def two_dim_wright_fisher_figure(N, m, mu=0.01, incentive_func=replicator):
"""
Plot relative entropies and stationary distribution for the Wright-Fisher
process.
"""
n = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
incentive = incentive_func(fitness_landscape)
if not mu:
mu = 1./ N
edge_func = wright_fisher.multivariate_transitions(N, incentive, mu=mu, num_types=n)
states = list(simplex_generator(N, d=n-1))
s = stationary_distribution(edge_func, states=states, iterations=4*N)
s0 = expected_divergence(edge_func, states=states, q_d=0)
s1 = expected_divergence(edge_func, states=states, q_d=1)
# Set up plots
gs = gridspec.GridSpec(2, 1)
ax2 = pyplot.subplot(gs[0, 0])
ax2.set_title("Relative Entropy")
plot_dictionary(s0, ax=ax2)
plot_dictionary(s1, ax=ax2)
ax3 = pyplot.subplot(gs[1, 0])
ax3.set_title("Stationary Distribution")
plot_dictionary(s, ax=ax3)
ax3.set_xlabel("Number of A individuals (i)")
def graphical_abstract_figures(N=60, q=1, beta=0.1):
"""
Three dimensional process examples.
"""
a = 0
b = 1
m = [[a, b, b], [b, a, b], [b, b, a]]
mu = (3. / 2 ) * 1. / N
fitness_landscape = linear_fitness_landscape(m)
incentive = fermi(fitness_landscape, beta=beta, q=q)
edges = incentive_process.multivariate_transitions(N, incentive, num_types=3, mu=mu)
d = stationary_distribution(edges, iterations=None)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d, scale=N)
tax.savefig(filename="ga_stationary.eps", dpi=600)
d = expected_divergence(edges, q_d=0)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d, scale=N)
tax.savefig(filename="ga_d_0.eps", dpi=600)
d = expected_divergence(edges, q_d=1)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d, scale=N)
tax.savefig(filename="ga_d_1.eps", dpi=600)
def graphical_abstract_figures(N=60, q=1, beta=0.1):
"""
Three dimensional process examples.
"""
a = 0
b = 1
m = [[a, b, b], [b, a, b], [b, b, a]]
mu = (3. / 2) * 1. / N
fitness_landscape = linear_fitness_landscape(m)
incentive = fermi(fitness_landscape, beta=beta, q=q)
edges = incentive_process.multivariate_transitions(N,
incentive,
num_types=3,
mu=mu)
d = stationary_distribution(edges, iterations=None)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d, scale=N)
tax.savefig(filename="ga_stationary.eps", dpi=600)
d = expected_divergence(edges, q_d=0)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d, scale=N)
tax.savefig(filename="ga_d_0.eps", dpi=600)
d = expected_divergence(edges, q_d=1)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d, scale=N)
tax.savefig(filename="ga_d_1.eps", dpi=600)
def two_dim_transitions_figure(N, m, mu=0.01, incentive_func=replicator):
"""
Plot transition entropies and stationary distributions.
"""
n = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
incentive = incentive_func(fitness_landscape)
if not mu:
mu = 1./ N
edges = incentive_process.multivariate_transitions(N, incentive, num_types=n, mu=mu)
s = stationary_distribution(edges, exact=True)
d = edges_to_edge_dict(edges)
# Set up plots
gs = gridspec.GridSpec(3, 1)
ax1 = pyplot.subplot(gs[0, 0])
ax1.set_title("Transition Probabilities")
ups, downs, _ = two_dim_transitions(edges)
xs = range(0, N+1)
ax1.plot(xs, ups)
ax1.plot(xs, downs)
ax2 = pyplot.subplot(gs[1, 0])
ax2.set_title("Relative Entropy")
divs1 = expected_divergence(edges)
divs2 = expected_divergence(edges, q_d=0)
plot_dictionary(divs1, ax=ax2)
plot_dictionary(divs2, ax=ax2)
ax3 = pyplot.subplot(gs[2, 0])
ax3.set_title("Stationary Distribution")
plot_dictionary(s, ax=ax3)
ax3.set_xlabel("Number of A individuals (i)")
def two_dim_wright_fisher_figure(N, m, mu=0.01, incentive_func=replicator):
"""
Plot relative entropies and stationary distribution for the Wright-Fisher
process.
"""
n = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
incentive = incentive_func(fitness_landscape)
if not mu:
mu = 1. / N
edge_func = wright_fisher.multivariate_transitions(N,
incentive,
mu=mu,
num_types=n)
states = list(simplex_generator(N, d=n - 1))
s = stationary_distribution(edge_func, states=states, iterations=4 * N)
s0 = expected_divergence(edge_func, states=states, q_d=0)
s1 = expected_divergence(edge_func, states=states, q_d=1)
# Set up plots
gs = gridspec.GridSpec(2, 1)
ax2 = pyplot.subplot(gs[0, 0])
ax2.set_title("Relative Entropy")
plot_dictionary(s0, ax=ax2)
plot_dictionary(s1, ax=ax2)
ax3 = pyplot.subplot(gs[1, 0])
ax3.set_title("Stationary Distribution")
plot_dictionary(s, ax=ax3)
ax3.set_xlabel("Number of A individuals (i)")
def bomze_figures(N=60, beta=1, process="incentive", directory=None):
"""
Makes plots of the stationary distribution and expected divergence for each
of the plots in Bomze's classification.
"""
if not directory:
directory = "bomze_paper_figures_%s" % process
ensure_directory(directory)
for i, m in enumerate(bomze_matrices()):
mu = 3. / 2 * 1. / N
fitness_landscape = linear_fitness_landscape(m)
incentive = fermi(fitness_landscape, beta=beta)
edges = incentive_process.multivariate_transitions(N,
incentive,
num_types=3,
mu=mu)
d1 = stationary_distribution(edges)
filename = os.path.join(directory, "%s_%s_stationary.eps" % (i, N))
figure, tax = ternary.figure(scale=N)
tax.heatmap(d1)
tax.savefig(filename=filename)
pyplot.close(figure)
for q_d in [0., 1.]:
d2 = expected_divergence(edges, q_d=q_d)
filename = os.path.join(directory, "%s_%s_%s.eps" % (i, N, q_d))
figure, tax = ternary.figure(scale=N)
tax.heatmap(d2)
tax.savefig(filename=filename)
pyplot.close(figure)
def four_dim_figures(N=30, beta=1., q=1.):
"""
Four dimensional example. Three dimensional slices are plotted
for illustation.
"""
m = [[0, 1, 1, 1], [1, 0, 1, 1], [1, 1, 0, 1], [0, 0, 0, 1]]
num_types = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
mu = 4. / 3 * 1. / N
incentive = fermi(fitness_landscape, beta=beta, q=q)
edges = incentive_process.multivariate_transitions(N,
incentive,
num_types=num_types,
mu=mu)
d1 = expected_divergence(edges, q_d=0, boundary=True)
d2 = stationary_distribution(edges)
# We need to slice the 4dim dictionary into three-dim slices for plotting.
for slice_index in range(4):
for d in [d1, d2]:
slice_dict = slice_dictionary(d, N, slice_index=3)
figure, tax = ternary.figure(scale=N)
tax.heatmap(slice_dict, style="d")
pyplot.show()
def test_extrema_wf(lim=1e-10):
"""
For small mu, the Wright-Fisher process is minimal in the center.
Test that this happens.
"""
for n, N, mins in [(2, 40, [(20, 20)]), (3, 30, [(10, 10, 10)])]:
mu = 1. / N**3
m = numpy.ones((n, n)) # neutral landscape
fitness_landscape = linear_fitness_landscape(m)
incentive = replicator(fitness_landscape)
edge_func = wright_fisher.multivariate_transitions(N,
incentive,
mu=mu,
num_types=n)
states = list(simplex_generator(N, d=n - 1))
s = stationary_distribution(edge_func,
states=states,
iterations=4 * N,
lim=lim)
s2 = expected_divergence(edge_func, states=states, q_d=0)
assert_equal(find_local_minima(s), set(mins))
er = entropy_rate(edge_func, s, states=states)
assert_greater_equal(er, 0)
def test_extrema_moran_5(lim=1e-16):
"""
Test for extrema of the stationary distribution.
"""
n = 3
N = 60
mu = (3. / 2) * 1. / N
m = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
maxes = set([(20, 20, 20), (0, 0, 60), (0, 60, 0), (60, 0, 0), (30, 0, 30),
(0, 30, 30), (30, 30, 0)])
fitness_landscape = linear_fitness_landscape(m)
incentive = fermi(fitness_landscape, beta=0.1)
edges = incentive_process.multivariate_transitions(N,
incentive,
num_types=n,
mu=mu)
s = stationary_distribution(edges, lim=lim)
s2 = expected_divergence(edges, q_d=0)
flow = inflow_outflow(edges)
# These sets should all correspond
assert_equal(find_local_maxima(s), set(maxes))
assert_equal(find_local_minima(s2), set(maxes))
assert_equal(find_local_minima(flow), set(maxes))
# The minima are pathological
assert_equal(find_local_minima(s), set([(3, 3, 54), (3, 54, 3),
(54, 3, 3)]))
assert_equal(find_local_maxima(s2),
set([(4, 52, 4), (4, 4, 52), (52, 4, 4)]))
assert_equal(find_local_maxima(flow),
set([(1, 58, 1), (1, 1, 58), (58, 1, 1)]))
def test_extrema_moran_5(lim=1e-16):
"""
Test for extrema of the stationary distribution.
"""
n = 3
N = 60
mu = (3./2) * 1./N
m = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
maxes = set([(20, 20, 20), (0, 0, 60), (0, 60, 0), (60, 0, 0),
(30, 0, 30), (0, 30, 30), (30, 30, 0)])
fitness_landscape = linear_fitness_landscape(m)
incentive = fermi(fitness_landscape, beta=0.1)
edges = incentive_process.multivariate_transitions(
N, incentive, num_types=n, mu=mu)
s = stationary_distribution(edges, lim=lim)
s2 = expected_divergence(edges, q_d=0)
flow = inflow_outflow(edges)
# These sets should all correspond
assert_equal(find_local_maxima(s), set(maxes))
assert_equal(find_local_minima(s2), set(maxes))
assert_equal(find_local_minima(flow), set(maxes))
# The minima are pathological
assert_equal(find_local_minima(s),
set([(3, 3, 54), (3, 54, 3), (54, 3, 3)]))
assert_equal(find_local_maxima(s2),
set([(4, 52, 4), (4, 4, 52), (52, 4, 4)]))
assert_equal(find_local_maxima(flow),
set([(1, 58, 1), (1, 1, 58), (58, 1, 1)]))
def bomze_figures(N=60, beta=1, process="incentive", directory=None):
"""
Makes plots of the stationary distribution and expected divergence for each
of the plots in Bomze's classification.
"""
if not directory:
directory = "bomze_paper_figures_%s" % process
ensure_directory(directory)
for i, m in enumerate(bomze_matrices()):
mu = 3./2 * 1./N
fitness_landscape = linear_fitness_landscape(m)
incentive = fermi(fitness_landscape, beta=beta)
edges = incentive_process.multivariate_transitions(N, incentive,
num_types=3, mu=mu)
d1 = stationary_distribution(edges)
filename = os.path.join(directory, "%s_%s_stationary.eps" % (i, N))
figure, tax = ternary.figure(scale = N)
tax.heatmap(d1)
tax.savefig(filename=filename)
pyplot.close(figure)
for q_d in [0., 1.]:
d2 = expected_divergence(edges, q_d=q_d)
filename = os.path.join(directory, "%s_%s_%s.eps" % (i, N, q_d))
figure, tax = ternary.figure(scale = N)
tax.heatmap(d2)
tax.savefig(filename=filename)
pyplot.close(figure)
def two_dim_transitions_figure(N, m, mu=0.01, incentive_func=replicator):
"""
Plot transition entropies and stationary distributions.
"""
n = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
incentive = incentive_func(fitness_landscape)
if not mu:
mu = 1. / N
edges = incentive_process.multivariate_transitions(N,
incentive,
num_types=n,
mu=mu)
s = stationary_distribution(edges, exact=True)
d = edges_to_edge_dict(edges)
# Set up plots
gs = gridspec.GridSpec(3, 1)
ax1 = pyplot.subplot(gs[0, 0])
ax1.set_title("Transition Probabilities")
ups, downs, _ = two_dim_transitions(edges)
xs = range(0, N + 1)
ax1.plot(xs, ups)
ax1.plot(xs, downs)
ax2 = pyplot.subplot(gs[1, 0])
ax2.set_title("Relative Entropy")
divs1 = expected_divergence(edges)
divs2 = expected_divergence(edges, q_d=0)
plot_dictionary(divs1, ax=ax2)
plot_dictionary(divs2, ax=ax2)
ax3 = pyplot.subplot(gs[2, 0])
ax3.set_title("Stationary Distribution")
plot_dictionary(s, ax=ax3)
ax3.set_xlabel("Number of A individuals (i)")
def test_extrema_moran_2(lim=1e-16):
"""
Test for extrema of the stationary distribution.
"""
n = 2
N = 100
mu = 1. / 1000
m = [[1, 2], [3, 1]]
maxes = set([(33, 67), (100,0), (0, 100)])
fitness_landscape = linear_fitness_landscape(m)
incentive = replicator(fitness_landscape)
edges = incentive_process.multivariate_transitions(N, incentive, num_types=n, mu=mu)
s = stationary_distribution(edges, lim=lim)
s2 = expected_divergence(edges, q_d=0)
assert_equal(find_local_maxima(s), set(maxes))
assert_equal(find_local_minima(s2), set(maxes))
def tournament_stationary_3(N, mu=None):
"""
Example for a tournament selection matrix.
"""
if not mu:
mu = 3./2 * 1./N
m = [[1,1,1], [0,1,1], [0,0,1]]
num_types = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
incentive = replicator(fitness_landscape)
edges = incentive_process.multivariate_transitions(N, incentive, num_types=num_types, mu=mu)
s = stationary_distribution(edges)
ternary.heatmap(s, scale=N, scientific=True)
d = expected_divergence(edges, q_d=0)
ternary.heatmap(d, scale=N, scientific=True)
pyplot.show()
def tournament_stationary_3(N, mu=None):
"""
Example for a tournament selection matrix.
"""
if not mu:
mu = 3. / 2 * 1. / N
m = [[1, 1, 1], [0, 1, 1], [0, 0, 1]]
num_types = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
incentive = replicator(fitness_landscape)
edges = incentive_process.multivariate_transitions(N,
incentive,
num_types=num_types,
mu=mu)
s = stationary_distribution(edges)
ternary.heatmap(s, scale=N, scientific=True)
d = expected_divergence(edges, q_d=0)
ternary.heatmap(d, scale=N, scientific=True)
pyplot.show()
def test_extrema_moran_4(lim=1e-16):
"""
Test for extrema of the stationary distribution.
"""
n = 3
N = 60
mu = 3./ (2 * N)
m = [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
maxes = set([(20,20,20)])
mins = set([(0, 0, 60), (0, 60, 0), (60, 0, 0)])
fitness_landscape = linear_fitness_landscape(m)
incentive = logit(fitness_landscape, beta=0.1)
edges = incentive_process.multivariate_transitions(N, incentive, num_types=n, mu=mu)
s = stationary_distribution(edges, lim=lim)
s2 = expected_divergence(edges, q_d=0)
assert_equal(find_local_maxima(s), set(maxes))
assert_equal(find_local_minima(s), set(mins))
assert_equal(find_local_minima(s2), set(maxes))
assert_equal(find_local_maxima(s2), set(mins))
def test_extrema_wf(lim=1e-10):
"""
For small mu, the Wright-Fisher process is minimal in the center.
Test that this happens.
"""
for n, N, mins in [(2, 40, [(20, 20)]), (3, 30, [(10, 10, 10)])]:
mu = 1. / N ** 3
m = numpy.ones((n, n)) # neutral landscape
fitness_landscape = linear_fitness_landscape(m)
incentive = replicator(fitness_landscape)
edge_func = wright_fisher.multivariate_transitions(N, incentive, mu=mu, num_types=n)
states = list(simplex_generator(N, d=n-1))
s = stationary_distribution(edge_func, states=states, iterations=4*N, lim=lim)
s2 = expected_divergence(edge_func, states=states, q_d=0)
assert_equal(find_local_minima(s), set(mins))
er = entropy_rate(edge_func, s, states=states)
assert_greater_equal(er, 0)
def test_extrema_moran_3(lim=1e-12):
"""
Test for extrema of the stationary distribution.
"""
n = 2
N = 100
mu = 6./ 25
m = [[1, 0], [0, 1]]
maxes = set([(38, 62), (62, 38)])
mins = set([(50, 50), (100, 0), (0, 100)])
fitness_landscape = linear_fitness_landscape(m)
incentive = replicator(fitness_landscape)
edges = incentive_process.multivariate_transitions(N, incentive, num_types=n, mu=mu)
s = stationary_distribution(edges, lim=lim)
flow = inflow_outflow(edges)
for q_d in [0, 1]:
s2 = expected_divergence(edges, q_d=1)
assert_equal(find_local_maxima(s), set(maxes))
assert_equal(find_local_minima(s), set(mins))
assert_equal(find_local_minima(s2), set([(50,50), (40, 60), (60, 40)]))
assert_equal(find_local_maxima(flow), set(mins))
def four_dim_figures(N=30, beta=1., q=1.):
"""
Four dimensional example. Three dimensional slices are plotted
for illustation.
"""
m = [[0, 1, 1, 1], [1, 0, 1, 1], [1, 1, 0, 1], [0, 0, 0, 1]]
num_types = len(m[0])
fitness_landscape = linear_fitness_landscape(m)
mu = 4. / 3 * 1. / N
incentive = fermi(fitness_landscape, beta=beta, q=q)
edges = incentive_process.multivariate_transitions(N, incentive, num_types=num_types, mu=mu)
d1 = expected_divergence(edges, q_d=0, boundary=True)
d2 = stationary_distribution(edges)
# We need to slice the 4dim dictionary into three-dim slices for plotting.
for slice_index in range(4):
for d in [d1, d2]:
slice_dict = slice_dictionary(d, N, slice_index=3)
figure, tax = ternary.figure(scale=N)
tax.heatmap(slice_dict, style="d")
pyplot.show()
if __name__ == '__main__':
N = 40
mu = 3./2. * 1./N
m = [[1, 2], [2, 1]]
fitness_landscape = linear_fitness_landscape(m, normalize=False)
incentive = replicator(fitness_landscape)
death_probabilities = even_death(N)
edges = variable_population_transitions(
N, fitness_landscape, death_probabilities, incentive=incentive, mu=mu)
s = stationary_distribution(edges, iterations=10000)
# Print out the states with the highest stationary probabilities
vs = [(v, k) for (k, v) in s.items()]
vs.sort(reverse=True)
print(vs[:10])
# Plot the stationary distribution and expected divergence
figure, tax = ternary.figure(scale=N)
tax.heatmap(s)
d2 = expected_divergence(edges, q_d=0)
d = dict()
for k, v in d2.items():
d[k] = math.sqrt(v)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d)
pyplot.show()
if __name__ == '__main__':
N = 40
mu = 3./2. * 1./N
m = [[1, 2], [2, 1]]
fitness_landscape = linear_fitness_landscape(m, normalize=False)
incentive = replicator(fitness_landscape)
death_probabilities = even_death(N)
edges = variable_population_transitions(N, fitness_landscape, death_probabilities, incentive=incentive, mu=mu)
s = stationary_distribution(edges, iterations=10000)
# Print out the states with the highest stationary probabilities
vs = [(v,k) for (k,v) in s.items()]
vs.sort(reverse=True)
print vs[:10]
# Plot the stationary distributoin and expected divergence
figure, tax = ternary.figure(scale=N)
tax.heatmap(s)
d2 = expected_divergence(edges, q_d=0)
d = dict()
for k, v in d2.items():
d[k] = math.sqrt(v)
figure, tax = ternary.figure(scale=N)
tax.heatmap(d)
pyplot.show()
