PCA (Principal Component Analysis) › Visualizing PCA
Michael BrenndoerferJuly 21, 20253 min read
Machine Learning from ScratchAll chapters Part 6 of 9Explainability
PCA (Principal Component Analysis) › Visualizing PCAOne section · Sections in this chapter4 / 9 ↓
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The mathematics is clearer when you see it in action. Let's visualize PCA on a simple two-dimensional dataset where we can see what's happening.
In[3]:
Codeimport sys
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
for parent in [Path.cwd(), *Path.cwd().parents]:
helper_dir = (
parent / "apps/web/content/notebooks/1_machine-learning-from-scratch"
)
if (helper_dir / "mls_plot_style.py").exists():
if str(helper_dir) not in sys.path:
sys.path.insert(0, str(helper_dir))
break
else:
raise FileNotFoundError("Could not locate mls_plot_style.py")
from mls_plot_style import PALETTE, polish_axes, use_mls_style
# Set random seed for reproducibility
np.random.seed(42)
# Create correlated 2D data
n_samples = 200
mean = [0, 0]
cov = [[2, 1.5], [1.5, 2]]
X = np.random.multivariate_normal(mean, cov, n_samples)
# Standardize the data
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)
# Apply PCA
pca = PCA()
X_pca = pca.fit_transform(X_scaled)
use_mls_style(fixed_canvas=True)
# Original data
fig, ax1 = plt.subplots(figsize=(3, 2.8))
ax1.scatter(
X_scaled[:, 0],
X_scaled[:, 1],
alpha=0.66,
c=PALETTE["primary"],
s=18,
edgecolors=PALETTE["paper"],
linewidths=0.3,
)
ax1.set_xlabel("Feature 1")
ax1.set_ylabel("Feature 2")
ax1.set_title("Original Data (Standardized)")
ax1.set_aspect("equal")
# Add principal component directions
pc1 = pca.components_[0] * np.sqrt(pca.explained_variance_[0])
pc2 = pca.components_[1] * np.sqrt(pca.explained_variance_[1])
ax1.arrow(
0,
0,
pc1[0],
pc1[1],
head_width=0.1,
head_length=0.1,
fc=PALETTE["coral"],
ec=PALETTE["coral"],
linewidth=2,
)
ax1.arrow(
0,
0,
pc2[0],
pc2[1],
head_width=0.1,
head_length=0.1,
fc=PALETTE["teal"],
ec=PALETTE["teal"],
linewidth=2,
)
ax1.text(
pc1[0],
pc1[1],
"PC1",
fontsize=8,
color=PALETTE["coral"],
weight="bold",
)
ax1.text(
pc2[0],
pc2[1],
"PC2",
fontsize=8,
color=PALETTE["teal"],
weight="bold",
)
polish_axes(ax1)
fig.subplots_adjust(left=0.18, right=0.97, bottom=0.18, top=0.86)
plt.show()
# PCA transformed data
fig, ax2 = plt.subplots(figsize=(3, 2.8))
ax2.scatter(
X_pca[:, 0],
X_pca[:, 1],
alpha=0.66,
c=PALETTE["purple"],
s=18,
edgecolors=PALETTE["paper"],
linewidths=0.3,
)
ax2.set_xlabel("First Principal Component")
ax2.set_ylabel("Second Principal Component")
ax2.set_title("Data After PCA Transformation")
ax2.set_aspect("equal")
polish_axes(ax2)
fig.subplots_adjust(left=0.18, right=0.97, bottom=0.18, top=0.86)
plt.show()
# Print explained variance ratios
print("Explained variance ratios:")
for i, ratio in enumerate(pca.explained_variance_ratio_):
print(f"PC{i + 1}: {ratio:.3f} ({ratio * 100:.1f}%)")Out[3]:
Visualization
Explained variance ratios: PC1: 0.873 (87.3%) PC2: 0.127 (12.7%)
In[4]:
Codeuse_mls_style(fixed_canvas=True)
# Create scree plot
fig, ax = plt.subplots(figsize=(3, 2.8))
ax.plot(
range(1, len(pca.explained_variance_ratio_) + 1),
pca.explained_variance_ratio_,
color=PALETTE["primary"],
marker="o",
linewidth=2,
markersize=5,
)
ax.set_xlabel("Principal Component")
ax.set_ylabel("Explained Variance Ratio")
ax.set_title("Scree Plot")
ax.set_xticks(range(1, len(pca.explained_variance_ratio_) + 1))
polish_axes(ax)
fig.subplots_adjust(left=0.20, right=0.97, bottom=0.20, top=0.86)
plt.show()
# Cumulative variance explained
cumulative_variance = np.cumsum(pca.explained_variance_ratio_)
fig, ax = plt.subplots(figsize=(3, 2.8))
ax.plot(
range(1, len(cumulative_variance) + 1),
cumulative_variance,
color=PALETTE["rose"],
marker="o",
linewidth=2,
markersize=5,
)
ax.set_xlabel("Number of Components")
ax.set_ylabel("Cumulative Variance Explained")
ax.set_title("Cumulative Variance Explained")
ax.set_xticks(range(1, len(cumulative_variance) + 1))
ax.axhline(
y=0.95,
color=PALETTE["teal"],
linestyle="--",
alpha=0.85,
label="95% threshold",
)
ax.legend(loc="lower right")
polish_axes(ax)
fig.subplots_adjust(left=0.20, right=0.97, bottom=0.20, top=0.86)
plt.show()Out[4]:
Visualization
