_c = 0
for i in range(n):
for j in range(n):
if i != j:
prefix = (v[i] * v[j])**.5
a = 0
for k in range(n):
a += U[i, k] * U[j, k] * math.exp(t * w[k])
b = 0
for k in range(n - 1):
b += U[i, k] * U[j, k] * math.exp(t * w[k])
x1 = v[i] * v[j]
x2 = prefix * b
y1 = math.log(a)
y2 = -math.log(prefix)
accum_a += x1 * y1
accum_b += x1 * y2
accum_c += x2 * y1
terms = [
accum_diag_a, accum_diag_b, accum_diag_c, accum_a, accum_b, accum_c
]
return sum(terms)
def get_mutual_information_b(R, t):
"""
This uses some cancellation.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum_diag_a = 0
accum_diag_b = 0
accum_diag_c = 0
for i in range(n):
a = 0
b = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
for k in range(n-1):
prefix = U[i, k] * U[i, k]
b += prefix * math.exp(t * w[k])
x1 = v[i] * v[i]
x2 = v[i] * b
y1 = math.log(a)
y2 = -math.log(v[i])
accum_diag_a += x1 * y1
accum_diag_b += x1 * y2
accum_diag_c += x2 * y1
accum_a = 0
accum_b = 0
accum_c = 0
for i in range(n):
for j in range(n):
if i != j:
prefix = (v[i] * v[j]) ** .5
a = 0
for k in range(n):
a += U[i, k] * U[j, k] * math.exp(t * w[k])
b = 0
for k in range(n-1):
b += U[i, k] * U[j, k] * math.exp(t * w[k])
x1 = v[i] * v[j]
x2 = prefix * b
y1 = math.log(a)
y2 = -math.log(prefix)
accum_a += x1 * y1
accum_b += x1 * y2
accum_c += x2 * y1
terms = [
accum_diag_a, accum_diag_b, accum_diag_c,
accum_a, accum_b, accum_c]
return sum(terms)
def get_mutual_information_diff_approx_c(R, t):
"""
This is an approximation for large times.
It has been rewritten using orthogonality.
It has also been rewritten using orthonormality.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for k in range(n - 1):
accum += w[k] * math.exp(2 * t * w[k])
return accum
def get_mutual_information_diff_approx_c(R, t):
"""
This is an approximation for large times.
It has been rewritten using orthogonality.
It has also been rewritten using orthonormality.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for k in range(n-1):
accum += w[k]*math.exp(2*t*w[k])
return accum
def get_mutual_information_approx_b(R, t):
"""
This is an approximation for large times.
It has been rewritten using orthogonality.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
for j in range(n):
for k in range(n - 1):
accum += ((U[i, k] * U[j, k])**2) * math.exp(2 * t * w[k]) / 2
return accum
def get_mutual_information_approx_b(R, t):
"""
This is an approximation for large times.
It has been rewritten using orthogonality.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
for j in range(n):
for k in range(n-1):
accum += ((U[i,k]*U[j,k])**2) * math.exp(2*t*w[k]) / 2
return accum
def get_asymptotic_variance_d(R, t):
"""
Use a very aggressive approximation.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R * t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance approximation
accum = 0
for k in range(n - 1):
accum += w[k] * w[k] * math.exp(2 * t * w[k])
return 1 / accum
def get_asymptotic_variance_d(R, t):
"""
Use a very aggressive approximation.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R*t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance approximation
accum = 0
for k in range(n-1):
accum += w[k] * w[k] * math.exp(2 * t * w[k])
return 1 / accum
def get_mutual_information_small_approx_c(R, t):
"""
This is an approximation for small times.
This is an even more aggressive approximation.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
a = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
accum += -v[i] * math.log(v[i]) * a
return accum
def get_mutual_information_small_approx_c(R, t):
"""
This is an approximation for small times.
This is an even more aggressive approximation.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
a = 0
for k in range(n):
prefix = U[i, k] * U[i, k]
a += prefix * math.exp(t * w[k])
accum += - v[i] * math.log(v[i]) * a
return accum
def get_mutual_information_small_approx(R, t):
"""
This is an approximation for small times.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
a = 0
for k in range(n):
a += (U[i, k]**2) * math.exp(t * w[k])
accum += v[i] * a * math.log(a / v[i])
#print [R[i, i] for i in range(n)]
#print [sum(U[i, k] * U[i, k] * w[k] for k in range(n)) for i in range(n)]
#print [sum(U[i, k] * U[i, k] for k in range(n)) for i in range(n)]
return accum
def get_mutual_information_small_approx(R, t):
"""
This is an approximation for small times.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
a = 0
for k in range(n):
a += (U[i, k]**2) * math.exp(t * w[k])
accum += v[i] * a * math.log(a / v[i])
#print [R[i, i] for i in range(n)]
#print [sum(U[i, k] * U[i, k] * w[k] for k in range(n)) for i in range(n)]
#print [sum(U[i, k] * U[i, k] for k in range(n)) for i in range(n)]
return accum
def get_mutual_information_diff_approx(R, t):
"""
This is an approximation for large times.
It can be rewritten using orthogonality.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
for j in range(n):
b = 0
for k in range(n - 1):
b += U[i, k] * U[j, k] * math.exp(t * w[k])
c = 0
for k in range(n - 1):
c += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
accum += b * c
return accum
def get_mutual_information_diff_approx(R, t):
"""
This is an approximation for large times.
It can be rewritten using orthogonality.
"""
n = len(R)
v = mrate.R_to_distn(R)
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
accum = 0
for i in range(n):
for j in range(n):
b = 0
for k in range(n-1):
b += U[i,k]*U[j,k]*math.exp(t*w[k])
c = 0
for k in range(n-1):
c += U[i,k]*U[j,k]*w[k]*math.exp(t*w[k])
accum += b * c
return accum
def get_asymptotic_variance_c(R, t):
"""
Re-evaluate, this time throwing away the part that is structurally zero.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R * t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance
accum = 0
for i in range(n):
for j in range(n):
# define f
f = p[i] * P[i, j]
# define the first derivative of f
f_dt = 0
for k in range(n):
f_dt += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
f_dt *= (p[i] * p[j])**.5
accum -= (f_dt * f_dt) / f
return -1 / accum
def get_asymptotic_variance_c(R, t):
"""
Re-evaluate, this time throwing away the part that is structurally zero.
"""
# get non-spectral summaries
n = len(R)
P = scipy.linalg.expm(R*t)
p = mrate.R_to_distn(R)
# get spectral summaries
S = mrate.symmetrized(R)
w, U = np.linalg.eigh(S)
# compute the asymptotic variance
accum = 0
for i in range(n):
for j in range(n):
# define f
f = p[i] * P[i, j]
# define the first derivative of f
f_dt = 0
for k in range(n):
f_dt += U[i, k] * U[j, k] * w[k] * math.exp(t * w[k])
f_dt *= (p[i] * p[j])**.5
accum -= (f_dt * f_dt) / f
return - 1 / accum
def get_response_content(fs):
out = StringIO()
np.set_printoptions(linewidth=200)
# get the user defined variables
n = fs.nstates
t = fs.divtime
#h = fs.delta
# sample a random rate matrix
v = divtime.sample_distribution(n)
S = divtime.sample_symmetric_rate_matrix(n)
R = mrate.to_gtr_halpern_bruno(S, v)
# get some properties of the rate matrix
distn = mrate.R_to_distn(R)
spectrum = np.linalg.eigvalsh(mrate.symmetrized(R))
#spectrum, U = np.linalg.eigh(mrate.symmetrized(R))
#spectrum = np.linalg.eigvals(R)
# report some information about the mutual information curve
mi = ctmcmi.get_mutual_information(R, t)
mi_diff = ctmcmi.get_mutual_information_diff(R, t)
mi_diff_b = ctmcmi.get_mutual_information_diff_b(R, t)
mi_diff_c = ctmcmi.get_mutual_information_diff_c(R, t)
print >> out, 'arbitrary large-ish divergence time:'
print >> out, t
print >> out
print >> out, 'randomly sampled reversible rate matrix:'
print >> out, R
print >> out
print >> out, 'stationary distribution:'
print >> out, distn
print >> out
print >> out, 'spectrum of the rate matrix:'
print >> out, spectrum
print >> out
print >> out, 'mutual information at t = %f:' % t
print >> out, mi
print >> out
print >> out, 'mutual information at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_b(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_approx(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_approx_b(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f (ver. 3):' % t
print >> out, ctmcmi.cute_MI_alternate(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f (ver. 4):' % t
print >> out, ctmcmi.get_mutual_information_approx_c(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_small_approx(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_small_approx_b(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f (ver. 3):' % t
print >> out, ctmcmi.get_mutual_information_small_approx_c(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f (ver. 4):' % t
print >> out, ctmcmi.get_mutual_information_small_approx_d(R, t)
print >> out
print >> out, 'mutual information diff at t = %f:' % t
print >> out, mi_diff
print >> out
print >> out, 'mutual information diff at t = %f (ver. 2):' % t
print >> out, mi_diff_b
print >> out
print >> out, 'mutual information diff at t = %f (ver. 3):' % t
print >> out, mi_diff_c
print >> out
print >> out, 'large t approximation of MI diff at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_diff_approx(R, t)
print >> out
print >> out, 'large t approximation of MI diff at t = %f: (ver. 2)' % t
print >> out, ctmcmi.get_mutual_information_diff_approx_b(R, t)
print >> out
print >> out, 'large t approximation of MI diff at t = %f: (ver. 4)' % t
print >> out, ctmcmi.get_mutual_information_diff_approx_c(R, t)
print >> out
print >> out, 'log of mutual information at t = %f:' % t
print >> out, math.log(mi)
print >> out
#print >> out, 'estimated derivative',
#print >> out, 'of log of mutual information at t = %f:' % t
#print >> out, (math.log(mi_c) - math.log(mi_a)) / (2*h)
#print >> out
print >> out, 'estimated derivative of log of MI',
print >> out, 'at t = %f:' % t
print >> out, mi_diff / mi
print >> out
print >> out, 'large t approximation of derivative of log of MI',
print >> out, 'at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_diff_approx(R,
t) / ctmcmi.get_mutual_information_approx(R, t)
print >> out
print >> out, 'large t approximation of derivative of log of MI',
print >> out, 'at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_diff_approx_b(R,
t) / ctmcmi.get_mutual_information_approx_b(R, t)
print >> out
print >> out, 'twice the relevant eigenvalue:'
print >> out, 2 * spectrum[-2]
print >> out
print >> out
#print >> out, 'estimated derivative',
#print >> out, 'of mutual information at t = %f:' % t
#print >> out, (mi_c - mi_a) / (2*h)
#print >> out
#print >> out, '(estimated derivative of mutual information) /',
#print >> out, '(mutual information) at t = %f:' % t
#print >> out, (mi_c - mi_a) / (2*h*mi_b)
#print >> out
return out.getvalue()
def get_response_content(fs):
out = StringIO()
np.set_printoptions(linewidth=200)
# get the user defined variables
n = fs.nstates
t = fs.divtime
#h = fs.delta
# sample a random rate matrix
v = divtime.sample_distribution(n)
S = divtime.sample_symmetric_rate_matrix(n)
R = mrate.to_gtr_halpern_bruno(S, v)
# get some properties of the rate matrix
distn = mrate.R_to_distn(R)
spectrum = np.linalg.eigvalsh(mrate.symmetrized(R))
#spectrum, U = np.linalg.eigh(mrate.symmetrized(R))
#spectrum = np.linalg.eigvals(R)
# report some information about the mutual information curve
mi = ctmcmi.get_mutual_information(R, t)
mi_diff = ctmcmi.get_mutual_information_diff(R, t)
mi_diff_b = ctmcmi.get_mutual_information_diff_b(R, t)
mi_diff_c = ctmcmi.get_mutual_information_diff_c(R, t)
print >> out, 'arbitrary large-ish divergence time:'
print >> out, t
print >> out
print >> out, 'randomly sampled reversible rate matrix:'
print >> out, R
print >> out
print >> out, 'stationary distribution:'
print >> out, distn
print >> out
print >> out, 'spectrum of the rate matrix:'
print >> out, spectrum
print >> out
print >> out, 'mutual information at t = %f:' % t
print >> out, mi
print >> out
print >> out, 'mutual information at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_b(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_approx(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_approx_b(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f (ver. 3):' % t
print >> out, ctmcmi.cute_MI_alternate(R, t)
print >> out
print >> out, 'large t approximation of MI at t = %f (ver. 4):' % t
print >> out, ctmcmi.get_mutual_information_approx_c(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_small_approx(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_small_approx_b(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f (ver. 3):' % t
print >> out, ctmcmi.get_mutual_information_small_approx_c(R, t)
print >> out
print >> out, 'small t approximation of MI at t = %f (ver. 4):' % t
print >> out, ctmcmi.get_mutual_information_small_approx_d(R, t)
print >> out
print >> out, 'mutual information diff at t = %f:' % t
print >> out, mi_diff
print >> out
print >> out, 'mutual information diff at t = %f (ver. 2):' % t
print >> out, mi_diff_b
print >> out
print >> out, 'mutual information diff at t = %f (ver. 3):' % t
print >> out, mi_diff_c
print >> out
print >> out, 'large t approximation of MI diff at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_diff_approx(R, t)
print >> out
print >> out, 'large t approximation of MI diff at t = %f: (ver. 2)' % t
print >> out, ctmcmi.get_mutual_information_diff_approx_b(R, t)
print >> out
print >> out, 'large t approximation of MI diff at t = %f: (ver. 4)' % t
print >> out, ctmcmi.get_mutual_information_diff_approx_c(R, t)
print >> out
print >> out, 'log of mutual information at t = %f:' % t
print >> out, math.log(mi)
print >> out
#print >> out, 'estimated derivative',
#print >> out, 'of log of mutual information at t = %f:' % t
#print >> out, (math.log(mi_c) - math.log(mi_a)) / (2*h)
#print >> out
print >> out, 'estimated derivative of log of MI',
print >> out, 'at t = %f:' % t
print >> out, mi_diff / mi
print >> out
print >> out, 'large t approximation of derivative of log of MI',
print >> out, 'at t = %f:' % t
print >> out, ctmcmi.get_mutual_information_diff_approx(
R, t) / ctmcmi.get_mutual_information_approx(R, t)
print >> out
print >> out, 'large t approximation of derivative of log of MI',
print >> out, 'at t = %f (ver. 2):' % t
print >> out, ctmcmi.get_mutual_information_diff_approx_b(
R, t) / ctmcmi.get_mutual_information_approx_b(R, t)
print >> out
print >> out, 'twice the relevant eigenvalue:'
print >> out, 2 * spectrum[-2]
print >> out
print >> out
#print >> out, 'estimated derivative',
#print >> out, 'of mutual information at t = %f:' % t
#print >> out, (mi_c - mi_a) / (2*h)
#print >> out
#print >> out, '(estimated derivative of mutual information) /',
#print >> out, '(mutual information) at t = %f:' % t
#print >> out, (mi_c - mi_a) / (2*h*mi_b)
#print >> out
return out.getvalue()
def get_response_content(fs):
out = StringIO()
np.set_printoptions(linewidth=200)
# get the user defined variables
n = fs.nstates
# sample a random rate matrix
v = divtime.sample_distribution(n)
S = divtime.sample_symmetric_rate_matrix(n)
R = mrate.to_gtr_halpern_bruno(S, v)
# get some properties of the rate matrix and its re-symmetrization
S = mrate.symmetrized(R)
distn = mrate.R_to_distn(R)
w, U = np.linalg.eigh(S)
D = np.diag(U.T[-1])**2
D_inv = np.diag(np.reciprocal(U.T[-1]))**2
for t in (1.0, 2.0):
P = scipy.linalg.expm(R*t)
M = ndot(D**.5, scipy.linalg.expm(S*t), D**.5)
M_star = ndot(D_inv**.5, scipy.linalg.expm(S*t), D_inv**.5)
M_star_log = np.log(M_star)
M_star_log_w, M_star_log_U = np.linalg.eigh(M_star_log)
E = M * np.log(M_star)
E_w, E_U = np.linalg.eigh(E)
print >> out, 't:'
print >> out, t
print >> out
print >> out, 'randomly sampled rate matrix R'
print >> out, R
print >> out
print >> out, 'symmetrized matrix S'
print >> out, S
print >> out
print >> out, 'stationary distribution diagonal D'
print >> out, D
print >> out
print >> out, 'R = D^-1/2 S D^1/2'
print >> out, ndot(D_inv**.5, S, D**.5)
print >> out
print >> out, 'probability matrix e^(R*t) = P'
print >> out, P
print >> out
print >> out, 'P = D^-1/2 e^(S*t) D^1/2'
print >> out, ndot(D_inv**.5, scipy.linalg.expm(S*t), D**.5)
print >> out
print >> out, 'pairwise distribution matrix M'
print >> out, 'M = D^1/2 e^(S*t) D^1/2'
print >> out, M
print >> out
print >> out, 'sum of entries of M'
print >> out, np.sum(M)
print >> out
print >> out, 'M_star = D^-1/2 e^(S*t) D^-1/2'
print >> out, M_star
print >> out
print >> out, 'entrywise logarithm logij(M_star)'
print >> out, np.log(M_star)
print >> out
print >> out, 'Hadamard product M o logij(M_star) = E'
print >> out, E
print >> out
print >> out, 'spectrum of M:'
print >> out, np.linalg.eigvalsh(M)
print >> out
print >> out, 'spectrum of logij(M_star):'
print >> out, M_star_log_w
print >> out
print >> out, 'corresponding eigenvectors of logij(M_star) as columns:'
print >> out, M_star_log_U
print >> out
print >> out, 'spectrum of E:'
print >> out, E_w
print >> out
print >> out, 'corresponding eigenvectors of E as columns:'
print >> out, E_U
print >> out
print >> out, 'entrywise square roots of stationary distribution:'
print >> out, np.sqrt(v)
print >> out
print >> out, 'sum of entries of E:'
print >> out, np.sum(E)
print >> out
print >> out, 'mutual information:'
print >> out, ctmcmi.get_mutual_information(R, t)
print >> out
print >> out
return out.getvalue()
def get_response_content(fs):
out = StringIO()
np.set_printoptions(linewidth=200)
# get the user defined variables
n = fs.nstates
# sample a random rate matrix
v = divtime.sample_distribution(n)
S = divtime.sample_symmetric_rate_matrix(n)
R = mrate.to_gtr_halpern_bruno(S, v)
# get some properties of the rate matrix and its re-symmetrization
S = mrate.symmetrized(R)
distn = mrate.R_to_distn(R)
w, U = np.linalg.eigh(S)
D = np.diag(U.T[-1])**2
D_inv = np.diag(np.reciprocal(U.T[-1]))**2
for t in (1.0, 2.0):
P = scipy.linalg.expm(R * t)
M = ndot(D**.5, scipy.linalg.expm(S * t), D**.5)
M_star = ndot(D_inv**.5, scipy.linalg.expm(S * t), D_inv**.5)
M_star_log = np.log(M_star)
M_star_log_w, M_star_log_U = np.linalg.eigh(M_star_log)
E = M * np.log(M_star)
E_w, E_U = np.linalg.eigh(E)
print >> out, 't:'
print >> out, t
print >> out
print >> out, 'randomly sampled rate matrix R'
print >> out, R
print >> out
print >> out, 'symmetrized matrix S'
print >> out, S
print >> out
print >> out, 'stationary distribution diagonal D'
print >> out, D
print >> out
print >> out, 'R = D^-1/2 S D^1/2'
print >> out, ndot(D_inv**.5, S, D**.5)
print >> out
print >> out, 'probability matrix e^(R*t) = P'
print >> out, P
print >> out
print >> out, 'P = D^-1/2 e^(S*t) D^1/2'
print >> out, ndot(D_inv**.5, scipy.linalg.expm(S * t), D**.5)
print >> out
print >> out, 'pairwise distribution matrix M'
print >> out, 'M = D^1/2 e^(S*t) D^1/2'
print >> out, M
print >> out
print >> out, 'sum of entries of M'
print >> out, np.sum(M)
print >> out
print >> out, 'M_star = D^-1/2 e^(S*t) D^-1/2'
print >> out, M_star
print >> out
print >> out, 'entrywise logarithm logij(M_star)'
print >> out, np.log(M_star)
print >> out
print >> out, 'Hadamard product M o logij(M_star) = E'
print >> out, E
print >> out
print >> out, 'spectrum of M:'
print >> out, np.linalg.eigvalsh(M)
print >> out
print >> out, 'spectrum of logij(M_star):'
print >> out, M_star_log_w
print >> out
print >> out, 'corresponding eigenvectors of logij(M_star) as columns:'
print >> out, M_star_log_U
print >> out
print >> out, 'spectrum of E:'
print >> out, E_w
print >> out
print >> out, 'corresponding eigenvectors of E as columns:'
print >> out, E_U
print >> out
print >> out, 'entrywise square roots of stationary distribution:'
print >> out, np.sqrt(v)
print >> out
print >> out, 'sum of entries of E:'
print >> out, np.sum(E)
print >> out
print >> out, 'mutual information:'
print >> out, ctmcmi.get_mutual_information(R, t)
print >> out
print >> out
return out.getvalue()
