\u4e0a\u7bc7\u6587\u7ae0\u5df2\u7ecf\u5b9e\u73b0\u4e86\u4e8c\u7ef4\u76f4\u7ebf\\(Ax+By+C=0\\)\u7684\u62df\u5408\u7b97\u6cd5\u3002\u5982\u679c\u8981\u62df\u5408\u4e09\u7ef4\u76f4\u7ebf\u600e\u4e48\u529e\uff1f\u9996\u5148\uff0c\u65b9\u7a0b\\(Ax+By+Cz+D=0\\)\u662f\u4e0d\u53ef\u4ee5\u7684\uff0c\u56e0\u4e3a\u4ed6\u662f\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5e73\u9762\u3002\u5982\u679c\u8981\u8868\u8fbe\u4e00\u6761\u76f4\u7ebf\uff0c\u9700\u8981\u4e24\u4e2a\u4e09\u7ef4\u5e73\u9762\u7684\u8054\u7acb\uff0c\u4f3c\u4e4e\u4e5f\u4e0d\u662f\u4e2a\u597d\u529e\u6cd5\u3002\u8fd9\u91cc\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u76f4\u7ebf\u7684\u201c\u70b9+\u5411\u91cf\u201d\u65b9\u7a0b\\(l = A + dD\\)\u7684\u65b9\u5f0f\u3002\u5176\u4e2dA\u4e3a\u76f4\u7ebf\u4e0a\u4e00\u70b9\uff0cD\u4e3a\u76f4\u7ebf\u65b9\u5411\u5411\u91cf\u3002\u5728\u8ba1\u7b97\u51e0\u4f55\u4e2d\uff0c\u8fd9\u662f\u4e00\u79cd\u5e38\u7528\u7684\u8868\u8fbe\u65b9\u5f0f\u3002<\/p>\n
\u5df2\u77e5\uff0c\u91c7\u6837\u70b9$$\\{(x_1,y_1),(x_2,y_2),…,(x_n,y_n)\\}$$\u6b32\u6c42\u76f4\u7ebf\\(l = A + dD\\)\uff0c\u4f7f\u5f97\u5404\u70b9\u5230\u76f4\u7ebf\u7684\u8ddd\u79bb\u7684\u5e73\u65b9\u548c\u6700\u5c0f\u3002<\/p>\n
\u91c7\u6837\u70b9\\(X_i(x_i,y_i)\\)\u5230\u76f4\u7ebf\\(l\\)\u7684\u8ddd\u79bb\u53ef\u4ee5\u7528\\(X_i\\)\u5230\\(A\\)\u7684\u5411\u91cf \\(Y_i = X_i -A\\) \u4e0e\\(Y_i\\)\u5728\u76f4\u7ebf\u4e0a\u7684\u6295\u5f71\\((Y_i^TD)D\\)\u505a\u5dee\u6c42\u5f97\u3002\u56e0\u6b64\u53ef\u4ee5\u5217\u76ee\u6807\u65b9\u7a0b\u4e3a<\/p>\n
$$f = \\sum (Yi^2 – (D\\cdot Y_i)^2)$$<\/p>\n
\u89e3\u9898\u7684\u6280\u5de7\u5728\u4e0e\u5bf9\u4e0a\u5f0f\u7684\u91cd\u7ec4\u3002\u5148\u6c42 A\u3002\\( \\frac {\\partial f}{\\partial A} = \\frac {\\partial f}{\\partial Y} \\cdot \\frac {\\partial Y}{\\partial A} = – \\frac {\\partial f}{\\partial Y} =0 \\)<\/p>\n
$$f = \\sum (Y_i^2 – (D^TY_i)^2) $$<\/p>\n
$$\\frac {\\partial f}{\\partial Y} = \\sum(2Y_i – 2DD^TY_i ) =2 (I-DD^T) \\sum Y_i =0$$<\/p>\n
\u53ef\u4ee5\u5f97\u51fa \\(\\sum Y_i =\\sum (X_i-A)=0\\),\u89e3\u51fa \\( A = \\frac {\\sum X_i }{n} = \\bar X\\)\u3002\u5373\u76f4\u7ebf\u7ecf\u8fc7\u6240\u6709\u70b9\u7684\u4e2d\u5fc3\u70b9\u3002<\/p>\n
\u5c06\\(f\\)\u6539\u5199\u4e3a$$f = \\sum (Y_i^2 – (Y_i^TD)^2) =\\sum (D^TY_i^TY_iD –\u00a0D^TY_iY_i^TD ) = D^T \\sum ((Y_i^TY_i)I – Y_iY_i^T)D$$<\/p>\n
\u8fd9\u91cc\u8981\u89e3\u91ca\u4e00\u4e0b \u56e0\u4e3a\u7ea6\u675f\u4e86 \\(D^TD=1\\),\u6240\u4ee5\\(Y_i^2 = D^TY_i^TY_iD^T\\)\u3002<\/p>\n
\u4ee4 \\(S =\\sum ((Y_i^TY_i)I – Y_iY_i^T) \\), \u7b80\u5316\u6210<\/p>\n
$$f = D^T SD; ||D||=1$$<\/p>\n
\u8fd9\u4e2a\u51fd\u6570\u7684\u5f62\u5f0f\u4e0e\u524d\u4e00\u7bc7\u6587\u7ae0<\/a>\u4e00\u6837\uff0cf\u7684\u6700\u5c0f\u503c\u4e3aS\u5bf9\u5e94\u6700\u5c0f\u7279\u5f81\u503c\u65f6\u7684\u7279\u5f81\u5411\u91cf\u3002\u8fd9\u91cc\u4e0d\u518d\u91cd\u590d\u8bf4\u660e\u3002<\/p>\n \u603b\u7ed3\u4e00\u4e0b\uff0c\u672c\u6587\u63d0\u51fa\u7684\u76f4\u7ebf\u65b9\u7a0b\u5177\u6709\u4e00\u822c\u6027\uff0c\u4e0d\u4ec5\u9002\u7528\u4e8e\u4e8c\u7ef4\u3001\u4e09\u7ef4\u7a7a\u95f4\u76f4\u7ebf\uff0c\u66f4\u80fd\u9002\u7528\u4e8ek-D\u76f4\u7ebf\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u5e94\u7528\u4e0e\u5176\u4ed6\u4e1a\u52a1\u9886\u57df\uff0c\u5982\u7edf\u8ba1\u3001\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7b49\u7b49\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u4e0a\u7bc7\u6587\u7ae0\u5df2\u7ecf\u5b9e\u73b0\u4e86\u4e8c\u7ef4\u76f4\u7ebf\\(Ax+By+C=0\\)\u7684\u62df\u5408\u7b97\u6cd5\u3002\u5982\u679c\u8981\u62df\u5408\u4e09\u7ef4\u76f4 … Continue reading