Total and explained variance
import pandas as pd
df_wine = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/wine/wine.data', header=None)
from sklearn.cross_validation import train_test_split
from sklearn.preprocessing import StandardScaler
X, y = df_wine.iloc[:, 1:].values, df_wine.iloc[:, 0].values
X_train, X_test, y_train, y_test = \
train_test_split(X, y,
test_size=0.3, random_state=0)
sc = StandardScaler()
X_train_std = sc.fit_transform(X_train)
X_test_std = sc.fit_transform(X_test)
import numpy as np
cov_mat = np.cov(X_train_std.T)
eigen_vals, eigen_vecs = np.linalg.eig(cov_mat)
print('\nEigenvalues \n%s' % eigen_vals)
tot = sum(eigen_vals)
var_exp = [(i / tot) for i in
sorted(eigen_vals, reverse=True)]
cum_var_exp = np.cumsum(var_exp)
import matplotlib.pyplot as plt
plt.bar(range(1,14), var_exp, alpha=0.5, align='center',
label='individual explained variance')
plt.step(range(1,14), cum_var_exp, where='mid',
label='cumulative explained variance')
plt.ylabel('Explained variance ratio')
plt.xlabel('Principal components')
plt.legend(loc='best')
plt.show()
Feature transformation
eigen_pairs =[(np.abs(eigen_vals[i]),eigen_vecs[:,i]) for i inrange(len(eigen_vals))]
eigen_pairs.sort(reverse=True)
w= np.hstack((eigen_pairs[0][1][:, np.newaxis], eigen_pairs[1][1][:, np.newaxis]))
print('Matrix W:\n',w)
X_train_std[0].dot(w)
X_train_pca = X_train_std.dot(w)
colors = ['r', 'b', 'g']
markers = ['s', 'x', 'o']
for l, c, m in zip(np.unique(y_train), colors, markers):
plt.scatter(X_train_pca[y_train==l, 0],
X_train_pca[y_train==l, 1],
c=c, label=l, marker=m)
plt.xlabel('PC 1')
plt.ylabel('PC 2')
plt.legend(loc='lower left')
plt.show()
Principal component analysis in scikit-learn
from matplotlib.colors import ListedColormap
def plot_decision_regions(X, y, classifier, resolution=0.02):
# setup marker generator and color map
markers = ('s', 'x', 'o', '^', 'v')
colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
cmap = ListedColormap(colors[:len(np.unique(y))])
# plot the decision surface
x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),
np.arange(x2_min, x2_max, resolution))
Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
Z = Z.reshape(xx1.shape)
plt.contourf(xx1, xx2, Z, alpha=0.4, cmap=cmap)
plt.xlim(xx1.min(), xx1.max())
plt.ylim(xx2.min(), xx2.max())
# plot class samples
for idx, cl in enumerate(np.unique(y)):
plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1],
alpha=0.8, c=cmap(idx),
marker=markers[idx], label=cl)
from sklearn.linear_model import LogisticRegression
from sklearn.decomposition import PCA
pca = PCA(n_components=2)
lr = LogisticRegression()
X_train_pca = pca.fit_transform(X_train_std)
X_test_pca = pca.transform(X_test_std)
lr.fit(X_train_pca, y_train)
plot_decision_regions(X_train_pca, y_train, classifier=lr)
plt.xlabel('PC1')
plt.ylabel('PC2')
plt.legend(loc='lower left')
plt.show()
pca = PCA(n_components=None)
X_train_pca = pca.fit_transform(X_train_std)
pca.explained_variance_ratio_
-- Supervised data compression vis linear discriminant analysis
-- Computing the scatter matrices
np.set_printoptions(precision=4)
mean_vecs = []
for label in range(1,4):
mean_vecs.append(np.mean(
X_train_std[y_train==label], axis=0))
print('MV %s: %s\n' %(label, mean_vecs[label-1]))
d = 13 # number of features
S_W = np.zeros((d, d))
for label,mv in zip(range(1,4), mean_vecs):
class_scatter = np.zeros((d, d))
for row in X[y == label]:
row, mv = row.reshape(d, 1), mv.reshape(d, 1)
class_scatter += (row-mv).dot((row-mv).T)
S_W += class_scatter
print('Within-class scatter matrix: %sx%s' % (S_W.shape[0], S_W.shape[1]))
print('Class label distribution: %s' % np.bincount(y_train)[1:])
mean_overall = np.mean(X_train_std, axis=0)
d = 13 # number of features
S_B = np.zeros((d, d))
for i,mean_vec in enumerate(mean_vecs):
n = X[y==i+1, :].shape[0]
mean_vec = mean_vec.reshape(d, 1)
mean_overall = mean_overall.reshape(d, 1)
S_B += n * (mean_vec - mean_overall).dot(
(mean_vec - mean_overall).T)
print('Between-class scatter matrix: %sx%s'
% (S_B.shape[0], S_B.shape[1]))
Selecting linear discriminants for the new feature subspace
eigen_vals, eigen_vecs =np.linalg.eig(np.linalg.inv(S_W).dot(S_B))
eigen_pairs = [(np.abs(eigen_vals[i]), eigen_vecs[:,i])
for i in range(len(eigen_vals))]
eigen_pairs = sorted(eigen_pairs,
key=lambda k: k[0], reverse=True)
print('Eigenvalues in decreasing order:\n')
for eigen_val in eigen_pairs:
print(eigen_val[0])
tot = sum(eigen_vals.real)
discr = [(i / tot) for i in sorted(eigen_vals.real, reverse=True)]
cum_discr = np.cumsum(discr)
plt.bar(range(1, 14), discr, alpha=0.5, align='center',
label='individual "discriminability"')
plt.step(range(1, 14), cum_discr, where='mid',
label='cumulative "discriminability"')
plt.ylabel('"discriminability" ratio')
plt.xlabel('Linear Discriminants')
plt.ylim([-0.1, 1.1])
plt.legend(loc='best')
plt.show()
w = np.hstack((eigen_pairs[0][1][:, np.newaxis].real,
eigen_pairs[1][1][:, np.newaxis].real))
print('Matrix W:\n', w)
Projecting samples onto the new feature space
X_train_lda = X_train_std.dot(w)
colors = ['r', 'b', 'g']
markers = ['s', 'x', 'o']
for l, c, m in zip(np.unique(y_train), colors, markers):
plt.scatter(X_train_lda[y_train==l, 0],
X_train_lda[y_train==l, 1],
c=c, label=l, marker=m)
plt.xlabel('LD 1')
plt.ylabel('LD 2')
plt.legend(loc='upper right')
plt.show()
LDA via scikit-learn
from sklearn.lda import LDA
lda = LDA(n_components=2)
X_train_lda = lda.fit_transform(X_train_std, y_train)
lr = LogisticRegression()
lr = lr.fit(X_train_lda, y_train)
plot_decision_regions(X_train_lda, y_train, classifier=lr)
plt.xlabel('LD 1')
plt.ylabel('LD 2')
plt.legend(loc='lower left')
plt.show()
X_test_lda = lda.transform(X_test_std)
plot_decision_regions(X_test_lda, y_test, classifier=lr)
plt.xlabel('LD 1')
plt.ylabel('LD 2')
plt.legend(loc='lower left')
plt.show()
-- Using kernel principal component analysis for nonlinear mappings
Kernel functions and the kernel trick
Implementing a kernel principal component analysis in Python
from scipy.spatial.distance import pdist, squareform
from scipy import exp
from scipy.linalg import eigh
import numpy as np
def rbf_kernel_pca(X, gamma, n_components):
"""
RBF kernel PCA implementation.
Parameters
------------
X: {NumPy ndarray}, shape = [n_samples, n_features]
gamma: float
Tuning parameter of the RBF kernel
n_components: int
Number of principal components to return
Returns
------------
X_pc: {NumPy ndarray}, shape = [n_samples, k_features]
Projected dataset
"""
# 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)
X_pc = np.column_stack((eigvecs[:, -i]
for i in range(1, n_components + 1)))
return X_pc
Example 1 ñ separating half-moon shapes
from sklearn.datasets import make_moons
X, y = make_moons(n_samples=100, random_state=123)
plt.scatter(X[y==0, 0], X[y==0, 1],
color='red', marker='^', alpha=0.5)
plt.scatter(X[y==1, 0], X[y==1, 1],
color='blue', marker='o', alpha=0.5)
plt.show()
from sklearn.decomposition import PCA
scikit_pca = PCA(n_components=2)
X_spca = scikit_pca.fit_transform(X)
fig, ax = plt.subplots(nrows=1,ncols=2, figsize=(7,3))
ax[0].scatter(X_spca[y==0, 0], X_spca[y==0, 1],
color='red', marker='^', alpha=0.5)
ax[0].scatter(X_spca[y==1, 0], X_spca[y==1, 1],
color='blue', marker='o', alpha=0.5)
ax[1].scatter(X_spca[y==0, 0], np.zeros((50,1))+0.02,
color='red', marker='^', alpha=0.5)
ax[1].scatter(X_spca[y==1, 0], np.zeros((50,1))-0.02,
color='blue', marker='o', alpha=0.5)
ax[0].set_xlabel('PC1')
ax[0].set_ylabel('PC2')
ax[1].set_ylim([-1, 1])
ax[1].set_yticks([])
ax[1].set_xlabel('PC1')
plt.show()
from matplotlib.ticker import FormatStrFormatter
X_kpca = rbf_kernel_pca(X, gamma=15, n_components=2)
fig, ax = plt.subplots(nrows=1,ncols=2, figsize=(7,3))
ax[0].scatter(X_kpca[y==0, 0], X_kpca[y==0, 1],
color='red', marker='^', alpha=0.5)
ax[0].scatter(X_kpca[y==1, 0], X_kpca[y==1, 1],
color='blue', marker='o', alpha=0.5)
ax[1].scatter(X_kpca[y==0, 0], np.zeros((50,1))+0.02,
color='red', marker='^', alpha=0.5)
ax[1].scatter(X_kpca[y==1, 0], np.zeros((50,1))-0.02,
color='blue', marker='o', alpha=0.5)
ax[0].set_xlabel('PC1')
ax[0].set_ylabel('PC2')
ax[1].set_ylim([-1, 1])
ax[1].set_yticks([])
ax[1].set_xlabel('PC1')
ax[0].xaxis.set_major_formatter(FormatStrFormatter('%0.1f'))
ax[1].xaxis.set_major_formatter(FormatStrFormatter('%0.1f'))
plt.show()
Example 2 ñ separating concentric circles
from sklearn.datasets import make_circles
X, y = make_circles(n_samples=1000,
random_state=123, noise=0.1, factor=0.2)
plt.scatter(X[y==0, 0], X[y==0, 1],
color='red', marker='^', alpha=0.5)
plt.scatter(X[y==1, 0], X[y==1, 1],
color='blue', marker='o', alpha=0.5)
plt.show()
scikit_pca = PCA(n_components=2)
X_spca = scikit_pca.fit_transform(X)
fig, ax = plt.subplots(nrows=1,ncols=2, figsize=(7,3))
ax[0].scatter(X_spca[y==0, 0], X_spca[y==0, 1],
color='red', marker='^', alpha=0.5)
ax[0].scatter(X_spca[y==1, 0], X_spca[y==1, 1],
color='blue', marker='o', alpha=0.5)
ax[1].scatter(X_spca[y==0, 0], np.zeros((500,1))+0.02,
color='red', marker='^', alpha=0.5)
ax[1].scatter(X_spca[y==1, 0], np.zeros((500,1))-0.02,
color='blue', marker='o', alpha=0.5)
ax[0].set_xlabel('PC1')
ax[0].set_ylabel('PC2')
ax[1].set_ylim([-1, 1])
ax[1].set_yticks([])
ax[1].set_xlabel('PC1')
plt.show()
X_kpca = rbf_kernel_pca(X, gamma=15, n_components=2)
fig, ax = plt.subplots(nrows=1,ncols=2, figsize=(7,3))
ax[0].scatter(X_kpca[y==0, 0], X_kpca[y==0, 1],
color='red', marker='^', alpha=0.5)
ax[0].scatter(X_kpca[y==1, 0], X_kpca[y==1, 1],
color='blue', marker='o', alpha=0.5)
ax[1].scatter(X_kpca[y==0, 0], np.zeros((500,1))+0.02,
color='red', marker='^', alpha=0.5)
ax[1].scatter(X_kpca[y==1, 0], np.zeros((500,1))-0.02,
color='blue', marker='o', alpha=0.5)
ax[0].set_xlabel('PC1')
ax[0].set_ylabel('PC2')
ax[1].set_ylim([-1, 1])
ax[1].set_yticks([])
ax[1].set_xlabel('PC1')
plt.show()
from scipy.spatial.distance import pdist, squareform
from scipy import exp
from scipy.linalg import eigh
import numpy as np
def rbf_kernel_pca(X, gamma, n_components):
"""
RBF kernel PCA implementation.
Parameters
------------
X: {NumPy ndarray}, shape = [n_samples, n_features]
gamma: float
Tuning parameter of the RBF kernel
n_components: int
Number of principal components to return
Returns
------------
X_pc: {NumPy ndarray}, shape = [n_samples, k_features]
Projected dataset
lambdas: list
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]
x_new
x_proj = alphas[25] # original projection
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)
x_reproj = project_x(x_new, X,
gamma=15, alphas=alphas, lambdas=lambdas)
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.show()
Kernel principal component analysis in scikit-learn
from sklearn.decomposition import KernelPCA
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.show()
No comments:
Post a Comment