Notebook source code: notebooks/06_practical_methods__riemannian_frechet_mean_and_tangent_pca.ipynb
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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.

 In [1]:

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 [2]:

from geomstats.geometry.hypersphere import Hypersphere

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

 In [3]:

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();
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_5_0.png

Fréchet mean#

We compute the Fréchet mean of the simulated data points.

 In [4]:

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

mean_estimate = mean.estimate_

 In [5]:

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();
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_9_0.png

Tangent PCA (at the Fréchet mean)#

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

 In [6]:

tpca = TangentPCA(sphere, n_components=2)
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 [7]:

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 [8]:

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_);
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_15_0.png 
 In [9]:

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]);
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_16_0.png

In the Hyperbolic plane#

Generate data on the hyperbolic plane#

 In [10]:

from geomstats.geometry.hyperboloid import Hyperboloid

hyperbolic_plane = Hyperboloid(dim=2)

data = hyperbolic_plane.random_point(n_samples=140)

 In [11]:

fig = plt.figure(figsize=(8, 8))
ax = visualization.plot(
    data, space="H2_poincare_disk", color="black", alpha=0.7, label="Data points"
)
ax.legend();
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_20_0.png

Fréchet mean#

We compute the Fréchet mean of the data points.

 In [12]:

mean = FrechetMean(hyperbolic_plane)
mean.fit(data)

mean_estimate = mean.estimate_

 In [13]:

fig = plt.figure(figsize=(8, 8))
ax = visualization.plot(
    data, space="H2_poincare_disk", color="black", alpha=0.2, label="Data points"
)
ax = visualization.plot(
    mean_estimate,
    space="H2_poincare_disk",
    color="red",
    ax=ax,
    s=200,
    label="Fréchet mean",
)
ax.legend();
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_24_0.png

Tangent PCA (at the Fréchet mean)#

We perform tangent PCA at the Fréchet mean.

 In [14]:

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

We compute the geodesics corresponding to the first components of the tangent PCA.

 In [15]:

geodesic_0 = hyperbolic_plane.metric.geodesic(
    initial_point=mean_estimate, initial_tangent_vec=tpca.components_[0]
)
geodesic_1 = hyperbolic_plane.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 [16]:

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_);
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_30_0.png 
 In [17]:

fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(111)

ax = visualization.plot(
    geodesic_points_0,
    ax,
    space="H2_poincare_disk",
    linewidth=2,
    label="First component",
)
ax = visualization.plot(
    geodesic_points_1,
    ax,
    space="H2_poincare_disk",
    linewidth=2,
    label="Second component",
)
ax = visualization.plot(
    data, ax, space="H2_poincare_disk", color="black", alpha=0.2, label="Data points"
)
ax = visualization.plot(
    mean_estimate,
    ax,
    space="H2_poincare_disk",
    color="red",
    s=200,
    label="Fréchet mean",
)
ax.legend();
../_images/notebooks_06_practical_methods__riemannian_frechet_mean_and_tangent_pca_31_0.png