pca_projected_variance.ipynb

Kernel: Unknown Kernel

# pca 1d : visualize variacnce of projecton
# Code is based on chapter 8 of
# https://github.com/ageron/handson-ml2


import numpy as np
import matplotlib.pyplot as plt

try:
    from sklearn.decomposition import PCA
except ModuleNotFoundError:
    %pip install -qq scikit-learn
    from sklearn.decomposition import PCA

try:
    from probml_utils import savefig
except ModuleNotFoundError:
    %pip install -qq git+https://github.com/probml/probml-utils.git
    from probml_utils import savefig

angle = np.pi / 5
stretch = 5
m = 200

np.random.seed(3)
X = np.random.randn(m, 2) / 10
X = X.dot(np.array([[stretch, 0], [0, 1]]))  # stretch
X = X.dot([[np.cos(angle), np.sin(angle)], [-np.sin(angle), np.cos(angle)]])  # rotate

u1 = np.array([np.cos(angle), np.sin(angle)])
u2 = np.array([np.cos(angle - 2 * np.pi / 6), np.sin(angle - 2 * np.pi / 6)])
u3 = np.array([np.cos(angle - np.pi / 2), np.sin(angle - np.pi / 2)])

X_proj1 = X.dot(u1.reshape(-1, 1))
X_proj2 = X.dot(u2.reshape(-1, 1))
X_proj3 = X.dot(u3.reshape(-1, 1))

plt.figure(figsize=(8, 4))
plt.subplot2grid((3, 2), (0, 0), rowspan=3)
plt.plot([-1.4, 1.4], [-1.4 * u1[1] / u1[0], 1.4 * u1[1] / u1[0]], "k-", linewidth=1)
plt.plot([-1.4, 1.4], [-1.4 * u2[1] / u2[0], 1.4 * u2[1] / u2[0]], "k--", linewidth=1)
plt.plot([-1.4, 1.4], [-1.4 * u3[1] / u3[0], 1.4 * u3[1] / u3[0]], "k:", linewidth=2)
plt.plot(X[:, 0], X[:, 1], "bo", alpha=0.5)
plt.axis([-1.4, 1.4, -1.4, 1.4])
plt.arrow(0, 0, u1[0], u1[1], head_width=0.1, linewidth=5, length_includes_head=True, head_length=0.1, fc="k", ec="k")
plt.arrow(0, 0, u3[0], u3[1], head_width=0.1, linewidth=5, length_includes_head=True, head_length=0.1, fc="k", ec="k")
plt.text(u1[0] + 0.1, u1[1] - 0.05, r"$v_1$", fontsize=22)
plt.text(u2[0] + 0.1, u2[1], r"$v'$", fontsize=22)  # KPM
plt.text(u3[0] + 0.1, u3[1], r"$v_2$", fontsize=22)
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$x_2$", fontsize=18, rotation=0)
plt.grid(True)

plt.subplot2grid((3, 2), (0, 1))
plt.plot([-2, 2], [0, 0], "k-", linewidth=1)
plt.plot(X_proj1[:, 0], np.zeros(m), "bo", alpha=0.3)
plt.gca().get_yaxis().set_ticks([])
plt.gca().get_xaxis().set_ticklabels([])
plt.axis([-2, 2, -1, 1])
plt.grid(True)

plt.subplot2grid((3, 2), (1, 1))
plt.plot([-2, 2], [0, 0], "k--", linewidth=1)
plt.plot(X_proj2[:, 0], np.zeros(m), "bo", alpha=0.3)
plt.gca().get_yaxis().set_ticks([])
plt.gca().get_xaxis().set_ticklabels([])
plt.axis([-2, 2, -1, 1])
plt.grid(True)

plt.subplot2grid((3, 2), (2, 1))
plt.plot([-2, 2], [0, 0], "k:", linewidth=2)
plt.plot(X_proj3[:, 0], np.zeros(m), "bo", alpha=0.3)
plt.gca().get_yaxis().set_ticks([])
plt.axis([-2, 2, -1, 1])
plt.xlabel("$z_1$", fontsize=18)
plt.grid(True)

plt.tight_layout()
savefig("pca_projected_variance.pdf", dpi=300)
plt.show()