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plot_expectation_maximization_ball.py
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plot_expectation_maximization_ball.py
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"""Apply Expectation Maximization on manifolds and plots the results.
Random data is generated in separate regions of the
manifold. Then Expectation Maximization deduces a Gaussian Mixture Model
that best fits the random data. For the moment
the example works on the Poincaré Ball hyperbolic space.
"""
import os
import matplotlib.pyplot as plt
import mpl_toolkits.mplot3d.art3d as art3d
from matplotlib.patches import Circle
import geomstats.backend as gs
from geomstats.geometry.poincare_ball import PoincareBall
from geomstats.learning.expectation_maximization import (
GaussianMixtureModel,
RiemannianEM,
)
DEFAULT_PLOT_PRECISION = 100
def plot_gaussian_mixture_distribution(
space,
data,
mixture_coefficients,
means,
variances,
plot_precision=DEFAULT_PLOT_PRECISION,
save_path="",
):
"""Plot Gaussian Mixture Model."""
x_axis_samples = gs.linspace(-1, 1, plot_precision)
y_axis_samples = gs.linspace(-1, 1, plot_precision)
x_axis_samples, y_axis_samples = gs.meshgrid(x_axis_samples, y_axis_samples)
z_axis_samples = gs.zeros((plot_precision, plot_precision))
for z_index, _ in enumerate(z_axis_samples):
x_y_plane_mesh = gs.concatenate(
(
gs.expand_dims(x_axis_samples[z_index], -1),
gs.expand_dims(y_axis_samples[z_index], -1),
),
axis=-1,
)
mesh_probabilities = GaussianMixtureModel(space, means, variances).weighted_pdf(
mixture_coefficients, x_y_plane_mesh
)
z_axis_samples[z_index] = mesh_probabilities.sum(-1)
fig = plt.figure(
"Learned Gaussian Mixture Model "
"via Expectation Maximization on Poincaré Disc"
)
ax = fig.add_subplot(projection="3d")
ax.plot_surface(
x_axis_samples,
y_axis_samples,
z_axis_samples,
rstride=1,
cstride=1,
linewidth=1,
antialiased=True,
cmap=plt.get_cmap("viridis"),
)
z_circle = -0.8
p = Circle((0, 0), 1, edgecolor="b", lw=1, facecolor="none")
ax.add_patch(p)
art3d.pathpatch_2d_to_3d(p, z=z_circle, zdir="z")
for data_index, _ in enumerate(data):
ax.scatter(
data[data_index][0], data[data_index][1], z_circle, c="b", marker="."
)
for means_index, _ in enumerate(means):
ax.scatter(
means[means_index][0], means[means_index][1], z_circle, c="r", marker="D"
)
ax.set_xlim(-1.2, 1.2)
ax.set_ylim(-1.2, 1.2)
ax.set_zlim(-0.8, 0.4)
ax.set_xlabel("X")
ax.set_ylabel("Y")
ax.set_zlabel("P")
plt.savefig(save_path, format="pdf")
return plt
def expectation_maximisation_poincare_ball():
"""Apply EM algorithm on three random data clusters."""
dim = 2
n_samples = 5
cluster_1 = gs.random.uniform(low=0.2, high=0.6, size=(n_samples, dim))
cluster_2 = gs.random.uniform(low=-0.6, high=-0.2, size=(n_samples, dim))
cluster_3 = gs.random.uniform(low=-0.3, high=0, size=(n_samples, dim))
cluster_3[:, 0] = -cluster_3[:, 0]
data = gs.concatenate((cluster_1, cluster_2, cluster_3), axis=0)
n_clusters = 3
manifold = PoincareBall(dim=2)
EM = RiemannianEM(manifold, n_gaussians=n_clusters, initialisation_method="random")
EM.fit(X=data)
means = EM.means_
variances = EM.variances_
mixture_coefficients = EM.mixture_coefficients_
# Plot result
plot = plot_gaussian_mixture_distribution(
manifold,
data,
mixture_coefficients,
means,
variances,
plot_precision=100,
save_path="result.png",
)
return plot
def main():
"""Apply Expectation Maximization on random data.
Fits three randomly generated clusters into a
Gaussian Mixture Model on Poincaré Ball.
Then a plot function computes the probability density
function of the GMM for visualization.
"""
plots = expectation_maximisation_poincare_ball()
plots.show()
if __name__ == "__main__":
if os.environ.get("GEOMSTATS_BACKEND", "numpy") != "numpy":
print(
"Expectation Maximization example\n"
"works with\n"
"numpy backend.\n"
"To change backend, write: "
"export GEOMSTATS_BACKEND = 'numpy'."
)
else:
main()