def test_sample_von_mises_fisher(self):
"""
Check that the maximum likelihood estimates of the mean and
concentration parameter are close to the real values. A first
estimation of the concentration parameter is obtained by a
closed-form expression and improved through the Newton method.
"""
dim = 2
n_points = 1000000
sphere = Hypersphere(dim)
# check mean value for concentrated distribution
kappa = 1000000
points = sphere.random_von_mises_fisher(kappa, n_points)
sum_points = gs.sum(points, axis=0)
mean = gs.array([0., 0., 1.])
mean_estimate = sum_points / gs.linalg.norm(sum_points)
expected = mean
result = mean_estimate
self.assertTrue(
gs.allclose(result, expected, atol=MEAN_ESTIMATION_TOL)
)
# check concentration parameter for dispersed distribution
kappa = 1
points = sphere.random_von_mises_fisher(kappa, n_points)
sum_points = gs.sum(points, axis=0)
mean_norm = gs.linalg.norm(sum_points) / n_points
kappa_estimate = (mean_norm * (dim + 1. - mean_norm**2)
/ (1. - mean_norm**2))
p = dim + 1
n_steps = 100
for i in range(n_steps):
bessel_func_1 = scipy.special.iv(p/2., kappa_estimate)
bessel_func_2 = scipy.special.iv(p/2.-1., kappa_estimate)
ratio = bessel_func_1 / bessel_func_2
denominator = 1. - ratio**2 - (p-1.)*ratio/kappa_estimate
kappa_estimate = kappa_estimate - (ratio-mean_norm)/denominator
expected = kappa
result = kappa_estimate
self.assertTrue(
gs.allclose(result, expected, atol=KAPPA_ESTIMATION_TOL))
def test_optimal_quantization(self):
"""
Check that optimal quantization yields the same result as
the karcher flow algorithm when we look for one center.
"""
dim = 2
n_points = 1000
n_centers = 1
sphere = Hypersphere(dim)
points = sphere.random_von_mises_fisher(kappa=10, n_samples=n_points)
mean = sphere.metric.mean(points)
centers, weights, clusters, n_iterations = sphere.metric.\
optimal_quantization(points=points, n_centers=n_centers)
error = sphere.metric.dist(mean, centers)
diameter = sphere.metric.diameter(points)
result = error / diameter
expected = 0.0
self.assertTrue(
gs.allclose(result, expected, atol=OPTIMAL_QUANTIZATION_TOL))
def main():
fig = plt.figure(figsize=(15, 5))
sphere = Hypersphere(dimension=2)
data = sphere.random_von_mises_fisher(kappa=15, n_samples=140)
mean = sphere.metric.mean(data)
tpca = TangentPCA(metric=sphere.metric, n_components=2)
tpca = tpca.fit(data, base_point=mean)
tangent_projected_data = tpca.transform(data)
geodesic_0 = sphere.metric.geodesic(
initial_point=mean, initial_tangent_vec=tpca.components_[0])
geodesic_1 = sphere.metric.geodesic(
initial_point=mean, initial_tangent_vec=tpca.components_[1])
n_steps = 100
t = np.linspace(-1, 1, n_steps)
geodesic_points_0 = geodesic_0(t)
geodesic_points_1 = geodesic_1(t)
print('Coordinates of the Log of the first 5 data points at the mean, '
'projected on the principal components:')
print(tangent_projected_data[:5])
ax_var = fig.add_subplot(121)
xticks = np.arange(1, 2 + 1, 1)
ax_var.xaxis.set_ticks(xticks)
ax_var.set_title('Explained variance')
ax_var.set_xlabel('Number of Principal Components')
ax_var.set_ylim((0, 1))
ax_var.plot(xticks, tpca.explained_variance_ratio_)
ax = fig.add_subplot(122, projection="3d")
visualization.plot(mean, ax, space='S2', color='darkgreen', s=10)
visualization.plot(geodesic_points_0, ax, space='S2', linewidth=2)
visualization.plot(geodesic_points_1, ax, space='S2', linewidth=2)
visualization.plot(data, ax, space='S2', color='black', alpha=0.7)
plt.show()