def _check_laplace_transform_equilibrium_solution(tau):
# The tau parameter must be positive for this solution to work.
assert_array_less(0, tau)
# Define the structural properties of the process.
n = 5
state_space_shape = (n, n)
nstates = np.prod(state_space_shape)
bivariate_states = list(product(range(n), repeat=2))
bivariate_state_pairs = list(permutations(bivariate_states, 2))
I_univariate = np.identity(n)
# Sample a random time-reversible univariate rate matrix
# together with its stationary distribution.
Q, d = sample_time_reversible_rate_matrix(n)
# Compute a decomposition of Q.
a = np.sqrt(d)
b = np.reciprocal(a)
S = Q * np.outer(a, b)
assert_allclose(S, S.T, atol=1e-12)
w, U = scipy.linalg.eigh(S)
assert_allclose(U.dot(U.T), I_univariate, atol=1e-12)
assert_allclose(U.dot(np.diag(w)).dot(U.T), S)
# Check properties of the Kronecker sum of Q and Q.
# This corresponds to joint independent evolution of two variables.
# First check detailed balance.
# Then check the eigendecomposition.
R_Q = _kronecker_sum(Q, Q)
R_d = np.kron(d, d)
assert_allclose(R_d.sum(), 1)
assert_symmetric_matrix(np.diag(R_d).dot(R_Q))
R_U = np.kron(U, U)
R_w = _kronecker_sum_1d(w, w)
R_S = R_U.dot(np.diag(R_w)).dot(R_U.T)
R_a = np.sqrt(R_d)
R_b = np.reciprocal(R_a)
assert_allclose(R_S * np.outer(R_b, R_a), R_Q, atol=1e-12)
# Get the bivariate gene-conversion process rate matrix directly.
# Define the gene conversion mask.
T = np.ones((n, n)) - np.identity(n)
mask = _kronecker_sum(T, T) * _vec(np.identity(n))
T_Q = R_Q + tau * mask
T_Q = T_Q - np.diag(T_Q.sum(axis=1))
T_d_brute = _brute_force_equilibrium(T_Q)
assert_equal(T_d_brute.shape, (n*n, ))
# Check the shortcut for computing the equilibrium distribution
# of the bivariate process with gene conversion.
# This involves the Laplace transform.
s = 2 * tau
T_u = 1 / (1 - R_w / s)
T_S = R_U.dot(np.diag(T_u)).dot(R_U.T)
laplace_thing = T_S * np.outer(R_b, R_a)
T_d_clever = _vec(np.diag(d)).dot(laplace_thing)
assert_equal(T_d_clever.shape, (n*n, ))
assert_allclose(T_d_clever, T_d_brute)
# Check separate calculations of expectations.
M_u = 1 / (1 - w / tau)
yet_another_laplace_thing = U.dot(np.diag(M_u)).dot(U.T) * np.outer(b, a)
T_d_yet_another = np.diag(d).dot(yet_another_laplace_thing).flatten()
assert_equal(T_d_yet_another.shape, (n*n, ))
assert_allclose(T_d_yet_another, T_d_brute)
# Check some transition probabilities.
P = U.dot(np.diag(np.exp(w))).dot(U.T) * np.outer(b, a)
R_P = R_U.dot(np.diag(np.exp(R_w))).dot(R_U.T) * np.outer(R_b, R_a)
desired = np.array([np.kron(r, r) for r in P])
actual = R_P[_vec(np.identity(n, dtype=bool)), :]
assert_allclose(actual, desired)
def _check_laplace_transform_equilibrium_solution(tau):
# The tau parameter must be positive for this solution to work.
assert_array_less(0, tau)
# Define the structural properties of the process.
n = 5
state_space_shape = (n, n)
nstates = np.prod(state_space_shape)
bivariate_states = list(product(range(n), repeat=2))
bivariate_state_pairs = list(permutations(bivariate_states, 2))
I_univariate = np.identity(n)
# Sample a random time-reversible univariate rate matrix
# together with its stationary distribution.
Q, d = sample_time_reversible_rate_matrix(n)
# Compute a decomposition of Q.
a = np.sqrt(d)
b = np.reciprocal(a)
S = Q * np.outer(a, b)
assert_allclose(S, S.T, atol=1e-12)
w, U = scipy.linalg.eigh(S)
assert_allclose(U.dot(U.T), I_univariate, atol=1e-12)
assert_allclose(U.dot(np.diag(w)).dot(U.T), S)
# Check properties of the Kronecker sum of Q and Q.
# This corresponds to joint independent evolution of two variables.
# First check detailed balance.
# Then check the eigendecomposition.
R_Q = _kronecker_sum(Q, Q)
R_d = np.kron(d, d)
assert_allclose(R_d.sum(), 1)
assert_symmetric_matrix(np.diag(R_d).dot(R_Q))
R_U = np.kron(U, U)
R_w = _kronecker_sum_1d(w, w)
R_S = R_U.dot(np.diag(R_w)).dot(R_U.T)
R_a = np.sqrt(R_d)
R_b = np.reciprocal(R_a)
assert_allclose(R_S * np.outer(R_b, R_a), R_Q, atol=1e-12)
# Get the bivariate gene-conversion process rate matrix directly.
# Define the gene conversion mask.
T = np.ones((n, n)) - np.identity(n)
mask = _kronecker_sum(T, T) * _vec(np.identity(n))
T_Q = R_Q + tau * mask
T_Q = T_Q - np.diag(T_Q.sum(axis=1))
T_d_brute = _brute_force_equilibrium(T_Q)
assert_equal(T_d_brute.shape, (n * n, ))
# Check the shortcut for computing the equilibrium distribution
# of the bivariate process with gene conversion.
# This involves the Laplace transform.
s = 2 * tau
T_u = 1 / (1 - R_w / s)
T_S = R_U.dot(np.diag(T_u)).dot(R_U.T)
laplace_thing = T_S * np.outer(R_b, R_a)
T_d_clever = _vec(np.diag(d)).dot(laplace_thing)
assert_equal(T_d_clever.shape, (n * n, ))
assert_allclose(T_d_clever, T_d_brute)
# Check separate calculations of expectations.
M_u = 1 / (1 - w / tau)
yet_another_laplace_thing = U.dot(np.diag(M_u)).dot(U.T) * np.outer(b, a)
T_d_yet_another = np.diag(d).dot(yet_another_laplace_thing).flatten()
assert_equal(T_d_yet_another.shape, (n * n, ))
assert_allclose(T_d_yet_another, T_d_brute)
# Check some transition probabilities.
P = U.dot(np.diag(np.exp(w))).dot(U.T) * np.outer(b, a)
R_P = R_U.dot(np.diag(np.exp(R_w))).dot(R_U.T) * np.outer(R_b, R_a)
desired = np.array([np.kron(r, r) for r in P])
actual = R_P[_vec(np.identity(n, dtype=bool)), :]
assert_allclose(actual, desired)
def test_empirical():
nsamples = 100
n = 3
I_univariate = np.identity(n)
# Sample a random time-reversible univariate rate matrix
# together with its stationary distribution.
Q, d = sample_time_reversible_rate_matrix(n)
# Compute a decomposition of Q.
a = np.sqrt(d)
b = np.reciprocal(a)
S = Q * np.outer(a, b)
assert_allclose(S, S.T, atol=1e-12)
w, U = scipy.linalg.eigh(S)
assert_allclose(U.dot(U.T), I_univariate, atol=1e-12)
assert_allclose(U.dot(np.diag(w)).dot(U.T), S)
# Check properties of the Kronecker sum of Q and Q.
# This corresponds to joint independent evolution of two variables.
# First check detailed balance.
# Then check the eigendecomposition.
R_Q = _kronecker_sum(Q, Q)
R_d = np.kron(d, d)
assert_allclose(R_d.sum(), 1)
assert_symmetric_matrix(np.diag(R_d).dot(R_Q))
R_U = np.kron(U, U)
R_w = _kronecker_sum_1d(w, w)
R_S = R_U.dot(np.diag(R_w)).dot(R_U.T)
R_a = np.sqrt(R_d)
R_b = np.reciprocal(R_a)
assert_allclose(R_S * np.outer(R_b, R_a), R_Q, atol=1e-12)
# Sample times independently per branch.
P_A = np.zeros((n, n*n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=1)
P0 = U.dot(np.diag(np.exp(t0*w))).dot(U.T) * np.outer(b, a)
t1 = np.random.exponential(scale=1)
P1 = U.dot(np.diag(np.exp(t1*w))).dot(U.T) * np.outer(b, a)
P_A += np.array([np.kron(P0[i], P1[i]) for i in range(n)])
P_A /= nsamples
# Sample one time shared across both branches.
P_B = np.zeros((n, n*n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=1)
P0 = U.dot(np.diag(np.exp(t0*w))).dot(U.T) * np.outer(b, a)
P_B += np.array([np.kron(P0[i], P0[i]) for i in range(n)])
P_B /= nsamples
# Sample one longer time shared across both branches.
P_C = np.zeros((n, n*n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=2)
P0 = U.dot(np.diag(np.exp(t0*w))).dot(U.T) * np.outer(b, a)
P_C += np.array([np.kron(P0[i], P0[i]) for i in range(n)])
P_C /= nsamples
# Sample one shorter time shared across both branches.
P_D = np.zeros((n, n*n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=0.5)
P0 = U.dot(np.diag(np.exp(t0*w))).dot(U.T) * np.outer(b, a)
P_D += np.array([np.kron(P0[i], P0[i]) for i in range(n)])
P_D /= nsamples
# Use the Laplace transform to exactly compute
# one of the empirically estimated transition matrices.
P0 = U.dot(np.diag(1 / (1 - w))).dot(U.T) * np.outer(b, a)
P_E = np.array([np.kron(P0[i], P0[i]) for i in range(n)])
assert_allclose(P_E.sum(axis=1), 1)
# Use the Laplace transform to exactly compute
# one of the empirically estimated transition matrices.
P0 = R_U.dot(np.diag(1 / (1 - R_w))).dot(R_U.T) * np.outer(R_b, R_a)
P_F = P0[_vec(np.identity(n, dtype=bool)), :]
assert_allclose(P_F.sum(axis=1), 1)
def test_empirical():
nsamples = 100
n = 3
I_univariate = np.identity(n)
# Sample a random time-reversible univariate rate matrix
# together with its stationary distribution.
Q, d = sample_time_reversible_rate_matrix(n)
# Compute a decomposition of Q.
a = np.sqrt(d)
b = np.reciprocal(a)
S = Q * np.outer(a, b)
assert_allclose(S, S.T, atol=1e-12)
w, U = scipy.linalg.eigh(S)
assert_allclose(U.dot(U.T), I_univariate, atol=1e-12)
assert_allclose(U.dot(np.diag(w)).dot(U.T), S)
# Check properties of the Kronecker sum of Q and Q.
# This corresponds to joint independent evolution of two variables.
# First check detailed balance.
# Then check the eigendecomposition.
R_Q = _kronecker_sum(Q, Q)
R_d = np.kron(d, d)
assert_allclose(R_d.sum(), 1)
assert_symmetric_matrix(np.diag(R_d).dot(R_Q))
R_U = np.kron(U, U)
R_w = _kronecker_sum_1d(w, w)
R_S = R_U.dot(np.diag(R_w)).dot(R_U.T)
R_a = np.sqrt(R_d)
R_b = np.reciprocal(R_a)
assert_allclose(R_S * np.outer(R_b, R_a), R_Q, atol=1e-12)
# Sample times independently per branch.
P_A = np.zeros((n, n * n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=1)
P0 = U.dot(np.diag(np.exp(t0 * w))).dot(U.T) * np.outer(b, a)
t1 = np.random.exponential(scale=1)
P1 = U.dot(np.diag(np.exp(t1 * w))).dot(U.T) * np.outer(b, a)
P_A += np.array([np.kron(P0[i], P1[i]) for i in range(n)])
P_A /= nsamples
# Sample one time shared across both branches.
P_B = np.zeros((n, n * n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=1)
P0 = U.dot(np.diag(np.exp(t0 * w))).dot(U.T) * np.outer(b, a)
P_B += np.array([np.kron(P0[i], P0[i]) for i in range(n)])
P_B /= nsamples
# Sample one longer time shared across both branches.
P_C = np.zeros((n, n * n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=2)
P0 = U.dot(np.diag(np.exp(t0 * w))).dot(U.T) * np.outer(b, a)
P_C += np.array([np.kron(P0[i], P0[i]) for i in range(n)])
P_C /= nsamples
# Sample one shorter time shared across both branches.
P_D = np.zeros((n, n * n), dtype=float)
for sample_index in range(nsamples):
t0 = np.random.exponential(scale=0.5)
P0 = U.dot(np.diag(np.exp(t0 * w))).dot(U.T) * np.outer(b, a)
P_D += np.array([np.kron(P0[i], P0[i]) for i in range(n)])
P_D /= nsamples
# Use the Laplace transform to exactly compute
# one of the empirically estimated transition matrices.
P0 = U.dot(np.diag(1 / (1 - w))).dot(U.T) * np.outer(b, a)
P_E = np.array([np.kron(P0[i], P0[i]) for i in range(n)])
assert_allclose(P_E.sum(axis=1), 1)
# Use the Laplace transform to exactly compute
# one of the empirically estimated transition matrices.
P0 = R_U.dot(np.diag(1 / (1 - R_w))).dot(R_U.T) * np.outer(R_b, R_a)
P_F = P0[_vec(np.identity(n, dtype=bool)), :]
assert_allclose(P_F.sum(axis=1), 1)
