Requirements：
Python环境部署：http://blog.csdn.net/luzhangting/article/details/61414485
PCA原理：http://blog.csdn.net/zhongkelee/article/details/44064401
第一步：生成三维的样本数据
生成40个三维数据，分两类，每一类20个
第一类：均值[0,0,0],方差[1,0,0],[0,1,0],[0,0,1]；
第一类：均值[1,1,1],方差[1,0,0],[0,1,0],[0,0,1]。
import numpy as np
np.random.seed(1)
mu_vec1 = np.array([0,0,0])
cov_mat1 = np.array([[1,0,0],[0,1,0],[0,0,1]])
class1_sample = np.random.multivariate_normal(mu_vec1, cov_mat1, 20).T
assert class1_sample.shape == (3,20), "The matrix has not the dimensions 3x20"
mu_vec2 = np.array([1,1,1])
cov_mat2 = np.array([[1,0,0],[0,1,0],[0,0,1]])
class2_sample = np.random.multivariate_normal(mu_vec2, cov_mat2, 20).T
assert class2_sample.shape == (3,20), "The matrix has not the dimensions 3x20"画图查看生成好的数据
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from mpl_toolkits.mplot3d import proj3d
fig = plt.figure(figsize=(8,8))
ax = fig.add_subplot(111, projection='3d')
plt.rcParams['legend.fontsize'] = 10
ax.plot(class1_sample[0,:], class1_sample[1,:], class1_sample[2,:], 'o', markersize=8, color='blue', alpha=0.5, label='class1')
ax.plot(class2_sample[0,:], class2_sample[1,:], class2_sample[2,:], '^', markersize=8, alpha=0.5, color='red', label='class2')
plt.title('Samples for class 1 and class 2')
ax.legend(loc='upper right')
plt.show()
整理数据，因为PCA方法不对数据进行分类处理，这也是PCA的一个缺点
all_samples = np.concatenate((class1_sample, class2_sample), axis=1)
assert all_samples.shape == (3,40), "The matrix has not the dimensions 3x40"第二步：计算均值向量
mean_x = np.mean(all_samples[0,:])
mean_y = np.mean(all_samples[1,:])
mean_z = np.mean(all_samples[2,:])
mean_vector = np.array([[mean_x],[mean_y],[mean_z]])
print('Mean Vector:\n', mean_vector)Mean Vector:
[[ 0.41667492]
[ 0.69848315]
[ 0.49242335]]
第三步：计算离散度度矩阵Scatter Matrix或者协方差矩阵Covariance Matrix
scatter_matrix = np.zeros((3,3))
for i in range(all_samples.shape[1]):
scatter_matrix += (all_samples[:,i].reshape(3,1) - mean_vector).dot((all_samples[:,i].reshape(3,1) - mean_vector).T)
print('Scatter Matrix:\n', scatter_matrix)Scatter Matrix:
[[ 38.4878051 10.50787213 11.13746016]
[10.50787213 36.23651274 11.96598642 ]
[11.13746016 11.96598642 49.73596619 ]]
cov_mat = np.cov([all_samples[0,:],all_samples[1,:],all_samples[2,:]])
print('Covariance Matrix:\n', cov_mat)Covariance Matrix:
[[ 0.9868668 0.26943262 0.2855759]
[ 0.26943262 0.92914135 0.30682016]
[ 0.2855759 0.30682016 1.27528118]]
第四步：计算特征值和特征向量
eig_val_sc, eig_vec_sc = np.linalg.eig(scatter_matrix)
eig_val_cov, eig_vec_cov = np.linalg.eig(cov_mat)
for i in range(len(eig_val_sc)):
eigvec_sc = eig_vec_sc[:,i].reshape(1,3).T
eigvec_cov = eig_vec_cov[:,i].reshape(1,3).T
assert eigvec_sc.all() == eigvec_cov.all(), 'Eigenvectors are not identical'
print('Eigenvector {}: \n{}'.format(i+1, eigvec_sc))
print('Eigenvalue {} from scatter matrix: {}'.format(i+1, eig_val_sc[i]))
print('Eigenvalue {} from covariance matrix: {}'.format(i+1, eig_val_cov[i]))
print('Scaling factor: ', eig_val_sc[i]/eig_val_cov[i])
print(40 * '-')Eigenvector 1:
[[-0.49210223]
[-0.47927902]
[-0.72672348]]
Eigenvalue 1 from scatter matrix: 65.1693677908
Eigenvalue 1 from covariance matrix: 1.67100943053
(‘Scaling factor: ‘, 39.000000000000021)
Eigenvector 2:
[[-0.64670286]
[-0.35756937]
[ 0.67373552]]
Eigenvalue 2 from scatter matrix: 32.6947129632
Eigenvalue 2 from covariance matrix: 0.838325973416
(‘Scaling factor: ‘, 39.000000000000021)
Eigenvector 3:
[[ 0.58276136]
[-0.8015209 ]
[ 0.13399043]]
Eigenvalue 3 from scatter matrix: 26.5962032821
Eigenvalue 3 from covariance matrix: 0.68195393031
(‘Scaling factor: ‘, 39.000000000000028)
为了满足好奇心，将特征向量画出来看看。
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from mpl_toolkits.mplot3d import proj3d
from matplotlib.patches import FancyArrowPatch
class Arrow3D(FancyArrowPatch):
def __init__(self, xs, ys, zs, *args, **kwargs):
FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs)
self._verts3d = xs, ys, zs
def draw(self, renderer):
xs3d, ys3d, zs3d = self._verts3d
xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)
self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))
FancyArrowPatch.draw(self, renderer)
fig = plt.figure(figsize=(7,7))
ax = fig.add_subplot(111, projection='3d')
ax.plot(all_samples[0,:], all_samples[1,:], all_samples[2,:], 'o', markersize=8, color='green', alpha=0.2)
ax.plot([mean_x], [mean_y], [mean_z], 'o', markersize=10, color='red', alpha=0.5)
for v in eig_vec_sc.T:
a = Arrow3D([mean_x, v[0]], [mean_y, v[1]], [mean_z, v[2]], mutation_scale=20, lw=3, arrowstyle="-|>", color="r")
ax.add_artist(a)
ax.set_xlabel('x_values')
ax.set_ylabel('y_values')
ax.set_zlabel('z_values')
plt.title('Eigenvectors')
plt.show()
第五步：根据特征值对特征向量进行排序；选择Top-k个特征向量
eig_pairs = [(np.abs(eig_val_sc[i]), eig_vec_sc[:,i]) for i in range(len(eig_val_sc))]
eig_pairs.sort(key=lambda x: x[0], reverse=True)
for i in eig_pairs:
print(i[0])65.1693677908
32.6947129632
26.5962032821
本例子中将三维降到两维
matrix_w = np.hstack((eig_pairs[0][1].reshape(3,1), eig_pairs[1][1].reshape(3,1)))
print('Matrix W:\n', matrix_w)Matrix W:
[[-0.49210223, -0.64670286],
[-0.47927902, -0.35756937],
[-0.72672348, 0.67373552]]
第六步：将数据映射到子空间
transformed = matrix_w.T.dot(all_samples)
plt.plot(transformed[0,0:20], transformed[1,0:20], 'o', markersize=7, color='blue', alpha=0.5, label='class1')
plt.plot(transformed[0,20:40], transformed[1,20:40], '^', markersize=7, color='red', alpha=0.5, label='class2')
plt.xlim([-4,4])
plt.ylim([-4,4])
plt.xlabel('x_values')
plt.ylabel('y_values')
plt.legend()
plt.title('Transformed samples with class labels')
plt.show()
第七步：其他实现方式
使用matplotlib.mlab.PCA()
from matplotlib.mlab import PCA as mlabPCA
mlab_pca = mlabPCA(all_samples.T)
print('PC axes in terms of the measurement axes scaled by the standard deviations:\n', mlab_pca.Wt)
plt.plot(mlab_pca.Y[0:20,0],mlab_pca.Y[0:20,1], 'o', markersize=7, color='blue', alpha=0.5, label='class1')
plt.plot(mlab_pca.Y[20:40,0], mlab_pca.Y[20:40,1], '^', markersize=7, color='red', alpha=0.5, label='class2')
plt.xlabel('x_values')
plt.ylabel('y_values')
plt.xlim([-4,4])
plt.ylim([-4,4])
plt.legend()
plt.title('Transformed samples with class labels from matplotlib.mlab.PCA()')
plt.show()PC axes in terms of the measurement axes scaled by the standard deviations:
[[-0.57087338, -0.58973911, -0.5712367 ],
[-0.71046744, 0.006114 , 0.70370352],
[ 0.41150894, -0.80757068, 0.42248075]]
计算结果与上面的有点不同，原因在于在计算协方差矩阵之前进行了标准化的处理。
使用sklearn.decomposition
from sklearn.decomposition import PCA as sklearnPCA
sklearn_pca = sklearnPCA(n_components=2)
sklearn_transf = sklearn_pca.fit_transform(all_samples.T)
sklearn_transf = sklearn_transf * (-1)
# sklearn.decomposition.PCA
plt.plot(sklearn_transf[0:20,0],sklearn_transf[0:20,1], 'o', markersize=7, color='blue', alpha=0.5, label='class1')
plt.plot(sklearn_transf[20:40,0], sklearn_transf[20:40,1], '^', markersize=7, color='red', alpha=0.5, label='class2')
plt.xlabel('x_values')
plt.ylabel('y_values')
plt.xlim([-4,4])
plt.ylim([-4,4])
plt.legend()
plt.title('Transformed samples via sklearn.decomposition.PCA')
plt.show()