kasa\u65b9\u6cd5\u5706\u62df\u5408\u4f5c\u4e3a\u6700\u5e38\u89c1\u7684\u5706\u62df\u5408\u65b9\u6cd5\uff0c\u867d\u7136\u8ba1\u7b97\u65b9\u6cd5\u7b80\u5355\uff0c\u6548\u7387\u5feb\uff0c\u4f46\u662f\u62df\u5408\u7ed3\u679c\u5b58\u5728\u8f83\u5927\u504f\u5dee(Heavy bias)\u3002\u901a\u8fc7\u5c06\\(D=\\sqrt{(x-a)^2+(y-b)^2}\\)\u4e0e\u534a\u5f84R\u7684\u5dee\u503c\\(D-R\\)\u8f6c\u6362\u4e3a\\(D^2-R^2\\)\uff0c\u5c06\u975e\u7ebf\u6027\u95ee\u9898\u8f6c\u6362\u4e3a\u7ebf\u6027\u95ee\u9898\u3002\u4f46\u662f\u56e0\u4e3a\\(D^2-R^2 = d(d+2R) (\u4ee4d=D-R)\\),\u5f53\u504f\u79bb\u503cd\u8f83\u5927\u65f6\uff0ckasa\u65b9\u6cd5\u5bfc\u81f4R\u660e\u663e\u53d8\u5c0f\u3002Pratt\u901a\u8fc7\u5c06kasa\u65b9\u6cd5\u7684\u76ee\u6807\u9664\u4ee5\\((2R)^2\\)\u7684\u65b9\u5f0f\uff0c\u53d6\u5f97\u66f4\u51c6\u786e\u7684\u7ed3\u679c\u3002<\/p>\n
$$f=\u00a0 \\frac{ \\sum_{i=1}^{n}((x_i-a)^2+(y_i-b)^2-R^2)^2}{(2R)^2}$$\u7c7b\u4f3ckasa method\uff0cpratt\u65b9\u6cd5\u5c06\u5706\u65b9\u7a0b\u63cf\u8ff0\u4e3a \\(A(x^2+y^2) + Bx + Cy+D=0\\)\u3002\u8fd9\u6837\u5706\u5fc3\\((a,b) = (-\\frac{B}{2A},-\\frac{C}{2A})\\)\u534a\u5f84\\(R = \\frac{\\sqrt{B^2+C^2-4AD}}{2A}\\)\u3002\u6ce8\u610f\uff0cA\u6709\u53ef\u80fd\u4e3a0\uff0c\u6b64\u65f6\u5706\u62df\u5408\u65b9\u6cd5\u5f97\u5230\u7684\u662f\u4e00\u6761\u76f4\u7ebf\uff0c\u6216\u8005\u8bf4\u4e00\u4e2a\u66f2\u7387\u4e3a0\u7684\u5706\u3002\u4f46\u662f\u5728\u8fd9\u91cc\uff0c\u56e0\u4e3aA,B,C,D\u7684\u503c\u4e58\u4ee5\u4e00\u4e2a\u6807\u91cf\u4e5f\u4e0d\u4f1a\u6539\u53d8\u5706\u7684\u65b9\u7a0b\uff0c\u53ef\u4ee5\u8ba9A=1,\u6240\u4ee5\u76ee\u6807\u65b9\u7a0b\u53ef\u4ee5\u6539\u5199\u4e3a\uff1a<\/p>\n
$$f= (\\frac{\\sum A(x_i^2+y_i^2) + Bx_i + Cy_i+D}{B^2+C^2-4AD})^2 $$<\/p>\n
Pratt\u5c06\u8fd9\u4e2a\u65b9\u7a0b\u8f6c\u5316\u4e3a\u6c42$$g=(\\sum A(x_i^2+y_i^2) + Bx_i + Cy_i+D)^2$$\u7684\u6700\u5c0f\u503c\uff0c\u5e76\u4e14\u5728\u7ea6\u675f\u6761\u4ef6$$B^2+C^2-4AD=1$$\u7684\u6761\u4ef6\u4e0b\u3002<\/p>\n
\u60ef\u4f8b\u5c06\u4e0a\u5f0f\u5199\u6210\u77e9\u9635\u5f62\u5f0f:$$g= A^TMA\u00a0 (\\vec A=[A,B,C,D]^T ; M=\\begin{bmatrix}\\sum z^2 & \\sum xz & \\sum yz & \\sum z \\\\ \\sum xz & \\sum x^2 & \\sum xy & \\sum x \\\\ \\sum yz & \\sum xy &\\sum y^2 & \\sum y \\\\ \\sum z & \\sum x & \\sum y & n \\end{bmatrix}; z_i = x_i^2 + y_i^2)$$<\/p>\n
$$A^TBA = \\begin{bmatrix}A&B&C&D\\end{bmatrix}\\begin{bmatrix}0&0&0&-2\\\\0&1&0&0\\\\0&0&1&0\\\\-2&0&0&0 \\end{bmatrix}\\begin{bmatrix}A\\\\B\\\\C\\\\D\\end{bmatrix}=1 $$<\/p>\n
\u901a\u8fc7\u5f15\u5165\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u89e3\u51b3\u6700\u5c0f\u503c\u95ee\u9898\uff1a<\/p>\n
$$g(A,\\eta) = A^TMA – \\eta(A^TBA-1)$$<\/p>\n
\u5bf9\\(\\vec A\\)\u6c42\u504f\u5bfc: \\(MA – \\eta BA=0\\)\u3002\u56e0\u4e3aB\u53ef\u9006\uff0c\u53ef\u4ee5\u53d8\u6362\u4e3a \\(B^{-1})MA = \\eta A\\)\u3002\u53ef\u77e5 \\(\\eta\\)\u4e3a\u77e9\u9635\\(B^{-1}M\\)\u7684\u7279\u5f81\u503c\uff0cA\u4e3a\u5176\u7279\u5f81\u5411\u91cf\u3002<\/p>\n
\u7531\\(MA – \\eta BA=0\\)\u53ef\u5f97\\(A^TMA – \\eta A^TBA=ATMA-\\eta=0\\)\u5373\\(A^TMA\\)\u7684\u6781\u503c\u4e3a\\(\\eta\\)\uff0c\u90a3\u4e48\u77e9\u9635\\(B^{-1}M\\)\u6709\u56db\u4e2a\u7279\u5f81\u503c\uff0c\u6700\u5c0f\u7684\u7279\u5f81\u503c\u5c31\u662f\\(A^TMA\\)\u7684\u6700\u5c0f\u503c\uff1f\u4e0d\u7136\u3002\u5982\u679c\\(\\eta<0\\) ,\u90a3\u4e48\u62df\u5408\u51fa\u7684\u5706\u5c06\u6ca1\u6709\u610f\u4e49\uff0c\u56e0\u4e3a\\(A^TMA \\ge 0\\)\u3002\uff08Nikolai Chernov<\/a> \u6559\u6388\u7ecf\u8fc7\u590d\u6742\u7684\u8bc1\u660e\uff0c\\(B^{-1}M\\) \u5b58\u5728\u4e00\u4e2a\u8d1f\u7279\u5f81\u503c\u3002\uff09\u56e0\u6b64\uff0cA\u7684\u503c\u5e94\u8be5\u662f\u5bf9\u5e94\u77e9\u9635\\(B^{-1}M\\)\u6700\u5c0f\u975e\u8d1f\u7279\u5f81\u503c\u7684\u7279\u5f81\u5411\u91cf\u3002<\/p>\n \u6700\u540e\uff0c\u901a\u8fc7\u91c7\u7528Eigen\u5e93\u7684EigenSolver<\/a>\u53ef\u4ee5\u8ba1\u7b97\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u3002<\/p>\n \u7b97\u6cd5\u5206\u6790\uff1a<\/strong><\/p>\n pratt \u65b9\u6cd5\u6709\u4e24\u4e2a\u660e\u663e\u4f18\u52bf\uff0c\u4e00\u4e2a\u662f\u6d88\u9664\u4e86\u534a\u5f84\u7684\u5f71\u54cd\uff0c\u4f7f\u5f97\u66f2\u7387\u534a\u5f84\u975e\u5e38\u5927\u3001\u566a\u97f3\u5927\u7684\u60c5\u51b5\u4e0b\u7b97\u6cd5\u4e5f\u80fd\u6b63\u786e\u8fd0\u884c\u3002<\/p>\n \u4e00\u4e2a\u662f\u5728\u65b9\u7a0b\u5f0f\u4e0a\u878d\u5408\u4e86\u76f4\u7ebf\u65b9\u7a0b\uff0c\u5f53\u7cfb\u6570A=0\u65f6\uff0c\u6c42\u5f97\u7684\u66f2\u7ebf\u662f\u76f4\u7ebf\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u4e0d\u6e05\u695a\u5f85\u62df\u5408\u66f2\u7ebf\u4e3a\u76f4\u7ebf\u8fd8\u662f\u66f2\u7ebf\u7684\u60c5\u51b5\u4e0b\u975e\u5e38\u6709\u6548\u3002<\/p>\n \u8fd8\u6709\u4e00\u70b9\u5c31\u662fpratt\u65b9\u6cd5\u7684\u6548\u7387\u4e5f\u6bd4\u8f83\u9ad8\uff0c\u56e0\u6b64\u662f\u4e00\u79cd\u975e\u5e38\u5b9e\u7528\u7684\u65b9\u6cd5\u3002<\/p>\n \u672c\u6587\u53c2\u8003\u6750\u6599 Nikolai Chernov<\/a> \u6559\u6388\u7684 < Circular and Linear Regression > Chernov<\/a> \u6559\u6388\u4e3b\u9875\u4e0a\u4e5f\u7ed9\u51fa\u4e86\u7f16\u7a0b\u5b9e\u73b0\u3002\u611f\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u76f4\u63a5\u4e0b\u8f7d\u8bd5\u7528\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" kasa\u65b9\u6cd5\u5706\u62df\u5408\u4f5c\u4e3a\u6700\u5e38\u89c1\u7684\u5706\u62df\u5408\u65b9\u6cd5\uff0c\u867d\u7136\u8ba1\u7b97\u65b9\u6cd5\u7b80\u5355\uff0c\u6548\u7387\u5feb\uff0c\u4f46\u662f\u62df\u5408\u7ed3\u679c … Continue reading
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