wehub-resource-sync
640 lines
19 KiB
Python
640 lines
19 KiB
Python
# Sebastian Raschka, 2015 (http://sebastianraschka.com)
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# Python Machine Learning - Code Examples
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#
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# Chapter 5 - Compressing Data via Dimensionality Reduction
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#
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# S. Raschka. Python Machine Learning. Packt Publishing Ltd., 2015.
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# GitHub Repo: https://github.com/rasbt/python-machine-learning-book
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#
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# License: MIT
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# https://github.com/rasbt/python-machine-learning-book/blob/master/LICENSE.txt
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import pandas as pd
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import numpy as np
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from sklearn.preprocessing import StandardScaler
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from sklearn.decomposition import PCA
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import matplotlib.pyplot as plt
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from matplotlib.colors import ListedColormap
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from sklearn.linear_model import LogisticRegression
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from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA
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from sklearn.datasets import make_moons
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from sklearn.datasets import make_circles
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from sklearn.decomposition import KernelPCA
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from scipy.spatial.distance import pdist, squareform
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from scipy import exp
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from scipy.linalg import eigh
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from matplotlib.ticker import FormatStrFormatter
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# for sklearn 0.18's alternative syntax
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from distutils.version import LooseVersion as Version
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from sklearn import __version__ as sklearn_version
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if Version(sklearn_version) < '0.18':
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from sklearn.grid_search import train_test_split
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from sklearn.lda import LDA
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else:
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from sklearn.model_selection import train_test_split
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from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA
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#############################################################################
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print(50 * '=')
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print('Section: Unsupervised dimensionality reduction'
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' via principal component analysis')
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print(50 * '-')
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df_wine = pd.read_csv('https://archive.ics.uci.edu/ml/'
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'machine-learning-databases/wine/wine.data',
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header=None)
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df_wine.columns = ['Class label', 'Alcohol', 'Malic acid', 'Ash',
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'Alcalinity of ash', 'Magnesium', 'Total phenols',
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'Flavanoids', 'Nonflavanoid phenols', 'Proanthocyanins',
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'Color intensity', 'Hue',
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'OD280/OD315 of diluted wines', 'Proline']
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print('Wine data excerpt:\n\n:', df_wine.head())
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X, y = df_wine.iloc[:, 1:].values, df_wine.iloc[:, 0].values
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X_train, X_test, y_train, y_test = \
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train_test_split(X, y, test_size=0.3, random_state=0)
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sc = StandardScaler()
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X_train_std = sc.fit_transform(X_train)
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X_test_std = sc.transform(X_test)
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cov_mat = np.cov(X_train_std.T)
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eigen_vals, eigen_vecs = np.linalg.eig(cov_mat)
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print('\nEigenvalues \n%s' % eigen_vals)
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#############################################################################
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print(50 * '=')
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print('Section: Total and explained variance')
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print(50 * '-')
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tot = sum(eigen_vals)
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var_exp = [(i / tot) for i in sorted(eigen_vals, reverse=True)]
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cum_var_exp = np.cumsum(var_exp)
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plt.bar(range(1, 14), var_exp, alpha=0.5, align='center',
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label='individual explained variance')
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plt.step(range(1, 14), cum_var_exp, where='mid',
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label='cumulative explained variance')
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plt.ylabel('Explained variance ratio')
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plt.xlabel('Principal components')
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plt.legend(loc='best')
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# plt.tight_layout()
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# plt.savefig('./figures/pca1.png', dpi=300)
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plt.show()
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#############################################################################
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print(50 * '=')
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print('Section: Feature Transformation')
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print(50 * '-')
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# Make a list of (eigenvalue, eigenvector) tuples
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eigen_pairs = [(np.abs(eigen_vals[i]), eigen_vecs[:, i])
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for i in range(len(eigen_vals))]
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# Sort the (eigenvalue, eigenvector) tuples from high to low
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eigen_pairs.sort(reverse=True)
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w = np.hstack((eigen_pairs[0][1][:, np.newaxis],
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eigen_pairs[1][1][:, np.newaxis]))
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print('Matrix W:\n', w)
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X_train_pca = X_train_std.dot(w)
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colors = ['r', 'b', 'g']
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markers = ['s', 'x', 'o']
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for l, c, m in zip(np.unique(y_train), colors, markers):
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plt.scatter(X_train_pca[y_train == l, 0],
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X_train_pca[y_train == l, 1],
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c=c, label=l, marker=m)
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plt.xlabel('PC 1')
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plt.ylabel('PC 2')
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plt.legend(loc='lower left')
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# plt.tight_layout()
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# plt.savefig('./figures/pca2.png', dpi=300)
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plt.show()
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print('Dot product:\n', X_train_std[0].dot(w))
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#############################################################################
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print(50 * '=')
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print('Section: Principal component analysis in scikit-learn')
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print(50 * '-')
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pca = PCA()
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X_train_pca = pca.fit_transform(X_train_std)
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print('Variance explained ratio:\n', pca.explained_variance_ratio_)
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plt.bar(range(1, 14), pca.explained_variance_ratio_, alpha=0.5, align='center')
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plt.step(range(1, 14), np.cumsum(pca.explained_variance_ratio_), where='mid')
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plt.ylabel('Explained variance ratio')
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plt.xlabel('Principal components')
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plt.show()
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pca = PCA(n_components=2)
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X_train_pca = pca.fit_transform(X_train_std)
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X_test_pca = pca.transform(X_test_std)
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plt.scatter(X_train_pca[:, 0], X_train_pca[:, 1])
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plt.xlabel('PC 1')
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plt.ylabel('PC 2')
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plt.show()
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def plot_decision_regions(X, y, classifier, resolution=0.02):
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# setup marker generator and color map
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markers = ('s', 'x', 'o', '^', 'v')
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colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
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cmap = ListedColormap(colors[:len(np.unique(y))])
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# plot the decision surface
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x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
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x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
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xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),
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np.arange(x2_min, x2_max, resolution))
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Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
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Z = Z.reshape(xx1.shape)
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plt.contourf(xx1, xx2, Z, alpha=0.4, cmap=cmap)
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plt.xlim(xx1.min(), xx1.max())
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plt.ylim(xx2.min(), xx2.max())
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# plot class samples
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for idx, cl in enumerate(np.unique(y)):
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plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1],
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alpha=0.8, c=cmap(idx),
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marker=markers[idx], label=cl)
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lr = LogisticRegression()
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lr = lr.fit(X_train_pca, y_train)
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plot_decision_regions(X_train_pca, y_train, classifier=lr)
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plt.xlabel('PC 1')
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plt.ylabel('PC 2')
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plt.legend(loc='lower left')
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# plt.tight_layout()
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# plt.savefig('./figures/pca3.png', dpi=300)
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plt.show()
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plot_decision_regions(X_test_pca, y_test, classifier=lr)
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plt.xlabel('PC 1')
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plt.ylabel('PC 2')
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plt.legend(loc='lower left')
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# plt.tight_layout()
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# plt.savefig('./figures/pca4.png', dpi=300)
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plt.show()
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pca = PCA(n_components=None)
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X_train_pca = pca.fit_transform(X_train_std)
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print('Explaind variance ratio:\n', pca.explained_variance_ratio_)
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#############################################################################
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print(50 * '=')
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print('Section: Supervised data compression via linear discriminant analysis'
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' - Computing the scatter matrices')
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print(50 * '-')
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np.set_printoptions(precision=4)
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mean_vecs = []
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for label in range(1, 4):
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mean_vecs.append(np.mean(X_train_std[y_train == label], axis=0))
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print('MV %s: %s\n' % (label, mean_vecs[label - 1]))
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d = 13 # number of features
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S_W = np.zeros((d, d))
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for label, mv in zip(range(1, 4), mean_vecs):
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class_scatter = np.zeros((d, d)) # scatter matrix for each class
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for row in X_train_std[y_train == label]:
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row, mv = row.reshape(d, 1), mv.reshape(d, 1) # make column vectors
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class_scatter += (row - mv).dot((row - mv).T)
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S_W += class_scatter # sum class scatter matrices
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print('Within-class scatter matrix: %sx%s' % (S_W.shape[0], S_W.shape[1]))
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print('Class label distribution: %s'
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% np.bincount(y_train)[1:])
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d = 13 # number of features
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S_W = np.zeros((d, d))
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for label, mv in zip(range(1, 4), mean_vecs):
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class_scatter = np.cov(X_train_std[y_train == label].T)
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S_W += class_scatter
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print('Scaled within-class scatter matrix: %sx%s' % (S_W.shape[0],
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S_W.shape[1]))
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mean_overall = np.mean(X_train_std, axis=0)
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d = 13 # number of features
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S_B = np.zeros((d, d))
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for i, mean_vec in enumerate(mean_vecs):
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n = X_train[y_train == i + 1, :].shape[0]
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mean_vec = mean_vec.reshape(d, 1) # make column vector
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mean_overall = mean_overall.reshape(d, 1) # make column vector
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S_B += n * (mean_vec - mean_overall).dot((mean_vec - mean_overall).T)
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print('Between-class scatter matrix: %sx%s' % (S_B.shape[0], S_B.shape[1]))
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#############################################################################
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print(50 * '=')
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print('Section: Selecting linear discriminants for the new feature subspace')
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print(50 * '-')
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eigen_vals, eigen_vecs = np.linalg.eig(np.linalg.inv(S_W).dot(S_B))
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# Make a list of (eigenvalue, eigenvector) tuples
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eigen_pairs = [(np.abs(eigen_vals[i]), eigen_vecs[:, i])
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for i in range(len(eigen_vals))]
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# Sort the (eigenvalue, eigenvector) tuples from high to low
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eigen_pairs = sorted(eigen_pairs, key=lambda k: k[0], reverse=True)
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# Visually confirm that the list is correctly sorted by decreasing eigenvalues
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print('Eigenvalues in decreasing order:\n')
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for eigen_val in eigen_pairs:
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print(eigen_val[0])
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tot = sum(eigen_vals.real)
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discr = [(i / tot) for i in sorted(eigen_vals.real, reverse=True)]
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cum_discr = np.cumsum(discr)
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plt.bar(range(1, 14), discr, alpha=0.5, align='center',
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label='individual "discriminability"')
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plt.step(range(1, 14), cum_discr, where='mid',
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label='cumulative "discriminability"')
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plt.ylabel('"discriminability" ratio')
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plt.xlabel('Linear Discriminants')
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plt.ylim([-0.1, 1.1])
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plt.legend(loc='best')
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# plt.tight_layout()
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# plt.savefig('./figures/lda1.png', dpi=300)
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plt.show()
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w = np.hstack((eigen_pairs[0][1][:, np.newaxis].real,
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eigen_pairs[1][1][:, np.newaxis].real))
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print('Matrix W:\n', w)
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#############################################################################
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print(50 * '=')
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print('Section: Projecting samples onto the new feature space')
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print(50 * '-')
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X_train_lda = X_train_std.dot(w)
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colors = ['r', 'b', 'g']
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markers = ['s', 'x', 'o']
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for l, c, m in zip(np.unique(y_train), colors, markers):
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plt.scatter(X_train_lda[y_train == l, 0] * (-1),
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X_train_lda[y_train == l, 1] * (-1),
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c=c, label=l, marker=m)
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plt.xlabel('LD 1')
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plt.ylabel('LD 2')
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plt.legend(loc='lower right')
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# plt.tight_layout()
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# plt.savefig('./figures/lda2.png', dpi=300)
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plt.show()
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#############################################################################
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print(50 * '=')
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print('Section: LDA via scikit-learn')
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print(50 * '-')
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lda = LDA(n_components=2)
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X_train_lda = lda.fit_transform(X_train_std, y_train)
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lr = LogisticRegression()
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lr = lr.fit(X_train_lda, y_train)
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plot_decision_regions(X_train_lda, y_train, classifier=lr)
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plt.xlabel('LD 1')
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plt.ylabel('LD 2')
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plt.legend(loc='lower left')
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# plt.tight_layout()
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# plt.savefig('./images/lda3.png', dpi=300)
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plt.show()
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X_test_lda = lda.transform(X_test_std)
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plot_decision_regions(X_test_lda, y_test, classifier=lr)
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plt.xlabel('LD 1')
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plt.ylabel('LD 2')
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plt.legend(loc='lower left')
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# plt.tight_layout()
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# plt.savefig('./images/lda4.png', dpi=300)
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plt.show()
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#############################################################################
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print(50 * '=')
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print('Section: Implementing a kernel principal component analysis in Python')
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print(50 * '-')
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def rbf_kernel_pca(X, gamma, n_components):
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"""
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RBF kernel PCA implementation.
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Parameters
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------------
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X: {NumPy ndarray}, shape = [n_samples, n_features]
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gamma: float
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Tuning parameter of the RBF kernel
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n_components: int
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Number of principal components to return
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Returns
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------------
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X_pc: {NumPy ndarray}, shape = [n_samples, k_features]
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Projected dataset
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"""
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# Calculate pairwise squared Euclidean distances
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# in the MxN dimensional dataset.
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sq_dists = pdist(X, 'sqeuclidean')
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# Convert pairwise distances into a square matrix.
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mat_sq_dists = squareform(sq_dists)
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# Compute the symmetric kernel matrix.
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K = exp(-gamma * mat_sq_dists)
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# Center the kernel matrix.
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N = K.shape[0]
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one_n = np.ones((N, N)) / N
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K = K - one_n.dot(K) - K.dot(one_n) + one_n.dot(K).dot(one_n)
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# Obtaining eigenpairs from the centered kernel matrix
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# numpy.eigh returns them in sorted order
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eigvals, eigvecs = eigh(K)
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# Collect the top k eigenvectors (projected samples)
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X_pc = np.column_stack((eigvecs[:, -i]
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for i in range(1, n_components + 1)))
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return X_pc
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#############################################################################
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print(50 * '=')
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print('Section: Example 1: Separating half-moon shapes')
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print(50 * '-')
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X, y = make_moons(n_samples=100, random_state=123)
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plt.scatter(X[y == 0, 0], X[y == 0, 1], color='red', marker='^', alpha=0.5)
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plt.scatter(X[y == 1, 0], X[y == 1, 1], color='blue', marker='o', alpha=0.5)
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# plt.tight_layout()
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# plt.savefig('./figures/half_moon_1.png', dpi=300)
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plt.show()
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scikit_pca = PCA(n_components=2)
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X_spca = scikit_pca.fit_transform(X)
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fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(7, 3))
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ax[0].scatter(X_spca[y == 0, 0], X_spca[y == 0, 1],
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color='red', marker='^', alpha=0.5)
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ax[0].scatter(X_spca[y == 1, 0], X_spca[y == 1, 1],
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color='blue', marker='o', alpha=0.5)
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ax[1].scatter(X_spca[y == 0, 0], np.zeros((50, 1)) + 0.02,
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color='red', marker='^', alpha=0.5)
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ax[1].scatter(X_spca[y == 1, 0], np.zeros((50, 1)) - 0.02,
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color='blue', marker='o', alpha=0.5)
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ax[0].set_xlabel('PC1')
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ax[0].set_ylabel('PC2')
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ax[1].set_ylim([-1, 1])
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ax[1].set_yticks([])
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ax[1].set_xlabel('PC1')
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# plt.tight_layout()
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# plt.savefig('./figures/half_moon_2.png', dpi=300)
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plt.show()
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X_kpca = rbf_kernel_pca(X, gamma=15, n_components=2)
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fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(7, 3))
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ax[0].scatter(X_kpca[y == 0, 0], X_kpca[y == 0, 1],
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color='red', marker='^', alpha=0.5)
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ax[0].scatter(X_kpca[y == 1, 0], X_kpca[y == 1, 1],
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color='blue', marker='o', alpha=0.5)
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ax[1].scatter(X_kpca[y == 0, 0], np.zeros((50, 1)) + 0.02,
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color='red', marker='^', alpha=0.5)
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ax[1].scatter(X_kpca[y == 1, 0], np.zeros((50, 1)) - 0.02,
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color='blue', marker='o', alpha=0.5)
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ax[0].set_xlabel('PC1')
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ax[0].set_ylabel('PC2')
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ax[1].set_ylim([-1, 1])
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ax[1].set_yticks([])
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ax[1].set_xlabel('PC1')
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ax[0].xaxis.set_major_formatter(FormatStrFormatter('%0.1f'))
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ax[1].xaxis.set_major_formatter(FormatStrFormatter('%0.1f'))
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# plt.tight_layout()
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# plt.savefig('./figures/half_moon_3.png', dpi=300)
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plt.show()
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#############################################################################
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print(50 * '=')
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print('Section: Example 2: Separating concentric circles')
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print(50 * '-')
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X, y = make_circles(n_samples=1000, random_state=123, noise=0.1, factor=0.2)
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plt.scatter(X[y == 0, 0], X[y == 0, 1], color='red', marker='^', alpha=0.5)
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plt.scatter(X[y == 1, 0], X[y == 1, 1], color='blue', marker='o', alpha=0.5)
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# plt.tight_layout()
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# plt.savefig('./figures/circles_1.png', dpi=300)
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plt.show()
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scikit_pca = PCA(n_components=2)
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X_spca = scikit_pca.fit_transform(X)
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fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(7, 3))
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ax[0].scatter(X_spca[y == 0, 0], X_spca[y == 0, 1],
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color='red', marker='^', alpha=0.5)
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ax[0].scatter(X_spca[y == 1, 0], X_spca[y == 1, 1],
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color='blue', marker='o', alpha=0.5)
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ax[1].scatter(X_spca[y == 0, 0], np.zeros((500, 1)) + 0.02,
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color='red', marker='^', alpha=0.5)
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ax[1].scatter(X_spca[y == 1, 0], np.zeros((500, 1)) - 0.02,
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color='blue', marker='o', alpha=0.5)
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ax[0].set_xlabel('PC1')
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ax[0].set_ylabel('PC2')
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ax[1].set_ylim([-1, 1])
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ax[1].set_yticks([])
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ax[1].set_xlabel('PC1')
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# plt.tight_layout()
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# plt.savefig('./figures/circles_2.png', dpi=300)
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plt.show()
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|
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X_kpca = rbf_kernel_pca(X, gamma=15, n_components=2)
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fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(7, 3))
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ax[0].scatter(X_kpca[y == 0, 0], X_kpca[y == 0, 1],
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color='red', marker='^', alpha=0.5)
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ax[0].scatter(X_kpca[y == 1, 0], X_kpca[y == 1, 1],
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color='blue', marker='o', alpha=0.5)
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|
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ax[1].scatter(X_kpca[y == 0, 0], np.zeros((500, 1)) + 0.02,
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color='red', marker='^', alpha=0.5)
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ax[1].scatter(X_kpca[y == 1, 0], np.zeros((500, 1)) - 0.02,
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color='blue', marker='o', alpha=0.5)
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ax[0].set_xlabel('PC1')
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ax[0].set_ylabel('PC2')
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ax[1].set_ylim([-1, 1])
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ax[1].set_yticks([])
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ax[1].set_xlabel('PC1')
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# plt.tight_layout()
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# plt.savefig('./figures/circles_3.png', dpi=300)
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plt.show()
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|
|
|
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|
#############################################################################
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print(50 * '=')
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|
print('Section: Projecting new data points')
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|
print(50 * '-')
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|
|
|
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|
def rbf_kernel_pca(X, gamma, n_components):
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|
"""
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|
RBF kernel PCA implementation.
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|
|
|
Parameters
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|
------------
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|
X: {NumPy ndarray}, shape = [n_samples, n_features]
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|
|
|
gamma: float
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|
Tuning parameter of the RBF kernel
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|
|
|
n_components: int
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|
Number of principal components to return
|
|
|
|
Returns
|
|
------------
|
|
X_pc: {NumPy ndarray}, shape = [n_samples, k_features]
|
|
Projected dataset
|
|
|
|
lambdas: list
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|
Eigenvalues
|
|
|
|
"""
|
|
# Calculate pairwise squared Euclidean distances
|
|
# in the MxN dimensional dataset.
|
|
sq_dists = pdist(X, 'sqeuclidean')
|
|
|
|
# Convert pairwise distances into a square matrix.
|
|
mat_sq_dists = squareform(sq_dists)
|
|
|
|
# Compute the symmetric kernel matrix.
|
|
K = exp(-gamma * mat_sq_dists)
|
|
|
|
# Center the kernel matrix.
|
|
N = K.shape[0]
|
|
one_n = np.ones((N, N)) / N
|
|
K = K - one_n.dot(K) - K.dot(one_n) + one_n.dot(K).dot(one_n)
|
|
|
|
# Obtaining eigenpairs from the centered kernel matrix
|
|
# numpy.eigh returns them in sorted order
|
|
eigvals, eigvecs = eigh(K)
|
|
|
|
# Collect the top k eigenvectors (projected samples)
|
|
alphas = np.column_stack((eigvecs[:, -i]
|
|
for i in range(1, n_components + 1)))
|
|
|
|
# Collect the corresponding eigenvalues
|
|
lambdas = [eigvals[-i] for i in range(1, n_components + 1)]
|
|
|
|
return alphas, lambdas
|
|
|
|
|
|
X, y = make_moons(n_samples=100, random_state=123)
|
|
alphas, lambdas = rbf_kernel_pca(X, gamma=15, n_components=1)
|
|
|
|
|
|
x_new = X[25]
|
|
print('New data point x_new:', x_new)
|
|
|
|
x_proj = alphas[25] # original projection
|
|
print('Original projection x_proj:', x_proj)
|
|
|
|
|
|
def project_x(x_new, X, gamma, alphas, lambdas):
|
|
pair_dist = np.array([np.sum((x_new - row)**2) for row in X])
|
|
k = np.exp(-gamma * pair_dist)
|
|
return k.dot(alphas / lambdas)
|
|
|
|
|
|
# projection of the "new" datapoint
|
|
x_reproj = project_x(x_new, X, gamma=15, alphas=alphas, lambdas=lambdas)
|
|
print('Reprojection x_reproj:', x_reproj)
|
|
|
|
plt.scatter(alphas[y == 0, 0], np.zeros((50)),
|
|
color='red', marker='^', alpha=0.5)
|
|
plt.scatter(alphas[y == 1, 0], np.zeros((50)),
|
|
color='blue', marker='o', alpha=0.5)
|
|
plt.scatter(x_proj, 0, color='black',
|
|
label='original projection of point X[25]', marker='^', s=100)
|
|
plt.scatter(x_reproj, 0, color='green',
|
|
label='remapped point X[25]', marker='x', s=500)
|
|
plt.legend(scatterpoints=1)
|
|
|
|
# plt.tight_layout()
|
|
# plt.savefig('./figures/reproject.png', dpi=300)
|
|
plt.show()
|
|
|
|
|
|
#############################################################################
|
|
print(50 * '=')
|
|
print('Section: Kernel principal component analysis in scikit-learn')
|
|
print(50 * '-')
|
|
|
|
|
|
X, y = make_moons(n_samples=100, random_state=123)
|
|
scikit_kpca = KernelPCA(n_components=2, kernel='rbf', gamma=15)
|
|
X_skernpca = scikit_kpca.fit_transform(X)
|
|
|
|
plt.scatter(X_skernpca[y == 0, 0], X_skernpca[y == 0, 1],
|
|
color='red', marker='^', alpha=0.5)
|
|
plt.scatter(X_skernpca[y == 1, 0], X_skernpca[y == 1, 1],
|
|
color='blue', marker='o', alpha=0.5)
|
|
|
|
plt.xlabel('PC1')
|
|
plt.ylabel('PC2')
|
|
# plt.tight_layout()
|
|
# plt.savefig('./figures/scikit_kpca.png', dpi=300)
|
|
plt.show()
|