def get_plot_array(N_diploid, Nr, theta_values, Ns_values):
"""
Compute expected hitting times.
Theta is 4*N*mu, and the units of time are 4*N*mu generations.
@param N_diploid: diploid population size
@param Nr: recombination rate
@param theta_values: mutation rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2 * N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2 * N_diploid) ** 2))
#
arr = []
for theta in theta_values:
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2 * 2 * N_diploid))
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
# compute the stationary distribution
v = MatrixUtil.get_stationary_distribution(P)
# compute the transition matrix limit at time infinity
# P_inf = np.outer(np.ones_like(v), v)
# compute the fundamental matrix Z
# Z = linalg.inv(np.eye(nstates) - (P - P_inf)) - P_inf
#
# Use broadcasting instead of constructing P_inf.
Z = linalg.inv(np.eye(nstates) - (P - v)) - v
# compute the hitting time from state AB to state ab.
i = 0
j = 3
hitting_time_generations = (Z[j, j] - Z[i, j]) / v[j]
hitting_time = hitting_time_generations * theta
row.append(hitting_time)
arr.append(row)
return arr
def get_plot_array(N_diploid, Nr, theta_values, Ns_values):
"""
Compute expected hitting times.
Theta is 4*N*mu, and the units of time are 4*N*mu generations.
@param N_diploid: diploid population size
@param Nr: recombination rate
@param theta_values: mutation rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2 * N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2 * N_diploid)**2))
#
arr = []
for theta in theta_values:
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2 * 2 * N_diploid))
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
# compute the stationary distribution
v = MatrixUtil.get_stationary_distribution(P)
# compute the transition matrix limit at time infinity
#P_inf = np.outer(np.ones_like(v), v)
# compute the fundamental matrix Z
#Z = linalg.inv(np.eye(nstates) - (P - P_inf)) - P_inf
#
# Use broadcasting instead of constructing P_inf.
Z = linalg.inv(np.eye(nstates) - (P - v)) - v
# compute the hitting time from state AB to state ab.
i = 0
j = 3
hitting_time_generations = (Z[j, j] - Z[i, j]) / v[j]
hitting_time = hitting_time_generations * theta
row.append(hitting_time)
arr.append(row)
return arr
def get_plot_array(N_diploid, Nr, theta_values, Ns_values):
"""
@param N_diploid: diploid population size
@param Nr: recombination rate
@param theta_values: mutation rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2 * N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2 * N_diploid)**2))
#
arr = []
for theta in theta_values:
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2 * 2 * N_diploid))
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
# compute the stochastic complement
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
# condition on not looping
np.fill_diagonal(H, 0)
v = np.sum(H, axis=1)
H /= v[:, np.newaxis]
# let ab be an absorbing state
# and compute the expected number of returns to AB
Q = H[:3, :3]
I = np.eye(3)
N = linalg.inv(I - Q)
#
row.append(N[0, 0] - 1)
arr.append(row)
return arr
def get_plot_array(N_diploid, Nr, theta_values, Ns_values):
"""
@param N_diploid: diploid population size
@param Nr: recombination rate
@param theta_values: mutation rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2 * N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2 * N_diploid) ** 2))
#
arr = []
for theta in theta_values:
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2 * 2 * N_diploid))
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
# compute the stochastic complement
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
# condition on not looping
np.fill_diagonal(H, 0)
v = np.sum(H, axis=1)
H /= v[:, np.newaxis]
# let ab be an absorbing state
# and compute the expected number of returns to AB
Q = H[:3, :3]
I = np.eye(3)
N = linalg.inv(I - Q)
#
row.append(N[0, 0] - 1)
arr.append(row)
return arr
def get_plot_array(N_diploid, Nr, theta_values, Ns_values):
"""
@param N_diploid: diploid population size
@param Nr: recombination rate
@param theta_values: mutation rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2*N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2*N_diploid)**2))
#
arr = []
for theta in theta_values:
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2*2*N_diploid))
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
# compute the stochastic complement
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
# compute a conditional transition probability
i = 0
j = 3
row.append(H[i, j] / (1-H[i,i]))
arr.append(row)
return arr
def get_p2(N_diploid, theta, Nr, Ns):
"""
Compute the probability of compensatory substitution pathway type 2.
@param N_diploid: diploid population size
@param theta: a mutation rate
@param Nr: a recombination rate
@param Ns: a selection value
@return: p_t2
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2 * N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# compute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# compute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2 * N_diploid)**2))
# compute the expected number of mutation events per generation
mu = theta / 2
# compute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2 * 2 * N_diploid))
MR_prob = np.dot(M_prob, R_prob)
# compute the selection coefficient
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
#
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
p_direct = H[0, 3] / (1 - H[0, 0])
# The following line is Equation (1) of the Nasrallah manuscript.
p_t2 = (2 * p_direct) / (1 + p_direct)
p_t1 = 1 - p_t2
#
return p_t2
def get_p2(N_diploid, theta, Nr, Ns):
"""
Compute the probability of compensatory substitution pathway type 2.
@param N_diploid: diploid population size
@param theta: a mutation rate
@param Nr: a recombination rate
@param Ns: a selection value
@return: p_t2
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2*N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# compute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# compute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2*N_diploid)**2))
# compute the expected number of mutation events per generation
mu = theta / 2
# compute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2*2*N_diploid))
MR_prob = np.dot(M_prob, R_prob)
# compute the selection coefficient
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
#
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
p_direct = H[0, 3] / (1 - H[0,0])
# The following line is Equation (1) of the Nasrallah manuscript.
p_t2 = (2*p_direct) / (1 + p_direct)
p_t1 = 1 - p_t2
#
return p_t2
def get_backward_info(N_diploid, theta, Nr, Ns):
"""
Compute expectations and variances for the two substitution pathways.
Here backward is somewhat of a misnomer; it is meant as a contrast
to forward simulation.
@param N_diploid: diploid population size
@param theta: a mutation rate
@param Nr: a recombination rate
@param Ns: a selection value
@return: (t1, v1), (t2, v2)
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2 * N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# compute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# compute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2 * N_diploid)**2))
# compute the expected number of mutation events per generation
mu = theta / 2
# compute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2 * 2 * N_diploid))
MR_prob = np.dot(M_prob, R_prob)
# compute the selection coefficient
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
#
t1, v1 = wfbckcompens.get_type_1_info(P)
t2, v2 = wfbckcompens.get_type_2_info(P)
return (t1, v1), (t2, v2)
def get_backward_info(N_diploid, theta, Nr, Ns):
"""
Compute expectations and variances for the two substitution pathways.
Here backward is somewhat of a misnomer; it is meant as a contrast
to forward simulation.
@param N_diploid: diploid population size
@param theta: a mutation rate
@param Nr: a recombination rate
@param Ns: a selection value
@return: (t1, v1), (t2, v2)
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2*N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# compute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# compute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2*N_diploid)**2))
# compute the expected number of mutation events per generation
mu = theta / 2
# compute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2*2*N_diploid))
MR_prob = np.dot(M_prob, R_prob)
# compute the selection coefficient
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
#
t1, v1 = wfbckcompens.get_type_1_info(P)
t2, v2 = wfbckcompens.get_type_2_info(P)
return (t1, v1), (t2, v2)
def get_plot_array(N_diploid, theta, Nr_values, Ns_values):
"""
@param N_diploid: diploid population size
@param theta: mutation rate
@param Nr_values: recombination rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# define the haplotypes
AB, Ab, aB, ab = 0, 1, 2, 3
# initialize the state space
N_hap = 2 * N_diploid
k = 4
M = multinomstate.get_sorted_states(N_hap, k)
nstates = M.shape[0]
# compute the inverse map
T = multinomstate.get_inverse_map(M)
#
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
M_prob = linalg.expm(mu * M_rate / float(2*2*N_diploid))
#
arr = []
for Nr in Nr_values:
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2*N_diploid)**2))
# precompute the product of mutation and recombination.
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
#
t1, v1 = wfbckcompens.get_type_1_info(P)
t2, v2 = wfbckcompens.get_type_2_info(P)
#
"""
# What is the distribution over next fixed states
# from the current state?
# This question can be answered
# by hacking with transience and absorption.
Q = P[:-k, :-k]
R = P[:-k, -k:]
B = linalg.solve(np.eye(nstates-k) - Q, R)
# At this point B is the matrix whose nstates-k rows give
# distributions over the k fixed states.
# Next construct the transition matrix that is conditional
# upon first hitting the ab fixed state.
w = np.zeros(nstates)
w[:-k] = R[:, -1]
w[-k:] = np.array([0, 0, 0, 1])
P_t2 = P * w
# normalize after scaling by the weights
v = P_t2.sum(axis=1)
P_t2 /= v[:, np.newaxis]
# Get the hitting time from state AB to state ab.
# Because of the conditioning, this should be the same
# as the expected time to reach state ab given that state ab
# is the first reached fixed state.
# Note that this means that the first step is away from AB.
# Or actually we can just use expected time to absorption.
Q = P_t2[:-1, :-1]
c = np.ones(nstates-1)
t = linalg.lstsq(np.eye(nstates-1) - Q, c)
t2 = t[-4]
# Now do type 1 events.
w = np.zeros(nstates)
w[:-k] = 1 - R[:, 0]
w[-k:] = np.array([0, 0, 0, 1])
P_t2 = P * w
#
"""
# Get the probability of type 2.
# This uses the stochastic complement.
# Wait this is wrong.
# This is the probability of a direct transition.
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
p_direct = H[0, 3] / (1 - H[0,0])
# The following line is Equation (1) of the Nasrallah manuscript.
p_t2 = (2*p_direct) / (1 + p_direct)
p_t1 = 1 - p_t2
"""
expectation_of_variance = p_t2*v2 + p_t1*v1
variance_of_expectation = p_t2*p_t1*(t1 - t2)*(t1 - t2)
pooled_variance = (
expectation_of_variance + variance_of_expectation)
"""
#
# Just do a simulation,
# assuming that the wait times are normally distributed.
nsamples = 500
n1 = np.random.binomial(nsamples, p_t1)
n2 = nsamples - n1
X1 = np.random.normal(t1, math.sqrt(v1), n1)
X2 = np.random.normal(t2, math.sqrt(v2), n2)
X_pooled = np.hstack((X1, X2))
x = np.mean(X1) - np.mean(X2)
s_pooled = math.sqrt(np.var(X_pooled) / nsamples)
t_statistic = x / s_pooled
row.append(t_statistic)
#
#x = (t1 - t2) / math.sqrt(variance / 200.0)
#x = (t1 - t2) / math.sqrt((v1 + v2) / 200.0)
#x = (t1 - t2) / math.sqrt(pooled_variance)
#x = (t1 - t2)
#row.append(math.log(t1) - math.log(t2))
#row.append(x)
#row.append(v2)
arr.append(row)
return arr
def get_plot_array(N_diploid, Nr, theta_values, Ns_values):
"""
@param N_diploid: diploid population size
@param Nr: recombination rate
@param theta_values: mutation rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2 * N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2 * N_diploid)**2))
#
arr = []
for theta in theta_values:
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2 * 2 * N_diploid))
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
# compute the stochastic complement
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
# condition on a transition change
#np.fill_diagonal(H, 0)
#H = (H.T / np.sum(H, axis=1)).T
#
#v = MatrixUtil.get_stationary_distribution(H)
#print 'reversibility check (0 if reversible):'
#print H/v - H.T/v
#print
#H_rev = (H*v).T/v
#if not np.allclose(
#[v[i]*H[i,j]-v[j]*H[j,i] for i in range(k) for j in range(k)],
#np.zeros(k)):
#raise ValueError('not reversible')
# break the last state into two states
AB, Ab, aB, ab, abx = 0, 1, 2, 3, 4
J = np.zeros((k + 1, k + 1))
J[:k, :k] = H
# force ab and abx to be absorbing states
J[ab] = np.zeros(k + 1)
J[abx] = np.zeros(k + 1)
J[ab, ab] = 1
J[abx, abx] = 1
# connect AB to the new ab state
J[AB, abx] = J[AB, ab]
J[AB, ab] = 0
# Now transition matrix J is in "canonical form"
# because all of the absorbing states are at the end.
B = linalg.solve(
np.eye(k - 1) - J[:k - 1, :k - 1], J[:k - 1, k - 1:])
# compute the absorbing state distribution
row.append(B[0, 1])
arr.append(row)
return arr
def get_plot_array(N_diploid, Nr, theta_values, Ns_values):
"""
@param N_diploid: diploid population size
@param Nr: recombination rate
@param theta_values: mutation rates
@param Ns_values: selection values
@return: arr[i][j] gives time for Ns_values[i] and theta_values[j]
"""
# set up the state space
k = 4
M = multinomstate.get_sorted_states(2*N_diploid, k)
T = multinomstate.get_inverse_map(M)
nstates = M.shape[0]
lmcs = wfengine.get_lmcs(M)
# precompute rate matrices
R_rate = wfcompens.create_recomb(M, T)
M_rate = wfcompens.create_mutation(M, T)
# precompute a recombination probability matrix
R_prob = linalg.expm(Nr * R_rate / float((2*N_diploid)**2))
#
arr = []
for theta in theta_values:
# Compute the expected number of mutation events per generation.
mu = theta / 2
# Precompute the mutation matrix
# and the product of mutation and recombination.
M_prob = linalg.expm(mu * M_rate / float(2*2*N_diploid))
MR_prob = np.dot(M_prob, R_prob)
#
row = []
for Ns in Ns_values:
s = Ns / float(N_diploid)
lps = wfcompens.create_selection(s, M)
S_prob = np.exp(wfengine.create_genic(lmcs, lps, M))
P = np.dot(MR_prob, S_prob)
# compute the stochastic complement
X = linalg.solve(np.eye(nstates - k) - P[k:, k:], P[k:, :k])
H = P[:k, :k] + np.dot(P[:k, k:], X)
# condition on a transition change
#np.fill_diagonal(H, 0)
#H = (H.T / np.sum(H, axis=1)).T
#
#v = MatrixUtil.get_stationary_distribution(H)
#print 'reversibility check (0 if reversible):'
#print H/v - H.T/v
#print
#H_rev = (H*v).T/v
#if not np.allclose(
#[v[i]*H[i,j]-v[j]*H[j,i] for i in range(k) for j in range(k)],
#np.zeros(k)):
#raise ValueError('not reversible')
# break the last state into two states
AB, Ab, aB, ab, abx = 0, 1, 2, 3, 4
J = np.zeros((k+1, k+1))
J[:k, :k] = H
# force ab and abx to be absorbing states
J[ab] = np.zeros(k+1)
J[abx] = np.zeros(k+1)
J[ab, ab] = 1
J[abx, abx] = 1
# connect AB to the new ab state
J[AB, abx] = J[AB, ab]
J[AB, ab] = 0
# Now transition matrix J is in "canonical form"
# because all of the absorbing states are at the end.
B = linalg.solve(np.eye(k-1) - J[:k-1, :k-1], J[:k-1, k-1:])
# compute the absorbing state distribution
row.append(B[0, 1])
arr.append(row)
return arr
