def plot_posterior(data: np.ndarray):
rng_key = random.PRNGKey(0)
# PYMM parameters
T = 20
t = np.arange(T + 1)
for Npoints in (500, 1000, 2000):
y = np.zeros(T+1)
for _ in range(REPEATS):
idx = np.random.choice(len(data), size=Npoints, replace=False)
data_sub = data[idx]
z = sample_posterior(rng_key, multivariate_gaussian_DPMM_isotropic, data_sub, N_SAMPLES, alpha=1, sigma=0, T=T,
# Uncomment the line below to use HMCGibbs
# gibbs_fn=make_multivariate_gaussian_DPMM_gibbs_fn(data_sub), gibbs_sites=['z'],
)
y = compute_n_clusters_distribution(z, T)
y /= REPEATS
plt.plot(t, y, label=f"N={Npoints}")
plt.ylabel(r"$P(T_n=t|X_{1:N})$")
plt.xlabel("$t$")
plt.title("Posterior distribution of the number of clusters")
plt.legend()
plt.show()
def make_synthetic_experiment_sigma(sample_data,
model,
make_gibbs_fn,
explicit_ub=None):
rng_key = random.PRNGKey(0)
# Sampling parameters
Npoints = 1000
data = sample_data(rng_key, Npoints)
# DPMM/PYMM parameters
T = 20
t = np.arange(T + 1)
for sigma in [0, 0.1, .5]:
z = sample_posterior(
rng_key,
model,
data,
N_SAMPLES,
alpha=1,
sigma=sigma,
T=T,
gibbs_fn=make_gibbs_fn(data) if USE_GIBBS else None,
gibbs_sites=['z'] if USE_GIBBS else None,
)
cluster_count = compute_n_clusters_distribution(z, T)
plt.plot(t, cluster_count, label=r"$\sigma={}$".format(sigma))
color = plt.gca().lines[-1].get_color()
# Upper bound
if explicit_ub is not None:
upper_bound = explicit_ub(data, t, params_PY=(1, sigma))
plt.plot(t[1:],
upper_bound[1:],
label=r"Upper bound $\sigma={}$".format(sigma),
color=color,
linestyle="dotted",
lw=1)
# Prior
prior = compute_PY_prior(1, sigma, [Npoints])[0]
plt.plot(t,
prior[:T + 1],
label=r"Prior $\sigma={}$".format(sigma),
color=color,
linestyle="dashed",
lw=1)
# plt.axvline(alpha*np.log(Npoints), color=color, lw=1, linestyle="dashed")
plt.legend()
plt.title(r"Impact of $\sigma$")
plt.ylabel(r"$P(T_n=t|X_{1:N})$")
plt.xlabel("Number of clusters")
plt.show()
def plot_clusters(data: np.ndarray):
T = 10
t = np.arange(T + 1)
Npoints = 100
rng_key = random.PRNGKey(0)
x = data[:Npoints]
z = sample_posterior(rng_key, multivariate_gaussian_DPMM, x, Nsamples=1, T=T, alpha=1)
pca = PCA(n_components=2)
x_pca = np.ascontiguousarray(pca.fit_transform(x))
for c in np.unique(z):
xc = x_pca[z == c]
plt.scatter(xc[:, 0], xc[:, 1], alpha=.5)
plt.show()
def make_synthetic_experiment(sample_data,
model,
make_gibbs_fn,
explicit_ub=None):
"""
Template for small synthetic experiments with varying size of observed data.
Args:
sample_data: sampling function
model: model function
"""
rng_key = random.PRNGKey(0)
# Sampling parameters
n_values = [100, 1000, 10000]
# DPMM/PYMM parameters
T = 20 # max number of component in the truncated stick breaking representation
t = np.arange(T + 1)
alpha = 1
sigma = 0
# Plotting parameters
fig, (ax0, ax1) = plt.subplots(1, 2)
priors = compute_PY_prior(alpha, sigma, n_values)
for Npoints, prior in zip(n_values, priors):
cluster_count = np.zeros(T + 1) # cluster count histogram
upper_bound = np.zeros(T + 1)
cluster_size = np.zeros(Npoints + 1)
# Repeat the experiment
for _ in range(REPEATS):
data = sample_data(rng_key, Npoints)
z = sample_posterior(
rng_key,
model,
data,
N_SAMPLES,
T=T,
alpha=1,
gibbs_fn=make_gibbs_fn(data) if USE_GIBBS else None,
gibbs_sites=['z'] if USE_GIBBS else None,
)
cluster_count += compute_n_clusters_distribution(z, T)
if explicit_ub is not None:
upper_bound += explicit_ub(data, t, params_PY=(alpha, sigma))
cluster_size += compute_cluster_size_distribution(z)
cluster_count /= REPEATS
cluster_size /= REPEATS
# Plot cluster count histograms (ax0)
ax0.plot(t, cluster_count, label=f"N={Npoints}")
color = ax0.lines[-1].get_color()
ax0.plot(t,
prior[:T + 1],
label=f"Prior N={Npoints}",
color=color,
linestyle='dashed',
lw=1)
if explicit_ub is not None:
upper_bound /= REPEATS
ax0.plot(t[1:],
upper_bound[1:],
label=f"Upper bound N={Npoints}",
color=color,
linestyle='dotted',
lw=1)
# Plot cluster size histograms (ax1)
bins = np.linspace(0, 1, 10, endpoint=True)
frac = np.arange(0, Npoints + 1) / Npoints
# TODO : use an actual histogram ?
# Overlaying histograms doesn't really look good.
hist, edges = np.histogram(frac,
bins,
density=True,
weights=cluster_size)
ax1.plot(0.5 * (edges[1:] + edges[:-1]),
hist,
color=color,
label=f"N={Npoints}")
ax0.axhline(y=1, color='black', linewidth=0.3, linestyle='dotted')
ax0.set(title=r"Number of clusters",
xlabel="$t$",
ylabel=r"$P(T_n=t|X_{1:N})$")
ax0.legend()
ax1.set(xlabel="Fraction of total size", title="Size of clusters")
ax1.legend()
plt.show()