Fréchet Mean and Tangent PCA¶

Lead author: Nina Miolane.

This notebook shows how to compute the Fréchet mean of a data set. Then it performs tangent PCA at the mean.

Setup¶

In [1]:

import os
import sys
import warnings

sys.path.append(os.path.dirname(os.getcwd()))
warnings.filterwarnings("ignore")

In [2]:

import matplotlib.pyplot as plt

import geomstats.backend as gs
import geomstats.visualization as visualization

from geomstats.learning.frechet_mean import FrechetMean
from geomstats.learning.pca import TangentPCA

INFO: Using numpy backend

On the sphere¶

Generate data on the sphere¶

In [3]:

from geomstats.geometry.hypersphere import Hypersphere

sphere = Hypersphere(dim=2)
data = sphere.random_von_mises_fisher(kappa=15, n_samples=140)

In [4]:

fig = plt.figure(figsize=(8, 8))
ax = visualization.plot(data, space="S2", color="black", alpha=0.7, label="Data points")
ax.set_box_aspect([1, 1, 1])
ax.legend();

Fréchet mean¶

We compute the Fréchet mean of the simulated data points.

In [5]:

mean = FrechetMean(metric=sphere.metric)
mean.fit(data)

mean_estimate = mean.estimate_

In [6]:

fig = plt.figure(figsize=(8, 8))
ax = visualization.plot(data, space="S2", color="black", alpha=0.2, label="Data points")
ax = visualization.plot(
    mean_estimate, space="S2", color="red", ax=ax, s=200, label="Fréchet mean"
)
ax.set_box_aspect([1, 1, 1])
ax.legend();

Tangent PCA (at the Fréchet mean)¶

We perform tangent PCA at the Fréchet mean, with two principal components.

In [7]:

tpca = TangentPCA(metric=sphere.metric, n_components=2)
tpca = tpca.fit(data, base_point=mean_estimate)
tangent_projected_data = tpca.transform(data)

We compute the geodesics on the sphere corresponding to the two principal components.

In [8]:

geodesic_0 = sphere.metric.geodesic(
    initial_point=mean_estimate, initial_tangent_vec=tpca.components_[0]
)
geodesic_1 = sphere.metric.geodesic(
    initial_point=mean_estimate, initial_tangent_vec=tpca.components_[1]
)

n_steps = 100
t = gs.linspace(-1.0, 1.0, n_steps)
geodesic_points_0 = geodesic_0(t)
geodesic_points_1 = geodesic_1(t)

In [10]:

fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(111)
xticks = gs.arange(1, 2 + 1, 1)
ax.xaxis.set_ticks(xticks)
ax.set_title("Explained variance")
ax.set_xlabel("Number of Principal Components")
ax.set_ylim((0, 1))
ax.bar(xticks, tpca.explained_variance_ratio_);

In [18]:

fig = plt.figure(figsize=(10, 10))
ax = fig.add_subplot(111, projection="3d")

ax = visualization.plot(
    geodesic_points_0, ax, space="S2", linewidth=2, label="First component"
)
ax = visualization.plot(
    geodesic_points_1, ax, space="S2", linewidth=2, label="Second component"
)
ax = visualization.plot(
    data, ax, space="S2", color="black", alpha=0.2, label="Data points"
)
ax = visualization.plot(
    mean_estimate, ax, space="S2", color="red", s=200, label="Fréchet mean"
)
ax.legend()
ax.set_box_aspect([1, 1, 1])
plt.show()