LDA | Linear Discriminant Analysis in Python

1. PCA vs LDA #

• PCA: unsupervised, keeps the directions of largest variance irrespective of class labels.
• LDA: supervised, searches for directions that maximise the ratio of between-class variance to within-class variance.

2. Formulation #

With labelled classes (C_1, \dots, C_k):

1. Within-class scatter $$ S_W = \sum_{j=1}^k \sum_{x_i \in C_j} (x_i - \mu_j)(x_i - \mu_j)^\top $$
2. Between-class scatter $$ S_B = \sum_{j=1}^k n_j (\mu_j - \mu)(\mu_j - \mu)^\top $$
3. Optimisation $$ J(w) = \frac{w^\top S_B w}{w^\top S_W w} $$ The eigenvectors of (S_W^{-1} S_B) give the discriminant directions. At most (k-1) components carry information.

3. Build a dataset #

import numpy as np
import matplotlib.pyplot as plt
import japanize_matplotlib
from sklearn.datasets import make_blobs

X, y = make_blobs(
    n_samples=600,
    n_features=3,
    random_state=11711,
    cluster_std=4,
    centers=3,
)

fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(projection="3d")
ax.scatter(X[:, 0], X[:, 1], X[:, 2], c=y)
ax.set_xlabel("$x_1$")
ax.set_ylabel("$x_2$")
ax.set_zlabel("$x_3$")

4. Apply LDA #

from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA

lda = LDA(n_components=2).fit(X, y)
X_lda = lda.transform(X)

plt.figure(figsize=(8, 8))
plt.scatter(X_lda[:, 0], X_lda[:, 1], c=y, alpha=0.5)
plt.xlabel("LD1")
plt.ylabel("LD2")
plt.title("2-D embedding via LDA")
plt.show()

5. Compare with PCA #

from sklearn.decomposition import PCA

pca = PCA(n_components=2)
X_pca = pca.fit_transform(X)

plt.figure(figsize=(8, 8))
plt.scatter(X_pca[:, 0], X_pca[:, 1], c=y, alpha=0.5)
plt.xlabel("PC1")
plt.ylabel("PC2")
plt.title("2-D embedding via PCA")
plt.show()

PCA mixes the classes because it ignores labels; LDA keeps them separated.

6. Practical notes #

• The number of useful discriminants is at most n_classes - 1.
• Standardise features before fitting, especially when different units are mixed.
• LDA assumes roughly equal covariance within classes; when that is violated, consider QDA or regularised LDA.

Summary #

• LDA maximises (S_B) while minimising (S_W), yielding projections suitable for visualisation or as input to classifiers.
• It complements PCA: PCA preserves variance, LDA preserves class structure.
• Combine both ideas (e.g., PCA for denoising then LDA) when the number of features is huge.