主成分分析(Principal Components Analysis, PCA)简介可以参考：/fengbingchun/article/details/78977202
这里是参照 /Articles/_pca_step_by_step.html文章中的code整理的Python代码，实现过程为：
1.随机生成3行*40列的数据集，每一列代表一个样本，前20列属于类1，后20列属于类2；每一个样本特征长度为3；
2.计算每行均值；
3.计算协方差矩阵，产生一个3行*3列的矩阵；
4.由协方差矩阵计算特征向量和特征值；
5.按降序排序特征值和特征向量；
6.选择第一主成分和第二主成分组成一个新的3行*2列的矩阵；
7.根据产生的3行*2列矩阵重建原有数据集。
Python代码如下：
# reference: /Articles/_pca_step_by_step.htmlimport numpy as npfrom matplotlib import pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Dfrom mpl_toolkits.mplot3d import proj3dfrom matplotlib.patches import FancyArrowPatch# 1. generate 40 3-dimensional samples randomly drawn from a multivariate Gaussian distributionnp.random.seed(1) # random seed for consistencymu_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).Tassert class1_sample.shape == (3,20), "The matrix has not the dimensions 3x20"#print("class1_sample:\n", class1_sample)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).Tassert class2_sample.shape == (3,20), "The matrix has not the dimensions 3x20"fig = plt.figure(figsize=(8,8))ax = fig.add_subplot(111, projection='3d')plt.rcParams['legend.fontsize'] = 10ax.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()# Taking the whole dataset ignoring the class labelsall_samples = np.concatenate((class1_sample, class2_sample), axis=1)assert all_samples.shape == (3,40), "The matrix has not the dimensions 3x40"# 2. Computing the d-dimensional mean vectormean_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)# 3. Computing the Covariance Matrixcov_mat = np.cov([all_samples[0,:],all_samples[1,:],all_samples[2,:]])print('Covariance Matrix:\n', cov_mat)# 4. Computing eigenvectors and corresponding eigenvalueseig_val_cov, eig_vec_cov = np.linalg.eig(cov_mat)for i in range(len(eig_val_cov)):eigvec_cov = eig_vec_cov[:,i].reshape(1,3).Tprint('Eigenvector {}: \n{}'.format(i+1, eigvec_cov))print('Eigenvalue {} from covariance matrix: {}'.format(i+1, eig_val_cov[i]))print(40 * '-')# Visualizing the eigenvectorsclass Arrow3D(FancyArrowPatch):def __init__(self, xs, ys, zs, *args, **kwargs):FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs)self._verts3d = xs, ys, zsdef draw(self, renderer):xs3d, ys3d, zs3d = self._verts3dxs, 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_cov.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()# 5 Sorting the eigenvectors by decreasing eigenvalues# Make a list of (eigenvalue, eigenvector) tupleseig_pairs = [(np.abs(eig_val_cov[i]), eig_vec_cov[:,i]) for i in range(len(eig_val_cov))]# Sort the (eigenvalue, eigenvector) tuples from high to loweig_pairs.sort(key=lambda x: x[0], reverse=True)print("eig_pairs:\n", eig_pairs)# Visually confirm that the list is correctly sorted by decreasing eigenvaluesprint("sorted by decreasing eigenvalues:")for i in eig_pairs:print(i[0])# 6 Choosing k eigenvectors with the largest eigenvaluesmatrix_w = np.hstack((eig_pairs[0][1].reshape(3,1), eig_pairs[1][1].reshape(3,1)))print('Matrix W:\n', matrix_w)# 7 Transforming the samples onto the new subspacetransformed = matrix_w.T.dot(all_samples)assert transformed.shape == (2,40), "The matrix is not 2x40 dimensional."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()
执行结果如下：
GitHub：/fengbingchun/NN_Test
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