{"id":57891,"date":"2025-08-15T22:19:56","date_gmt":"2025-08-15T14:19:56","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/57891.html"},"modified":"2025-08-15T22:19:56","modified_gmt":"2025-08-15T14:19:56","slug":"%e6%9c%ba%e5%99%a8%e5%ad%a6%e4%b9%a0%e4%b8%ad%e7%9a%84pca%e9%99%8d%e7%bb%b4","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/57891.html","title":{"rendered":"\u673a\u5668\u5b66\u4e60\u4e2d\u7684PCA\u964d\u7ef4"},"content":{"rendered":"

\u5728\u673a\u5668\u5b66\u4e60\u548c\u6570\u636e\u79d1\u5b66\u7684\u65e5\u5e38\u5de5\u4f5c\u4e2d，\u6211\u4eec\u5e38\u5e38\u4f1a\u9047\u5230\u8fd9\u6837\u7684\u56f0\u5883：\u624b\u5934\u7684\u6570\u636e\u96c6\u7ef4\u5ea6\u9ad8\u8fbe\u51e0\u5341\u751a\u81f3\u4e0a\u767e\u7ef4（\u6bd4\u5982\u56fe\u50cf\u7684\u50cf\u7d20\u7279\u5f81\u3001\u6587\u672c\u7684\u8bcd\u888b\u6a21\u578b），\u4f46\u8ba1\u7b97\u6548\u7387\u4f4e\u3001\u6a21\u578b\u5bb9\u6613\u8fc7\u62df\u5408，\u751a\u81f3\u8fde\u53ef\u89c6\u5316\u90fd\u6210\u4e86\u96be\u9898\u3002\u8fd9\u65f6\u5019，\u964d\u7ef4（Dimensionality Reduction） \u6280\u672f\u5c31\u6210\u4e3a\u4e86\u6211\u4eec\u7684\u201c\u6551\u661f\u201d\u3002\u800c\u5728\u4f17\u591a\u964d\u7ef4\u65b9\u6cd5\u4e2d，\u4e3b\u6210\u5206\u5206\u6790（Principal Component Analysis, PCA） \u56e0\u5176\u7b80\u5355\u9ad8\u6548\u3001\u65e0\u9700\u6807\u7b7e\u7684\u7279\u6027，\u6210\u4e3a\u4e86\u6700\u7ecf\u5178\u7684\u964d\u7ef4\u7b97\u6cd5\u4e4b\u4e00\u3002<\/p>\n

\u672c\u6587\u5c06\u4ece\u201c\u4e3a\u4ec0\u4e48\u9700\u8981\u964d\u7ef4\u201d\u8bb2\u8d77，\u9010\u6b65\u62c6\u89e3PCA\u7684\u6838\u5fc3\u539f\u7406，\u5e76\u901a\u8fc7\u5b9e\u9645\u6848\u4f8b\u6f14\u793a\u5982\u4f55\u7528\u4ee3\u7801\u5b9e\u73b0PCA\u964d\u7ef4\u3002\u65e0\u8bba\u4f60\u662f\u521a\u5165\u95e8\u673a\u5668\u5b66\u4e60\u7684\u65b0\u624b，\u8fd8\u662f\u9700\u8981\u4f18\u5316\u6a21\u578b\u7684\u4ece\u4e1a\u8005，\u8fd9\u7bc7\u6587\u7ae0\u90fd\u80fd\u5e2e\u4f60\u5feb\u901f\u638c\u63e1PCA\u7684\u7cbe\u9ad3\u3002<\/p>\n

\n

\u4e00\u3001\u4e3a\u4ec0\u4e48\u9700\u8981\u964d\u7ef4？\u9ad8\u7ef4\u6570\u636e\u7684\u201c\u4e09\u5927\u5929\u654c\u201d<\/h3>\n

\u5728\u7406\u89e3PCA\u4e4b\u524d，\u6211\u4eec\u9700\u8981\u5148\u660e\u786e：\u4e3a\u4ec0\u4e48\u9ad8\u7ef4\u6570\u636e\u9700\u8981\u88ab\u201c\u538b\u7f29\u201d？ \u9ad8\u7ef4\u6570\u636e\u5e26\u6765\u7684\u95ee\u9898，\u88ab\u5f62\u8c61\u5730\u79f0\u4e3a\u201c\u7ef4\u5ea6\u707e\u96be（Curse of Dimensionality）\u201d，\u4e3b\u8981\u4f53\u73b0\u5728\u4ee5\u4e0b\u4e09\u4e2a\u65b9\u9762：<\/p>\n

1. \u8ba1\u7b97\u6548\u7387\u66b4\u8dcc<\/h4>\n

\u5047\u8bbe\u4e00\u4e2a\u6570\u636e\u96c6\u6709 nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u4e2a\u6837\u672c，\u6bcf\u4e2a\u6837\u672c\u6709 dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u7ef4\u7279\u5f81\u3002\u5b58\u50a8\u8fd9\u6837\u7684\u6570\u636e\u96c6\u9700\u8981 O(nd)O(nd)<\/span><\/span>O<\/span>(<\/span>n<\/span>d<\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u7a7a\u95f4，\u800c\u8bb8\u591a\u673a\u5668\u5b66\u4e60\u7b97\u6cd5（\u5982KNN\u3001SVM）\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u968f\u7ef4\u5ea6 dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span> \u5448\u6307\u6570\u7ea7\u589e\u957f\u3002\u4f8b\u5982，KNN\u7684\u9884\u6d4b\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(nd)O(nd)<\/span><\/span>O<\/span>(<\/span>n<\/span>d<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5f53 d=1000d=1000<\/span><\/span>d<\/span><\/span>=<\/span><\/span><\/span><\/span>1000<\/span><\/span><\/span><\/span><\/span> \u65f6，\u8ba1\u7b97\u91cf\u53ef\u80fd\u662f d=100d=100<\/span><\/span>d<\/span><\/span>=<\/span><\/span><\/span><\/span>100<\/span><\/span><\/span><\/span><\/span> \u65f6\u768410\u500d\u4ee5\u4e0a\u3002<\/p>\n

2. \u4fe1\u606f\u5197\u4f59\u4e0e\u566a\u58f0\u653e\u5927<\/h4>\n

\u9ad8\u7ef4\u6570\u636e\u4e2d，\u5f88\u591a\u7279\u5f81\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u9ad8\u5ea6\u76f8\u5173\u6027（\u6bd4\u5982\u4eba\u7684\u8eab\u9ad8\u548c\u4f53\u91cd），\u6216\u8005\u67d0\u4e9b\u7279\u5f81\u5bf9\u4efb\u52a1\u7684\u8d21\u732e\u6781\u5c0f（\u6bd4\u5982\u56fe\u50cf\u4e2d\u7684\u968f\u673a\u566a\u58f0）\u3002\u8fd9\u4e9b\u201c\u5197\u4f59\u7279\u5f81\u201d\u4e0d\u4ec5\u6d6a\u8d39\u8ba1\u7b97\u8d44\u6e90，\u8fd8\u53ef\u80fd\u5e72\u6270\u6a21\u578b\u5b66\u4e60\u5173\u952e\u6a21\u5f0f\u3002<\/p>\n

3. \u53ef\u89c6\u5316\u4e0e\u7406\u89e3\u56f0\u96be<\/h4>\n

\u4eba\u7c7b\u7684\u5927\u8111\u6700\u591a\u53ea\u80fd\u76f4\u89c2\u7406\u89e33\u7ef4\u7a7a\u95f4\u3002\u5f53\u6570\u636e\u7ef4\u5ea6\u8d85\u8fc73\u7ef4\u65f6，\u6211\u4eec\u65e0\u6cd5\u901a\u8fc7\u56fe\u8868\u76f4\u63a5\u89c2\u5bdf\u6570\u636e\u7684\u5206\u5e03\u89c4\u5f8b（\u6bd4\u5982\u805a\u7c7b\u6548\u679c\u3001\u7c7b\u522b\u8fb9\u754c），\u8fd9\u4f7f\u5f97\u6a21\u578b\u8c03\u4f18\u548c\u7ed3\u679c\u5206\u6790\u53d8\u5f97\u5f02\u5e38\u56f0\u96be\u3002<\/p>\n

\u964d\u7ef4\u7684\u76ee\u6807，\u5c31\u662f\u5728\u5c3d\u53ef\u80fd\u4fdd\u7559\u539f\u59cb\u6570\u636e\u5173\u952e\u4fe1\u606f\u7684\u524d\u63d0\u4e0b，\u5c06\u9ad8\u7ef4\u6570\u636e\u6620\u5c04\u5230\u4e00\u4e2a\u4f4e\u7ef4\u7a7a\u95f4（\u901a\u5e38\u662f2\u7ef4\u62163\u7ef4），\u4ece\u800c\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898\u3002<\/p>\n

\n

\u4e8c\u3001PCA\u7684\u6838\u5fc3\u601d\u60f3：\u7528\u201c\u4e3b\u6210\u5206\u201d\u91cd\u6784\u6570\u636e<\/h3>\n

PCA\u662f\u4e00\u79cd\u65e0\u76d1\u7763\u964d\u7ef4\u65b9\u6cd5（\u4e0d\u9700\u8981\u6807\u7b7e），\u5176\u6838\u5fc3\u601d\u60f3\u53ef\u4ee5\u6982\u62ec\u4e3a：\u627e\u5230\u6570\u636e\u4e2d\u65b9\u5dee\u6700\u5927\u7684\u65b9\u5411（\u4e3b\u6210\u5206），\u5e76\u5c06\u6570\u636e\u6295\u5f71\u5230\u8fd9\u4e9b\u65b9\u5411\u4e0a，\u4f7f\u5f97\u6295\u5f71\u540e\u7684\u6570\u636e\u65b9\u5dee\u6700\u5927（\u5373\u4fdd\u7559\u6700\u591a\u4fe1\u606f）\u3002<\/p>\n

1. \u65b9\u5dee：\u6570\u636e\u7684\u201c\u4fe1\u606f\u91cf\u201d\u6307\u6807<\/h4>\n

\u5728\u7edf\u8ba1\u5b66\u4e2d，\u65b9\u5dee\u8861\u91cf\u7684\u662f\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\u3002\u5bf9\u4e8e\u4e00\u7ec4\u6570\u636e x1,x2,…,xnx_1, x_2, …, x_n<\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>…<\/span>,<\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u65b9\u5dee\u5b9a\u4e49\u4e3a：
\nVar(x)=1n\u2211i=1n(xi\u2212\u03bc)2\\\\text{Var}(x) = \\\\frac{1}{n} \\\\sum_{i=1}^n (x_i – \\\\mu)^2<\/span><\/span>Var<\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03bc<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\n\u5176\u4e2d \u03bc\\\\mu<\/span><\/span>\u03bc<\/span><\/span><\/span><\/span><\/span> \u662f\u5747\u503c\u3002\u65b9\u5dee\u8d8a\u5927，\u6570\u636e\u5728\u67d0\u4e2a\u65b9\u5411\u4e0a\u7684\u201c\u53d8\u5316\u201d\u8d8a\u5267\u70c8，\u610f\u5473\u7740\u8fd9\u4e2a\u65b9\u5411\u5305\u542b\u7684\u4fe1\u606f\u8d8a\u4e30\u5bcc\u3002<\/p>\n

\u4e3e\u4e2a\u4f8b\u5b50：\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u7ec4\u4e8c\u7ef4\u6570\u636e（\u5982\u56fe1\u5de6），\u5176\u4e2dx\u8f74\u65b9\u5411\u7684\u65b9\u5dee\u5f88\u5927（\u6570\u636e\u70b9\u6cbfx\u8f74\u5206\u6563），y\u8f74\u65b9\u5411\u7684\u65b9\u5dee\u5f88\u5c0f（\u6570\u636e\u70b9\u6cbfy\u8f74\u96c6\u4e2d）\u3002\u6b64\u65f6，y\u8f74\u65b9\u5411\u7684\u4fe1\u606f\u91cf\u5f88\u4f4e，\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4e22\u5f03y\u8f74，\u4ec5\u7528x\u8f74\u8868\u793a\u6570\u636e（\u5982\u56fe1\u53f3），\u8fd9\u6837\u51e0\u4e4e\u4e0d\u4f1a\u4e22\u5931\u5173\u952e\u4fe1\u606f\u3002<\/p>\n

!https:\/\/miro.medium.com\/v2\/resize:fit:1400\/1*qg276-6QZJq3q3QZJq3QZQ.png
\n（\u5de6：\u539f\u59cb\u4e8c\u7ef4\u6570\u636e；\u53f3：\u6295\u5f71\u5230x\u8f74\u540e\u7684\u4e00\u7ef4\u6570\u636e）<\/p>\n

2. \u4e3b\u6210\u5206：\u6b63\u4ea4\u7684\u201c\u4fe1\u606f\u6700\u5927\u5316\u201d\u65b9\u5411<\/h4>\n

\u5728\u66f4\u9ad8\u7ef4\u7684\u573a\u666f\u4e2d（\u6bd4\u5982d\u7ef4），PCA\u4f1a\u5bfb\u627e\u4e00\u7ec4**\u6b63\u4ea4（\u4e0d\u76f8\u5173）**\u7684\u65b9\u5411\u5411\u91cf w1,w2,…,wd\\\\mathbf{w}_1, \\\\mathbf{w}_2, …, \\\\mathbf{w}_d<\/span><\/span>w<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>w<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>…<\/span>,<\/span><\/span>w<\/span><\/span>d<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d\u6bcf\u4e2a\u65b9\u5411\u5411\u91cf wi\\\\mathbf{w}_i<\/span><\/span>w<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6ee1\u8db3：<\/p>\n