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 compute_entropy_rate(N=30, n=2, m=None, incentive_func=None, beta=1., mu=None, exact=False, lim=1e-13, logspace=False): if not m: m = np.ones((n, n)) if not incentive_func: incentive_func = incentives.fermi if not mu: # mu = (n-1.)/n * 1./(N+1) mu = 1. / N fitness_landscape = incentives.linear_fitness_landscape(m) incentive = incentive_func(fitness_landscape, beta=beta, q=1) edges = incentive_process.multivariate_transitions(N, incentive, num_types=n, mu=mu) s = stationary.stationary_distribution(edges, exact=exact, lim=lim, logspace=logspace) e = stationary.entropy_rate(edges, s) return e, s
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 = incentive_process.kl(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 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 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 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 test_incentive_process_k(lim=1e-14): """ Compare stationary distribution computations to known analytic form for neutral landscape for the Moran process. """ for k in [1, 2, 10,]: for n, N in [(2, 20), (2, 50), (3, 10), (3, 20)]: mu = (n-1.)/n * 1./(N+1) m = numpy.ones((n, n)) # neutral landscape fitness_landscape = linear_fitness_landscape(m) incentive = replicator(fitness_landscape) # Neutral landscape is the default edges = incentive_process.k_fold_incentive_transitions( N, incentive, num_types=n, mu=mu, k=k) stationary_1 = stationary_distribution(edges, lim=lim) # Check that the stationary distribution satisfies balance # conditions check_detailed_balance(edges, stationary_1) check_global_balance(edges, stationary_1) check_eigenvalue(edges, stationary_1) # Also check edge_func calculation edges = incentive_process.multivariate_transitions( N, incentive, num_types=n, mu=mu) states = states_from_edges(edges) edge_func = power_transitions(edges, k) stationary_2 = stationary_distribution( edge_func, states=states, lim=lim) for key in stationary_1.keys(): assert_almost_equal( stationary_1[key], stationary_2[key], places=5)
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 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 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 compute_entropy_rate(N=30, n=2, m=None, incentive_func=None, beta=1., mu=None, exact=False, lim=1e-13, logspace=False): if not m: m = np.ones((n, n)) if not incentive_func: incentive_func = incentives.fermi if not mu: # mu = (n-1.)/n * 1./(N+1) mu = 1. / N fitness_landscape = incentives.linear_fitness_landscape(m) incentive = incentive_func(fitness_landscape, beta=beta, q=1) edges = incentive_process.multivariate_transitions( N, incentive, num_types=n, mu=mu) s = stationary.stationary_distribution(edges, exact=exact, lim=lim, logspace=logspace) e = stationary.entropy_rate(edges, s) return e, s
def rps_figures(N=60, q=1, beta=1.): """ Three rock-paper-scissors examples. """ m = [[0, -1, 1], [1, 0, -1], [-1, 1, 0]] num_types = len(m[0]) fitness_landscape = linear_fitness_landscape(m) for i, mu in enumerate([1./math.sqrt(N), 1./N, 1./N**(3./2)]): # Approximate calculation mu = 3/2. * mu incentive = fermi(fitness_landscape, beta=beta, q=q) edges = incentive_process.multivariate_transitions(N, incentive, num_types=num_types, mu=mu) d = stationary_distribution(edges, lim=1e-10) figure, tax = ternary.figure() tax.heatmap(d, scale=N) tax.savefig(filename="rsp_mu_" + str(i) + ".eps", dpi=600)
def moran(N, game_matrix=None, mu=None, incentive_func=replicator, exact=False, logspace=False): """ A convenience function for the Moran process with mutation. Computes the transition probabilities and the stationary distribution. The number of types is determined from the dimensions of the game_matrix. Parameters ---------- N: int The population size game_matrix: list of lists or numpy matrix, None The game matrix of the process, e.g. [[1, 2], [2, 1]] for the two-type Hawk-Dove game. If not specified, the 2-type neutral landscape is used. mu: float, None The mutation rate, if None then `mu` is set to 1 / N incentive_func: function, replicator A function defining the process, e.g. the Moran process, logit, Fermi, etc. Incentives functions are in stationary.processes.incentives exact: bool, False Use the approximate or exact calculation function logspace: bool, False Compute in log-space or not Returns ------- edges, s, er: the list of transitions, the stationary distribution, and the entropy rate. """ if not game_matrix: game_matrix = [[1, 1], [1, 1]] if not mu: mu = 1. / N num_types = len(game_matrix[0]) fitness_landscape = linear_fitness_landscape(game_matrix) incentive = incentive_func(fitness_landscape) edges = incentive_process.multivariate_transitions( N, incentive, num_types=num_types, mu=mu) s = stationary_distribution(edges, exact=exact, logspace=logspace) er = entropy_rate(edges, s) return edges, s, er
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 rps_figures(N=60, q=1, beta=1.): """ Three rock-paper-scissors examples. """ m = [[0, -1, 1], [1, 0, -1], [-1, 1, 0]] num_types = len(m[0]) fitness_landscape = linear_fitness_landscape(m) for i, mu in enumerate([1. / math.sqrt(N), 1. / N, 1. / N**(3. / 2)]): # Approximate calculation mu = 3 / 2. * mu incentive = fermi(fitness_landscape, beta=beta, q=q) edges = incentive_process.multivariate_transitions(N, incentive, num_types=num_types, mu=mu) d = stationary_distribution(edges, lim=1e-10) figure, tax = ternary.figure() tax.heatmap(d, scale=N) tax.savefig(filename="rsp_mu_" + str(i) + ".eps", dpi=600)
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 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 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()