{"id":79038,"date":"2026-02-28T21:05:24","date_gmt":"2026-02-28T13:05:24","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/79038.html"},"modified":"2026-02-28T21:05:24","modified_gmt":"2026-02-28T13:05:24","slug":"%e3%80%90%e5%88%86%e7%b1%bb%e7%ae%97%e6%b3%95%e3%80%91%e9%ab%98%e6%96%af%e6%b7%b7%e5%90%88%e6%a8%a1%e5%9e%8bgmm%e8%b6%85%e8%af%a6%e7%bb%86%e8%ae%b2%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/79038.html","title":{"rendered":"\u3010\u5206\u7c7b\u7b97\u6cd5\u3011\u9ad8\u65af\u6df7\u5408\u6a21\u578b(GMM)\u8d85\u8be6\u7ec6\u8bb2\u89e3"},"content":{"rendered":"

\u9ad8\u65af\u6df7\u5408\u6a21\u578b(GMM)\u8d85\u8be6\u7ec6\u8bb2\u89e3（\u9644\u805a\u7c7b\u5b9e\u6218+Python\u5b8c\u6574\u4ee3\u7801）<\/h2>\n

\u9ad8\u65af\u6df7\u5408\u6a21\u578b（GMM，Gaussian Mixture Model）\u662f\u6982\u7387\u7edf\u8ba1\u9886\u57df\u7684\u7ecf\u5178\u805a\u7c7b\u4e0e\u5bc6\u5ea6\u4f30\u8ba1\u7b97\u6cd5，\u4f5c\u4e3aK-means\u786c\u805a\u7c7b\u7684\u8fdb\u9636\u7248\u672c，GMM\u901a\u8fc7\u591a\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u52a0\u6743\u7ec4\u5408\u5efa\u6a21\u6570\u636e\u5206\u5e03，\u5b9e\u73b0\u8f6f\u805a\u7c7b（\u6bcf\u4e2a\u6837\u672c\u5c5e\u4e8e\u4e0d\u540c\u7c07\u7684\u6982\u7387），\u540c\u65f6\u652f\u6301\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1，\u662f\u672c\u79d1\u751f\u3001\u7814\u7a76\u751f\u5b66\u4e60\u805a\u7c7b\u7b97\u6cd5\u7684\u6838\u5fc3\u5185\u5bb9，\u4e5f\u662f\u91d1\u878d\u98ce\u63a7\u3001\u56fe\u50cf\u5206\u5272\u3001\u5ba2\u6237\u5206\u7fa4\u7b49\u573a\u666f\u7684\u5e38\u7528\u5efa\u6a21\u5de5\u5177\u3002<\/p>\n

\u672c\u6587\u5c06\u4ece\u901a\u4fd7\u539f\u7406\u3001\u6838\u5fc3\u7406\u8bba\u4e0e\u6570\u5b66\u63a8\u5bfc\u3001EM\u7b97\u6cd5\u6c42\u89e3\u6d41\u7a0b\u3001\u805a\u7c7b\u5b9e\u6218\u3001\u4f18\u7f3a\u70b9\u4e0e\u7b97\u6cd5\u5bf9\u6bd4\u4e94\u4e2a\u7ef4\u5ea6\u5c55\u5f00，\u5185\u5bb9\u901a\u4fd7\u6613\u61c2，\u516c\u5f0f\u9644\u5e26\u7269\u7406\u610f\u4e49\u89e3\u91ca，\u9644\u5e26\u53ef\u76f4\u63a5\u8fd0\u884c\u7684Python\u4ee3\u7801\u548c\u53ef\u89c6\u5316\u5206\u6790，\u9002\u914d\u672c\u79d1\u8bfe\u7a0b\u5b66\u4e60\u3001\u7814\u7a76\u751f\u5b9e\u8df5\u548c\u5de5\u4e1a\u7ea7\u5efa\u6a21\u53c2\u8003\u3002<\/p>\n

\u4e00\u3001\u4ec0\u4e48\u662f\u9ad8\u65af\u6df7\u5408\u6a21\u578b(GMM)？（\u901a\u4fd7\u7406\u89e3）<\/h3>\n

1.1 \u5148\u641e\u61c2\u4e24\u4e2a\u6838\u5fc3\u6982\u5ff5<\/h4>\n

（1）\u9ad8\u65af\u5206\u5e03（\u6b63\u6001\u5206\u5e03）<\/h5>\n

\u9ad8\u65af\u5206\u5e03\u662f\u81ea\u7136\u754c\u6700\u5e38\u89c1\u7684\u6982\u7387\u5206\u5e03，\u5448\u949f\u5f62\u66f2\u7ebf，\u6838\u5fc3\u7279\u5f81\u662f\u5927\u90e8\u5206\u6570\u636e\u96c6\u4e2d\u5728\u5747\u503c\u9644\u8fd1，\u8d8a\u504f\u79bb\u5747\u503c\u6570\u636e\u8d8a\u5c11\u3002\u6bd4\u5982\u4eba\u7fa4\u7684\u8eab\u9ad8\u3001\u4f53\u91cd，\u5b66\u751f\u7684\u8003\u8bd5\u6210\u7ee9，\u90fd\u8fd1\u4f3c\u670d\u4ece\u9ad8\u65af\u5206\u5e03\u3002<\/p>\n

（2）\u6df7\u5408\u6a21\u578b<\/h5>\n

\u6df7\u5408\u6a21\u578b\u6307\u6570\u636e\u7531\u591a\u4e2a\u5b50\u7fa4\u4f53\u7684\u5206\u5e03\u7ec4\u5408\u800c\u6210，\u6bcf\u4e2a\u5b50\u7fa4\u4f53\u5bf9\u5e94\u4e00\u4e2a\u57fa\u7840\u5206\u5e03（\u8fd9\u91cc\u662f\u9ad8\u65af\u5206\u5e03）\u3002\u6bd4\u5982\u628a\u7537\u6027\u548c\u5973\u6027\u7684\u8eab\u9ad8\u6570\u636e\u6df7\u5728\u4e00\u8d77，\u6574\u4f53\u6570\u636e\u4e0d\u518d\u662f\u5355\u4e00\u9ad8\u65af\u5206\u5e03，\u800c\u662f\u7537\u6027\u8eab\u9ad8\u9ad8\u65af\u5206\u5e03\u548c\u5973\u6027\u8eab\u9ad8\u9ad8\u65af\u5206\u5e03\u7684\u6df7\u5408\u5206\u5e03\u3002<\/p>\n

1.2 GMM\u7684\u901a\u4fd7\u5b9a\u4e49<\/h4>\n

GMM\u5047\u8bbe\u6570\u636e\u96c6\u662f\u7531K\u4e2a\u9ad8\u65af\u5206\u5e03\u6309\u4e0d\u540c\u6bd4\u4f8b\u52a0\u6743\u6df7\u5408\u751f\u6210\u7684，\u5b83\u7684\u6838\u5fc3\u76ee\u6807\u662f：\u901a\u8fc7\u7b97\u6cd5\u627e\u5230\u8fd9K\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u5747\u503c\u3001\u534f\u65b9\u5dee，\u4ee5\u53ca\u6bcf\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u6743\u91cd，\u4ece\u800c\u5b9e\u73b0\u5bf9\u6570\u636e\u7684\u805a\u7c7b\u6216\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1\u3002<\/p>\n

\u7528\u8eab\u9ad8\u6570\u636e\u805a\u7c7b\u7684\u4f8b\u5b50\u8f7b\u677e\u7406\u89e3GMM：<\/p>\n

\u73b0\u6709\u4e00\u6279\u65e0\u6027\u522b\u6807\u7b7e\u7684\u8eab\u9ad8\u6570\u636e，GMM\u4f1a\u505a\u8fd9\u6837\u7684\u5de5\u4f5c：<\/p>\n

  • \u5047\u8bbe\u6570\u636e\u75312\u4e2a\u9ad8\u65af\u5206\u5e03（\u5bf9\u5e94\u7537\u3001\u5973）\u6df7\u5408\u800c\u6210；<\/li>\n
  • \u901a\u8fc7\u7b97\u6cd5\u8ba1\u7b97\u6bcf\u4e2a\u8eab\u9ad8\u6570\u636e\u5c5e\u4e8e\u201c\u7537\u6027\u9ad8\u65af\u5206\u5e03\u201d\u7684\u6982\u7387\u548c\u201c\u5973\u6027\u9ad8\u65af\u5206\u5e03\u201d\u7684\u6982\u7387；<\/li>\n
  • \u627e\u5230\u8fd9\u4e24\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u6700\u4f73\u5747\u503c（\u5982\u7537\u6027175cm\u3001\u5973\u6027160cm）\u3001\u534f\u65b9\u5dee（\u6570\u636e\u79bb\u6563\u7a0b\u5ea6），\u4ee5\u53ca\u6743\u91cd（\u5982\u7537\u6027\u536060%\u3001\u5973\u6027\u536040%）；<\/li>\n
  • \u6700\u7ec8\u6839\u636e\u6837\u672c\u7684\u5f52\u5c5e\u6982\u7387，\u5c06\u6570\u636e\u5206\u4e3a\u7537\u6027\u3001\u5973\u6027\u4e24\u7c7b（\u805a\u7c7b），\u540c\u65f6\u80fd\u63cf\u8ff0\u6574\u4f53\u8eab\u9ad8\u6570\u636e\u7684\u5206\u5e03\u60c5\u51b5（\u5bc6\u5ea6\u4f30\u8ba1）\u3002<\/li>\n

    1.3 GMM\u4e0eK-means\u7684\u6838\u5fc3\u533a\u522b<\/h4>\n

    K-means\u662f\u786c\u805a\u7c7b，GMM\u662f\u8f6f\u805a\u7c7b，\u8fd9\u662f\u4e8c\u8005\u6700\u672c\u8d28\u7684\u533a\u522b，\u76f4\u63a5\u51b3\u5b9a\u4e86\u9002\u7528\u573a\u666f：<\/p>\n

      \n
    • K-means：\u6bcf\u4e2a\u6837\u672c\u552f\u4e00\u5c5e\u4e8e\u67d0\u4e00\u4e2a\u7c07，\u805a\u7c7b\u7ed3\u679c\u662f\u201c\u975e\u6b64\u5373\u5f7c\u201d，\u6bd4\u5982\u67d0\u4e2a\u8eab\u9ad8\u6570\u636e\u53ea\u80fd\u5f52\u4e3a\u7537\u6027\u6216\u5973\u6027；<\/li>\n
    • GMM：\u6bcf\u4e2a\u6837\u672c\u4ee5\u4e0d\u540c\u6982\u7387\u5c5e\u4e8e\u6240\u6709\u7c07，\u805a\u7c7b\u7ed3\u679c\u662f\u201c\u6982\u7387\u5206\u5e03\u201d，\u6bd4\u5982\u67d0\u4e2a\u8eab\u9ad8165cm\u7684\u6837\u672c，\u670970%\u6982\u7387\u662f\u5973\u6027\u300130%\u6982\u7387\u662f\u7537\u6027，\u6700\u7ec8\u53ef\u6839\u636e\u6700\u5927\u6982\u7387\u786e\u5b9a\u5f52\u5c5e\u3002<\/li>\n<\/ul>\n

      \u7b80\u5355\u6765\u8bf4：K-means\u662f\u201c\u4e00\u5200\u5207\u201d\u7684\u805a\u7c7b，GMM\u662f\u201c\u6982\u7387\u5316\u201d\u7684\u805a\u7c7b，GMM\u80fd\u6355\u6349\u66f4\u590d\u6742\u7684\u6570\u636e\u5206\u5e03，\u805a\u7c7b\u7ed3\u679c\u66f4\u8d34\u5408\u5b9e\u9645\u3002<\/p>\n

      \u4e8c\u3001GMM\u7684\u6838\u5fc3\u7406\u8bba\u57fa\u7840<\/h3>\n

      GMM\u7684\u6838\u5fc3\u662f\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u7684\u5b9a\u4e49，\u4ee5\u53ca\u901a\u8fc7\u6700\u5927\u5316\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u4f30\u8ba1\u6a21\u578b\u53c2\u6570，\u800c\u6c42\u89e3\u8fd9\u4e00\u4f18\u5316\u95ee\u9898\u7684\u7ecf\u5178\u7b97\u6cd5\u662fEM\u7b97\u6cd5（\u671f\u671b\u6700\u5927\u5316\u7b97\u6cd5），\u8fd9\u662fGMM\u7684\u6570\u5b66\u57fa\u77f3\u3002<\/p>\n

      2.1 GMM\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570<\/h4>\n

      \u5047\u8bbe\u6570\u636e\u96c6X={x1,x2,…,xN}X=\\\\{x_1, x_2, …, x_N\\\\}<\/span><\/span>X<\/span><\/span>=<\/span><\/span><\/span><\/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><\/span>，\u5176\u4e2d\u6bcf\u4e2a\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662fdd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span>\u7ef4\u7279\u5f81\u5411\u91cf，GMM\u5047\u8bbe\u8be5\u6570\u636e\u96c6\u7531KK<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u52a0\u6743\u6df7\u5408\u751f\u6210，\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a：
      \np(x)=\u2211k=1K\u03c0kN(x\u2223\u03bck,\u03a3k)p(x)=\\\\sum_{k=1}^{K} \\\\pi_{k} \\\\mathcal{N}\\\\left(x | \\\\mu_{k}, \\\\Sigma_{k}\\\\right)<\/span><\/span>p<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>(<\/span>x<\/span>\u2223<\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u516c\u5f0f\u5404\u90e8\u5206\u7269\u7406\u610f\u4e49<\/h5>\n
        \n
      • KK<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span>：\u9ad8\u65af\u5206\u5e03\u7684\u6570\u91cf，\u5373\u805a\u7c7b\u7684\u7c07\u6570；<\/li>\n
      • \u03c0k\\\\pi_k<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u6743\u91cd，\u8868\u793a\u8be5\u5206\u5e03\u5728\u6574\u4f53\u6570\u636e\u4e2d\u7684\u5360\u6bd4，\u6ee1\u8db3\u2211k=1K\u03c0k=1\\\\sum_{k=1}^K \\\\pi_k=1<\/span><\/span>\u2211<\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u4e14\u03c0k\u22650\\\\pi_k \\\\geq 0<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>；<\/li>\n
      • N(x\u2223\u03bck,\u03a3k)\\\\mathcal{N}(x | \\\\mu_k, \\\\Sigma_k)<\/span><\/span>N<\/span>(<\/span>x<\/span>\u2223<\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570，\u03bck\\\\mu_k<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u5747\u503c\u5411\u91cf，\u03a3k\\\\Sigma_k<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u534f\u65b9\u5dee\u77e9\u9635；<\/li>\n
      • p(x)p(x)<\/span><\/span>p<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>：\u6837\u672cxx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u5728GMM\u6a21\u578b\u4e2d\u7684\u6982\u7387\u5bc6\u5ea6\u503c，\u53cd\u6620\u6837\u672c\u7b26\u5408\u8be5\u6df7\u5408\u5206\u5e03\u7684\u7a0b\u5ea6\u3002<\/li>\n<\/ul>\n
        \u5355\u9ad8\u65af\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570<\/h5>\n

        GMM\u7684\u57fa\u7840\u662f\u5355\u9ad8\u65af\u5206\u5e03，dd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span>\u7ef4\u5355\u9ad8\u65af\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a：
        \nN(x\u2223\u03bck,\u03a3k)=1(2\u03c0)d\/2\u2223\u03a3k\u22231\/2exp\u2061(\u221212(x\u2212\u03bck)T\u03a3k\u22121(x\u2212\u03bck))\\\\mathcal{N}\\\\left(x | \\\\mu_{k}, \\\\Sigma_{k}\\\\right)=\\\\frac{1}{(2 \\\\pi)^{d \/ 2}\\\\left|\\\\Sigma_{k}\\\\right|^{1 \/ 2}} \\\\exp \\\\left(-\\\\frac{1}{2}\\\\left(x-\\\\mu_{k}\\\\right)^{T} \\\\Sigma_{k}^{-1}\\\\left(x-\\\\mu_{k}\\\\right)\\\\right)<\/span><\/span>N<\/span><\/span>(<\/span>x<\/span>\u2223<\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>2<\/span>\u03c0<\/span>)<\/span><\/span>d<\/span>\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span><\/span>1\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>exp<\/span><\/span>(<\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

          \n
        • \u2223\u03a3k\u2223|\\\\Sigma_k|<\/span><\/span>\u2223<\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span><\/span><\/span><\/span>：\u534f\u65b9\u5dee\u77e9\u9635\u7684\u884c\u5217\u5f0f；<\/li>\n
        • \u03a3k\u22121\\\\Sigma_k^{-1}<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：\u534f\u65b9\u5dee\u77e9\u9635\u7684\u9006\u77e9\u9635；<\/li>\n
        • \u6838\u5fc3\u90e8\u5206(x\u2212\u03bck)T\u03a3k\u22121(x\u2212\u03bck)\\\\left(x-\\\\mu_k\\\\right)^T \\\\Sigma_k^{-1} \\\\left(x-\\\\mu_k\\\\right)<\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>：\u6837\u672cxx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u5230\u5747\u503c\u03bck\\\\mu_k<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u9a6c\u6c0f\u8ddd\u79bb，\u8003\u8651\u4e86\u7279\u5f81\u4e4b\u95f4\u7684\u76f8\u5173\u6027，\u6bd4\u6b27\u6c0f\u8ddd\u79bb\u66f4\u8d34\u5408\u5b9e\u9645\u3002<\/li>\n<\/ul>\n

          2.2 GMM\u7684\u6a21\u578b\u53c2\u6570\u4e0e\u4f18\u5316\u76ee\u6807<\/h4>\n

          （1）GMM\u7684\u5f85\u4f30\u53c2\u6570<\/h5>\n

          GMM\u7684\u6a21\u578b\u53c2\u6570\u96c6\u5408\u0398={\u03c0k,\u03bck,\u03a3k}k=1K\\\\Theta = \\\\{\\\\pi_k, \\\\mu_k, \\\\Sigma_k\\\\}_{k=1}^K<\/span><\/span>\u0398<\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5373\u9700\u8981\u4e3a\u6bcf\u4e2a\u9ad8\u65af\u5206\u5e03\u4f30\u8ba13\u4e2a\u53c2\u6570：<\/p>\n

            \n
          • \u6743\u91cd\u03c0k\\\\pi_k<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u7c07\u7684\u5360\u6bd4；<\/li>\n
          • \u5747\u503c\u03bck\\\\mu_k<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u7c07\u7684\u4e2d\u5fc3；<\/li>\n
          • \u534f\u65b9\u5dee\u03a3k\\\\Sigma_k<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u7c07\u7684\u6570\u636e\u79bb\u6563\u7a0b\u5ea6\u548c\u7279\u5f81\u76f8\u5173\u6027\u3002<\/li>\n<\/ul>\n
            （2）\u4f18\u5316\u76ee\u6807：\u6700\u5927\u5316\u5bf9\u6570\u4f3c\u7136\u51fd\u6570<\/h5>\n

            GMM\u7684\u8bad\u7ec3\u76ee\u6807\u662f\u627e\u5230\u4e00\u7ec4\u53c2\u6570\u0398\\\\Theta<\/span><\/span>\u0398<\/span><\/span><\/span><\/span><\/span>，\u8ba9\u6570\u636e\u96c6XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span>\u5728\u8be5\u6a21\u578b\u4e0b\u7684\u4f3c\u7136\u51fd\u6570\u6700\u5927，\u5373\u8ba9\u6a21\u578b\u5c3d\u53ef\u80fd\u8d34\u5408\u5b9e\u9645\u6570\u636e\u5206\u5e03\u3002<\/p>\n

            \u7531\u4e8e\u76f4\u63a5\u6700\u5927\u5316\u4f3c\u7136\u51fd\u6570\u8ba1\u7b97\u590d\u6742，\u901a\u5e38\u5bf9\u5176\u53d6\u5bf9\u6570（\u4e0d\u6539\u53d8\u5355\u8c03\u6027，\u4e14\u5c06\u4e58\u79ef\u8f6c\u4e3a\u6c42\u548c，\u7b80\u5316\u8ba1\u7b97），\u5f97\u5230\u5bf9\u6570\u4f3c\u7136\u51fd\u6570：
            \nln\u2061p(X\u2223\u0398)=\u2211i=1Nln\u2061(\u2211k=1K\u03c0kN(xi\u2223\u03bck,\u03a3k))\\\\ln p(X | \\\\Theta)=\\\\sum_{i=1}^{N} \\\\ln \\\\left(\\\\sum_{k=1}^{K} \\\\pi_{k} \\\\mathcal{N}\\\\left(x_{i} | \\\\mu_{k}, \\\\Sigma_{k}\\\\right)\\\\right)<\/span><\/span>ln<\/span><\/span>p<\/span>(<\/span>X<\/span>\u2223\u0398<\/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><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>ln<\/span><\/span>(<\/span><\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

              \n
            • \u8be5\u51fd\u6570\u8868\u793a\u6574\u4e2a\u6570\u636e\u96c6\u5728\u53c2\u6570\u0398\\\\Theta<\/span><\/span>\u0398<\/span><\/span><\/span><\/span><\/span>\u4e0b\u7684\u6574\u4f53\u6982\u7387，\u503c\u8d8a\u5927，\u6a21\u578b\u62df\u5408\u6548\u679c\u8d8a\u597d\u3002<\/li>\n<\/ul>\n

              2.3 \u4e3a\u4ec0\u4e48\u7528EM\u7b97\u6cd5\u6c42\u89e3GMM？<\/h4>\n

              \u76f4\u63a5\u5bf9\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u6c42\u5bfc\u6c42\u89e3\u6700\u4f18\u53c2\u6570\u0398\\\\Theta<\/span><\/span>\u0398<\/span><\/span><\/span><\/span><\/span>\u4f1a\u9047\u5230\u4e0d\u53ef\u89e3\u7684\u79ef\u5206\u95ee\u9898，\u56e0\u4e3a\u516c\u5f0f\u4e2d\u5b58\u5728\u201c\u6c42\u548c\u5185\u7684\u5bf9\u6570\u201d，\u65e0\u6cd5\u901a\u8fc7\u6c42\u5bfc\u5f97\u5230\u95ed\u5f0f\u89e3\u3002<\/p>\n

              \u800cEM\u7b97\u6cd5\u662f\u89e3\u51b3\u542b\u9690\u53d8\u91cf\u4f18\u5316\u95ee\u9898\u7684\u7ecf\u5178\u8fed\u4ee3\u7b97\u6cd5，GMM\u4e2d\u5b58\u5728\u9690\u53d8\u91cfziz_i<\/span><\/span>z<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>（\u8868\u793a\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c5e\u4e8e\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u6807\u7b7e），EM\u7b97\u6cd5\u901a\u8fc7\u4ea4\u66ff\u6267\u884c\u671f\u671b\u6b65（E\u6b65）\u548c\u6700\u5927\u5316\u6b65（M\u6b65），\u9010\u6b65\u903c\u8fd1\u6700\u4f18\u53c2\u6570，\u5b8c\u7f8e\u89e3\u51b3\u4e86GMM\u7684\u53c2\u6570\u4f30\u8ba1\u95ee\u9898\u3002<\/p>\n

              \u4e09\u3001EM\u7b97\u6cd5\u6c42\u89e3GMM\u7684\u6838\u5fc3\u63a8\u5bfc<\/h3>\n

              EM\u7b97\u6cd5\u7684\u6838\u5fc3\u601d\u60f3\u662f：\u901a\u8fc7\u8fed\u4ee3\u4f30\u8ba1\u9690\u53d8\u91cf\u7684\u540e\u9a8c\u6982\u7387（E\u6b65），\u518d\u5229\u7528\u540e\u9a8c\u6982\u7387\u66f4\u65b0\u6a21\u578b\u53c2\u6570（M\u6b65），\u76f4\u5230\u53c2\u6570\u6536\u655b（\u53d8\u5316\u5c0f\u4e8e\u9884\u8bbe\u9608\u503c），\u6700\u7ec8\u5f97\u5230\u6700\u4f18\u7684GMM\u6a21\u578b\u3002<\/p>\n

              3.1 \u9690\u53d8\u91cf\u4e0e\u540e\u9a8c\u6982\u7387（\u8d23\u4efb\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>）<\/h4>\n

              \u5b9a\u4e49\u9690\u53d8\u91cfzi\u2208{1,2,…,K}z_i \\\\in \\\\{1,2,…,K\\\\}<\/span><\/span>z<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>{<\/span>1<\/span>,<\/span><\/span>2<\/span>,<\/span><\/span>…<\/span>,<\/span><\/span>K<\/span>}<\/span><\/span><\/span><\/span><\/span>，\u8868\u793a\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u6240\u5c5e\u7684\u9ad8\u65af\u5206\u5e03\u7f16\u53f7，zi=kz_i=k<\/span><\/span>z<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u5373\u8868\u793axix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c5e\u4e8e\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u3002<\/p>\n

              \u5728E\u6b65\u4e2d，\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u7ed9\u5b9a\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u548c\u5f53\u524d\u53c2\u6570\u0398\\\\Theta<\/span><\/span>\u0398<\/span><\/span><\/span><\/span><\/span>\u65f6，\u9690\u53d8\u91cfzi=kz_i=k<\/span><\/span>z<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u7684\u540e\u9a8c\u6982\u7387，\u8bb0\u4e3a\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4e5f\u79f0\u4e3a\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u5bf9\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u8d23\u4efb：
              \n\u03b3ik=p(zi=k\u2223xi,\u0398)=\u03c0kN(xi\u2223\u03bck,\u03a3k)\u2211j=1K\u03c0jN(xi\u2223\u03bcj,\u03a3j)\\\\gamma_{i k}=p\\\\left(z_{i}=k | x_{i}, \\\\Theta\\\\right)=\\\\frac{\\\\pi_{k} \\\\mathcal{N}\\\\left(x_{i} | \\\\mu_{k}, \\\\Sigma_{k}\\\\right)}{\\\\sum_{j=1}^{K} \\\\pi_{j} \\\\mathcal{N}\\\\left(x_{i} | \\\\mu_{j}, \\\\Sigma_{j}\\\\right)}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>(<\/span>z<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>k<\/span>\u2223<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u0398<\/span>)<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span>j<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>\u03bc<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>j<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

              \u516c\u5f0f\u610f\u4e49<\/h5>\n
                \n
              • \u5206\u5b50：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u6743\u91cd\u03c0k\\\\pi_k<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e58\u4ee5\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5728\u8be5\u5206\u5e03\u4e0b\u7684\u6982\u7387\u5bc6\u5ea6，\u5373\u5148\u9a8c\u6982\u7387\u00d7\u4f3c\u7136；<\/li>\n
              • \u5206\u6bcd：\u6240\u6709\u9ad8\u65af\u5206\u5e03\u7684\u201c\u5148\u9a8c\u6982\u7387\u00d7\u4f3c\u7136\u201d\u4e4b\u548c，\u5373\u5168\u6982\u7387；<\/li>\n
              • \u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u7269\u7406\u610f\u4e49：\u5728\u5f53\u524d\u6a21\u578b\u4e0b，\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c5e\u4e8e\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u7c07\u7684\u6982\u7387，\u6ee1\u8db3\u2211k=1K\u03b3ik=1\\\\sum_{k=1}^K \\\\gamma_{ik}=1<\/span><\/span>\u2211<\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<\/ul>\n

                3.2 EM\u7b97\u6cd5\u7684\u4e24\u6b65\u8fed\u4ee3\u63a8\u5bfc<\/h4>\n

                EM\u7b97\u6cd5\u7684\u8fed\u4ee3\u8fc7\u7a0b\u5206\u4e3aE\u6b65\u548cM\u6b65，\u5148\u521d\u59cb\u5316\u53c2\u6570，\u518d\u4ea4\u66ff\u6267\u884c\u4e24\u6b65，\u76f4\u5230\u6536\u655b，\u4ee5\u4e0b\u662f\u5b8c\u6574\u7684\u53c2\u6570\u66f4\u65b0\u516c\u5f0f\u63a8\u5bfc（\u672c\u79d1\/\u7814\u7a76\u751f\u5165\u95e8\u9636\u6bb5\u91cd\u70b9\u638c\u63e1\u516c\u5f0f\u542b\u4e49\u548c\u4f7f\u7528，\u65e0\u9700\u6df1\u7a76\u590d\u6742\u7684\u6c42\u5bfc\u8fc7\u7a0b）\u3002<\/p>\n

                \u6b65\u9aa41：\u671f\u671b\u6b65（E\u6b65）<\/h5>\n

                \u6839\u636e\u5f53\u524d\u6a21\u578b\u53c2\u6570\u0398(t)\\\\Theta^{(t)}<\/span><\/span>\u0398<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>（\u4e0a\u4e00\u8f6e\u8fed\u4ee3\u7684\u53c2\u6570），\u8ba1\u7b97\u6bcf\u4e2a\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5c5e\u4e8e\u6bcf\u4e2a\u9ad8\u65af\u5206\u5e03kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u7684\u540e\u9a8c\u6982\u7387\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5373：
                \n\u03b3ik(t)=\u03c0k(t)N(xi\u2223\u03bck(t),\u03a3k(t))\u2211j=1K\u03c0j(t)N(xi\u2223\u03bcj(t),\u03a3j(t))\\\\gamma_{ik}^{(t)} = \\\\frac{\\\\pi_k^{(t)} \\\\mathcal{N}\\\\left(x_i | \\\\mu_k^{(t)}, \\\\Sigma_k^{(t)}\\\\right)}{\\\\sum_{j=1}^K \\\\pi_j^{(t)} \\\\mathcal{N}\\\\left(x_i | \\\\mu_j^{(t)}, \\\\Sigma_j^{(t)}\\\\right)}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span>j<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>j<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>(<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>\u03bc<\/span><\/span>j<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>j<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>(<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

                  \n
                • \u6838\u5fc3\u7ed3\u679c：\u5f97\u5230\u6240\u6709\u6837\u672c\u7684\u5f52\u5c5e\u6982\u7387\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4e3a\u540e\u7eed\u53c2\u6570\u66f4\u65b0\u63d0\u4f9b\u4f9d\u636e\u3002<\/li>\n<\/ul>\n
                  \u6b65\u9aa42：\u6700\u5927\u5316\u6b65（M\u6b65）<\/h5>\n

                  \u5229\u7528E\u6b65\u5f97\u5230\u7684\u540e\u9a8c\u6982\u7387\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u91cd\u65b0\u4f30\u8ba1\u6a21\u578b\u53c2\u6570\u0398(t+1)\\\\Theta^{(t+1)}<\/span><\/span>\u0398<\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8ba9\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u6700\u5927\u5316，\u4e09\u4e2a\u53c2\u6570\u7684\u66f4\u65b0\u516c\u5f0f\u5982\u4e0b：<\/p>\n

                  （1）\u66f4\u65b0\u6743\u91cd\u03c0k\\\\pi_k<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/h6>\n

                  \u03c0k(t+1)=NkN\\\\pi_k^{(t+1)} = \\\\frac{N_k}{N}<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
                  \n\u5176\u4e2dNk=\u2211i=1N\u03b3ikN_k = \\\\sum_{i=1}^N \\\\gamma_{ik}<\/span><\/span>N<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2211<\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8868\u793a\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u9ad8\u65af\u5206\u5e03\u7684\u6709\u6548\u6837\u672c\u6570（\u7531\u5f52\u5c5e\u6982\u7387\u52a0\u6743\u6c42\u548c\u5f97\u5230，\u800c\u975e\u6574\u6570），NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u4e3a\u603b\u6837\u672c\u6570\u3002<\/p>\n

                    \n
                  • \u610f\u4e49：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u7c07\u7684\u6743\u91cd\u7b49\u4e8e\u5176\u6709\u6548\u6837\u672c\u6570\u5360\u603b\u6837\u672c\u6570\u7684\u6bd4\u4f8b\u3002<\/li>\n<\/ul>\n
                    （2）\u66f4\u65b0\u5747\u503c\u03bck\\\\mu_k<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/h6>\n

                    \u03bck(t+1)=1Nk\u2211i=1N\u03b3ikxi\\\\mu_k^{(t+1)} = \\\\frac{1}{N_k} \\\\sum_{i=1}^N \\\\gamma_{ik} x_i<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/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>\u03b3<\/span><\/span>ik<\/span><\/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><\/span><\/span><\/span><\/span><\/p>\n

                      \n
                    • \u610f\u4e49：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u7c07\u7684\u5747\u503c\u662f\u6240\u6709\u6837\u672c\u7684\u52a0\u6743\u5e73\u5747，\u6743\u91cd\u4e3a\u6837\u672c\u5bf9\u8be5\u7c07\u7684\u5f52\u5c5e\u6982\u7387\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u76f8\u6bd4K-means\u7684\u7b80\u5355\u5e73\u5747，\u66f4\u8d34\u5408\u8f6f\u805a\u7c7b\u7684\u7279\u70b9\u3002<\/li>\n<\/ul>\n
                      （3）\u66f4\u65b0\u534f\u65b9\u5dee\u77e9\u9635\u03a3k\\\\Sigma_k<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/h6>\n

                      \u03a3k(t+1)=1Nk\u2211i=1N\u03b3ik(xi\u2212\u03bck(t+1))(xi\u2212\u03bck(t+1))T\\\\Sigma_k^{(t+1)} = \\\\frac{1}{N_k} \\\\sum_{i=1}^N \\\\gamma_{ik} \\\\left(x_i – \\\\mu_k^{(t+1)}\\\\right) \\\\left(x_i – \\\\mu_k^{(t+1)}\\\\right)^T<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/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>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/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>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/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>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

                        \n
                      • \u610f\u4e49：\u7b2ckk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e2a\u7c07\u7684\u534f\u65b9\u5dee\u77e9\u9635\u662f\u6837\u672c\u4e0e\u5747\u503c\u7684\u504f\u5dee\u7684\u52a0\u6743\u534f\u65b9\u5dee，\u6743\u91cd\u4ecd\u4e3a\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u80fd\u53cd\u6620\u8be5\u7c07\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\u548c\u7279\u5f81\u4e4b\u95f4\u7684\u76f8\u5173\u6027\u3002<\/li>\n<\/ul>\n

                        3.3 EM\u7b97\u6cd5\u7684\u6536\u655b\u5224\u65ad<\/h4>\n

                        \u6bcf\u6b21\u8fed\u4ee3（E\u6b65+M\u6b65）\u540e，\u8ba1\u7b97\u5f53\u524d\u53c2\u6570\u4e0b\u7684\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u503cL(\u0398(t+1))L(\\\\Theta^{(t+1)})<\/span><\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5e76\u4e0e\u4e0a\u4e00\u8f6e\u7684L(\u0398(t))L(\\\\Theta^{(t)})<\/span><\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u6bd4\u8f83：
                        \nL(\u0398)=\u2211i=1Nln\u2061(\u2211k=1K\u03c0kN(xi\u2223\u03bck,\u03a3k))L(\\\\Theta) = \\\\sum_{i=1}^N \\\\ln \\\\left(\\\\sum_{k=1}^K \\\\pi_k \\\\mathcal{N}\\\\left(x_i | \\\\mu_k, \\\\Sigma_k\\\\right)\\\\right)<\/span><\/span>L<\/span>(<\/span>\u0398<\/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>ln<\/span><\/span>(<\/span><\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>N<\/span><\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

                          \n
                        • \u82e5\u4e24\u6b21\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u7684\u5dee\u503c\u5c0f\u4e8e\u9884\u8bbe\u9608\u503c（\u598210\u2212610^{-6}<\/span><\/span>1<\/span>0<\/span><\/span>\u2212<\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>），\u5219\u8ba4\u4e3a\u6a21\u578b\u53c2\u6570\u6536\u655b，\u505c\u6b62\u8fed\u4ee3；<\/li>\n
                        • \u82e5\u672a\u6536\u655b，\u5219\u7528\u65b0\u53c2\u6570\u0398(t+1)\\\\Theta^{(t+1)}<\/span><\/span>\u0398<\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7ee7\u7eed\u6267\u884c\u4e0b\u4e00\u8f6eE\u6b65\u548cM\u6b65\u3002<\/li>\n<\/ul>\n

                          \u6838\u5fc3\u6027\u8d28：EM\u7b97\u6cd5\u7684\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u503c\u5355\u8c03\u4e0d\u51cf，\u5373L(\u0398(t+1))\u2265L(\u0398(t))L(\\\\Theta^{(t+1)}) \\\\geq L(\\\\Theta^{(t)})<\/span><\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u4fdd\u8bc1\u7b97\u6cd5\u6700\u7ec8\u4f1a\u6536\u655b（\u53ef\u80fd\u6536\u655b\u5230\u5c40\u90e8\u6700\u4f18，\u56e0\u6b64\u9700\u8981\u5408\u7406\u521d\u59cb\u5316\u53c2\u6570）\u3002<\/p>\n

                          \u56db\u3001GMM\u7684\u5b8c\u6574\u7b97\u6cd5\u6d41\u7a0b<\/h3>\n

                          GMM\u7684\u8bad\u7ec3\u8fc7\u7a0b\u662f\u53c2\u6570\u521d\u59cb\u5316+EM\u7b97\u6cd5\u8fed\u4ee3+\u6536\u655b\u5224\u65ad\u7684\u7ec4\u5408，\u6d41\u7a0b\u6e05\u6670，\u6bcf\u4e00\u6b65\u90fd\u6709\u660e\u786e\u7684\u6570\u5b66\u4f9d\u636e，\u540c\u65f6\u5408\u7406\u7684\u53c2\u6570\u521d\u59cb\u5316\u80fd\u907f\u514d\u7b97\u6cd5\u6536\u655b\u5230\u5c40\u90e8\u6700\u4f18\u3002<\/p>\n

                          \u6b65\u9aa41：\u6a21\u578b\u53c2\u6570\u521d\u59cb\u5316<\/h4>\n

                          \u9996\u5148\u786e\u5b9a\u9ad8\u65af\u5206\u5e03\u7684\u6570\u91cfKK<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span>（\u5373\u805a\u7c7b\u7c07\u6570），\u7136\u540e\u521d\u59cb\u5316\u6a21\u578b\u53c2\u6570\u0398(0)=[\u03c0k(0),\u03bck(0),\u03a3k(0)]\\\\Theta^{(0)} = [\\\\pi_k^{(0)}, \\\\mu_k^{(0)}, \\\\Sigma_k^{(0)}]<\/span><\/span>\u0398<\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span>，\u4e0d\u5efa\u8bae\u968f\u673a\u521d\u59cb\u5316（\u6613\u6536\u655b\u5230\u5c40\u90e8\u6700\u4f18），\u63a8\u8350\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u5f0f：<\/p>\n

                        • K-means\u521d\u59cb\u5316（\u6700\u5e38\u7528）：\u5148\u5bf9\u6570\u636e\u505aK-means\u805a\u7c7b，\u5c06K-means\u7684\u7c07\u4e2d\u5fc3\u4f5c\u4e3aGMM\u7684\u521d\u59cb\u5747\u503c\u03bck(0)\\\\mu_k^{(0)}<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u7c07\u7684\u6837\u672c\u6bd4\u4f8b\u4f5c\u4e3a\u521d\u59cb\u6743\u91cd\u03c0k(0)\\\\pi_k^{(0)}<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u7c07\u5185\u6837\u672c\u7684\u534f\u65b9\u5dee\u4f5c\u4e3a\u521d\u59cb\u534f\u65b9\u5dee\u03a3k(0)\\\\Sigma_k^{(0)}<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>；<\/li>\n
                        • \u968f\u673a\u91c7\u6837\u521d\u59cb\u5316：\u4ece\u6570\u636e\u4e2d\u968f\u673a\u91c7\u6837K\u4e2a\u6837\u672c\u4f5c\u4e3a\u521d\u59cb\u5747\u503c\u03bck(0)\\\\mu_k^{(0)}<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u6743\u91cd\u03c0k(0)\\\\pi_k^{(0)}<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u521d\u59cb\u5316\u4e3a1\/K1\/K<\/span><\/span>1\/<\/span>K<\/span><\/span><\/span><\/span><\/span>，\u534f\u65b9\u5dee\u03a3k(0)\\\\Sigma_k^{(0)}<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u521d\u59cb\u5316\u4e3a\u5355\u4f4d\u77e9\u9635\u3002<\/li>\n

                          \u6b65\u9aa42：\u6267\u884cEM\u7b97\u6cd5\u8fed\u4ee3<\/h4>\n

                          \u4ea4\u66ff\u6267\u884cE\u6b65\u548cM\u6b65，\u9010\u6b65\u66f4\u65b0\u6a21\u578b\u53c2\u6570：<\/p>\n

                        • E\u6b65：\u6839\u636e\u5f53\u524d\u53c2\u6570\u0398(t)\\\\Theta^{(t)}<\/span><\/span>\u0398<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8ba1\u7b97\u6240\u6709\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u5bf9\u6240\u6709\u7c07kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u7684\u540e\u9a8c\u6982\u7387（\u8d23\u4efb）\u03b3ik(t)\\\\gamma_{ik}^{(t)}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>；<\/li>\n
                        • M\u6b65：\u6839\u636e\u03b3ik(t)\\\\gamma_{ik}^{(t)}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5206\u522b\u66f4\u65b0\u6743\u91cd\u03c0k(t+1)\\\\pi_k^{(t+1)}<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u5747\u503c\u03bck(t+1)\\\\mu_k^{(t+1)}<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u534f\u65b9\u5dee\u03a3k(t+1)\\\\Sigma_k^{(t+1)}<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n

                          \u6b65\u9aa43：\u6536\u655b\u5224\u65ad<\/h4>\n

                          \u8ba1\u7b97\u8fed\u4ee3\u524d\u540e\u7684\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u503cL(\u0398(t))L(\\\\Theta^{(t)})<\/span><\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u548cL(\u0398(t+1))L(\\\\Theta^{(t+1)})<\/span><\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5224\u65ad\u662f\u5426\u6ee1\u8db3\u6536\u655b\u6761\u4ef6：<\/p>\n

                            \n
                          • \u82e5\u2223L(\u0398(t+1))\u2212L(\u0398(t))\u2223<\u03f5|L(\\\\Theta^{(t+1)}) – L(\\\\Theta^{(t)})| < \\\\epsilon<\/span><\/span>\u2223<\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>L<\/span>(<\/span>\u0398<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2223<\/span><\/span><<\/span><\/span><\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span>（\u03f5\\\\epsilon<\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span>\u4e3a\u9884\u8bbe\u9608\u503c，\u598210\u2212610^{-6}<\/span><\/span>1<\/span>0<\/span><\/span>\u2212<\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>），\u5219\u6536\u655b，\u505c\u6b62\u8fed\u4ee3；<\/li>\n
                          • \u82e5\u672a\u6536\u655b，\u4ee4\u0398(t)=\u0398(t+1)\\\\Theta^{(t)} = \\\\Theta^{(t+1)}<\/span><\/span>\u0398<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u0398<\/span><\/span>(<\/span>t<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8fd4\u56de\u6b65\u9aa42\u7ee7\u7eed\u8fed\u4ee3\u3002<\/li>\n<\/ul>\n

                            \u6b65\u9aa44：\u6a21\u578b\u8f93\u51fa\u4e0e\u5e94\u7528<\/h4>\n

                            \u8fed\u4ee3\u6536\u655b\u540e，\u5f97\u5230\u6700\u4f18\u6a21\u578b\u53c2\u6570\u0398\u2217\\\\Theta^*<\/span><\/span>\u0398<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u6b64\u65f6\u53ef\u5b9e\u73b0\u4e24\u5927\u6838\u5fc3\u529f\u80fd：<\/p>\n

                          • \u805a\u7c7b：\u5bf9\u6bcf\u4e2a\u6837\u672cxix_i<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8ba1\u7b97\u5176\u5bf9\u6240\u6709\u7c07\u7684\u540e\u9a8c\u6982\u7387\u03b3ik\\\\gamma_{ik}<\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5c06\u6837\u672c\u5f52\u4e3a\u6982\u7387\u6700\u5927\u7684\u7c07，\u5373k\u2217=arg\u2061max\u2061k\u03b3ikk^* = \\\\arg\\\\max_{k} \\\\gamma_{ik}<\/span><\/span>k<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>arg<\/span><\/span><\/span>max<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b3<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>；<\/li>\n
                          • \u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1：\u5229\u7528GMM\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570p(x)p(x)<\/span><\/span>p<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u8ba1\u7b97\u4efb\u610f\u6837\u672cxx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u7684\u5bc6\u5ea6\u503c，\u7528\u4e8e\u5f02\u5e38\u68c0\u6d4b（\u5bc6\u5ea6\u503c\u8fc7\u4f4e\u7684\u6837\u672c\u4e3a\u5f02\u5e38\u503c）\u3001\u6837\u672c\u751f\u6210\u7b49\u3002<\/li>\n

                            \u4e94\u3001GMM\u5b9e\u6218：\u6570\u636e\u805a\u7c7b\u4e0e\u6a21\u578b\u4f18\u5316（Python\u5b8c\u6574\u4ee3\u7801）<\/h3>\n

                            \u672c\u6b21\u5b9e\u6218\u57fa\u4e8e\u4eba\u5de5\u751f\u6210\u7684\u6df7\u5408\u9ad8\u65af\u6570\u636e\u96c6，\u5b9e\u73b0GMM\u805a\u7c7b\u3001\u805a\u7c7b\u7ed3\u679c\u53ef\u89c6\u5316\u3001\u6700\u4f18\u7c07\u6570\u9009\u62e9（AIC\/BIC\u51c6\u5219），\u4f7f\u7528sklearn\u7684GaussianMixture（\u5c01\u88c5\u597d\u7684GMM\u5b9e\u73b0，\u5165\u95e8\u53cb\u597d），\u4ee3\u7801\u53ef\u76f4\u63a5\u590d\u5236\u8fd0\u884c，\u9002\u914dPython3.8+\u3001scikit-learn1.0+\u3001matplotlib3.5+\u3002<\/p>\n

                            5.1 \u5b9e\u6218\u76ee\u6807<\/h4>\n
                          • \u638c\u63e1GMM\u7684\u57fa\u672c\u4f7f\u7528\u6d41\u7a0b：\u6570\u636e\u751f\u6210\u3001\u6807\u51c6\u5316\u3001\u6a21\u578b\u8bad\u7ec3\u3001\u805a\u7c7b\u9884\u6d4b；<\/li>\n
                          • \u5b66\u4e60GMM\u805a\u7c7b\u7ed3\u679c\u7684\u53ef\u89c6\u5316\u65b9\u6cd5（PCA\u964d\u7ef4）；<\/li>\n
                          • \u638c\u63e1\u6700\u4f18\u7c07\u6570KK<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span>\u7684\u9009\u62e9\u65b9\u6cd5：AIC\u51c6\u5219\u3001BIC\u51c6\u5219；<\/li>\n
                          • \u5bf9\u6bd4GMM\u805a\u7c7b\u7ed3\u679c\u4e0e\u771f\u5b9e\u6807\u7b7e，\u9a8c\u8bc1\u6a21\u578b\u6548\u679c；<\/li>\n
                          • \u638c\u63e1GMM\u7684\u6838\u5fc3\u8d85\u53c2\u6570\u8c03\u4f18\u65b9\u6cd5\u3002<\/li>\n

                            5.2 \u5b8c\u6574\u4ee3\u7801\u4e0e\u8be6\u7ec6\u6ce8\u91ca<\/h4>\n

                            # \u5bfc\u5165\u6240\u9700\u5e93<\/span>
                            \nimport<\/span> numpy as<\/span> np
                            \nimport<\/span> pandas as<\/span> pd
                            \nimport<\/span> matplotlib.<\/span>pyplot as<\/span> plt
                            \nimport<\/span> seaborn as<\/span> sns
                            \nfrom<\/span> sklearn.<\/span>mixture import<\/span> GaussianMixture
                            \nfrom<\/span> sklearn.<\/span>datasets import<\/span> make_blobs
                            \nfrom<\/span> sklearn.<\/span>preprocessing import<\/span> StandardScaler
                            \nfrom<\/span> sklearn.<\/span>decomposition import<\/span> PCA<\/p>\n

                            # \u89e3\u51b3\u4e2d\u6587\u663e\u793a\u95ee\u9898<\/span>
                            \nplt.<\/span>rcParams[<\/span>'font.sans-serif'<\/span>]<\/span> =<\/span> [<\/span>'SimHei'<\/span>]<\/span>
                            \nplt.<\/span>rcParams[<\/span>'axes.unicode_minus'<\/span>]<\/span> =<\/span> False<\/span>
                            \n# \u8bbe\u7f6e\u53ef\u89c6\u5316\u98ce\u683c<\/span>
                            \nsns.<\/span>set<\/span>(<\/span>style=<\/span>"white"<\/span>,<\/span> palette=<\/span>"muted"<\/span>,<\/span> color_codes=<\/span>True<\/span>)<\/span><\/p>\n

                            # ===================== 1. \u751f\u6210\u6df7\u5408\u9ad8\u65af\u6570\u636e\u96c6 =====================<\/span>
                            \n# \u6570\u636e\u96c6\u53c2\u6570<\/span>
                            \nn_samples =<\/span> 5000<\/span> # \u603b\u6837\u672c\u6570<\/span>
                            \nn_features =<\/span> 4<\/span> # \u7279\u5f81\u7ef4\u5ea6<\/span>
                            \nn_clusters =<\/span> 4<\/span> # \u771f\u5b9e\u7c07\u6570（\u9ad8\u65af\u5206\u5e03\u6570\u91cf）<\/span>
                            \nrandom_state =<\/span> 42<\/span> # \u968f\u673a\u79cd\u5b50，\u4fdd\u8bc1\u7ed3\u679c\u53ef\u590d\u73b0<\/span><\/p>\n

                            # \u751f\u6210\u6df7\u5408\u9ad8\u65af\u6570\u636e：make_blobs\u751f\u6210\u591a\u7c07\u6570\u636e，\u7b26\u5408GMM\u5047\u8bbe<\/span>
                            \nX,<\/span> y_true =<\/span> make_blobs(<\/span>
                            \n n_samples=<\/span>n_samples,<\/span>
                            \n n_features=<\/span>n_features,<\/span>
                            \n centers=<\/span>n_clusters,<\/span>
                            \n cluster_std=<\/span>0.60<\/span>,<\/span> # \u7c07\u5185\u6807\u51c6\u5dee<\/span>
                            \n random_state=<\/span>random_state
                            \n)<\/span>
                            \nprint<\/span>(<\/span>f"\u6570\u636e\u96c6\u5f62\u72b6：<\/span>{<\/span>X.<\/span>shape}<\/span><\/span>，\u771f\u5b9e\u7c07\u6570：<\/span>{<\/span>n_clusters}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

                            # ===================== 2. \u6570\u636e\u9884\u5904\u7406 =====================<\/span>
                            \n# GMM\u5bf9\u7279\u5f81\u5c3a\u5ea6\u654f\u611f，\u9700\u6807\u51c6\u5316（\u5747\u503c0，\u65b9\u5dee1）<\/span>
                            \nscaler =<\/span> StandardScaler(<\/span>)<\/span>
                            \nX_scaled =<\/span> scaler.<\/span>fit_transform(<\/span>X)<\/span><\/p>\n

                            # \u8f6c\u6362\u4e3aDataFrame，\u65b9\u4fbf\u540e\u7eed\u5904\u7406<\/span>
                            \ndf =<\/span> pd.<\/span>DataFrame(<\/span>
                            \n X_scaled,<\/span>
                            \n columns=<\/span>[<\/span>f'\u7279\u5f81<\/span>{<\/span>i+<\/span>1<\/span>}<\/span><\/span>'<\/span><\/span> for<\/span> i in<\/span> range<\/span>(<\/span>n_features)<\/span>]<\/span>
                            \n)<\/span>
                            \ndf[<\/span>'\u771f\u5b9e\u7c07\u6807\u7b7e'<\/span>]<\/span> =<\/span> y_true # \u6dfb\u52a0\u771f\u5b9e\u7c07\u6807\u7b7e<\/span><\/p>\n

                            # ===================== 3. \u57fa\u7840GMM\u6a21\u578b\u8bad\u7ec3\u4e0e\u805a\u7c7b =====================<\/span>
                            \n# \u521d\u59cb\u5316GMM\u6a21\u578b<\/span>
                            \ngmm_base =<\/span> GaussianMixture(<\/span>
                            \n n_components=<\/span>n_clusters,<\/span> # \u9ad8\u65af\u5206\u5e03\u6570\u91cf（\u7c07\u6570），\u4e0e\u771f\u5b9e\u503c\u4e00\u81f4<\/span>
                            \n covariance_type=<\/span>'full'<\/span>,<\/span> # \u534f\u65b9\u5dee\u7c7b\u578b：full\u8868\u793a\u6bcf\u4e2a\u7c07\u6709\u72ec\u7acb\u7684\u534f\u65b9\u5dee\u77e9\u9635<\/span>
                            \n random_state=<\/span>random_state,<\/span>
                            \n max_iter=<\/span>100<\/span>,<\/span> # \u6700\u5927\u8fed\u4ee3\u6b21\u6570<\/span>
                            \n tol=<\/span>1e-6<\/span> # \u6536\u655b\u9608\u503c<\/span>
                            \n)<\/span>
                            \n# \u8bad\u7ec3\u6a21\u578b\u5e76\u9884\u6d4b\u805a\u7c7b\u7ed3\u679c<\/span>
                            \ngmm_base.<\/span>fit(<\/span>X_scaled)<\/span>
                            \ny_pred_base =<\/span> gmm_base.<\/span>predict(<\/span>X_scaled)<\/span> # \u786c\u805a\u7c7b\u7ed3\u679c（\u6700\u5927\u6982\u7387\u7c07）<\/span>
                            \ny_pred_proba =<\/span> gmm_base.<\/span>predict_proba(<\/span>X_scaled)<\/span> # \u8f6f\u805a\u7c7b\u7ed3\u679c（\u5f52\u5c5e\u6982\u7387）<\/span><\/p>\n

                            # \u5c06\u7ed3\u679c\u6dfb\u52a0\u5230DataFrame<\/span>
                            \ndf[<\/span>'GMM\u805a\u7c7b\u6807\u7b7e'<\/span>]<\/span> =<\/span> y_pred_base
                            \n# \u6dfb\u52a0\u6837\u672c\u5bf9\u7b2c\u4e00\u7c07\u7684\u5f52\u5c5e\u6982\u7387（\u793a\u4f8b）<\/span>
                            \ndf[<\/span>'\u7b2c\u4e00\u7c07\u5f52\u5c5e\u6982\u7387'<\/span>]<\/span> =<\/span> y_pred_proba[<\/span>:<\/span>,<\/span> 0<\/span>]<\/span><\/p>\n

                            print<\/span>(<\/span>f"GMM\u6a21\u578b\u8fed\u4ee3\u6b21\u6570：<\/span>{<\/span>gmm_base.<\/span>n_iter_}<\/span><\/span>"<\/span><\/span>)<\/span>
                            \nprint<\/span>(<\/span>f"\u6a21\u578b\u6700\u7ec8\u5bf9\u6570\u4f3c\u7136\u503c：<\/span>{<\/span>gmm_base.<\/span>score(<\/span>X_scaled)<\/span>:<\/span>.4f<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

                            # ===================== 4. \u805a\u7c7b\u7ed3\u679c\u53ef\u89c6\u5316（PCA\u964d\u7ef4\u52302D） =====================<\/span>
                            \n# PCA\u964d\u7ef4：\u5c064\u7ef4\u7279\u5f81\u964d\u52302\u7ef4，\u65b9\u4fbf\u53ef\u89c6\u5316<\/span>
                            \npca =<\/span> PCA(<\/span>n_components=<\/span>2<\/span>,<\/span> random_state=<\/span>random_state)<\/span>
                            \nX_pca =<\/span> pca.<\/span>fit_transform(<\/span>X_scaled)<\/span>
                            \ndf[<\/span>'PCA1'<\/span>]<\/span> =<\/span> X_pca[<\/span>:<\/span>,<\/span> 0<\/span>]<\/span>
                            \ndf[<\/span>'PCA2'<\/span>]<\/span> =<\/span> X_pca[<\/span>:<\/span>,<\/span> 1<\/span>]<\/span><\/p>\n

                            # \u7ed8\u5236\u53ef\u89c6\u5316\u56fe：\u771f\u5b9e\u6807\u7b7e vs GMM\u805a\u7c7b\u6807\u7b7e<\/span>
                            \nplt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>16<\/span>,<\/span> 7<\/span>)<\/span>)<\/span><\/p>\n

                            # \u5b50\u56fe1：\u771f\u5b9e\u7c07\u6807\u7b7e\u5206\u5e03<\/span>
                            \nplt.<\/span>subplot(<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 1<\/span>)<\/span>
                            \nsns.<\/span>scatterplot(<\/span>
                            \n x=<\/span>'PCA1'<\/span>,<\/span> y=<\/span>'PCA2'<\/span>,<\/span> hue=<\/span>'\u771f\u5b9e\u7c07\u6807\u7b7e'<\/span>,<\/span>
                            \n data=<\/span>df,<\/span> palette=<\/span>'viridis'<\/span>,<\/span> s=<\/span>20<\/span>,<\/span> edgecolor=<\/span>'k'<\/span>,<\/span> alpha=<\/span>0.8<\/span>
                            \n)<\/span>
                            \nplt.<\/span>title(<\/span>'\u771f\u5b9e\u7c07\u6807\u7b7e\u5206\u5e03'<\/span>,<\/span> fontsize=<\/span>16<\/span>)<\/span>
                            \nplt.<\/span>xlabel(<\/span>'\u4e3b\u6210\u52061（PCA1）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>ylabel(<\/span>'\u4e3b\u6210\u52062（PCA2）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>legend(<\/span>title=<\/span>'\u771f\u5b9e\u7c07'<\/span>,<\/span> fontsize=<\/span>12<\/span>)<\/span>
                            \nplt.<\/span>grid(<\/span>alpha=<\/span>0.3<\/span>)<\/span><\/p>\n

                            # \u5b50\u56fe2：GMM\u805a\u7c7b\u6807\u7b7e\u5206\u5e03<\/span>
                            \nplt.<\/span>subplot(<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 2<\/span>)<\/span>
                            \nsns.<\/span>scatterplot(<\/span>
                            \n x=<\/span>'PCA1'<\/span>,<\/span> y=<\/span>'PCA2'<\/span>,<\/span> hue=<\/span>'GMM\u805a\u7c7b\u6807\u7b7e'<\/span>,<\/span>
                            \n data=<\/span>df,<\/span> palette=<\/span>'viridis'<\/span>,<\/span> s=<\/span>20<\/span>,<\/span> edgecolor=<\/span>'k'<\/span>,<\/span> alpha=<\/span>0.8<\/span>
                            \n)<\/span>
                            \nplt.<\/span>title(<\/span>'GMM\u805a\u7c7b\u7ed3\u679c\u5206\u5e03'<\/span>,<\/span> fontsize=<\/span>16<\/span>)<\/span>
                            \nplt.<\/span>xlabel(<\/span>'\u4e3b\u6210\u52061（PCA1）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>ylabel(<\/span>'\u4e3b\u6210\u52062（PCA2）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>legend(<\/span>title=<\/span>'GMM\u7c07'<\/span>,<\/span> fontsize=<\/span>12<\/span>)<\/span>
                            \nplt.<\/span>grid(<\/span>alpha=<\/span>0.3<\/span>)<\/span><\/p>\n

                            plt.<\/span>tight_layout(<\/span>)<\/span>
                            \nplt.<\/span>show(<\/span>)<\/span><\/p>\n

                            # ===================== 5. \u6700\u4f18\u7c07\u6570K\u7684\u9009\u62e9（AIC\/BIC\u51c6\u5219） =====================<\/span>
                            \n# \u6d4b\u8bd5\u7c07\u6570\u8303\u56f4：1~9<\/span>
                            \nn_components_range =<\/span> range<\/span>(<\/span>1<\/span>,<\/span> 10<\/span>)<\/span>
                            \n# \u5b58\u50a8\u4e0d\u540c\u7c07\u6570\u7684AIC\u548cBIC\u503c<\/span>
                            \naic_list =<\/span> [<\/span>]<\/span>
                            \nbic_list =<\/span> [<\/span>]<\/span><\/p>\n

                            for<\/span> n in<\/span> n_components_range:<\/span>
                            \n gmm =<\/span> GaussianMixture(<\/span>
                            \n n_components=<\/span>n,<\/span>
                            \n covariance_type=<\/span>'full'<\/span>,<\/span>
                            \n random_state=<\/span>random_state
                            \n )<\/span>
                            \n gmm.<\/span>fit(<\/span>X_scaled)<\/span>
                            \n aic_list.<\/span>append(<\/span>gmm.<\/span>aic(<\/span>X_scaled)<\/span>)<\/span> # AIC\u51c6\u5219<\/span>
                            \n bic_list.<\/span>append(<\/span>gmm.<\/span>bic(<\/span>X_scaled)<\/span>)<\/span> # BIC\u51c6\u5219<\/span><\/p>\n

                            # \u7ed8\u5236AIC\/BIC\u66f2\u7ebf<\/span>
                            \nplt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>12<\/span>,<\/span> 6<\/span>)<\/span>)<\/span>
                            \nplt.<\/span>plot(<\/span>n_components_range,<\/span> aic_list,<\/span> marker=<\/span>'o'<\/span>,<\/span> linewidth=<\/span>2<\/span>,<\/span> label=<\/span>'AIC'<\/span>)<\/span>
                            \nplt.<\/span>plot(<\/span>n_components_range,<\/span> bic_list,<\/span> marker=<\/span>'s'<\/span>,<\/span> linewidth=<\/span>2<\/span>,<\/span> label=<\/span>'BIC'<\/span>)<\/span>
                            \nplt.<\/span>xlabel(<\/span>'\u9ad8\u65af\u5206\u5e03\u6570\u91cf（\u7c07\u6570K）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>ylabel(<\/span>'AIC\/BIC\u503c'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>title(<\/span>'GMM\u6700\u4f18\u7c07\u6570\u9009\u62e9（AIC\/BIC\u51c6\u5219）'<\/span>,<\/span> fontsize=<\/span>16<\/span>)<\/span>
                            \nplt.<\/span>legend(<\/span>fontsize=<\/span>12<\/span>)<\/span>
                            \nplt.<\/span>xticks(<\/span>n_components_range)<\/span>
                            \nplt.<\/span>grid(<\/span>alpha=<\/span>0.3<\/span>)<\/span>
                            \nplt.<\/span>show(<\/span>)<\/span><\/p>\n

                            # \u9009\u62e9\u6700\u4f18\u7c07\u6570：BIC\u503c\u6700\u5c0f\u7684\u7c07\u6570（BIC\u5bf9\u590d\u6742\u6a21\u578b\u60e9\u7f5a\u66f4\u91cd，\u66f4\u9002\u5408\u5c0f\u6837\u672c）<\/span>
                            \nbest_n =<\/span> n_components_range[<\/span>np.<\/span>argmin(<\/span>bic_list)<\/span>]<\/span>
                            \nprint<\/span>(<\/span>f"\u6839\u636eBIC\u51c6\u5219，\u6700\u4f18\u7c07\u6570\u4e3a：<\/span>{<\/span>best_n}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

                            # ===================== 6. \u6700\u4f18\u7c07\u6570GMM\u6a21\u578b\u8bad\u7ec3\u4e0e\u53ef\u89c6\u5316 =====================<\/span>
                            \n# \u521d\u59cb\u5316\u6700\u4f18GMM\u6a21\u578b<\/span>
                            \ngmm_best =<\/span> GaussianMixture(<\/span>
                            \n n_components=<\/span>best_n,<\/span>
                            \n covariance_type=<\/span>'full'<\/span>,<\/span>
                            \n random_state=<\/span>random_state
                            \n)<\/span>
                            \ngmm_best.<\/span>fit(<\/span>X_scaled)<\/span>
                            \ny_pred_best =<\/span> gmm_best.<\/span>predict(<\/span>X_scaled)<\/span>
                            \ndf[<\/span>'GMM\u6700\u4f18\u805a\u7c7b\u6807\u7b7e'<\/span>]<\/span> =<\/span> y_pred_best<\/p>\n

                            # \u53ef\u89c6\u5316\u6700\u4f18\u805a\u7c7b\u7ed3\u679c<\/span>
                            \nplt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>10<\/span>,<\/span> 8<\/span>)<\/span>)<\/span>
                            \nsns.<\/span>scatterplot(<\/span>
                            \n x=<\/span>'PCA1'<\/span>,<\/span> y=<\/span>'PCA2'<\/span>,<\/span> hue=<\/span>'GMM\u6700\u4f18\u805a\u7c7b\u6807\u7b7e'<\/span>,<\/span>
                            \n data=<\/span>df,<\/span> palette=<\/span>'viridis'<\/span>,<\/span> s=<\/span>20<\/span>,<\/span> edgecolor=<\/span>'k'<\/span>,<\/span> alpha=<\/span>0.8<\/span>
                            \n)<\/span>
                            \nplt.<\/span>title(<\/span>f'GMM\u6700\u4f18\u805a\u7c7b\u7ed3\u679c（K=<\/span>{<\/span>best_n}<\/span><\/span>）'<\/span><\/span>,<\/span> fontsize=<\/span>16<\/span>)<\/span>
                            \nplt.<\/span>xlabel(<\/span>'\u4e3b\u6210\u52061（PCA1）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>ylabel(<\/span>'\u4e3b\u6210\u52062（PCA2）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>legend(<\/span>title=<\/span>'GMM\u6700\u4f18\u7c07'<\/span>,<\/span> fontsize=<\/span>12<\/span>)<\/span>
                            \nplt.<\/span>grid(<\/span>alpha=<\/span>0.3<\/span>)<\/span>
                            \nplt.<\/span>show(<\/span>)<\/span><\/p>\n

                            # ===================== 7. GMM\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1\u4e0e\u5f02\u5e38\u68c0\u6d4b（\u62d3\u5c55） =====================<\/span>
                            \n# \u8ba1\u7b97\u6bcf\u4e2a\u6837\u672c\u7684\u6982\u7387\u5bc6\u5ea6\u503c（\u5bf9\u6570\u5f62\u5f0f）<\/span>
                            \nlog_density =<\/span> gmm_best.<\/span>score_samples(<\/span>X_scaled)<\/span>
                            \ndf[<\/span>'\u5bf9\u6570\u5bc6\u5ea6\u503c'<\/span>]<\/span> =<\/span> log_density<\/p>\n

                            # \u5f02\u5e38\u68c0\u6d4b：\u5bc6\u5ea6\u503c\u4f4e\u4e8e\u7b2c5\u767e\u5206\u4f4d\u6570\u7684\u6837\u672c\u4e3a\u5f02\u5e38\u503c<\/span>
                            \nthreshold =<\/span> np.<\/span>percentile(<\/span>log_density,<\/span> 5<\/span>)<\/span>
                            \ndf[<\/span>'\u662f\u5426\u5f02\u5e38'<\/span>]<\/span> =<\/span> (<\/span>log_density <<\/span> threshold)<\/span>.<\/span>astype(<\/span>int<\/span>)<\/span><\/p>\n

                            # \u53ef\u89c6\u5316\u5f02\u5e38\u68c0\u6d4b\u7ed3\u679c<\/span>
                            \nplt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>10<\/span>,<\/span> 8<\/span>)<\/span>)<\/span>
                            \n# \u6b63\u5e38\u6837\u672c<\/span>
                            \nsns.<\/span>scatterplot(<\/span>
                            \n x=<\/span>'PCA1'<\/span>,<\/span> y=<\/span>'PCA2'<\/span>,<\/span> data=<\/span>df[<\/span>df[<\/span>'\u662f\u5426\u5f02\u5e38'<\/span>]<\/span>==<\/span>0<\/span>]<\/span>,<\/span>
                            \n color=<\/span>'lightblue'<\/span>,<\/span> s=<\/span>20<\/span>,<\/span> edgecolor=<\/span>'k'<\/span>,<\/span> alpha=<\/span>0.6<\/span>,<\/span> label=<\/span>'\u6b63\u5e38\u6837\u672c'<\/span>
                            \n)<\/span>
                            \n# \u5f02\u5e38\u6837\u672c<\/span>
                            \nsns.<\/span>scatterplot(<\/span>
                            \n x=<\/span>'PCA1'<\/span>,<\/span> y=<\/span>'PCA2'<\/span>,<\/span> data=<\/span>df[<\/span>df[<\/span>'\u662f\u5426\u5f02\u5e38'<\/span>]<\/span>==<\/span>1<\/span>]<\/span>,<\/span>
                            \n color=<\/span>'red'<\/span>,<\/span> s=<\/span>50<\/span>,<\/span> edgecolor=<\/span>'k'<\/span>,<\/span> alpha=<\/span>0.8<\/span>,<\/span> label=<\/span>'\u5f02\u5e38\u6837\u672c'<\/span>
                            \n)<\/span>
                            \nplt.<\/span>title(<\/span>'GMM\u5bc6\u5ea6\u4f30\u8ba1-\u5f02\u5e38\u68c0\u6d4b\u7ed3\u679c'<\/span>,<\/span> fontsize=<\/span>16<\/span>)<\/span>
                            \nplt.<\/span>xlabel(<\/span>'\u4e3b\u6210\u52061（PCA1）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>ylabel(<\/span>'\u4e3b\u6210\u52062（PCA2）'<\/span>,<\/span> fontsize=<\/span>14<\/span>)<\/span>
                            \nplt.<\/span>legend(<\/span>fontsize=<\/span>12<\/span>)<\/span>
                            \nplt.<\/span>grid(<\/span>alpha=<\/span>0.3<\/span>)<\/span>
                            \nplt.<\/span>show(<\/span>)<\/span><\/p>\n

                            print<\/span>(<\/span>f"\u68c0\u6d4b\u5230\u5f02\u5e38\u6837\u672c\u6570：<\/span>{<\/span>df[<\/span>'\u662f\u5426\u5f02\u5e38'<\/span>]<\/span>.<\/span>sum<\/span>(<\/span>)<\/span>}<\/span><\/span>，\u5360\u6bd4：<\/span>{<\/span>df[<\/span>'\u662f\u5426\u5f02\u5e38'<\/span>]<\/span>.<\/span>mean(<\/span>)<\/span>*<\/span>100<\/span>:<\/span>.2f<\/span>}<\/span><\/span>%"<\/span><\/span>)<\/span><\/p>\n

                            5.3 \u8fd0\u884c\u7ed3\u679c\u8bf4\u660e<\/h4>\n
                          • \u6570\u636e\u96c6\u57fa\u672c\u4fe1\u606f：\u751f\u6210\u7684\u6570\u636e\u96c6\u4e3a4\u7ef4\u7279\u5f81\u30015000\u4e2a\u6837\u672c\u30014\u4e2a\u771f\u5b9e\u7c07，\u7b26\u5408GMM\u7684\u6df7\u5408\u9ad8\u65af\u5206\u5e03\u5047\u8bbe，\u662f\u5178\u578b\u7684GMM\u9002\u7528\u573a\u666f；<\/li>\n
                          • \u57fa\u7840GMM\u6a21\u578b：\u8fed\u4ee3\u6b21\u6570\u7ea610\u6b21\u5de6\u53f3\u5373\u53ef\u6536\u655b，\u5bf9\u6570\u4f3c\u7136\u503c\u4e3a\u6b63\u4e14\u8f83\u5927，\u8bf4\u660e\u6a21\u578b\u62df\u5408\u6548\u679c\u597d，\u805a\u7c7b\u7ed3\u679c\u4e0e\u771f\u5b9e\u6807\u7b7e\u9ad8\u5ea6\u543b\u5408，\u53ef\u89c6\u5316\u56fe\u4e2d\u7c07\u7684\u5206\u5e03\u51e0\u4e4e\u4e00\u81f4；<\/li>\n
                          • AIC\/BIC\u66f2\u7ebf：\u5f53\u7c07\u6570K=4K=4<\/span><\/span>K<\/span><\/span>=<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span>\u65f6，AIC\u548cBIC\u503c\u5747\u8fbe\u5230\u6700\u5c0f\u503c，\u4e0e\u771f\u5b9e\u7c07\u6570\u4e00\u81f4，\u9a8c\u8bc1\u4e86AIC\/BIC\u51c6\u5219\u9009\u62e9\u6700\u4f18\u7c07\u6570\u7684\u6709\u6548\u6027；<\/li>\n
                          • \u5f02\u5e38\u68c0\u6d4b：\u901a\u8fc7GMM\u7684\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1，\u80fd\u6709\u6548\u68c0\u6d4b\u51fa\u6570\u636e\u96c6\u4e2d\u7684\u79bb\u7fa4\u70b9（\u5f02\u5e38\u6837\u672c），\u5360\u6bd4\u7ea65%，\u7b26\u5408\u9884\u8bbe\u7684\u5206\u4f4d\u6570\u9608\u503c；<\/li>\n
                          • \u8f6f\u805a\u7c7b\u7ed3\u679c：predict_proba\u65b9\u6cd5\u8fd4\u56de\u6bcf\u4e2a\u6837\u672c\u5bf9\u6240\u6709\u7c07\u7684\u5f52\u5c5e\u6982\u7387，\u53ef\u6839\u636e\u4e1a\u52a1\u9700\u6c42\u8bbe\u7f6e\u6982\u7387\u9608\u503c，\u5b9e\u73b0\u66f4\u7075\u6d3b\u7684\u805a\u7c7b\u3002<\/li>\n

                            5.4 GMM\u6838\u5fc3\u8d85\u53c2\u6570\u8c03\u4f18\u6307\u5357<\/h4>\n

                            sklearn\u7684GaussianMixture\u5c01\u88c5\u4e86GMM\u7684\u6838\u5fc3\u5b9e\u73b0，\u5b9e\u6218\u4e2d\u53ea\u9700\u8c03\u4f18\u6838\u5fc3\u8d85\u53c2\u6570\u5373\u53ef\u8fbe\u5230\u8f83\u597d\u7684\u6548\u679c，\u6309\u8c03\u4f18\u4f18\u5148\u7ea7\u6392\u5e8f\u5982\u4e0b：<\/p>\n\n\u8d85\u53c2\u6570\u4f5c\u7528\u53ef\u9009\u503c\/\u63a8\u8350\u8303\u56f4\u8c03\u4f18\u6280\u5de7<\/tr>\n\n\n\n\n\n\n
                            n_components<\/td>\n\u9ad8\u65af\u5206\u5e03\u6570\u91cf（\u805a\u7c7b\u7c07\u6570K）<\/td>\n1~10（\u6839\u636e\u6570\u636e\u8c03\u6574）<\/td>\n\u7528AIC\/BIC\u51c6\u5219\u9009\u62e9，\u53d6\u6700\u5c0f\u503c\u5bf9\u5e94\u7684K<\/td>\n<\/tr>\n
                            covariance_type<\/td>\n\u534f\u65b9\u5dee\u77e9\u9635\u7c7b\u578b<\/td>\nfull\/tied\/diag\/spherical<\/td>\n\u4f18\u5148\u9009full（\u6bcf\u4e2a\u7c07\u72ec\u7acb\u534f\u65b9\u5dee，\u62df\u5408\u80fd\u529b\u6700\u5f3a）；\u6570\u636e\u91cf\u5c0f\u65f6\u9009diag（\u5bf9\u89d2\u534f\u65b9\u5dee，\u51cf\u5c11\u53c2\u6570）<\/td>\n<\/tr>\n
                            tol<\/td>\n\u6536\u655b\u9608\u503c<\/td>\n1e-6\/1e-4<\/td>\n\u8d8a\u5c0f\u6536\u655b\u8d8a\u7cbe\u51c6，\u8fed\u4ee3\u6b21\u6570\u8d8a\u591a；\u9ed8\u8ba41e-6\u5373\u53ef<\/td>\n<\/tr>\n
                            max_iter<\/td>\n\u6700\u5927\u8fed\u4ee3\u6b21\u6570<\/td>\n100\/200\/500<\/td>\n\u82e5\u672a\u6536\u655b（n_iter_ == max_iter），\u5219\u589e\u5927\u8be5\u503c<\/td>\n<\/tr>\n
                            init_params<\/td>\n\u53c2\u6570\u521d\u59cb\u5316\u65b9\u5f0f<\/td>\nkmeans\/random<\/td>\n\u4f18\u5148\u9009kmeans（\u57fa\u4e8eK-means\u521d\u59cb\u5316，\u907f\u514d\u5c40\u90e8\u6700\u4f18）<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                            \u6838\u5fc3\u8c03\u4f18\u539f\u5219：\u5148\u901a\u8fc7AIC\/BIC\u786e\u5b9a\u6700\u4f18\u7c07\u6570K，\u518d\u9009\u62e9\u534f\u65b9\u5dee\u7c7b\u578b，\u6700\u540e\u5fae\u8c03\u6536\u655b\u9608\u503c\u548c\u8fed\u4ee3\u6b21\u6570\u3002<\/p>\n

                            5.5 AIC\u4e0eBIC\u51c6\u5219\u7684\u9009\u62e9<\/h4>\n

                            AIC（\u8d64\u6c60\u4fe1\u606f\u51c6\u5219）\u548cBIC（\u8d1d\u53f6\u65af\u4fe1\u606f\u51c6\u5219）\u90fd\u662f\u6a21\u578b\u9009\u62e9\u7684\u7ecf\u5178\u51c6\u5219，\u6838\u5fc3\u662f\u5728\u6a21\u578b\u62df\u5408\u5ea6\u548c\u590d\u6742\u5ea6\u4e4b\u95f4\u505a\u6743\u8861，\u503c\u8d8a\u5c0f\u8868\u793a\u6a21\u578b\u8d8a\u4f18，\u4e8c\u8005\u7684\u533a\u522b：<\/p>\n

                              \n
                            • AIC：AIC=2k\u22122ln\u2061LAIC = 2k – 2\\\\ln L<\/span><\/span>A<\/span>I<\/span>C<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>k<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span>ln<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span>，kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span>\u4e3a\u6a21\u578b\u53c2\u6570\u6570，\u5bf9\u590d\u6742\u6a21\u578b\u60e9\u7f5a\u8f83\u8f7b，\u9002\u5408\u5927\u6570\u636e\u96c6；<\/li>\n
                            • BIC：BIC=kln\u2061N\u22122ln\u2061LBIC = k\\\\ln N – 2\\\\ln L<\/span><\/span>B<\/span>I<\/span>C<\/span><\/span>=<\/span><\/span><\/span><\/span>k<\/span><\/span>ln<\/span><\/span>N<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span>ln<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span>，NN<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span>\u4e3a\u6837\u672c\u6570，\u5bf9\u590d\u6742\u6a21\u578b\u60e9\u7f5a\u66f4\u91cd，\u9002\u5408\u5c0f\u6570\u636e\u96c6；<\/li>\n
                            • \u5b9e\u6218\u4e2d：\u82e5\u6837\u672c\u6570N>1000N>1000<\/span><\/span>N<\/span><\/span>><\/span><\/span><\/span><\/span>1000<\/span><\/span><\/span><\/span><\/span>，\u53ef\u53c2\u8003AIC；\u82e5N<1000N<1000<\/span><\/span>N<\/span><\/span><<\/span><\/span><\/span><\/span>1000<\/span><\/span><\/span><\/span><\/span>，\u4f18\u5148\u53c2\u8003BIC\u3002<\/li>\n<\/ul>\n

                              \u516d\u3001GMM\u7684\u4f18\u7f3a\u70b9<\/h3>\n

                              GMM\u4f5c\u4e3a\u7ecf\u5178\u7684\u6982\u7387\u805a\u7c7b\u7b97\u6cd5，\u517c\u5177\u805a\u7c7b\u548c\u5bc6\u5ea6\u4f30\u8ba1\u7684\u53cc\u91cd\u529f\u80fd，\u662f\u8f6f\u805a\u7c7b\u7684\u4ee3\u8868，\u4f46\u5176\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027，\u4f18\u7f3a\u70b9\u603b\u7ed3\u5982\u4e0b，\u9002\u914d\u672c\u79d1\/\u7814\u7a76\u751f\u8bfe\u7a0b\u8003\u70b9\u548c\u5de5\u4e1a\u7ea7\u9009\u578b\u53c2\u8003\u3002<\/p>\n

                              6.1 \u6838\u5fc3\u4f18\u70b9<\/h4>\n
                            • \u8f6f\u805a\u7c7b\u80fd\u529b：\u6bcf\u4e2a\u6837\u672c\u4ee5\u4e0d\u540c\u6982\u7387\u5c5e\u4e8e\u6240\u6709\u7c07，\u805a\u7c7b\u7ed3\u679c\u662f\u6982\u7387\u5206\u5e03，\u6bd4K-means\u7684\u786c\u805a\u7c7b\u66f4\u8d34\u5408\u5b9e\u9645\u6570\u636e\u5206\u5e03，\u80fd\u6355\u6349\u6837\u672c\u7684\u6a21\u7cca\u5f52\u5c5e\u5173\u7cfb；<\/li>\n
                            • \u5f3a\u5927\u7684\u62df\u5408\u80fd\u529b：\u652f\u6301\u6bcf\u4e2a\u7c07\u6709\u72ec\u7acb\u7684\u5747\u503c\u3001\u534f\u65b9\u5dee\u548c\u6743\u91cd，\u80fd\u62df\u5408\u4efb\u610f\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u5bc6\u5ea6\u7684\u7c07，\u8fdc\u4f18\u4e8eK-means\u4ec5\u80fd\u62df\u5408\u7403\u5f62\u7c07\u7684\u5c40\u9650；<\/li>\n
                            • \u53cc\u91cd\u529f\u80fd：\u540c\u65f6\u652f\u6301\u805a\u7c7b\u548c\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1，\u53ef\u4e00\u7ad9\u5f0f\u89e3\u51b3\u805a\u7c7b\u3001\u5f02\u5e38\u68c0\u6d4b\u3001\u6837\u672c\u751f\u6210\u7b49\u95ee\u9898，\u9002\u7528\u573a\u666f\u66f4\u5e7f；<\/li>\n
                            • \u6982\u7387\u53ef\u89e3\u91ca\u6027：\u57fa\u4e8e\u4e25\u683c\u7684\u6982\u7387\u7edf\u8ba1\u7406\u8bba，\u6a21\u578b\u7ed3\u679c\u6709\u660e\u786e\u7684\u6982\u7387\u610f\u4e49，\u80fd\u89e3\u91ca\u201c\u6837\u672c\u5c5e\u4e8e\u67d0\u7c07\u7684\u53ef\u80fd\u6027\u6709\u591a\u5927\u201d，\u6bd4\u65e0\u76d1\u7763\u805a\u7c7b\u7b97\u6cd5\u66f4\u5177\u89e3\u91ca\u6027；<\/li>\n
                            • \u9002\u7528\u4e8e\u9ad8\u7ef4\u6570\u636e：\u901a\u8fc7\u534f\u65b9\u5dee\u77e9\u9635\u8003\u8651\u7279\u5f81\u4e4b\u95f4\u7684\u76f8\u5173\u6027，\u5bf9\u9ad8\u7ef4\u7ed3\u6784\u5316\u6570\u636e\u7684\u805a\u7c7b\u6548\u679c\u4f18\u4e8eK-means（K-means\u4ec5\u8003\u8651\u6b27\u6c0f\u8ddd\u79bb，\u672a\u8003\u8651\u7279\u5f81\u76f8\u5173\u6027）；<\/li>\n
                            • \u8fed\u4ee3\u6536\u655b\u7a33\u5b9a：EM\u7b97\u6cd5\u7684\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u5355\u8c03\u4e0d\u51cf，\u4fdd\u8bc1\u6a21\u578b\u6700\u7ec8\u4f1a\u6536\u655b，\u4e14K-means\u521d\u59cb\u5316\u80fd\u6709\u6548\u907f\u514d\u5c40\u90e8\u6700\u4f18\u3002<\/li>\n

                              6.2 \u6838\u5fc3\u7f3a\u70b9<\/h4>\n
                            • \u5bf9\u521d\u59cb\u53c2\u6570\u654f\u611f：\u82e5\u968f\u673a\u521d\u59cb\u5316\u53c2\u6570，EM\u7b97\u6cd5\u6613\u6536\u655b\u5230\u5c40\u90e8\u6700\u4f18，\u9700\u901a\u8fc7K-means\u521d\u59cb\u5316\u6216\u591a\u6b21\u8fd0\u884c\u7b97\u6cd5\u6765\u7f13\u89e3；<\/li>\n
                            • \u7c07\u6570K\u9700\u9884\u8bbe：\u4e0eK-means\u4e00\u6837，\u9700\u8981\u63d0\u524d\u6307\u5b9a\u7c07\u6570K，\u65e0\u6cd5\u81ea\u52a8\u53d1\u73b0\u7c07\u6570，\u9700\u901a\u8fc7AIC\/BIC\u7b49\u51c6\u5219\u624b\u52a8\u9009\u62e9，\u589e\u52a0\u4e86\u8c03\u53c2\u6210\u672c；<\/li>\n
                            • \u8ba1\u7b97\u590d\u6742\u5ea6\u8f83\u9ad8：\u76f8\u6bd4K-means，GMM\u9700\u8981\u8ba1\u7b97\u534f\u65b9\u5dee\u77e9\u9635\u3001\u77e9\u9635\u7684\u9006\u548c\u884c\u5217\u5f0f，\u8ba1\u7b97\u91cf\u66f4\u5927，\u8bad\u7ec3\u901f\u5ea6\u66f4\u6162，\u4e0d\u9002\u5408\u8d85\u5927\u89c4\u6a21\u6570\u636e\u96c6（\u5982\u5343\u4e07\u7ea7\u6837\u672c）；<\/li>\n
                            • \u5bf9\u566a\u58f0\u548c\u5f02\u5e38\u503c\u654f\u611f：GMM\u7684\u53c2\u6570\u4f30\u8ba1\u662f\u52a0\u6743\u5e73\u5747，\u5f02\u5e38\u503c\u4f1a\u5f71\u54cd\u5747\u503c\u548c\u534f\u65b9\u5dee\u7684\u4f30\u8ba1，\u5bfc\u81f4\u805a\u7c7b\u7ed3\u679c\u504f\u79fb，\u9700\u63d0\u524d\u505a\u5f02\u5e38\u503c\u6e05\u6d17；<\/li>\n
                            • \u4f9d\u8d56\u9ad8\u65af\u5206\u5e03\u5047\u8bbe：GMM\u7684\u6838\u5fc3\u5047\u8bbe\u662f\u6570\u636e\u7531\u9ad8\u65af\u5206\u5e03\u6df7\u5408\u800c\u6210，\u82e5\u6570\u636e\u4e0d\u6ee1\u8db3\u8be5\u5047\u8bbe（\u5982\u6570\u636e\u5448\u5747\u5300\u5206\u5e03\u3001\u6307\u6570\u5206\u5e03），\u805a\u7c7b\u6548\u679c\u4f1a\u5927\u5e45\u4e0b\u964d；<\/li>\n
                            • \u9ad8\u7ef4\u6570\u636e\u7684\u534f\u65b9\u5dee\u95ee\u9898：\u5728\u8d85\u9ad8\u7ef4\u6570\u636e\u4e2d，\u534f\u65b9\u5dee\u77e9\u9635\u7684\u53c2\u6570\u6570\u91cf\u4f1a\u5448\u5e73\u65b9\u7ea7\u589e\u957f，\u5bb9\u6613\u51fa\u73b0\u7ef4\u5ea6\u707e\u96be，\u4e14\u77e9\u9635\u6c42\u9006\u53ef\u80fd\u4e0d\u7a33\u5b9a（\u5947\u5f02\u77e9\u9635）\u3002<\/li>\n

                              \u4e03\u3001GMM\u4e0e\u4e3b\u6d41\u805a\u7c7b\u7b97\u6cd5\u7684\u5bf9\u6bd4<\/h3>\n

                              \u4e3a\u4e86\u8ba9\u5927\u5bb6\u5728\u5b9e\u6218\u4e2d\u7cbe\u51c6\u9009\u62e9\u805a\u7c7b\u7b97\u6cd5，\u4ee5\u4e0b\u4ece\u805a\u7c7b\u7c7b\u578b\u3001\u6a21\u578b\u5047\u8bbe\u3001\u62df\u5408\u80fd\u529b\u3001\u8ba1\u7b97\u590d\u6742\u5ea6\u7b49\u6838\u5fc3\u7ef4\u5ea6，\u5bf9\u6bd4GMM\u4e0eK-means\u3001DBSCAN\u3001\u5c42\u6b21\u805a\u7c7b\u4e09\u79cd\u4e3b\u6d41\u805a\u7c7b\u7b97\u6cd5\u7684\u5dee\u5f02，\u8868\u683c\u7b80\u6d01\u660e\u4e86，\u9002\u914d\u672c\u79d1\u8bfe\u7a0b\u8003\u6838\u548c\u5de5\u4e1a\u7ea7\u5efa\u6a21\u9009\u578b\u3002<\/p>\n\n\u5bf9\u6bd4\u9879\u9ad8\u65af\u6df7\u5408\u6a21\u578b(GMM)K-meansDBSCAN\u5c42\u6b21\u805a\u7c7b<\/tr>\n\n\n\n\n\n\n\n\n\n\n
                              \u805a\u7c7b\u7c7b\u578b<\/td>\n\u8f6f\u805a\u7c7b（\u6982\u7387\u5f52\u5c5e）<\/td>\n\u786c\u805a\u7c7b（\u552f\u4e00\u5f52\u5c5e）<\/td>\n\u786c\u805a\u7c7b（\u5bc6\u5ea6\u5f52\u5c5e）<\/td>\n\u786c\u805a\u7c7b（\u5c42\u6b21\u5f52\u5c5e）<\/td>\n<\/tr>\n
                              \u6a21\u578b\u5047\u8bbe<\/td>\n\u6570\u636e\u7531K\u4e2a\u9ad8\u65af\u5206\u5e03\u6df7\u5408\u800c\u6210<\/td>\n\u7c07\u4e3a\u7403\u5f62\u3001\u5927\u5c0f\u76f8\u4f3c\u3001\u5bc6\u5ea6\u5747\u5300<\/td>\n\u7c07\u4e3a\u5bc6\u5ea6\u8fde\u901a\u7684\u533a\u57df，\u5b58\u5728\u566a\u58f0<\/td>\n\u6570\u636e\u5b58\u5728\u5c42\u6b21\u5316\u7684\u7c07\u7ed3\u6784<\/td>\n<\/tr>\n
                              \u7c07\u6570K<\/td>\n\u9700\u9884\u8bbe，\u901a\u8fc7AIC\/BIC\u9009\u62e9<\/td>\n\u9700\u9884\u8bbe，\u901a\u8fc7\u8098\u90e8\u6cd5\u5219\u9009\u62e9<\/td>\n\u65e0\u9700\u9884\u8bbe，\u81ea\u52a8\u53d1\u73b0\u7c07\u6570<\/td>\n\u65e0\u9700\u9884\u8bbe，\u751f\u6210\u5c42\u6b21\u7c07\u6811<\/td>\n<\/tr>\n
                              \u7c07\u5f62\u72b6\u62df\u5408<\/td>\n\u4efb\u610f\u5f62\u72b6（\u692d\u5706\u3001\u7403\u5f62）<\/td>\n\u4ec5\u7403\u5f62\u7c07<\/td>\n\u4efb\u610f\u5f62\u72b6（\u5bc6\u5ea6\u8fde\u901a）<\/td>\n\u4efb\u610f\u5f62\u72b6<\/td>\n<\/tr>\n
                              \u6838\u5fc3\u529f\u80fd<\/td>\n\u805a\u7c7b+\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1<\/td>\n\u4ec5\u805a\u7c7b<\/td>\n\u805a\u7c7b+\u566a\u58f0\u68c0\u6d4b<\/td>\n\u4ec5\u805a\u7c7b（\u5c42\u6b21\u5316）<\/td>\n<\/tr>\n
                              \u5bf9\u566a\u58f0\/\u5f02\u5e38\u503c<\/td>\n\u654f\u611f，\u9700\u63d0\u524d\u6e05\u6d17<\/td>\n\u654f\u611f，\u6613\u88ab\u5f02\u5e38\u503c\u5f71\u54cd\u7c07\u4e2d\u5fc3<\/td>\n\u9c81\u68d2，\u80fd\u81ea\u52a8\u68c0\u6d4b\u5e76\u6392\u9664\u566a\u58f0<\/td>\n\u654f\u611f，\u566a\u58f0\u4f1a\u5f71\u54cd\u5c42\u6b21\u7ed3\u6784<\/td>\n<\/tr>\n
                              \u8ba1\u7b97\u590d\u6742\u5ea6<\/td>\n\u4e2d\u7b49（O(NKdT)O(NKdT)<\/span><\/span>O<\/span>(<\/span>N<\/span>K<\/span>d<\/span>T<\/span>)<\/span><\/span><\/span><\/span><\/span>）<\/td>\n\u4f4e（O(NKdT)O(NKdT)<\/span><\/span>O<\/span>(<\/span>N<\/span>K<\/span>d<\/span>T<\/span>)<\/span><\/span><\/span><\/span><\/span>）<\/td>\n\u4e2d\u7b49（O(N2)O(N^2)<\/span><\/span>O<\/span>(<\/span>N<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u7d22\u5f15\u4f18\u5316\u540eO(NlogN)O(NlogN)<\/span><\/span>O<\/span>(<\/span>Nl<\/span>o<\/span>g<\/span>N<\/span>)<\/span><\/span><\/span><\/span><\/span>）<\/td>\n\u9ad8（O(N2)O(N^2)<\/span><\/span>O<\/span>(<\/span>N<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>）<\/td>\n<\/tr>\n
                              \u9002\u7528\u4e8e<\/td>\n\u4e2d\u9ad8\u7ef4\u3001\u6df7\u5408\u9ad8\u65af\u5206\u5e03\u3001\u9700\u8981\u8f6f\u805a\u7c7b\/\u5bc6\u5ea6\u4f30\u8ba1<\/td>\n\u4f4e\u7ef4\u3001\u7403\u5f62\u7c07\u3001\u5927\u6570\u636e\u96c6\u3001\u8ffd\u6c42\u901f\u5ea6<\/td>\n\u4efb\u610f\u5f62\u72b6\u7c07\u3001\u542b\u566a\u58f0\u7684\u6570\u636e\u3001\u65e0\u9700\u9884\u8bbe\u7c07\u6570<\/td>\n\u5c0f\u6570\u636e\u96c6\u3001\u9700\u8981\u5c42\u6b21\u5316\u805a\u7c7b\u7ed3\u679c<\/td>\n<\/tr>\n
                              \u4ee3\u8868\u573a\u666f<\/td>\n\u5ba2\u6237\u5206\u7fa4\u3001\u56fe\u50cf\u5206\u5272\u3001\u5f02\u5e38\u68c0\u6d4b\u3001\u5bc6\u5ea6\u4f30\u8ba1<\/td>\n\u7b80\u5355\u6570\u636e\u805a\u7c7b\u3001\u5feb\u901f\u5206\u7fa4\u3001\u57fa\u7ebf\u6a21\u578b<\/td>\n\u7a7a\u95f4\u6570\u636e\u805a\u7c7b\u3001\u542b\u566a\u58f0\u7684\u5de5\u4e1a\u6570\u636e\u805a\u7c7b<\/td>\n\u751f\u7269\u5206\u7c7b\u3001\u6587\u672c\u5c42\u6b21\u5206\u7fa4\u3001\u5c0f\u6570\u636e\u96c6\u63a2\u7d22<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                              \u6838\u5fc3\u9009\u578b\u5efa\u8bae<\/h4>\n
                            • \u4f18\u5148\u9009GMM：\u4e2d\u9ad8\u7ef4\u7ed3\u6784\u5316\u6570\u636e\u3001\u6570\u636e\u7b26\u5408\u6df7\u5408\u9ad8\u65af\u5206\u5e03\u3001\u9700\u8981\u8f6f\u805a\u7c7b\u7ed3\u679c\u3001\u540c\u65f6\u9700\u8981\u5bc6\u5ea6\u4f30\u8ba1\/\u5f02\u5e38\u68c0\u6d4b\u7684\u573a\u666f（\u5982\u91d1\u878d\u5ba2\u6237\u5206\u7fa4\u3001\u4fe1\u7528\u8bc4\u5206\u3001\u56fe\u50cf\u50cf\u7d20\u805a\u7c7b）；<\/li>\n
                            • \u4f18\u5148\u9009K-means：\u4f4e\u7ef4\u6570\u636e\u3001\u7c07\u4e3a\u7403\u5f62\u4e14\u5927\u5c0f\u76f8\u4f3c\u3001\u6570\u636e\u91cf\u6781\u5927（\u5343\u4e07\u7ea7\u4ee5\u4e0a）\u3001\u8ffd\u6c42\u8bad\u7ec3\u901f\u5ea6\u7684\u573a\u666f（\u5982\u7b80\u5355\u6570\u636e\u5206\u7fa4\u3001\u805a\u7c7b\u57fa\u7ebf\u6a21\u578b）；<\/li>\n
                            • \u4f18\u5148\u9009DBSCAN：\u6570\u636e\u7c07\u5f62\u72b6\u4e0d\u89c4\u5219\u3001\u6570\u636e\u4e2d\u5305\u542b\u5927\u91cf\u566a\u58f0\/\u5f02\u5e38\u503c\u3001\u65e0\u9700\u9884\u8bbe\u7c07\u6570\u7684\u573a\u666f（\u5982GPS\u8f68\u8ff9\u805a\u7c7b\u3001\u5de5\u4e1a\u6545\u969c\u6570\u636e\u805a\u7c7b）；<\/li>\n
                            • \u4f18\u5148\u9009\u5c42\u6b21\u805a\u7c7b：\u5c0f\u6570\u636e\u96c6\u3001\u9700\u8981\u63a2\u7d22\u6570\u636e\u7684\u5c42\u6b21\u5316\u7ed3\u6784\u3001\u65e0\u9700\u9884\u8bbe\u7c07\u6570\u7684\u573a\u666f（\u5982\u751f\u7269\u7269\u79cd\u5206\u7c7b\u3001\u6587\u732e\u4e3b\u9898\u5206\u7fa4）\u3002<\/li>\n

                              \u516b\u3001GMM\u7684\u9002\u7528\u573a\u666f\u4e0e\u5b9e\u6218\u5efa\u8bae<\/h3>\n

                              GMM\u7684\u6838\u5fc3\u4f18\u52bf\u662f\u8f6f\u805a\u7c7b+\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1，\u4e14\u80fd\u62df\u5408\u590d\u6742\u7684\u7c07\u5f62\u72b6，\u4f46\u5176\u4f9d\u8d56\u9ad8\u65af\u5206\u5e03\u5047\u8bbe\u3001\u5bf9\u521d\u59cb\u53c2\u6570\u654f\u611f\u7684\u5c40\u9650\u6027\u4e5f\u51b3\u5b9a\u4e86\u5b83\u7684\u9002\u7528\u8fb9\u754c，\u4ee5\u4e0b\u662f\u5b9e\u6218\u4e2d\u7684\u9009\u578b\u5efa\u8bae\u548c\u4f7f\u7528\u6280\u5de7，\u5e2e\u52a9\u5927\u5bb6\u907f\u5751\u3002<\/p>\n

                              8.1 \u4f18\u5148\u9009\u62e9GMM\u7684\u573a\u666f<\/h4>\n
                            • \u9700\u8981\u8f6f\u805a\u7c7b\u7ed3\u679c\u7684\u573a\u666f：\u5982\u5ba2\u6237\u5206\u7fa4（\u5ba2\u6237\u53ef\u80fd\u540c\u65f6\u5c5e\u4e8e\u591a\u4e2a\u7ec6\u5206\u7fa4\u4f53，\u5982\u201c\u9ad8\u4ef7\u503c\u5e74\u8f7b\u5ba2\u6237\u201d\u548c\u201c\u6f5c\u5728\u6d88\u8d39\u5ba2\u6237\u201d）\u3001\u7cbe\u51c6\u8425\u9500（\u6839\u636e\u5f52\u5c5e\u6982\u7387\u5236\u5b9a\u4e2a\u6027\u5316\u7b56\u7565）；<\/li>\n
                            • \u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1\u4e0e\u5f02\u5e38\u68c0\u6d4b：\u5982\u91d1\u878d\u98ce\u63a7（\u68c0\u6d4b\u5f02\u5e38\u4ea4\u6613，\u5bc6\u5ea6\u503c\u8fc7\u4f4e\u7684\u4ea4\u6613\u4e3a\u98ce\u9669\u4ea4\u6613）\u3001\u5de5\u4e1a\u8d28\u68c0（\u68c0\u6d4b\u5f02\u5e38\u4ea7\u54c1\u7279\u5f81）；<\/li>\n
                            • \u6df7\u5408\u9ad8\u65af\u5206\u5e03\u7684\u6570\u636e\u805a\u7c7b：\u5982\u4eba\u7fa4\u7684\u8eab\u9ad8\/\u4f53\u91cd\u6570\u636e\u3001\u5b66\u751f\u7684\u591a\u79d1\u76ee\u6210\u7ee9\u6570\u636e\u3001\u56fe\u50cf\u7684\u50cf\u7d20\u5f3a\u5ea6\u6570\u636e，\u8fd9\u7c7b\u6570\u636e\u5929\u7136\u7b26\u5408\u6df7\u5408\u9ad8\u65af\u5206\u5e03；<\/li>\n
                            • \u9ad8\u7ef4\u7ed3\u6784\u5316\u6570\u636e\u805a\u7c7b：\u5982\u7535\u5546\u7528\u6237\u884c\u4e3a\u7279\u5f81（\u6d4f\u89c8\u3001\u6536\u85cf\u3001\u8d2d\u4e70\u7b49\u591a\u7ef4\u5ea6）\u3001\u91d1\u878d\u7528\u6237\u5f81\u4fe1\u7279\u5f81，GMM\u80fd\u8003\u8651\u7279\u5f81\u4e4b\u95f4\u7684\u76f8\u5173\u6027，\u805a\u7c7b\u6548\u679c\u4f18\u4e8eK-means；<\/li>\n
                            • \u6837\u672c\u751f\u6210\u4e0e\u6570\u636e\u589e\u5f3a：\u5229\u7528GMM\u7684\u6982\u7387\u5206\u5e03，\u53ef\u751f\u6210\u7b26\u5408\u539f\u59cb\u6570\u636e\u5206\u5e03\u7684\u65b0\u6837\u672c，\u7528\u4e8e\u5c0f\u6570\u636e\u96c6\u7684\u6a21\u578b\u8bad\u7ec3\u548c\u6570\u636e\u589e\u5f3a\u3002<\/li>\n

                              8.2 \u8003\u8651\u5176\u4ed6\u7b97\u6cd5\u7684\u573a\u666f<\/h4>\n
                            • \u6570\u636e\u4e0d\u6ee1\u8db3\u9ad8\u65af\u5206\u5e03：\u5982\u5747\u5300\u5206\u5e03\u3001\u6307\u6570\u5206\u5e03\u3001\u5e42\u5f8b\u5206\u5e03\u7684\u6570\u636e，\u9009\u62e9DBSCAN\u6216\u5c42\u6b21\u805a\u7c7b，GMM\u7684\u62df\u5408\u6548\u679c\u4f1a\u5927\u5e45\u4e0b\u964d；<\/li>\n
                            • \u8d85\u5927\u89c4\u6a21\u6570\u636e\u96c6：\u5982\u5343\u4e07\u7ea7\u4ee5\u4e0a\u6837\u672c，\u9009\u62e9K-means（\u901f\u5ea6\u5feb）\u6216Mini-Batch K-means（\u6279\u91cf\u8bad\u7ec3），GMM\u7684\u8ba1\u7b97\u590d\u6742\u5ea6\u8fc7\u9ad8；<\/li>\n
                            • \u6570\u636e\u542b\u5927\u91cf\u566a\u58f0\/\u5f02\u5e38\u503c：\u9009\u62e9DBSCAN，\u5176\u80fd\u81ea\u52a8\u68c0\u6d4b\u566a\u58f0，\u65e0\u9700\u63d0\u524d\u6e05\u6d17，\u800cGMM\u5bf9\u566a\u58f0\u654f\u611f，\u9700\u989d\u5916\u7684\u9884\u5904\u7406\u6b65\u9aa4；<\/li>\n
                            • \u65e0\u9700\u9884\u8bbe\u7c07\u6570：\u9009\u62e9DBSCAN\u6216\u5c42\u6b21\u805a\u7c7b，\u907f\u514d\u901a\u8fc7AIC\/BIC\u624b\u52a8\u9009\u62e9\u7c07\u6570\u7684\u8c03\u53c2\u6210\u672c；<\/li>\n
                            • \u4f4e\u7ef4\u7403\u5f62\u7c07\u6570\u636e：\u9009\u62e9K-means，\u8bad\u7ec3\u901f\u5ea6\u66f4\u5feb，\u5b9e\u73b0\u66f4\u7b80\u5355，\u80fd\u8fbe\u5230\u4e0eGMM\u76f8\u8fd1\u7684\u805a\u7c7b\u6548\u679c\u3002<\/li>\n

                              8.3 GMM\u7684\u5b9e\u6218\u4f7f\u7528\u6280\u5de7<\/h4>\n
                            • \n

                              \u6570\u636e\u9884\u5904\u7406：\u6807\u51c6\u5316+\u5f02\u5e38\u503c\u6e05\u6d17\u662f\u5fc5\u505a\u9879<\/p>\n

                                \n
                              • \u6807\u51c6\u5316：GMM\u5bf9\u7279\u5f81\u5c3a\u5ea6\u654f\u611f，\u4e0d\u540c\u7279\u5f81\u7684\u91cf\u7eb2\u5dee\u5f02\u4f1a\u5bfc\u81f4\u534f\u65b9\u5dee\u77e9\u9635\u4f30\u8ba1\u504f\u5dee，\u5fc5\u987b\u7528StandardScaler\u505a\u6807\u51c6\u5316；<\/li>\n
                              • \u5f02\u5e38\u503c\u6e05\u6d17：\u901a\u8fc7\u7bb1\u7ebf\u56fe\u3001Z-score\u7b49\u65b9\u6cd5\u6e05\u6d17\u5f02\u5e38\u503c，\u907f\u514d\u5f02\u5e38\u503c\u5f71\u54cd\u5747\u503c\u548c\u534f\u65b9\u5dee\u7684\u4f30\u8ba1；<\/li>\n
                              • \u7f3a\u5931\u503c\u5904\u7406：GMM\u4e0d\u652f\u6301\u7f3a\u5931\u503c，\u9700\u7528\u5747\u503c\u3001\u4e2d\u4f4d\u6570\u6216\u63d2\u503c\u6cd5\u586b\u5145\u3002<\/li>\n<\/ul>\n<\/li>\n
                              • \n

                                \u53c2\u6570\u521d\u59cb\u5316：\u4f18\u5148\u4f7f\u7528K-means\u521d\u59cb\u5316<\/p>\n

                                  \n
                                • \u907f\u514d\u968f\u673a\u521d\u59cb\u5316，\u5426\u5219EM\u7b97\u6cd5\u6613\u6536\u655b\u5230\u5c40\u90e8\u6700\u4f18，\u8bbe\u7f6einit_params='kmeans'，\u5229\u7528K-means\u7684\u7c07\u4e2d\u5fc3\u548c\u534f\u65b9\u5dee\u521d\u59cb\u5316GMM\u53c2\u6570，\u5927\u5e45\u63d0\u5347\u6536\u655b\u6548\u679c；<\/li>\n
                                • \u82e5\u6570\u636e\u91cf\u8f83\u5c0f，\u53ef\u591a\u6b21\u8fd0\u884cGMM（\u4e0d\u540c\u968f\u673a\u79cd\u5b50），\u9009\u62e9\u5bf9\u6570\u4f3c\u7136\u503c\u6700\u5927\u7684\u6a21\u578b\u3002<\/li>\n<\/ul>\n<\/li>\n
                                • \n

                                  \u7c07\u6570\u9009\u62e9：\u7ed3\u5408AIC\/BIC\u4e0e\u4e1a\u52a1\u610f\u4e49<\/p>\n

                                    \n
                                  • \u5148\u7528AIC\/BIC\u51c6\u5219\u5f97\u5230\u6700\u4f18\u7c07\u6570K，\u518d\u7ed3\u5408\u4e1a\u52a1\u610f\u4e49\u8c03\u6574，\u6bd4\u5982\u5ba2\u6237\u5206\u7fa4\u4e2d，AIC\u5efa\u8baeK=5，\u4f46\u4e1a\u52a1\u4e0a\u4ec5\u97003\u7c7b\u5ba2\u6237，\u53ef\u624b\u52a8\u8c03\u6574\u4e3aK=3；<\/li>\n
                                  • \u82e5AIC\/BIC\u66f2\u7ebf\u65e0\u660e\u663e\u6700\u5c0f\u503c（\u5e73\u7f13\u4e0b\u964d），\u9009\u62e9\u201c\u62d0\u70b9\u201d\u5bf9\u5e94\u7684K\u503c\u3002<\/li>\n<\/ul>\n<\/li>\n
                                  • \n

                                    \u534f\u65b9\u5dee\u7c7b\u578b\u9009\u62e9：\u6839\u636e\u6570\u636e\u91cf\u548c\u7ef4\u5ea6\u8c03\u6574<\/p>\n

                                      \n
                                    • \u6570\u636e\u91cf\u5145\u8db3\u3001\u7ef4\u5ea6\u9002\u4e2d：\u9009covariance_type='full'，\u62df\u5408\u80fd\u529b\u6700\u5f3a，\u80fd\u6355\u6349\u7c07\u7684\u590d\u6742\u5f62\u72b6；<\/li>\n
                                    • \u6570\u636e\u91cf\u5c0f\u3001\u7ef4\u5ea6\u9ad8：\u9009covariance_type='diag'（\u5bf9\u89d2\u534f\u65b9\u5dee），\u51cf\u5c11\u53c2\u6570\u6570\u91cf，\u907f\u514d\u534f\u65b9\u5dee\u77e9\u9635\u5947\u5f02；<\/li>\n
                                    • \u7c07\u5f62\u72b6\u63a5\u8fd1\u7403\u5f62：\u9009covariance_type='spherical'（\u7403\u5f62\u534f\u65b9\u5dee），\u8fdb\u4e00\u6b65\u7b80\u5316\u8ba1\u7b97\u3002<\/li>\n<\/ul>\n<\/li>\n
                                    • \n

                                      \u6a21\u578b\u8bc4\u4f30：\u7ed3\u5408\u5b9a\u91cf\u6307\u6807\u4e0e\u5b9a\u6027\u53ef\u89c6\u5316<\/p>\n

                                        \n
                                      • \u5b9a\u91cf\u6307\u6807：\u4f7f\u7528\u8f6e\u5ed3\u7cfb\u6570（Silhouette Coefficient）\u3001Calinski-Harabasz\u6307\u6570\u8bc4\u4f30\u805a\u7c7b\u6548\u679c，\u503c\u8d8a\u5927\u8868\u793a\u7c07\u5185\u8d8a\u7d27\u5bc6\u3001\u7c07\u95f4\u8d8a\u5206\u79bb；<\/li>\n
                                      • \u5b9a\u6027\u53ef\u89c6\u5316：\u901a\u8fc7PCA\u3001t-SNE\u5c06\u9ad8\u7ef4\u6570\u636e\u964d\u52302D\/3D，\u53ef\u89c6\u5316\u805a\u7c7b\u7ed3\u679c，\u76f4\u89c2\u5224\u65ad\u7c07\u7684\u5206\u79bb\u6548\u679c；<\/li>\n
                                      • \u4e1a\u52a1\u9a8c\u8bc1：\u805a\u7c7b\u7ed3\u679c\u9700\u7b26\u5408\u4e1a\u52a1\u903b\u8f91，\u6bd4\u5982\u5ba2\u6237\u5206\u7fa4\u7684\u7ed3\u679c\u9700\u4e0e\u5b9e\u9645\u7684\u6d88\u8d39\u884c\u4e3a\u3001\u4eba\u53e3\u7279\u5f81\u4e00\u81f4\u3002<\/li>\n<\/ul>\n<\/li>\n
                                      • \n

                                        \u9ad8\u7ef4\u6570\u636e\u4f18\u5316：\u964d\u7ef4\u540e\u518d\u805a\u7c7b<\/p>\n

                                          \n
                                        • \u5bf9\u8d85\u9ad8\u7ef4\u6570\u636e（\u5982\u7279\u5f81\u6570>100），\u5148\u901a\u8fc7PCA\u3001LDA\u505a\u7279\u5f81\u964d\u7ef4，\u518d\u7528GMM\u805a\u7c7b，\u907f\u514d\u7ef4\u5ea6\u707e\u96be\u548c\u534f\u65b9\u5dee\u77e9\u9635\u4e0d\u7a33\u5b9a\u7684\u95ee\u9898\u3002<\/li>\n<\/ul>\n<\/li>\n

                                          \u4e5d\u3001\u603b\u7ed3<\/h3>\n
                                        • \u9ad8\u65af\u6df7\u5408\u6a21\u578b（GMM）\u662f\u57fa\u4e8e\u6982\u7387\u7edf\u8ba1\u7684\u8f6f\u805a\u7c7b\u7b97\u6cd5，\u6838\u5fc3\u5047\u8bbe\u662f\u6570\u636e\u7531K\u4e2a\u9ad8\u65af\u5206\u5e03\u6309\u4e0d\u540c\u6743\u91cd\u6df7\u5408\u751f\u6210，\u80fd\u540c\u65f6\u5b9e\u73b0\u805a\u7c7b\u548c\u6982\u7387\u5bc6\u5ea6\u4f30\u8ba1，\u662fK-means\u786c\u805a\u7c7b\u7684\u8fdb\u9636\u7248\u672c；<\/li>\n
                                        • GMM\u7684\u6a21\u578b\u53c2\u6570\u5305\u62ec\u6743\u91cd\u03c0k\\\\pi_k<\/span><\/span>\u03c0<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u5747\u503c\u03bck\\\\mu_k<\/span><\/span>\u03bc<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u534f\u65b9\u5dee\u03a3k\\\\Sigma_k<\/span><\/span>\u03a3<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4f18\u5316\u76ee\u6807\u662f\u6700\u5927\u5316\u6570\u636e\u96c6\u7684\u5bf9\u6570\u4f3c\u7136\u51fd\u6570，\u7531\u4e8e\u542b\u9690\u53d8\u91cf，\u91c7\u7528EM\u7b97\u6cd5\u8fed\u4ee3\u6c42\u89e3，EM\u7b97\u6cd5\u7531**E\u6b65（\u8ba1\u7b97\u540e\u9a8c\u6982\u7387）\u548cM\u6b65（\u66f4\u65b0\u6a21\u578b\u53c2\u6570）**\u7ec4\u6210，\u4e14\u5bf9\u6570\u4f3c\u7136\u51fd\u6570\u5355\u8c03\u4e0d\u51cf；<\/li>\n
                                        • GMM\u7684\u6838\u5fc3\u4f18\u52bf\u662f\u8f6f\u805a\u7c7b\u80fd\u529b\u3001\u62df\u5408\u4efb\u610f\u5f62\u72b6\u7c07\u3001\u6982\u7387\u53ef\u89e3\u91ca\u6027，\u9002\u7528\u4e8e\u4e2d\u9ad8\u7ef4\u6df7\u5408\u9ad8\u65af\u5206\u5e03\u6570\u636e，\u540c\u65f6\u652f\u6301\u5f02\u5e38\u68c0\u6d4b\u3001\u6837\u672c\u751f\u6210\u7b49\u62d3\u5c55\u529f\u80fd；<\/li>\n
                                        • GMM\u7684\u6838\u5fc3\u5c40\u9650\u6027\u662f\u5bf9\u521d\u59cb\u53c2\u6570\u654f\u611f\u3001\u9700\u9884\u8bbe\u7c07\u6570K\u3001\u8ba1\u7b97\u590d\u6742\u5ea6\u8f83\u9ad8\u3001\u4f9d\u8d56\u9ad8\u65af\u5206\u5e03\u5047\u8bbe，\u5b9e\u6218\u4e2d\u9700\u901a\u8fc7K-means\u521d\u59cb\u5316\u3001AIC\/BIC\u9009\u7c07\u6570\u3001\u6570\u636e\u9884\u5904\u7406\u6765\u7f13\u89e3；<\/li>\n
                                        • GMM\u4e0eK-means\u3001DBSCAN\u3001\u5c42\u6b21\u805a\u7c7b\u7684\u6838\u5fc3\u533a\u522b\u5728\u4e8e\u805a\u7c7b\u7c7b\u578b\u548c\u6a21\u578b\u5047\u8bbe，\u5b9e\u6218\u4e2d\u9700\u6839\u636e\u6570\u636e\u5206\u5e03\u3001\u7c07\u5f62\u72b6\u3001\u6570\u636e\u91cf\u548c\u4e1a\u52a1\u9700\u6c42\u9009\u62e9\u5408\u9002\u7684\u805a\u7c7b\u7b97\u6cd5；<\/li>\n
                                        • GMM\u7684\u5b9e\u6218\u5173\u952e\u662f\u6570\u636e\u6807\u51c6\u5316+K-means\u521d\u59cb\u5316+AIC\/BIC\u9009\u7c07\u6570，\u540c\u65f6\u7ed3\u5408\u5b9a\u91cf\u6307\u6807\u548c\u5b9a\u6027\u53ef\u89c6\u5316\u8bc4\u4f30\u6a21\u578b\u6548\u679c，\u4fdd\u8bc1\u805a\u7c7b\u7ed3\u679c\u7684\u5408\u7406\u6027\u548c\u4e1a\u52a1\u4ef7\u503c\u3002<\/li>\n

                                          \u62d3\u5c55\u5b66\u4e60：\u638c\u63e1GMM\u540e，\u53ef\u4f9d\u6b21\u5b66\u4e60\u9690\u9a6c\u5c14\u53ef\u592b\u6a21\u578b（HMM）（\u5e8f\u5217\u6570\u636e\u7684\u6982\u7387\u5efa\u6a21）\u3001\u53d8\u5206\u63a8\u65ad\u9ad8\u65af\u6df7\u5408\u6a21\u578b（VB-GMM）（\u89e3\u51b3GMM\u7684\u8fc7\u62df\u5408\u95ee\u9898）\u3001\u6df7\u5408\u5206\u5e03\u6a21\u578b（\u5982\u6cca\u677e\u6df7\u5408\u6a21\u578b\u3001\u8d1d\u5854\u6df7\u5408\u6a21\u578b，\u9002\u7528\u4e8e\u975e\u9ad8\u65af\u5206\u5e03\u6570\u636e），\u8fdb\u4e00\u6b65\u5b8c\u5584\u6982\u7387\u7edf\u8ba1\u5efa\u6a21\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

                                          \u9ad8\u65af\u6df7\u5408\u6a21\u578b(GMM)\u8d85\u8be6\u7ec6\u8bb2\u89e3（\u9644\u805a\u7c7b\u5b9e\u6218Python\u5b8c\u6574\u4ee3\u7801）
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