Linear and Quadratic Discriminant Analysis with covariance ellipsoid

This example plots the covariance ellipsoids of each class and decision boundary learned by LDA and QDA. The ellipsoids display the double standard deviation for each class. With LDA, the standard deviation is the same for all the classes, while each class has its own standard deviation with QDA.

Colormap

import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib import colors

cmap = colors.LinearSegmentedColormap(
    "red_blue_classes",
    {
        "red": [(0, 1, 1), (1, 0.7, 0.7)],
        "green": [(0, 0.7, 0.7), (1, 0.7, 0.7)],
        "blue": [(0, 0.7, 0.7), (1, 1, 1)],
    },
)
plt.cm.register_cmap(cmap=cmap)

Datasets generation functions

import numpy as np


def dataset_fixed_cov():
    """Generate 2 Gaussians samples with the same covariance matrix"""
    n, dim = 300, 2
    np.random.seed(0)
    C = np.array([[0.0, -0.23], [0.83, 0.23]])
    X = np.r_[
        np.dot(np.random.randn(n, dim), C),
        np.dot(np.random.randn(n, dim), C) + np.array([1, 1]),
    ]
    y = np.hstack((np.zeros(n), np.ones(n)))
    return X, y


def dataset_cov():
    """Generate 2 Gaussians samples with different covariance matrices"""
    n, dim = 300, 2
    np.random.seed(0)
    C = np.array([[0.0, -1.0], [2.5, 0.7]]) * 2.0
    X = np.r_[
        np.dot(np.random.randn(n, dim), C),
        np.dot(np.random.randn(n, dim), C.T) + np.array([1, 4]),
    ]
    y = np.hstack((np.zeros(n), np.ones(n)))
    return X, y

Plot functions

from scipy import linalg


def plot_data(lda, X, y, y_pred, fig_index):
    splot = plt.subplot(2, 2, fig_index)
    if fig_index == 1:
        plt.title("Linear Discriminant Analysis")
        plt.ylabel("Data with\n fixed covariance")
    elif fig_index == 2:
        plt.title("Quadratic Discriminant Analysis")
    elif fig_index == 3:
        plt.ylabel("Data with\n varying covariances")

    tp = y == y_pred  # True Positive
    tp0, tp1 = tp[y == 0], tp[y == 1]
    X0, X1 = X[y == 0], X[y == 1]
    X0_tp, X0_fp = X0[tp0], X0[~tp0]
    X1_tp, X1_fp = X1[tp1], X1[~tp1]

    # class 0: dots
    plt.scatter(X0_tp[:, 0], X0_tp[:, 1], marker=".", color="red")
    plt.scatter(X0_fp[:, 0], X0_fp[:, 1], marker="x", s=20, color="#990000")  # dark red

    # class 1: dots
    plt.scatter(X1_tp[:, 0], X1_tp[:, 1], marker=".", color="blue")
    plt.scatter(
        X1_fp[:, 0], X1_fp[:, 1], marker="x", s=20, color="#000099"
    )  # dark blue

    # class 0 and 1 : areas
    nx, ny = 200, 100
    x_min, x_max = plt.xlim()
    y_min, y_max = plt.ylim()
    xx, yy = np.meshgrid(np.linspace(x_min, x_max, nx), np.linspace(y_min, y_max, ny))
    Z = lda.predict_proba(np.c_[xx.ravel(), yy.ravel()])
    Z = Z[:, 1].reshape(xx.shape)
    plt.pcolormesh(
        xx, yy, Z, cmap="red_blue_classes", norm=colors.Normalize(0.0, 1.0), zorder=0
    )
    plt.contour(xx, yy, Z, [0.5], linewidths=2.0, colors="white")

    # means
    plt.plot(
        lda.means_[0][0],
        lda.means_[0][1],
        "*",
        color="yellow",
        markersize=15,
        markeredgecolor="grey",
    )
    plt.plot(
        lda.means_[1][0],
        lda.means_[1][1],
        "*",
        color="yellow",
        markersize=15,
        markeredgecolor="grey",
    )

    return splot


def plot_ellipse(splot, mean, cov, color):
    v, w = linalg.eigh(cov)
    u = w[0] / linalg.norm(w[0])
    angle = np.arctan(u[1] / u[0])
    angle = 180 * angle / np.pi  # convert to degrees
    # filled Gaussian at 2 standard deviation
    ell = mpl.patches.Ellipse(
        mean,
        2 * v[0] ** 0.5,
        2 * v[1] ** 0.5,
        angle=180 + angle,
        facecolor=color,
        edgecolor="black",
        linewidth=2,
    )
    ell.set_clip_box(splot.bbox)
    ell.set_alpha(0.2)
    splot.add_artist(ell)
    splot.set_xticks(())
    splot.set_yticks(())


def plot_lda_cov(lda, splot):
    plot_ellipse(splot, lda.means_[0], lda.covariance_, "red")
    plot_ellipse(splot, lda.means_[1], lda.covariance_, "blue")


def plot_qda_cov(qda, splot):
    plot_ellipse(splot, qda.means_[0], qda.covariance_[0], "red")
    plot_ellipse(splot, qda.means_[1], qda.covariance_[1], "blue")

Plot

plt.figure(figsize=(10, 8), facecolor="white")
plt.suptitle(
    "Linear Discriminant Analysis vs Quadratic Discriminant Analysis",
    y=0.98,
    fontsize=15,
)

from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis

for i, (X, y) in enumerate([dataset_fixed_cov(), dataset_cov()]):
    # Linear Discriminant Analysis
    lda = LinearDiscriminantAnalysis(solver="svd", store_covariance=True)
    y_pred = lda.fit(X, y).predict(X)
    splot = plot_data(lda, X, y, y_pred, fig_index=2 * i + 1)
    plot_lda_cov(lda, splot)
    plt.axis("tight")

    # Quadratic Discriminant Analysis
    qda = QuadraticDiscriminantAnalysis(store_covariance=True)
    y_pred = qda.fit(X, y).predict(X)
    splot = plot_data(qda, X, y, y_pred, fig_index=2 * i + 2)
    plot_qda_cov(qda, splot)
    plt.axis("tight")

plt.tight_layout()
plt.subplots_adjust(top=0.92)
plt.show()