Sphering (whitening) data

import numpy as np
import matplotlib.pyplot as plt

from amica.linalg import pre_whiten

This example heavily borrows from this great tutorial.

Toy Data

Generate toy 2D Gaussian data with correlated dimensions

rng = np.random.default_rng(seed=2)
mu = [0, 0]
sigma = [[5, 4], [4, 5]]  # must be positive semi-definite
n = 1000
X = rng.multivariate_normal(mu, sigma, size=n).T # shape (n, 2)

Highlight 20 farthest points from the origin

idx_far = np.argsort(np.linalg.norm(X, axis=0))[-20:]
idx_rest = list(set(range(n)) - set(idx_far))

Utility plotting function

def plot_data(data, title):
    fig, ax = plt.subplots()
    ax.scatter(data[0, idx_rest], data[1, idx_rest], s=20, alpha=0.2)
    ax.scatter(data[0, idx_far], data[1, idx_far], s=20, c="red", alpha=0.2)
    ax.set_aspect("equal")
    ax.set_xlim(-8, 8)
    ax.set_ylim(-8, 8)
    ax.set_xlabel("$x_1$")
    ax.set_ylabel("$x_2$")
    ax.spines["top"].set_visible(False)
    ax.spines["right"].set_visible(False)
    ax.set_title(title)

Original correlated data

plot_data(X, "Original")

Whitening with different options

for mode, title in [
    ({"do_approx_sphere": True}, "ZCA"),
    ({"do_approx_sphere": False}, "PCA"),
    ({"do_sphere": False}, "Normalize by Variance"),
]:
    Z, *_ = pre_whiten(X=X.T, inplace=False, **mode)
    plot_data(Z.T, title)

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