{"id":173,"date":"2017-11-14T21:40:18","date_gmt":"2017-11-14T13:40:18","guid":{"rendered":"http:\/\/119.29.206.68\/?p=173"},"modified":"2017-11-28T23:03:13","modified_gmt":"2017-11-28T15:03:13","slug":"%e6%9c%80%e5%b0%8f%e4%ba%8c%e4%b9%98%e6%b3%95%e5%9c%86%e6%8b%9f%e5%90%88%ef%bc%9akasas-method","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=173","title":{"rendered":"\u6700\u5c0f\u4e8c\u4e58\u6cd5\u5706\u62df\u5408\uff1akasa’s method"},"content":{"rendered":"

\u5728\u5706\u62df\u5408\u65b9\u6cd5\u4e2d\uff0c\u6700\u5e38\u89c1\u7684\u662f\u4e00 \u79cd\u4ee3\u6570\u5706\u62df\u5408\u65b9\u6cd5\uff0c\u5728\u6211\u67e5\u9605\u7684\u8d44\u6599\u4e2d\uff0c\u8fd9\u79cd\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201ckasa’s method\u201d\u3002\u5df2\u77e5\u91c7\u6837\u70b9\u96c6\\( \\{(x_1,y_1),(x_2,y_2),…,(x_n,y_n)\\}\\)\u00a0\u6b32\u6c42\u5706$$(x-a)^2 + (y-b)^2 = r^2$$\u4f7f\u5f97\u91c7\u6837\u70b9\u5230\u5706\u7684\u8ddd\u79bb\u7684\u5e73\u65b9\u548c\u6700\u8fd1\u3002<\/p>\n

\u5373\u6b32\u6c42\\(a,b,r\\)\u4f7f$$f=\\sum_{i=1}^n (\\sqrt{(x_i-a) ^2+ (y-b)^2} – r)^2$$\u5f97\u5230\u6700\u5c0f\u503c\u3002\u4e3a\u4e86\u7b80\u5316\u63cf\u8ff0\uff0c\u53ef\u4ee5\u4ee4\\(d=\\sqrt{(x_i-a) ^2+ (y-b)^2}\\)\u3002\u56e0\u4e3ad\u4e2d\u5e26\u6709\u6839\u53f7\uff0c\u4ee4\u95ee\u9898\u79f0\u4e3a\u975e\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\uff0c\u8ba1\u7b97\u8d77\u6765\u5c31\u6ca1\u6709\u7ebf\u6027\u95ee\u9898\u90a3\u4e48\u5feb\u901f\u4e86\uff0c\u6240\u4ee5kasa\u00a0\u65b9\u6cd5\u63d0\u51fa\u4e86\u5c06\u6c42\u89e3 \\((d-r)^2\\)\u95ee\u9898\u8f6c\u6362\u4e3a\\((d^2-r^2)^2\\)\u6700\u5c0f\u503c\u95ee\u9898\u3002\u770b\u8d77\u6765\uff0c\\(d-r\\)\u53d6\u6700\u5c0f\u503c(\u63a5\u8fd1\u4e3a0\u65f6),\\(d-r\\)\u4e5f\u80fd\u53d6\u6700\u5c0f\u503c\u3002\u6211\u540e\u9762\u5206\u6790\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u548c\u9002\u7528\u8303\u56f4\u3002<\/p>\n

$$g=\\sum((x_i-a)^2+(y_i-b)^2-r^2)^2$$<\/p>\n

\u82e5\u4ee4\\(B=-2a, C=-2b , D=a^2+b^2-r^2\uff0cg=\\sum(x_i^2+y_i^2+Bx_i+Cy_i+D)^2 \\)<\/p>\n

\u6c42\u89e3B,C,D\u7684\u503c\u53ef\u4ee5\u89e3\u51faa,b,c\u3002\u4ee4 \\(z_i = x_i^2 + y_i^2\\)<\/p>\n

\\begin{equation}
\n\\left\\{
\n\\begin{aligned}
\n\\overset{.}\u00a0{\\frac {\\partial g}{\\partial D} =2(\\sum z+ \\sum x B + \\sum y C +\u00a0nD ) =0} \\\\
\n{\\frac {\\partial g}{\\partial B} =2(\\sum xz + \\sum x^2 B + \\sum xy C + \\sum x D ) =0} \\\\{\\frac {\\partial g}{\\partial C} =2(\\sum yz+ \\sum xy B + \\sum y^2 C + \\sum y D ) =0}
\n\\end{aligned}
\n\\right.
\n\\end{equation}<\/p>\n

\u5199\u6210\u77e9\u9635\u5f62\u5f0f<\/p>\n

$$\\begin{bmatrix} \\sum x& \\sum y& n\\\\\\sum x^2 &\u00a0\\sum xy &\u00a0\\sum x \\\\\u00a0\\sum xy &\u00a0\\sum y^2\u00a0 & \\sum y \\end{bmatrix} \\begin{bmatrix} B \\\\ C \\\\ D\\end{bmatrix} =\u00a0\\begin{bmatrix} -\\sum z \\\\ -\\sum xz \\\\ -\\sum yz\u00a0 \\end{bmatrix}\u00a0\u00a0$$<\/p>\n

\u91c7\u7528Eigen<\/a>\u53ef\u4ee5\u89e3\u7ebf\u6027\u65b9\u7a0b\uff0c\u6c42\u51faB,C,D;<\/p>\n

\\( a = -\\frac {B}{2} , b = -\\frac {C}{2},r = \\frac{\\sqrt{B^2+C^2-4D}}{2}\\)\u3002<\/p>\n

\u4e0b\u9762\u5206\u6790\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u548c\u4f7f\u7528\u8303\u56f4\u3002<\/p>\n

\\( d^2-r^2 = (d-r)(d+r) \\)\u5f53\u00a0d-r\u8f83\u5c0f\uff0c\u5373\u91c7\u6837\u70b9\u51e0\u4e4e\u90fd\u5206\u5e03\u5728\u540c\u4e00\u4e2a\u5706\u4e0a\u65f6\uff0c\u7b97\u6cd5\u6548\u679c\u826f\u597d\uff0c\u5f53d-r\u53d8\u5927\uff0c\u5373\u566a\u97f3\u8f83\u5927\u65f6\uff0c\\(min( d^2-r^2)\\)\u4f1a\u8bd5\u56fe\u51cf\u5c0f(d+r)\u9879\uff0c\u4f7f\u5f97\u7b97\u6cd5\u5f97\u51fa\u7684\u76f4\u5f84\u663e\u8457\u504f\u5c0f\u3002<\/p>\n

\u5176\u6b21\uff0c\u7b97\u6cd5\u7684\u6548\u679c\u8fd8\u548c\u91c7\u6837\u70b9\u7684\u5206\u5e03\u6709\u5173\u7cfb\uff0c\u5982\u679c\u91c7\u6837\u70b9\u5206\u5e03\u5728\u6574\u4e2a\u5706\u4e0a\uff0ckasa\u00a0\u65b9\u6cd5\u7684\u6548\u679c\u8f83\u597d\uff0c\u5982\u679c\u5206\u5e03\u5728\u534a\u5706\u4e0a\u6548\u679c\u8f83\u5dee\uff0c\u5982\u679c\u662f\u56db\u5206\u4e4b\u4e00\u5706\u548c\u516b\u5206\u4e4b\u4e00\u5706\uff0c\u6548\u679c\u66f4\u5dee\u3002\u5982\u4e0b\u56fe\u3002<\/p>\n

\"\"<\/p>\n

\u4e0a\u56fe\u5f15\u7528\u81eaNikolai Chernov\u00a0<\/a>\u6559\u6388\u7684 \u300aCircular and Linear Regression\u300b\uff0cChernov\u00a0\u6559\u6388\u662f\u83ab\u65af\u79d1\u5927\u5b66\u6570\u5b66\u7cfb\u535a\u58eb\uff0c\u957f\u671f\u4efb\u6559\u4e8eUAB(University of Alabama at Birmingham)\u3002\u4ed6\u7684\u7814\u7a76\u5e2e\u6211\u4eec\u89e3\u51b3\u4e86\u751f\u4ea7\u4e2d\u7684\u5b9e\u9645\u95ee\u9898\u3002\u4ed6\u4e8e2014\u5e74\u901d\u4e16\uff0cMay he rest in peace!<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5728\u5706\u62df\u5408\u65b9\u6cd5\u4e2d\uff0c\u6700\u5e38\u89c1\u7684\u662f\u4e00 \u79cd\u4ee3\u6570\u5706\u62df\u5408\u65b9\u6cd5\uff0c\u5728\u6211\u67e5\u9605\u7684\u8d44\u6599\u4e2d\uff0c\u8fd9\u79cd\u65b9\u6cd5\u88ab\u79f0\u4e3a … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[17,8],"_links":{"self":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/173"}],"collection":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=173"}],"version-history":[{"count":34,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/173\/revisions"}],"predecessor-version":[{"id":378,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/173\/revisions\/378"}],"wp:attachment":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=173"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}