Linear Discriminant Analysis (LDA)

Last updated 2020-03-11 Read time 3 min

Summary

LDA finds directions that maximize the ratio of between-class variance to within-class variance, serving both classification and dimensionality reduction.
The decision boundary takes the form $\mathbf{w}^\top \mathbf{x} + b = 0$, which becomes a line in 2D or a plane in 3D, giving a clear geometric interpretation.
Assuming each class is Gaussian with the same covariance matrix, LDA approaches the Bayes-optimal classifier.
scikit-learn’s LinearDiscriminantAnalysis makes it easy to visualise decision boundaries and to inspect the projected features.

Intuition #

This method should be interpreted through its assumptions, data conditions, and how parameter choices affect generalization.

Detailed Explanation #

Mathematical formulation #

For the two-class case, the projection direction $\mathbf{w}$ maximizes

$$ J(\mathbf{w}) = \frac{\mathbf{w}^\top \mathbf{S}_B \mathbf{w}}{\mathbf{w}^\top \mathbf{S}_W \mathbf{w}}, $$

where $\mathbf{S}_B$ is the between-class scatter matrix and $\mathbf{S}_W$ is the within-class scatter matrix. In the multi-class case we obtain up to $K-1$ projection directions, which can be used for dimensionality reduction.

Experiments with Python #

Below we apply LDA to a synthetic two-class data set, draw the decision boundary, and plot the projected 1D features. Calling transform returns the projected data directly.

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from __future__ import annotations

import japanize_matplotlib
import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import make_blobs
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.metrics import accuracy_score

def run_lda_demo(
    n_samples: int = 200,
    random_state: int = 42,
    title_boundary: str = "LDA decision boundary",
    title_projection: str = "One-dimensional projection by LDA",
    xlabel: str = "feature 1",
    ylabel: str = "feature 2",
    hist_xlabel: str = "projected feature",
    class0_label: str = "class 0",
    class1_label: str = "class 1",
) -> dict[str, float]:
    """Train LDA on synthetic blobs and plot boundary plus projection."""
    japanize_matplotlib.japanize()
    X, y = make_blobs(
        n_samples=n_samples,
        centers=2,
        n_features=2,
        cluster_std=2.0,
        random_state=random_state,
    )

    clf = LinearDiscriminantAnalysis(store_covariance=True)
    clf.fit(X, y)

    accuracy = float(accuracy_score(y, clf.predict(X)))
    w = clf.coef_[0]
    b = float(clf.intercept_[0])

    xs = np.linspace(X[:, 0].min() - 1, X[:, 0].max() + 1, 300)
    ys_boundary = -(w[0] / w[1]) * xs - b / w[1]

    fig, ax = plt.subplots(figsize=(7, 6))
    ax.set_title(title_boundary)
    ax.scatter(X[:, 0], X[:, 1], c=y, cmap="coolwarm", edgecolor="k", alpha=0.8)
    ax.plot(xs, ys_boundary, "k--", lw=1.2, label="w^T x + b = 0")
    ax.set_xlabel(xlabel)
    ax.set_ylabel(ylabel)
    ax.legend(loc="best")
    ax.grid(alpha=0.25)
    fig.tight_layout()
    plt.show()

    X_proj = clf.transform(X)[:, 0]

    fig, ax = plt.subplots(figsize=(8, 4))
    ax.set_title(title_projection)
    ax.hist(X_proj[y == 0], bins=20, alpha=0.7, label=class0_label)
    ax.hist(X_proj[y == 1], bins=20, alpha=0.7, label=class1_label)
    ax.set_xlabel(hist_xlabel)
    ax.legend(loc="best")
    ax.grid(alpha=0.25)
    fig.tight_layout()
    plt.show()

    return {"accuracy": accuracy}

metrics = run_lda_demo(
    title_boundary="LDA decision boundary",
    title_projection="One-dimensional projection by LDA",
    xlabel="feature 1",
    ylabel="feature 2",
    hist_xlabel="projected feature",
    class0_label="class 0",
    class1_label="class 1",
)
print(f"Training accuracy: {metrics['accuracy']:.3f}")

Calling transform returns the projected data directly figure

References #

Fisher, R. A. (1936). The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7(2), 179–188.
Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning. Springer.