一.PCA

PCA通過正交變換將一組由線性相關(guān)變量表示的數(shù)據(jù)轉(zhuǎn)換為少數(shù)幾個由線性無關(guān)變量表示的數(shù)據(jù)，這幾個線性無關(guān)的變量就是主成分。PCA通過將高維數(shù)據(jù)維度減少到少數(shù)幾個維度，本質(zhì)上屬于一種數(shù)據(jù)降維方法，也可以用來探索數(shù)據(jù)的內(nèi)在結(jié)構(gòu)。?

可看出如果兩個隨機(jī)變量的分布相似，它們的協(xié)方差很大。?

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import numpy as npclass PCA():# 計(jì)算協(xié)方差矩陣def calculate_covariance_matrix(self, X):m = X.shape[0]# 數(shù)據(jù)標(biāo)準(zhǔn)化X = X - np.mean(X, axis=0)#(x1 - Ex1) (x2 - Ex2)return 1 / m * np.matmul(X.T, X)def compute_feature(self, X, n_components):# 計(jì)算協(xié)方差矩陣covariance_matrix = self.calculate_covariance_matrix(X)print('==covariance_matrix:', covariance_matrix.shape)# 計(jì)算協(xié)方差矩陣的特征值和對應(yīng)特征向量eigenvalues, eigenvectors = np.linalg.eig(covariance_matrix)print('=eigenvectors.shape:', eigenvectors.shape)# 對特征值排序idx = eigenvalues.argsort()[::-1]# 取最大的前n_component組eigenvectors = eigenvectors[:, idx]eigenvectors = eigenvectors[:, :n_components]# Y=PX轉(zhuǎn)換return np.matmul(X, eigenvectors)#兩個樣本 一個樣本有三維 X = np.array([[4, 1, 3],[2, 2, 4]]) print('X:', X) pca = PCA() Y = pca.compute_feature(X, 2)#降到2維 print('Y:', Y)

import numpy as np import matplotlib.pyplot as plt from sklearn.datasets import load_digits from sklearn.decomposition import PCA from mpl_toolkits.mplot3d import Axes3Ddef plot_digits(images, y, max_n=10):"""顯示手寫數(shù)字的圖像"""# 設(shè)置圖像尺寸fig = plt.figure(figsize=(2. * n_col, 2.26 * n_row))i = 0while i < max_n and i < images.shape[0]:p = fig.add_subplot(n_row, n_col, i + 1, xticks=[], yticks=[])p.imshow(images[i], cmap=plt.cm.bone, interpolation='nearest')# 添加標(biāo)簽p.text(0, -1, str(y[i]))i = i + 1plt.show()# 顯示8*8數(shù)字 # plot_digits(digits.images, digits.target, max_n=10)def plot_pca_scatter_three(digits, X_pca, y_digits, class_nums):"""主成分顯示"""##顯示3Dfig = plt.figure()ax = fig.add_subplot(111, projection='3d')for i in range(len(class_nums)):# print('==y_digits == i:', y_digits == i)print(X_pca[:, 0][y_digits == i].shape)print(X_pca[:, 1][y_digits == i].shape)print(X_pca[:, 2][y_digits == i].shape)ax.scatter(X_pca[:, 0][y_digits == i], X_pca[:, 1][y_digits == i], X_pca[:, 2][y_digits == i], label = class_nums[i])plt.legend(digits.target_names)ax.set_xlabel('First Principal Component')ax.set_ylabel('Second Principal Component')ax.set_zlabel('Third Principal Component')plt.savefig('./three.jpg')plt.show()def plot_pca_scatter_two(digits, X_pca, y_digits, class_nums):# 顯示2Dfor i in range(len(class_nums)):px = X_pca[:, 0][y_digits == i]py = X_pca[:, 1][y_digits == i]plt.scatter(px, py, label=class_nums[i])plt.legend(digits.target_names)plt.xlabel('First Principal Component')plt.ylabel('Second Principal Component')plt.savefig('./two.jpg')plt.show()def pca_aysis_two_components():digits = load_digits()X_digits, y_digits = digits.data, digits.target# n_row, n_col = 2, 5print('=X_digits.shape:', X_digits.shape)print('==y_digits:', y_digits[:10])print(len(y_digits))class_nums = [i for i in range(10)]n_components = 2 # 取前2個主成分pca = PCA(n_components=n_components)# fit_transform表示做完訓(xùn)練以后并進(jìn)行轉(zhuǎn)換X_pca = pca.fit_transform(X_digits)print('==X_pca.shape:', X_pca.shape)plot_pca_scatter_two(digits, X_pca, y_digits, class_nums)def pca_aysis_three_components():digits = load_digits()X_digits, y_digits = digits.data, digits.targetprint('=X_digits.shape:', X_digits.shape)print('==y_digits:', y_digits[:10])print(len(y_digits))class_nums = [i for i in range(10)]n_components = 3 # 取前2個主成分pca = PCA(n_components=n_components)# fit_transform表示做完訓(xùn)練以后并進(jìn)行轉(zhuǎn)換X_pca = pca.fit_transform(X_digits)print('==X_pca.shape:', X_pca.shape)plot_pca_scatter_three(digits, X_pca, y_digits, class_nums)if __name__ == '__main__':# pca_aysis_two_components()pca_aysis_three_components()

可視化二維的時候：

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可視化三維的時候：

二.SVD

numpy案例

A = np.array([[4, 1, 3],[2, 2, 4]]) print('A:\n', A)# 對其進(jìn)行SVD分解 s是對角矩陣的值 u, s, vt = np.linalg.svd(A, full_matrices=True) print('u.shape, s.shape, vt.shape:', u.shape, s.shape, vt.shape) res = np.dot(u*s, vt[:2, :]) print('res:', res) #U UT=I print('np.dot(u, u.T):\n', np.dot(u, u.T)) #v vT=I print('np.dot(vt, vt.T):\n', np.dot(vt, vt.T))

將SVD用于圖像的壓縮算法。其原理就是保存像素矩陣的前k個奇異值，并在此基礎(chǔ)上做圖像恢復(fù)。由SVD的原理我們可以知道，在SVD分解中越靠前的奇異值越重要，代表的信息含量越大。?

import cv2 import numpy as np import os from tqdm import tqdm# 定義恢復(fù)函數(shù)，由分解后的矩陣恢復(fù)到原矩陣 def restore(u, s, v, K):'''u:左奇異矩陣 u uT=Iv:右奇異矩陣 v vT=Is:奇異值矩陣 對角矩陣K:奇異值個數(shù)'''m, n = len(u), len(v)a = np.zeros((m, n))for k in range(K):uk = u[:, k].reshape(m, 1)vk = v[k].reshape(1, n)# 前k個奇異值的加總a += np.dot(uk * s[k], vk)a = a.clip(0, 255)return np.rint(a).astype('uint8')img = cv2.imread("./111.png") print('==img.shape:', img.shape) h,w,_ = img.shape # 對RGB圖像進(jìn)行奇異值分解 u_b, s_b, v_b = np.linalg.svd(img[:, :, 0]) u_g, s_g, v_g = np.linalg.svd(img[:, :, 1]) u_r, s_r, v_r = np.linalg.svd(img[:, :, 2])print('==u_b.shape, s_b.shape, v_b.shape:', u_b.shape, s_b.shape, v_b.shape) B = np.dot(u_b * s_b, v_b[:h, :]) print('B:', B.shape)G = np.dot(u_g * s_g, v_g[:h, :]) print('G:', G.shape)R = np.dot(u_r * s_r, v_r[:h, :]) print('R:', R.shape)# # 使用前短邊個奇異值 K = min(img.shape[:2]) new_img = cv2.merge([B, G, R]) cv2.imwrite('./test_svd.jpg', new_img)# # 恢復(fù)圖像 ouput_path = './svd' os.makedirs(ouput_path, exist_ok=True) for k in tqdm(range(1, K + 1)):B = restore(u_b, s_b, v_b, k)R = restore(u_r, s_r, v_r, k)G = restore(u_g, s_g, v_g, k)new_img = cv2.merge([B, G, R])cv2.imwrite(os.path.join(ouput_path, '{}.jpg'.format(k)), new_img)

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