def test_plot_points_se2():
points = SpecialEuclidean(n=2, point_type='vector').random_point(4)
visu = visualization.SpecialEuclidean2(points, point_type='vector')
ax = visu.set_ax()
visu.draw(ax)
def main():
r"""Compute and visualize a geodesic regression on the SE(2).
The generative model of the data is:
:math:`Z = Exp_{\beta_0}(\beta_1.X)` and :math:`Y = Exp_Z(\epsilon)`
where:
- :math:`Exp` denotes the Riemannian exponential,
- :math:`\beta_0` is called the intercept,
- :math:`\beta_1` is called the coefficient,
- :math:`\epsilon \sim N(0, 1)` is a standard Gaussian noise,
- :math:`X` is the input, :math:`Y` is the target.
"""
# Generate noise-free data
n_samples = 20
X = gs.random.normal(size=(n_samples, ))
X -= gs.mean(X)
intercept = SPACE.random_point()
coef = SPACE.to_tangent(5.0 * gs.random.rand(3, 3), intercept)
y = METRIC.exp(X[:, None, None] * coef[None], intercept)
# Generate normal noise in the Lie algebra
normal_noise = gs.random.normal(size=(n_samples, 3))
normal_noise = SPACE.lie_algebra.matrix_representation(normal_noise)
noise = SPACE.tangent_translation_map(y)(normal_noise) / gs.pi
rss = gs.sum(METRIC.squared_norm(noise, y)) / n_samples
# Add noise
y = METRIC.exp(noise, y)
# True noise level and R2
estimator = FrechetMean(METRIC)
estimator.fit(y)
variance_ = variance(y, estimator.estimate_, metric=METRIC)
r2 = 1 - rss / variance_
# Fit geodesic regression
gr = GeodesicRegression(
SPACE,
metric=METRIC,
center_X=False,
method="riemannian",
max_iter=100,
init_step_size=0.1,
verbose=True,
initialization="frechet",
)
gr.fit(X, y, compute_training_score=True)
intercept_hat, beta_hat = gr.intercept_, gr.coef_
# Measure Mean Squared Error
mse_intercept = METRIC.squared_dist(intercept_hat, intercept)
mse_beta = METRIC.squared_norm(
METRIC.parallel_transport(beta_hat, intercept_hat,
METRIC.log(intercept_hat, intercept)) - coef,
intercept,
)
# Measure goodness of fit
r2_hat = gr.training_score_
print(f"MSE on the intercept: {mse_intercept:.2e}")
print(f"MSE on the initial velocity beta: {mse_beta:.2e}")
print(f"Determination coefficient: R^2={r2_hat:.2f}")
print(f"True R^2: {r2:.2f}")
# Plot
fitted_data = gr.predict(X)
fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(111)
sphere_visu = visualization.SpecialEuclidean2()
ax = sphere_visu.set_ax(ax=ax)
path = METRIC.geodesic(initial_point=intercept_hat,
initial_tangent_vec=beta_hat)
regressed_geodesic = path(gs.linspace(min(X), max(X), 100))
sphere_visu.draw_points(ax, y, marker="o", c="black")
sphere_visu.draw_points(ax, fitted_data, marker="o", c="gray")
sphere_visu.draw_points(ax, gs.array([intercept]), marker="x", c="r")
sphere_visu.draw_points(ax,
gs.array([intercept_hat]),
marker="o",
c="green")
ax.plot(regressed_geodesic[:, 0, 2], regressed_geodesic[:, 1, 2], c="gray")
plt.show()