- \n
- \u964d\u7ef4\u662f\u4fdd\u7559\u4e3b\u6210\u5206<\/li>\n
- \u964d\u566a\u662f\u820d\u5f03\u6b21\u8981\u6210\u5206（\u901a\u5e38\u542b\u566a\u58f0）<\/li>\n<\/ul>\n
2. \u4e3b\u6210\u5206\u5206\u6790（PCA）<\/h2>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u4e3b\u6210\u5206\u5206\u6790\u662f\u4e00\u79cd\u65e0\u76d1\u7763\u7684\u964d\u7ef4\u65b9\u6cd5，\u6838\u5fc3\u662f\u901a\u8fc7\u7ebf\u6027\u53d8\u6362\u5c06\u9ad8\u7ef4\u6570\u636e\u6620\u5c04\u5230\u4f4e\u7ef4\u7a7a\u95f4，\u540c\u65f6\u4fdd\u7559\u6570\u636e\u4e2d\u6700\u4e3b\u8981\u7684\u4fe1\u606f（\u65b9\u5dee\u6700\u5927\u7684\u65b9\u5411）\u3002<\/p>\n
2.1 \u6838\u5fc3\u601d\u60f3<\/h3>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u9ad8\u7ef4\u6570\u636e\u4e2d\u5b58\u5728\u5197\u4f59（\u7279\u5f81\u95f4\u76f8\u5173），PCA\u901a\u8fc7\u5bfb\u627e\u4e00\u7ec4\u6b63\u4ea4\u7684\u201c\u4e3b\u6210\u5206\u201d（\u65b0\u7279\u5f81），\u4f7f\u6570\u636e\u5728\u4e3b\u6210\u5206\u4e0a\u7684\u6295\u5f71\u65b9\u5dee\u6700\u5927，\u7528\u8f83\u5c11\u7684\u4e3b\u6210\u5206\u4fdd\u7559\u539f\u6570\u636e\u7684\u5927\u90e8\u5206\u4fe1\u606f\u3002<\/p>\n
2.2 \u6b65\u9aa4<\/h3>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01.\u00a0\u6570\u636e\u6807\u51c6\u5316：\u5bf9\u539f\u59cb\u6570\u636e\u8fdb\u884c\u4e2d\u5fc3\u5316（\u51cf\u53bb\u5747\u503c），\u4f7f\u5404\u7279\u5f81\u5747\u503c\u4e3a0（\u82e5\u7279\u5f81\u91cf\u7eb2\u4e0d\u540c，\u9700\u5148\u6807\u51c6\u5316，\u5982\u9664\u4ee5\u6807\u51c6\u5dee）\u3002<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a02.\u00a0\u8ba1\u7b97\u534f\u65b9\u5dee\u77e9\u9635：\u63cf\u8ff0\u7279\u5f81\u95f4\u7684\u76f8\u5173\u6027\u3002<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a03.\u00a0\u6c42\u89e3\u534f\u65b9\u5dee\u77e9\u9635\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf：\u7279\u5f81\u503c\u8868\u793a\u5bf9\u5e94\u7279\u5f81\u5411\u91cf\u65b9\u5411\u4e0a\u7684\u65b9\u5dee\u5927\u5c0f，\u7279\u5f81\u5411\u91cf\u4e3a\u6570\u636e\u6295\u5f71\u7684\u65b9\u5411（\u4e3b\u6210\u5206）\u3002<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a04.\u00a0\u9009\u62e9\u4e3b\u6210\u5206：\u6309\u7279\u5f81\u503c\u4ece\u5927\u5230\u5c0f\u6392\u5e8f，\u9009\u53d6\u524dk\u4e2a\u7279\u5f81\u5411\u91cf（\u7d2f\u8ba1\u65b9\u5dee\u8d21\u732e\u7387\u901a\u5e38\u226585%），\u6784\u6210\u6295\u5f71\u77e9\u9635\u3002<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a05.\u00a0\u6570\u636e\u964d\u7ef4：\u5c06\u539f\u59cb\u6570\u636e\u4e0e\u6295\u5f71\u77e9\u9635\u76f8\u4e58，\u5f97\u5230k\u7ef4\u7684\u4f4e\u7ef4\u6570\u636e\u3002<\/p>\n
2.3 \u5e94\u7528<\/h3>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u6570\u636e\u53ef\u89c6\u5316（\u5c06\u9ad8\u7ef4\u6570\u636e\u964d\u4e3a2D\/3D）\u3001\u53bb\u9664\u566a\u58f0（\u4fdd\u7559\u4e3b\u8981\u65b9\u5dee，\u8fc7\u6ee4\u5c0f\u65b9\u5dee\u7684\u566a\u58f0）\u3001\u52a0\u901f\u6a21\u578b\u8bad\u7ec3（\u51cf\u5c11\u7279\u5f81\u7ef4\u5ea6）\u3002<\/p>\n<\/p>\n
3. PCA\u6280\u672f\u8865\u5145<\/h2>\n
3.1\u6570\u5b66\u672c\u8d28<\/h3>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0PCA\u662f\u4e00\u79cd\u7ebf\u6027\u964d\u7ef4\u65b9\u6cd5，\u5176\u76ee\u6807\u662f\u6700\u5927\u5316\u4f4e\u7ef4\u6570\u636e\u7684\u65b9\u5dee（\u5373\u4fdd\u7559\u6570\u636e\u7684\u4e3b\u8981\u53d8\u5f02\u4fe1\u606f），\u7b49\u4ef7\u4e8e\u6700\u5c0f\u5316\u539f\u59cb\u6570\u636e\u4e0e\u4f4e\u7ef4\u6295\u5f71\u7684\u91cd\u6784\u8bef\u5dee（\u5747\u65b9\u8bef\u5dee）\u3002<\/p>\n
3.2 \u4f18\u7f3a\u70b9<\/h3>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0– \u4f18\u70b9：\u8ba1\u7b97\u7b80\u5355\u3001\u65e0\u53c2\u6570\u4f9d\u8d56\u3001\u53ef\u53bb\u9664\u7279\u5f81\u95f4\u7684\u5197\u4f59\u548c\u566a\u58f0\u3002<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0– \u7f3a\u70b9：\u4ec5\u80fd\u6355\u6349\u7ebf\u6027\u5173\u7cfb，\u5bf9\u975e\u7ebf\u6027\u6570\u636e\u964d\u7ef4\u6548\u679c\u5dee；\u4e3b\u6210\u5206\u7684\u7269\u7406\u610f\u4e49\u4e0d\u660e\u786e（\u96be\u4ee5\u89e3\u91ca）；\u5bf9\u5f02\u5e38\u503c\u654f\u611f\u3002<\/p>\n
3.3 \u6269\u5c55<\/h3>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u9488\u5bf9\u975e\u7ebf\u6027\u6570\u636e，\u53ef\u91c7\u7528\u6838PCA（\u901a\u8fc7\u6838\u51fd\u6570\u5c06\u6570\u636e\u6620\u5c04\u5230\u9ad8\u7ef4\u7a7a\u95f4，\u518d\u8fdb\u884cPCA\u964d\u7ef4）；\u9488\u5bf9\u9ad8\u7ef4\u7a00\u758f\u6570\u636e，\u53ef\u91c7\u7528\u7a00\u758fPCA（\u4f7f\u4e3b\u6210\u5206\u5177\u6709\u7a00\u758f\u6027，\u4fbf\u4e8e\u89e3\u91ca）\u3002<\/p>\n
3.4 \u4ee3\u7801\u793a\u4f8b<\/h3>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u901a\u8fc7PCA\u627e\u5230\u6570\u636e\u7684\u4e3b\u8981\u53d8\u5316\u65b9\u5411，\u4fdd\u7559\u91cd\u8981\u4fe1\u606f（\u4fe1\u53f7）\u7684\u540c\u65f6，\u820d\u5f03\u6b21\u8981\u65b9\u5411（\u901a\u5e38\u4e3a\u566a\u58f0），\u5b9e\u73b0\u6570\u636e\u964d\u7ef4\u4e0e\u964d\u566a\u7684\u534f\u540c\u5904\u7406\u3002<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7ebf\u6027\u53d8\u6362 \u2192 \u534f\u65b9\u5dee\u77e9\u9635\u5206\u89e3 \u2192 \u6295\u5f71\u5230\u6700\u5927\u65b9\u5dee\u65b9\u5411 \u2192 \u4e22\u5f03\u566a\u58f0\u4e3b\u5bfc\u7684\u5c0f\u65b9\u5dee\u6210\u5206<\/p>\n
import numpy as np
\nimport matplotlib.pyplot as plt
\nfrom sklearn.decomposition import PCA
\nfrom sklearn.preprocessing import StandardScaler<\/p>\n# 1. \u751f\u6210\u5e26\u566a\u58f0\u7684\u4e8c\u7ef4\u6570\u636e ======================================
\nnp.random.seed(42)
\nn_samples = 300
\ntrue_mean = [3, 2] # \u771f\u5b9e\u6570\u636e\u7684\u5747\u503c
\ncov = [[1, 0.8], [0.8, 1]] # \u534f\u65b9\u5dee\u77e9\u9635（\u5b58\u5728\u76f8\u5173\u6027）<\/p>\n# \u751f\u6210\u771f\u5b9e\u4fe1\u53f7（\u7ebf\u6027\u76f8\u5173\u7684\u4e8c\u7ef4\u6570\u636e）
\nX_true = np.random.multivariate_normal(true_mean, cov, n_samples)<\/p>\n# \u6dfb\u52a0\u9ad8\u65af\u566a\u58f0（\u6a21\u62df\u89c2\u6d4b\u566a\u58f0）
\nnoise = np.random.normal(0, 0.6, size=(n_samples, 2))
\nX_noisy = X_true + noise # \u542b\u566a\u58f0\u7684\u89c2\u6d4b\u6570\u636e<\/p>\n# 2. \u6570\u636e\u6807\u51c6\u5316 ==============================================
\nscaler = StandardScaler()
\nX_scaled = scaler.fit_transform(X_noisy) # \u6807\u51c6\u5316（\u5747\u503c\u4e3a0，\u65b9\u5dee\u4e3a1）<\/p>\n# 3. PCA\u964d\u7ef4\u4e0e\u91cd\u5efa ==========================================
\npca = PCA(n_components=1) # \u964d\u52301\u7ef4（\u4e3a\u4e86\u6e05\u6670\u5c55\u793a\u964d\u566a\u6548\u679c）
\nX_pca = pca.fit_transform(X_scaled) # \u964d\u7ef4\u540e\u7684\u6570\u636e
\nX_reconstructed = pca.inverse_transform(X_pca) # \u91cd\u5efa\u56de\u539f\u59cb\u7a7a\u95f4<\/p>\n# 4. \u53ef\u89c6\u5316\u7ed3\u679c ==============================================
\nplt.figure(figsize=(15, 5))<\/p>\n# 4.1 \u539f\u59cb\u6570\u636e\u4e0e\u566a\u58f0\u6570\u636e\u5bf9\u6bd4
\nplt.subplot(131)
\nplt.scatter(X_true[:, 0], X_true[:, 1], c='blue', alpha=0.6, label='\u771f\u5b9e\u4fe1\u53f7')
\nplt.scatter(X_noisy[:, 0], X_noisy[:, 1], c='red', alpha=0.2, label='\u542b\u566a\u89c2\u6d4b')
\nplt.title('\u6570\u636e\u5206\u5e03\u5bf9\u6bd4')
\nplt.xlabel('\u7279\u5f811')
\nplt.ylabel('\u7279\u5f812')
\nplt.legend()
\nplt.grid(True)<\/p>\n# 4.2 \u4e3b\u6210\u5206\u65b9\u5411\u53ef\u89c6\u5316
\nplt.subplot(132)
\n# \u7ed8\u5236\u4e3b\u6210\u5206\u65b9\u5411
\npc_direction = pca.components_[0] # \u7b2c\u4e00\u4e3b\u6210\u5206\u65b9\u5411\u5411\u91cf
\nplt.arrow(0, 0, pc_direction[0], pc_direction[1],
\n head_width=0.3, head_length=0.3, fc='black', ec='black')
\nplt.text(pc_direction[0]*1.2, pc_direction[1]*1.2,
\n '\u4e3b\u6210\u5206\u65b9\u5411', fontsize=12)<\/p>\n# \u7ed8\u5236\u6570\u636e\u70b9
\nplt.scatter(X_scaled[:, 0], X_scaled[:, 1], c='gray', alpha=0.4)
\nplt.title('\u4e3b\u6210\u5206\u65b9\u5411\u5206\u6790')
\nplt.xlabel('\u6807\u51c6\u5316\u7279\u5f811')
\nplt.ylabel('\u6807\u51c6\u5316\u7279\u5f812')
\nplt.axis('equal')
\nplt.grid(True)<\/p>\n# 4.3 PCA\u964d\u566a\u6548\u679c\u5bf9\u6bd4
\nplt.subplot(133)
\nplt.scatter(X_noisy[:, 0], X_noisy[:, 1], c='red', alpha=0.2, label='\u542b\u566a\u6570\u636e')
\nplt.scatter(X_reconstructed[:, 0], X_reconstructed[:, 1],
\n c='green', alpha=0.6, label='PCA\u964d\u566a')
\nplt.title('PCA\u964d\u566a\u6548\u679c')
\nplt.xlabel('\u7279\u5f811')
\nplt.ylabel('\u7279\u5f812')
\nplt.legend()
\nplt.grid(True)<\/p>\nplt.tight_layout()<\/p>\n
# 5. \u89e3\u91ca\u65b9\u5dee\u5206\u6790 ==========================================
\nprint(f"\u4e3b\u6210\u5206\u89e3\u91ca\u65b9\u5dee\u6bd4\u4f8b: {pca.explained_variance_ratio_[0]:.2%}")
\nplt.show()<\/p>\n\u4e3b\u6210\u5206\u89e3\u91ca\u65b9\u5dee\u6bd4\u4f8b: 78.20%<\/p>\n
\u8868\u793a\u8be5\u4e3b\u6210\u5206\u65b9\u5411\u643a\u5e26\u4e86\u539f\u59cb\u6570\u636e78.20%\u7684\u65b9\u5dee\u4fe1\u606f（\u566a\u58f0\u901a\u5e38\u5bf9\u5e94\u5269\u4f59\u7684\u5c0f\u90e8\u5206\u65b9\u5dee）<\/p>\n
\u5de5\u7a0b\u610f\u4e49\u200b\u200b：<\/p>\n
\u2022\u91cf\u5316\u964d\u7ef4\u8fc7\u7a0b\u4e2d\u7684\u4fe1\u606f\u4fdd\u7559\u7a0b\u5ea6<\/p>\n
\u2022\u6307\u5bfc\u9009\u62e9\u4e3b\u6210\u5206\u6570\u91cf（\u5982\u7d2f\u8ba1\u65b9\u5dee>78%）<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250816043746-68a00b1a7389c.png\")
<\/p>\n\u56fe1：\u6570\u636e\u5206\u5e03\u5bf9\u6bd4<\/h4>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u84dd\u8272\u70b9\u200b\u200b：\u539f\u59cb\u771f\u5b9e\u4fe1\u53f7（\u5448\u73b0\u660e\u663e\u7684\u7ebf\u6027\u76f8\u5173\u5206\u5e03）<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7ea2\u8272\u70b9\u200b\u200b：\u6dfb\u52a0\u566a\u58f0\u540e\u7684\u89c2\u6d4b\u6570\u636e（\u5728\u539f\u59cb\u4fe1\u53f7\u5468\u56f4\u968f\u673a\u6563\u5e03）<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u76f4\u89c2\u5c55\u793a\u566a\u58f0\u5982\u4f55\u7834\u574f\u539f\u59cb\u6570\u636e\u7684\u7ed3\u6784\u7279\u5f81<\/p>\n
\u56fe2：\u4e3b\u6210\u5206\u65b9\u5411\u5206\u6790<\/h4>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u200b\u200b\u9ed1\u8272\u7bad\u5934\u200b\u200b：PCA\u81ea\u52a8\u53d1\u73b0\u7684\u7b2c\u4e00\u4e3b\u6210\u5206\u65b9\u5411（\u6570\u636e\u65b9\u5dee\u6700\u5927\u7684\u65b9\u5411）<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u7070\u8272\u70b9\u4e3a\u6807\u51c6\u5316\u540e\u7684\u6570\u636e，\u7bad\u5934\u65b9\u5411\u4e0e\u6570\u636e\u5206\u5e03\u7684\u4e3b\u8981\u8d8b\u52bf\u4e00\u81f4<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u8bf4\u660ePCA\u80fd\u81ea\u52a8\u6355\u6349\u6570\u636e\u7684\u4e3b\u8981\u53d8\u5f02\u65b9\u5411<\/p>\n
\u56fe3：PCA\u964d\u566a\u6548\u679c<\/h4>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u200b\u200b\u7ea2\u8272\u70b9\u200b\u200b：\u539f\u59cb\u542b\u566a\u6570\u636e（\u6563\u5e03\u8303\u56f4\u5927）<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u200b\u200b\u7eff\u8272\u70b9\u200b\u200b：PCA\u964d\u566a\u7ed3\u679c（\u5168\u90e8\u6295\u5f71\u5230\u4e3b\u6210\u5206\u65b9\u5411\u540e\u91cd\u5efa）<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u53ef\u4ee5\u770b\u5230：<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u566a\u58f0\u88ab\u6709\u6548\u6291\u5236（\u70b9\u66f4\u96c6\u4e2d）<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u4fdd\u7559\u4e86\u539f\u59cb\u6570\u636e\u7684\u7ebf\u6027\u8d8b\u52bf<\/p>\n
\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u91cd\u5efa\u6570\u636e\u5b8c\u5168\u843d\u5728\u4e3b\u6210\u5206\u65b9\u5411\u4e0a（\u5b9e\u73b0\u964d\u7ef4+\u964d\u566a）<\/p>\n<\/p>\n
\u603b\u7ed3<\/h2>\n
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