def step_size_sigma(title,
samples,
time,
σ,
stepsize,
post,
plot,
legend_pos=None):
nplot = len(samples)
nsample = len(time)
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_xlabel("Time")
axis.set_ylabel(r"$σ$")
axis.set_title(title)
axis.set_xlim([10.0, nsample])
axis.semilogx(time,
numpy.full((len(time)), σ),
label="Target σ",
color="#000000")
for i in range(nplot):
axis.semilogx(time,
stats.cumsigma(samples[i]),
label=f"stepsize={format(stepsize[i], '2.2f')}")
if legend_pos is None:
axis.legend()
else:
axis.legend(bbox_to_anchor=legend_pos)
config.save_post_asset(figure, post, plot)
def pdf_samples_contour(pdf, p, q, xrange, yrange, contour_values, labels,
title, file):
npts = 500
fxy, x, y = grid_pdf(pdf, xrange, yrange, npts)
bins = [
numpy.linspace(xrange[0], xrange[1], 100),
numpy.linspace(yrange[0], yrange[1], 100)
]
figure, axis = pyplot.subplots(figsize=(11, 9))
axis.set_xlabel(labels[0])
axis.set_ylabel(labels[1])
axis.set_title(title)
hist, _, _, image = axis.hist2d(p,
q,
normed=True,
bins=bins,
cmap=config.alternate_color_map)
contour = axis.contour(x,
y,
fxy,
contour_values,
cmap=config.alternate_contour_color_map)
axis.clabel(contour,
contour.levels[::2],
fmt="%.3f",
inline=True,
fontsize=15)
figure.colorbar(image)
config.save_post_asset(figure, "hamiltonian_monte_carlo", file)
def integration_error_plot(p0, q0, nsteps, ε, plot_file):
contour_value = p0**2+ q0**2
p_leapfrog, q_leapfrog = leapfrog(p0, q0, nsteps, ε)
p_euler_cromer, q_euler_cromer = euler_cromer(p0, q0, nsteps, ε)
p_euler, q_euler = euler(p0, q0, nsteps, ε)
p_momentum_verlet, q_momentum_verlet = momentum_verlet(p0, q0, nsteps, ε)
error_leapfrog = 100.0*numpy.abs(numpy.sqrt(p_leapfrog[1:-1]**2 + q_leapfrog[1:-1]**2) - numpy.sqrt(contour_value))/numpy.sqrt(contour_value)
error_euler_cromer = 100.0*numpy.abs(numpy.sqrt(p_euler_cromer[1:-1]**2 + q_euler_cromer[1:-1]**2) - numpy.sqrt(2.0))/numpy.sqrt(2.0)
error_euler = 100.0*numpy.abs(numpy.sqrt(p_euler[1:-1]**2 + q_euler[1:-1]**2) - numpy.sqrt(2.0))/numpy.sqrt(2.0)
error_momentum_verlet = 100.0*numpy.abs(numpy.sqrt(p_momentum_verlet[1:-1]**2 + q_momentum_verlet[1:-1]**2) - numpy.sqrt(2.0))/numpy.sqrt(2.0)
t_plot = numpy.linspace(0.0, nsteps*ε, nsteps+1)
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_xlabel("t")
axis.set_ylabel("% Error")
axis.set_title(f"Hamiltion's Equations Integration Error: Δt={ε}, nsteps={nsteps}")
axis.set_xlim([0.0, t])
axis.plot(t_plot[1:-1], error_leapfrog, label="Leapfrog")
axis.plot(t_plot[1:-1], error_euler_cromer, label="Euler Cromer")
axis.plot(t_plot[1:-1], error_euler, label="Euler")
axis.plot(t_plot[1:-1], error_momentum_verlet, label="Momentum Verlet")
config.save_post_asset(figure, "hamiltonian_monte_carlo", plot_file)
axis.legend(bbox_to_anchor=(0.35, 0.9))
def parametric_contour_plot(μ1, μ2, σ1, σ2, γ, legend_box, contour_values, plot_name):
npts = 500
θ = numpy.linspace(0.0, 2.0 * numpy.pi, npts)
c = [pdf_contour_constant(μ1, μ2, σ1, σ2, γ, v) for v in contour_values]
f = pdf_parametric_contour(μ1, μ2, σ1, σ2, γ)
figure, axis = pyplot.subplots(figsize=(8, 8))
axis.set_xlabel(r"$u$")
axis.set_ylabel(r"$v$")
if (σ1 > σ2):
axis.set_xlim([-3.2*σ1, 3.2*σ1])
axis.set_ylim([-3.2*σ1, 3.2*σ1])
elif (σ2 > σ1):
axis.set_xlim([-3.2*σ2, 3.2*σ2])
axis.set_ylim([-3.2*σ2, 3.2*σ2])
else:
axis.set_xlim([-3.2*σ1, 3.2*σ1])
axis.set_ylim([-3.2*σ2, 3.2*σ2])
title = f"Bivariate Normal Distribution: γ={format(γ, '2.1f')}, " + \
r"$σ_u$=" + f"{format(σ1, '2.1f')}, " + r"$σ_v$=" + \
f"{format(σ2, '2.1f')}"
axis.set_title(title)
for n in range(len(c)):
y1, y2 = f(θ, c[n])
axis.plot(y1, y2, label=f"= {format(contour_values[n], '2.3f')}")
axis.legend(bbox_to_anchor=legend_box)
config.save_post_asset(figure, "bivariate_normal_distribution", plot_name)
def contour_plot_sigma_correlation_scan(μ1, μ2, σ1, σ2, γ, contour_value, legend, plot_name):
npts = 500
θ = numpy.linspace(0.0, 2.0 * numpy.pi, npts)
figure, axis = pyplot.subplots(figsize=(8, 8))
axis.set_xlabel(r"$u$")
axis.set_ylabel(r"$v$")
σ = numpy.amax([σ1, σ2])
axis.set_xlim([-2.5*σ, 2.5*σ])
axis.set_ylim([-2.5*σ, 2.5*σ])
title = f"Bivariate Normal Distribution: " + \
r"$σ_u$=" + f"{format(σ1, '2.1f')}, " + r"$σ_v$=" + \
f"{format(σ2, '2.1f')}, K={format(contour_value, '2.2f')}"
axis.set_title(title)
for i in range(len(γ)):
c = pdf_contour_constant(μ1, μ2, σ1, σ2, γ[i], contour_value)
f = pdf_parametric_contour(μ1, μ2, σ1, σ2, γ[i])
y1, y2 = f(θ, c)
axis.plot(y1, y2, label=f"γ = {format(γ[i], '2.2f')}")
x1 = numpy.linspace(-2.5*σ, 2.5*σ, npts)
slope = γ[-1]*σ2/σ1/abs(γ[-1])
axis.plot(x1, slope*x1, zorder = 6, color="#003B6F", alpha=0.1)
axis.legend(bbox_to_anchor=legend)
config.save_post_asset(figure, "bivariate_normal_distribution", plot_name)
def verification_time_series(title, p, q, time, ylim, file):
nplots = 2
figure, axis = pyplot.subplots(nrows=nplots, ncols=1, sharex=True, figsize=(10, 4*nplots))
axis[0].set_title(title)
axis[-1].set_xlabel("Time")
q1 = [q[0](t) for t in time]
q2 = [q[1](t) for t in time]
axis[0].set_xlim([time[0], time[-1]])
axis[0].set_ylim(ylim)
axis[0].set_ylabel("q")
axis[0].plot(time, q1, label=r"$q_1$")
axis[0].plot(time, q2, label=r"$q_2$")
axis[0].legend()
p1 = [p[0](t) for t in time]
p2 = [p[1](t) for t in time]
axis[1].set_xlim([time[0], time[-1]])
axis[1].set_ylim(ylim)
axis[1].set_ylabel("p")
axis[1].plot(time, p1, label=r"$p_1$")
axis[1].plot(time, p2, label=r"$p_2$")
axis[1].legend()
config.save_post_asset(figure, "hamiltonian_monte_carlo", file)
def time_series(title, samples, time, ylim, plot):
figure, axis = pyplot.subplots(figsize=(12, 4))
axis.set_title(title)
axis.set_xlabel("Time")
axis.set_xlim([time[0], time[-1]])
axis.set_ylim(ylim)
axis.plot(time, samples, lw="1")
config.save_post_asset(figure, "hamiltonian_monte_carlo", plot)
def univariate_pdf_plot(pdf, x, x_title, title, file):
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_xlabel(x_title)
axis.set_ylabel("PDF")
axis.set_xlim([x[0], x[-1]])
axis.set_title(title)
axis.plot(x, [pdf(j) for j in x])
config.save_post_asset(figure, "hamiltonian_monte_carlo", file)
def time_series(title, samples, time, ylim, post, plot):
nplots = len(samples)
figure, axis = pyplot.subplots(figsize=(12, 4))
axis.set_title(title)
axis.set_xlabel("Time")
axis.set_xlim([time[0], time[-1] + 1])
axis.set_ylim(ylim)
axis.plot(time, samples, lw="1")
config.save_post_asset(figure, post, plot)
def autocor(title, samples, max_lag, plot):
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_title(title)
axis.set_ylabel(r"$\gamma_{\tau}$")
axis.set_xlabel("Time Lag (τ)")
axis.set_xlim([0, max_lag])
axis.set_ylim([-0.05, 1.0])
ac = stats.autocorrelate(samples)
axis.plot(range(max_lag), numpy.real(ac[:max_lag]))
config.save_post_asset(figure, "hamiltonian_monte_carlo", plot)
def canonical_distribution_contour_plot(kinetic_energy, potential_energy, contour_values, title, plot_name):
npts = 500
x1_grid, x2_grid, f_x1_x2 = canonical_distribution_mesh(kinetic_energy, potential_energy, npts)
figure, axis = pyplot.subplots(figsize=(8, 8))
axis.set_xlabel(r"$q$")
axis.set_ylabel(r"$p$")
axis.set_xlim([-3.2, 3.2])
axis.set_ylim([-3.2, 3.2])
axis.set_title(title)
contour = axis.contour(x1_grid, x2_grid, f_x1_x2, contour_values, cmap=config.contour_color_map)
axis.clabel(contour, contour.levels[::2], fmt="%.3f", inline=True, fontsize=15)
config.save_post_asset(figure, "hamiltonian_monte_carlo", plot_name)
def cumulative_standard_deviation(title, samples, time, σ, ylim, file):
nsample = len(time)
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_xlabel("Time")
axis.set_ylabel(r"$σ$")
axis.set_title(title)
axis.set_ylim(ylim)
axis.set_xlim([10.0, nsample])
axis.semilogx(time,
numpy.full((len(time)), σ),
label="Target σ",
color="#000000")
axis.semilogx(time, stats.cumsigma(samples))
config.save_post_asset(figure, "hamiltonian_monte_carlo", file)
def plot_sampled_pdf(title, f, samples, plot_name):
figure, axis = pyplot.subplots(figsize=(10, 6))
axis.set_prop_cycle(config.distribution_sample_cycler)
axis.set_title(title)
axis.set_ylabel("PDF")
axis.set_xlabel(r"$X$")
axis.set_ylim([0, 2.0])
axis.set_yticks([0.2, 0.6, 1.0, 1.4, 1.8])
axis.set_xlim([0.0, 1.6])
axis.hist(samples, 30, density=True, rwidth=0.8, label=f"Samples", zorder=5)
x = numpy.linspace(0.0, 1.6, 500)
axis.plot(x, f(x), label=f"Target PDF", zorder=7)
axis.legend(bbox_to_anchor=(0.3, 0.85))
config.save_post_asset(figure, "rejection_sampling", plot_name)
def acceptance(title, x, y, xlim, example_idx, post, plot):
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_xlabel("Step Size")
axis.set_ylabel("Acceptance %")
axis.set_title(title)
axis.set_xlim(xlim)
axis.set_ylim([0.7, 200.0])
axis.set_prop_cycle(config.alternate_cycler)
axis.loglog(x,
y,
zorder=5,
marker='o',
markersize=15.0,
linestyle="None",
markeredgewidth=1.0)
axis.loglog(
x[example_idx[0]],
y[example_idx[0]],
zorder=5,
marker='o',
markersize=15.0,
linestyle="None",
markeredgewidth=1.0,
label=
f" Small Step Size: ({format(x[example_idx[0]], '.2f')}, {format(y[example_idx[0]], '2.0f')}%)"
)
axis.loglog(
x[example_idx[1]],
y[example_idx[1]],
zorder=5,
marker='o',
markersize=15.0,
linestyle="None",
markeredgewidth=1.0,
label=
f" Best Step Size: ({format(x[example_idx[1]], '.2f')}, {format(y[example_idx[1]], '2.0f')}%)"
)
axis.loglog(
x[example_idx[2]],
y[example_idx[2]],
zorder=5,
marker='o',
markersize=15.0,
linestyle="None",
markeredgewidth=1.0,
label=
f" Large Step Size: ({format(x[example_idx[2]], '.2f')}, {format(y[example_idx[2]], '2.0f')}%)"
)
axis.legend(bbox_to_anchor=(0.55, 0.6))
config.save_post_asset(figure, post, plot)
def step_size_autocor(title, samples, stepsize, npts, post, plot):
nplot = len(samples)
figure, axis = pyplot.subplots(figsize=(12, 9))
axis.set_title(title)
axis.set_ylabel(r"$\gamma_{\tau}$")
axis.set_xlabel("Time Lag (τ))")
axis.set_xlim([0, npts])
for i in range(nplot):
ac = stats.autocorrelate(samples[i])
axis.plot(range(npts),
numpy.real(ac[:npts]),
label=f"stepsize={format(stepsize[i], '2.2f')}")
axis.legend(bbox_to_anchor=(0.7, 0.6))
config.save_post_asset(figure, post, plot)
def relaxation_plot(πt, nsteps, plot_name):
steps = [i for i in range(0, nsteps + 1)]
figure, axis = pyplot.subplots(figsize=(10, 6))
axis.set_xlabel("Time")
axis.set_ylabel("Probability")
axis.set_title(r"$\pi_t$ Relaxation to Equilibrium")
axis.set_ylim([0, 0.55])
axis.set_yticks([0.1, 0.2, 0.3, 0.4, 0.5])
axis.set_xlim([0, nsteps])
axis.plot(steps, [πt[i][0, 0] for i in steps], label=f"0", lw="3", zorder=10)
axis.plot(steps, [πt[i][0, 1] for i in steps], label=f"1", lw="3", zorder=10)
axis.plot(steps, [πt[i][0, 2] for i in steps], label=f"2", lw="3", zorder=10)
axis.plot(steps, [πt[i][0, 3] for i in steps], label=f"3", lw="3", zorder=10)
axis.legend(bbox_to_anchor=(0.8, 0.15))
config.save_post_asset(figure, "discrete_state_markov_chain_equilibrium", plot_name)
def mean_convergence(title, samples, μ, plot_name):
nsamples = len(samples)
time = range(nsamples)
figure, axis = pyplot.subplots(figsize=(10, 6))
axis.set_xlabel("Sample Number")
axis.set_ylabel("μ")
axis.set_title(title)
axis.set_xlim([1.0, nsamples])
axis.set_ylim( [0.5, 1.5])
cmean = stats.cummean(samples)
μs = numpy.full(len(samples), μ)
axis.semilogx(time, μs, label="Target μ")
axis.semilogx(time, cmean, label="Sampled μ")
axis.legend(bbox_to_anchor=([0.9, 0.9]))
config.save_post_asset(figure, "rejection_sampling", plot_name)
def distribution_samples(x, y, xrange, yrange, labels, title, file):
bins = [
numpy.linspace(xrange[0], xrange[1], 100),
numpy.linspace(yrange[0], yrange[1], 100)
]
figure, axis = pyplot.subplots(figsize=(11, 9))
axis.set_xlabel(labels[0])
axis.set_ylabel(labels[1])
axis.set_title(title)
hist, _, _, image = axis.hist2d(x,
y,
normed=True,
bins=bins,
cmap=config.alternate_color_map)
figure.colorbar(image)
config.save_post_asset(figure, "hamiltonian_monte_carlo", file)
def sigma_convergence(title, samples, σ, plot_name):
nsamples = len(samples)
time = range(nsamples)
figure, axis = pyplot.subplots(figsize=(10, 6))
axis.set_xlabel("Sample Number")
axis.set_ylabel("σ")
axis.set_title(title)
axis.set_xlim([1.0, nsamples])
axis.set_ylim([0.0, 0.5])
axis.set_yticks([0.1, 0.2, 0.3, 0.4])
csigma = stats.cumsigma(samples)
σs = numpy.full(len(samples), σ)
axis.semilogx(time, σs, label="Target σ")
axis.semilogx(time, csigma, label="Sampled σ")
axis.legend(bbox_to_anchor=([0.95, 0.9]))
config.save_post_asset(figure, "rejection_sampling", plot_name)
def pdf_samples(title, pdf, samples, post, plot, xrange=None, ylimit=None):
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_xlabel(r"$X$")
axis.set_title(title)
axis.set_prop_cycle(config.distribution_sample_cycler)
_, bins, _ = axis.hist(samples, 50, rwidth=0.8, density=True, label=f"Samples", zorder=5)
if xrange is None:
delta = (bins[-1] - bins[0]) / 500.0
xrange = numpy.arange(bins[0], bins[-1], delta)
sample_distribution = [pdf(val) for val in xrange]
axis.set_xlim([xrange[0], xrange[-1]])
if ylimit is not None:
axis.set_ylim(ylimit)
axis.plot(xrange, sample_distribution, label=f"Target PDF", zorder=6)
axis.legend(bbox_to_anchor=(0.75, 0.9))
config.save_post_asset(figure, post, plot)
def hamiltons_equations_integration_plot(kinetic_energy, potential_energy, contour_value, p, q, title, legend_anchor, plot_name):
npts = 500
x1_grid, x2_grid, f_x1_x2 = canonical_distribution_mesh(kinetic_energy, potential_energy, npts)
figure, axis = pyplot.subplots(figsize=(8, 8))
axis.set_xlabel(r"$q$")
axis.set_ylabel(r"$p$")
axis.set_xlim([-3.2, 3.2])
axis.set_ylim([-3.2, 3.2])
axis.set_title(title)
contour = axis.contour(x1_grid, x2_grid, f_x1_x2, [contour_value], cmap=config.contour_color_map, alpha=0.3)
axis.clabel(contour, contour.levels[::2], fmt="%.3f", inline=True, fontsize=15)
axis.plot(q, p, lw=1, color="#320075")
axis.plot(q[0], p[0], marker='o', color="#FF9500", markersize=13.0, label="Start")
axis.plot(q[-1], p[-1], marker='o', color="#320075", markersize=13.0, label="End")
axis.legend(bbox_to_anchor=legend_anchor)
config.save_post_asset(figure, "hamiltonian_monte_carlo", plot_name)
def contour_axis_plot_sigma_scan(μ1, μ2, σ1, σ2, γ, contour_value, legend, plot_name):
npts = 500
θ = numpy.linspace(0.0, 2.0 * numpy.pi, npts)
figure, axis = pyplot.subplots(figsize=(8, 8))
title = f"Bivariate Normal Distribution: " + \
r"$σ_u$=" + f"{format(σ1, '2.1f')}, " + r"$\gamma$=" + \
f"{format(γ, '2.1f')}, K={format(contour_value, '2.2f')}"
axis.set_title(title)
axis.set_xlabel(r"$\theta$")
axis.set_ylabel(r"$r(\theta)$")
for i in range(len(σ2)):
c = pdf_contour_constant(μ1, μ2, σ1, σ2[i], γ, contour_value)
rf = r(μ1, μ2, σ1, σ2[i], γ)
axis.plot(θ, rf(θ, c), label=f"$σ_v$ = {format(σ2[i], '2.2f')}")
axis.legend(bbox_to_anchor=legend)
config.save_post_asset(figure, "bivariate_normal_distribution", plot_name)
def transform_plot(μ1, μ2, σ1, σ2, γ, xrange, yrange, xnudge, ynudge, unudge, anudge, legend, plot_name):
npts = 500
figure, axis = pyplot.subplots(figsize=(9, 9))
axis.set_xlabel("x")
axis.set_ylabel("y")
axis.set_ylim(yrange)
axis.set_xlim(xrange)
title = f"Bivariate Normal Transformation: γ={format(γ, '2.1f')}, " + \
r"$σ_u$=" + f"{format(σ1, '2.1f')}, " + r"$σ_v$=" + \
f"{format(σ2, '2.1f')}"
axis.set_title(title)
ar = (yrange[1] - yrange[0]) / (xrange[1] - xrange[0])
θ = (180.0 / numpy.pi) * numpy.arctan(-γ / (ar * numpy.sqrt(1-γ**2))) + anudge
c = [-4.0, -2.0, 0.0, 2.0, 4.0]
x1 = numpy.zeros(len(c))
for i in range(len(c)):
y2 = numpy.linspace(3.0 * yrange[0], 3.0 * yrange[1], npts)
transform_y1 = pdf_transform_y1_constant(μ1, μ2, σ1, σ2, γ, c[i])
x1_y1, x2_y1 = transform_y1(y2)
x1[i] = x1_y1[0]
axis.text(x1[i] - 0.35, unudge[i], f"u={format(c[i], '2.0f')}", fontsize=18, rotation=90.0)
if i == 0:
axis.plot(x1_y1, x2_y1, color="#0067C4", label=r"Constant $u$")
else:
axis.plot(x1_y1, x2_y1, color="#0067C4")
for i in range(len(c)):
y1 = numpy.linspace(3.0 * yrange[0], 3.0 * yrange[1], npts)
transform_y2 = pdf_transform_y2_constant(μ1, μ2, σ1, σ2, γ, c[i])
x1_y2, x2_y2 = transform_y2(y1)
if i == len(c) - 1:
xoffset = x1[i] - 2.0 * xnudge
else:
xoffset = x1[i] + xnudge
x2 = (γ * xoffset - (c[i] - μ2) / σ2) / numpy.sqrt(1.0 - γ**2)
axis.text(xoffset, x2 + ynudge[i], f"v={format(c[i], '2.0f')}", fontsize=18, rotation=θ)
if i == 0:
axis.plot(x1_y2, x2_y2, color="#FF9500", label=r"Constant $v$")
else:
axis.plot(x1_y2, x2_y2, color="#FF9500")
axis.legend(bbox_to_anchor=legend)
config.save_post_asset(figure, "bivariate_normal_distribution", plot_name)
def acceptance_plot(title, h, y_samples, ymax, xmax, legend_loc, plot_name):
nsamples = len(y_samples)
x_values = numpy.linspace(0.0, xmax, 500)
samples, u, accepted_mask = rejection_sample(h, y_samples, ymax)
rejected_mask = numpy.logical_not(accepted_mask)
efficiency = 100.0 * (len(samples) / nsamples)
figure, axis = pyplot.subplots(figsize=(10, 6))
axis.set_title(title + f", Accepted {format(efficiency, '2.0f')}%")
axis.set_ylim([0, 1.05])
axis.set_yticks([0.2, 0.4, 0.6, 0.8, 1.0])
axis.set_ylabel(r"$h(Y)/c$")
axis.set_xlabel(r"$Y$")
axis.set_xlim([0.0, 1.6])
axis.plot(x_values, h(x_values) / ymax, zorder=5, lw=3, color="#5600C9")
axis.scatter(y_samples[accepted_mask], u[accepted_mask], label="Accepted Samples", alpha=0.5, zorder=5)
axis.scatter(y_samples[rejected_mask], u[rejected_mask], label="Rejected Samples", alpha=0.4, zorder=5)
axis.legend(bbox_to_anchor=legend_loc, framealpha=0.9, edgecolor="#DDDDDD")
config.save_post_asset(figure, "rejection_sampling", plot_name)
def acceptance_plot(title, x, y, idx, text_pos, best_idx, xlim, plot):
slope, y0, r2 = acceptance_regress(x[idx:], y[idx:])
xfit = numpy.linspace(-1.0, 3.0, 100)
bbox = dict(boxstyle='square,pad=1', facecolor="#FFFFFF", edgecolor="white")
figure, axis = pyplot.subplots(figsize=(10, 7))
axis.set_xlabel("Relative Step Size")
axis.set_ylabel("Acceptance %")
axis.set_title(title)
axis.set_xlim(xlim)
axis.set_prop_cycle(config.alternate_cycler)
axis.set_ylim([0.05, 200.0])
axis.loglog(x, y, zorder=6, marker='o', markersize=15.0, linestyle="None", markeredgewidth=1.0, label="Simulations")
axis.loglog(x[best_idx], y[best_idx], zorder=6, marker='o', markersize=15.0, linestyle="None", markeredgewidth=1.0, label=f"({format(x[best_idx], '.3f')}, {format(y[best_idx], '2.0f')}%)")
axis.loglog(10**xfit, acceptance_fit(xfit, slope, y0), zorder=5, color="#320075", label="Fit")
axis.loglog(numpy.linspace(xlim[0], xlim[1], 100), numpy.full((100), 100.0), zorder=5, color="#320075")
axis.text(text_pos[0], text_pos[1], f"slope={format(slope, '2.2f')}", fontsize=16, bbox=bbox)
axis.legend(bbox_to_anchor=(0.5, 0.5))
config.save_post_asset(figure, "metropolis_hastings_sampling", plot)
def pdf_contour_plot(pdf, contour_values, xrange, yrange, labels, title,
plot_name):
npts = 500
f_x1_x2, x1_grid, x2_grid = grid_pdf(pdf, xrange, yrange, npts)
figure, axis = pyplot.subplots(figsize=(8, 8))
axis.set_xlabel(labels[0])
axis.set_ylabel(labels[1])
axis.set_title(title)
contour = axis.contour(x1_grid,
x2_grid,
f_x1_x2,
contour_values,
cmap=config.contour_color_map)
axis.clabel(contour,
contour.levels[::2],
fmt="%.3f",
inline=True,
fontsize=15)
config.save_post_asset(figure, "hamiltonian_monte_carlo", plot_name)
def phase_space_plot(p, q, title, labels, legend_anchor, plot_name):
figure, axis = pyplot.subplots(figsize=(9, 9))
axis.set_xlabel(labels[0])
axis.set_ylabel(labels[1])
axis.set_title(title)
axis.plot(q, p, lw=1, color="#320075")
axis.plot(q[0],
p[0],
marker='o',
color="#FF9500",
markersize=13.0,
label="Start")
axis.plot(q[-1],
p[-1],
marker='o',
color="#320075",
markersize=13.0,
label="End")
axis.legend(bbox_to_anchor=legend_anchor)
config.save_post_asset(figure, "hamiltonian_monte_carlo", plot_name)
def surface_plot(μ1, μ2, σ1, σ2, γ, zticks, plot_name):
x1_grid, x2_grid, f_x1_x2 = pdf_mesh(μ1, μ2, σ1, σ2, γ)
figure, axis = pyplot.subplots(figsize=(10, 10))
title = f"Bivariate Normal Distribution: γ={format(γ, '2.1f')}, " + \
r"$σ_u$=" + f"{format(σ1, '2.1f')}, " + r"$σ_v$=" + \
f"{format(σ2, '2.1f')}"
axis.set_title(title)
axis.set_yticklabels([])
axis.set_xticklabels([])
axis_z = figure.add_subplot(111, projection='3d')
axis_z.set_xlabel(r"$u$")
axis_z.set_ylabel(r"$v$")
axis_z.set_zticks(zticks)
axis_z.set_xticks([-σ1*3.0, -σ1*2.0, -σ1, 0.0, σ1, σ1*2.0, σ1*3.0])
axis_z.set_yticks([-σ2*3.0, -σ2*2.0, -σ2, 0.0, σ2, σ2*2.0, σ2*3.0])
axis_z.plot_wireframe(x1_grid, x2_grid, f_x1_x2, rstride=25, cstride=25)
axis_z.elev = 55.0
axis_z.azim = 45.0
config.save_post_asset(figure, "bivariate_normal_distribution", plot_name)
def phase_space_time_series(title, PQ, time, ylim, file):
nplots = 2
figure, axis = pyplot.subplots(nrows=nplots, ncols=1, sharex=True, figsize=(10, 4*nplots))
axis[0].set_title(title)
axis[-1].set_xlabel("Time")
axis[0].set_xlim([time[0], time[-1]])
axis[0].set_ylim(ylim)
axis[0].set_ylabel("q")
axis[0].plot(time, numpy.real(PQ[:, 2, 0]), label=r"$q_1$")
axis[0].plot(time, numpy.real(PQ[:, 3, 0]), label=r"$q_2$")
axis[0].legend()
axis[1].set_xlim([time[0], time[-1]])
axis[1].set_ylim(ylim)
axis[1].set_ylabel("p")
axis[1].plot(time, numpy.real(PQ[:, 0, 0]), label=r"$p_1$")
axis[1].plot(time, numpy.real(PQ[:, 1, 0]), label=r"$p_2$")
axis[1].legend()
config.save_post_asset(figure, "hamiltonian_monte_carlo", file)
def contour_plot_sigma1_scan(μ1, μ2, σ1, σ2, γ, contour_value, legend, plot_name):
npts = 500
θ = numpy.linspace(0.0, 2.0 * numpy.pi, npts)
figure, axis = pyplot.subplots(figsize=(8, 8))
axis.set_xlabel(r"$u$")
axis.set_ylabel(r"$v$")
axis.set_xlim([-5.0, 5.0])
axis.set_ylim([-5.0, 5.0])
title = f"Bivariate Normal Distribution: γ={format(γ, '2.1f')}, " + \
r"$σ_v$=" + f"{format(σ2, '2.1f')}, " + r"$K$=" + \
f"{format(contour_value, '2.2f')}"
axis.set_title(title)
for i in range(len(σ1)):
c = pdf_contour_constant(μ1, μ2, σ1[i], σ2, γ, contour_value)
f = pdf_parametric_contour(μ1, μ2, σ1[i], σ2, γ)
y1, y2 = f(θ, c)
axis.plot(y1, y2, label=r"$σ_u$" + f" = {format(σ1[i], '2.3f')}")
axis.legend(bbox_to_anchor=legend)
config.save_post_asset(figure, "bivariate_normal_distribution", plot_name)
