.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/classification/plot_lda_qda.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_classification_plot_lda_qda.py: ==================================================================== 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. .. GENERATED FROM PYTHON SOURCE LINES 15-17 Colormap -------- .. GENERATED FROM PYTHON SOURCE LINES 17-33 .. code-block:: default 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) .. GENERATED FROM PYTHON SOURCE LINES 34-36 Datasets generation functions ----------------------------- .. GENERATED FROM PYTHON SOURCE LINES 36-66 .. code-block:: default 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 .. GENERATED FROM PYTHON SOURCE LINES 67-69 Plot functions -------------- .. GENERATED FROM PYTHON SOURCE LINES 69-164 .. code-block:: default 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") .. GENERATED FROM PYTHON SOURCE LINES 165-167 Plot ---- .. GENERATED FROM PYTHON SOURCE LINES 167-196 .. code-block:: default 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() .. image-sg:: /auto_examples/classification/images/sphx_glr_plot_lda_qda_001.png :alt: Linear Discriminant Analysis vs Quadratic Discriminant Analysis, Linear Discriminant Analysis, Quadratic Discriminant Analysis :srcset: /auto_examples/classification/images/sphx_glr_plot_lda_qda_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.292 seconds) .. _sphx_glr_download_auto_examples_classification_plot_lda_qda.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/1.2.X?urlpath=lab/tree/notebooks/auto_examples/classification/plot_lda_qda.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_lda_qda.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_lda_qda.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_