Note
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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.
print(__doc__) from scipy import linalg import numpy as np import matplotlib.pyplot as plt import matplotlib as mpl from matplotlib import colors from sklearn.discriminant_analysis import LinearDiscriminantAnalysis from sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis # ############################################################################# # Colormap 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) # ############################################################################# # Generate datasets 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., -0.23], [0.83, .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., -1.], [2.5, .7]]) * 2. 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 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] alpha = 0.5 # class 0: dots plt.plot(X0_tp[:, 0], X0_tp[:, 1], 'o', alpha=alpha, color='red', markeredgecolor='k') plt.plot(X0_fp[:, 0], X0_fp[:, 1], '*', alpha=alpha, color='#990000', markeredgecolor='k') # dark red # class 1: dots plt.plot(X1_tp[:, 0], X1_tp[:, 1], 'o', alpha=alpha, color='blue', markeredgecolor='k') plt.plot(X1_fp[:, 0], X1_fp[:, 1], '*', alpha=alpha, color='#000099', markeredgecolor='k') # 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., 1.)) plt.contour(xx, yy, Z, [0.5], linewidths=2., colors='k') # means plt.plot(lda.means_[0][0], lda.means_[0][1], 'o', color='black', markersize=10, markeredgecolor='k') plt.plot(lda.means_[1][0], lda.means_[1][1], 'o', color='black', markersize=10, markeredgecolor='k') 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, 180 + angle, facecolor=color, edgecolor='yellow', linewidth=2, zorder=2) ell.set_clip_box(splot.bbox) ell.set_alpha(0.5) 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') 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.suptitle('Linear Discriminant Analysis vs Quadratic Discriminant' 'Analysis') plt.show()
Total running time of the script: ( 0 minutes 0.264 seconds)
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http://scikit-learn.org/stable/auto_examples/classification/plot_lda_qda.html