feature_dict = {i:label for i,label in zip(
range(4),
('sepal length in cm',
'sepal width in cm',
'petal length in cm',
'petal width in cm', ))}
import pandas as pd
df = pd.io.parsers.read_csv(
filepath_or_buffer='https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data',
header=None,
sep=',',
)
df.columns = [l for i,l in sorted(feature_dict.items())] + ['class label']
df.dropna(how="all", inplace=True) # to drop the empty line at file-end
df.tail()
from sklearn.preprocessing import LabelEncoder
X = df[['sepal length in cm','sepal width in cm','petal length in cm','petal width in cm']].values
y = df['class label'].values
enc = LabelEncoder()
label_encoder = enc.fit(y)
y = label_encoder.transform(y) + 1
#label_dict = {1: 'Setosa', 2: 'Versicolor', 3:'Virginica'}
分别求三种鸢尾花数据在不同特征维度上的均值向量 mi
import numpy as np
np.set_printoptions(precision=4)
mean_vectors = []
for cl in range(1,4):
mean_vectors.append(np.mean(X[y==cl], axis=0))
print('Mean Vector class %s: %s\n' %(cl, mean_vectors[cl-1]))
输出 ：
Mean Vector class 1: [ 5.006 3.418 1.464 0.244]
Mean Vector class 2: [ 5.936 2.77 4.26 1.326]
Mean Vector class 3: [ 6.588 2.974 5.552 2.026]
计算两个 4×4 维矩阵：类内散布矩阵和类间散布矩阵
S_W = np.zeros((4,4))
for cl,mv in zip(range(1,4), mean_vectors):
class_sc_mat = np.zeros((4,4)) # scatter matrix for every class
for row in X[y == cl]:
row, mv = row.reshape(4,1), mv.reshape(4,1) # make column vectors
class_sc_mat += (row-mv).dot((row-mv).T)
S_W += class_sc_mat # sum class scatter matrices
print('within-class Scatter Matrix:\n', S_W)
输出 ：
within-class Scatter Matrix:
[[ 38.9562 13.683 24.614 5.6556]
[ 13.683 17.035 8.12 4.9132]
[ 24.614 8.12 27.22 6.2536]
[ 5.6556 4.9132 6.2536 6.1756]]
overall_mean = np.mean(X, axis=0)
S_B = np.zeros((4,4))
for i,mean_vec in enumerate(mean_vectors):
n = X[y==i+1,:].shape[0]
mean_vec = mean_vec.reshape(4,1) # make column vector
overall_mean = overall_mean.reshape(4,1) # make column vector
S_B += n * (mean_vec - overall_mean).dot((mean_vec - overall_mean).T)
print('between-class Scatter Matrix:\n', S_B)
输出 ：
between-class Scatter Matrix:
[[ 63.2121 -19.534 165.1647 71.3631]
[ -19.534 10.9776 -56.0552 -22.4924]
[ 165.1647 -56.0552 436.6437 186.9081]
[ 71.3631 -22.4924 186.9081 80.6041]]
eig_vals, eig_vecs = np.linalg.eig(np.linalg.inv(S_W).dot(S_B))
for i in range(len(eig_vals)):
eigvec_sc = eig_vecs[:,i].reshape(4,1)
print('\nEigenvector {}: \n{}'.format(i+1, eigvec_sc.real))
print('Eigenvalue {:}: {:.2e}'.format(i+1, eig_vals[i].real))
输出 ：
Eigenvector 1:
[[ 0.2049]
[ 0.3871]
[-0.5465]
[-0.7138]]
Eigenvalue 1: 3.23e+01
Eigenvector 2:
[[-0.009 ]
[-0.589 ]
[ 0.2543]
[-0.767 ]]
Eigenvalue 2: 2.78e-01
Eigenvector 3:
[[-0.7113]
[ 0.0353]
[-0.0267]
[ 0.7015]]
Eigenvalue 3: -5.76e-15
Eigenvector 4:
[[ 0.422 ]
[-0.4364]
[-0.4851]
[ 0.6294]]
Eigenvalue 4: 7.80e-15
特征值与特征向量：
- 特征向量：表示映射方向
- 特征值：特征向量的重要程度
#Make a list of (eigenvalue, eigenvector) tuples
eig_pairs = [(np.abs(eig_vals[i]), eig_vecs[:,i]) for i in range(len(eig_vals))]
# Sort the (eigenvalue, eigenvector) tuples from high to low
eig_pairs = sorted(eig_pairs, key=lambda k: k[0], reverse=True)
# Visually confirm that the list is correctly sorted by decreasing eigenvalues
print('Eigenvalues in decreasing order:\n')
for i in eig_pairs:
print(i[0])
输出 ：
Eigenvalues in decreasing order:
32.2719577997
0.27756686384
7.7995841654e-15
5.76433252705e-15
print('Variance explained:\n')
eigv_sum = sum(eig_vals)
for i,j in enumerate(eig_pairs):
print('eigenvalue {0:}: {1:.2%}'.format(i+1, (j[0]/eigv_sum).real))
输出 ：
Variance explained:
eigenvalue 1: 99.15%
eigenvalue 2: 0.85%
eigenvalue 3: 0.00%
eigenvalue 4: 0.00%
选择前两维特征
W = np.hstack((eig_pairs[0][1].reshape(4,1), eig_pairs[1][1].reshape(4,1)))
print('Matrix W:\n', W.real)
输出 ：
Matrix W:
[[ 0.2049 -0.009 ]
[ 0.3871 -0.589 ]
[-0.5465 0.2543]
[-0.7138 -0.767 ]]
X_lda = X.dot(W)
assert X_lda.shape == (150,2), "The matrix is not 150x2 dimensional."
from matplotlib import pyplot as plt
def plot_step_lda():
ax = plt.subplot(111)
for label,marker,color in zip(
range(1,4),('^', 's', 'o'),('blue', 'red', 'green')):
plt.scatter(x=X_lda[:,0].real[y == label],
y=X_lda[:,1].real[y == label],
marker=marker,
color=color,
alpha=0.5,
label=label_dict[label]
)
plt.xlabel('LD1')
plt.ylabel('LD2')
leg = plt.legend(loc='upper right', fancybox=True)
leg.get_frame().set_alpha(0.5)
plt.title('LDA: Iris projection onto the first 2 linear discriminants')
# hide axis ticks
plt.tick_params(axis="both", which="both", bottom="off", top="off",
labelbottom="on", left="off", right="off", labelleft="on")
# remove axis spines
ax.spines["top"].set_visible(False)
ax.spines["right"].set_visible(False)
ax.spines["bottom"].set_visible(False)
ax.spines["left"].set_visible(False)
plt.grid()
plt.tight_layout
plt.show()
plot_step_lda()
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA
# LDA
sklearn_lda = LDA(n_components=2)
X_lda_sklearn = sklearn_lda.fit_transform(X, y)
def plot_scikit_lda(X, title):
ax = plt.subplot(111)
for label,marker,color in zip(
range(1,4),('^', 's', 'o'),('blue', 'red', 'green')):
plt.scatter(x=X[:,0][y == label],
y=X[:,1][y == label] * -1, # flip the figure
marker=marker,
color=color,
alpha=0.5,
label=label_dict[label])
plt.xlabel('LD1')
plt.ylabel('LD2')
leg = plt.legend(loc='upper right', fancybox=True)
leg.get_frame().set_alpha(0.5)
plt.title(title)
# hide axis ticks
plt.tick_params(axis="both", which="both", bottom="off", top="off",
labelbottom="on", left="off", right="off", labelleft="on")
# remove axis spines
ax.spines["top"].set_visible(False)
ax.spines["right"].set_visible(False)
ax.spines["bottom"].set_visible(False)
ax.spines["left"].set_visible(False)
plt.grid()
plt.tight_layout
plt.show()
plot_step_lda()
plot_scikit_lda(X_lda_sklearn, title='Default LDA via scikit-learn')
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