\u5bf9\u4e8e\u4e00\u4e2a\u5305\u542b\\(m\\)\u4e2a\u6837\u4f8b\u7684\u6570\u636e\u96c6\\(D=\\{(x_i,y_i)|i=1,2,\\cdots,m\\}\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5bf9\\(D\\)\u8fdb\u884c\u4e00\u5b9a\u5904\u7406\uff0c\u4ece\u4e2d\u4ea7\u751f\u8bad\u7ec3\u96c6\\(S\\)\u548c\u6d4b\u8bd5\u96c6\\(T\\)\u5206\u522b\u8fdb\u884c\u6d4b\u8bd5\u548c\u8bc4\u4f30.<\/p>\n\n\n\n
\u7559\u51fa\u6cd5\u5373\u5c06\u6570\u636e\u96c6\u5212\u5206\u4e3a\u4e24\u4e2a\u4e92\u65a5\u7684\u96c6\u5408\\(S\\)\uff08\u6570\u636e\u96c6\uff09\u4ee5\u53ca\\(T\\)\uff08\u6d4b\u8bd5\u96c6\uff09\uff0c\u6ee1\u8db3\uff1a
\\[
\\begin{cases}
S\\cup T=D\\\\
S\\cap T=\\emptyset
\\end{cases}
\\]
\u5728\u4f7f\u7528\\(S\\)\u8bad\u7ec3\u51fa\u6a21\u578b\u4e4b\u540e\u7528\\(T\\)\u6765\u8bc4\u4f30\u5176\u6d4b\u8bd5\u8bef\u5dee.
\u8be5\u65b9\u6cd5\u6700\u597d\u4f7f\u7528\u6570\u636e\u96c6\u4e2d\u5927\u7ea6\\(67\\%\\sim80\\%\\)\u6837\u672c\u4f5c\u4e3a\u8bad\u7ec3\u6570\u636e\uff0c\u5176\u4f59\u7528\u4e8e\u6d4b\u8bd5\uff0c\u5e76\u4e14\u5728\u4f7f\u7528\u8be5\u65b9\u6cd5\u65f6\u9700\u8981\u91c7\u7528\u82e5\u5e72\u6b21\u968f\u673a\u5212\u5206\u3001\u91cd\u590d\u8fdb\u884c\u5b9e\u9a8c\u8bc4\u4f30\u540e\u53d6\u5e73\u5747\u503c\u4f5c\u4e3a\u8bc4\u4f30\u7ed3\u679c.<\/p>\n\n\n\n
\u4ea4\u53c9\u9a8c\u8bc1\u6cd5\u5373\u5c06\u6570\u636e\u96c6\\(D\\)\u5212\u5206\u4e3a\\(k\\)\u4e2a\u5927\u5c0f\u76f8\u4f3c\u7684\u4e92\u65a5\u5b50\u96c6\\(D_i(i=1,2,\\cdots,k)\\)\uff0c\u6ee1\u8db3\uff1a
\\[
\\begin{cases}
D_1\\cup D_2\\cup\\cdots\\cup D_k=D\\\\
D_i\\cap D_j=\\emptyset, i\\neq j
\\end{cases}
\\]
\u6bcf\u4e2a\u5b50\u96c6\u90fd\u5c3d\u53ef\u80fd\u4fdd\u6301\u6570\u636e\u5206\u5e03\u7684\u4e00\u81f4\u6027\uff0c\u4e4b\u540e\u6bcf\u6b21\u91c7\u7528\\(k-1\\)\u4e2a\u5b50\u96c6\u7684\u5e76\u96c6\u4f5c\u4e3a\u8bad\u7ec3\u96c6\uff0c\u5e76\u7528\u5269\u4e0b\u4e00\u4e2a\u5b50\u96c6\u4f5c\u4e3a\u6d4b\u8bd5\u96c6\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5f97\u5230\\(k\\)\u7ec4\u8bad\u7ec3\u96c6\u548c\u6d4b\u8bd5\u96c6\uff1a
\\[
\\begin{cases}
\\text{\u7ec4\\(1\\)}:&
\\begin{cases}
S=D_2\\cup D_3\\cup\\cdots\\cup D_k\\\\
T=D_1
\\end{cases}\\\\
\\text{\u7ec4\\(2\\)}:&
\\begin{cases}
S=D_1\\cup D_3\\cup\\cdots\\cup D_k\\\\
T=D_2
\\end{cases}\\\\
\\text{\u7ec4\\(3\\)}:&
\\begin{cases}
S=D_1\\cup D_2\\cup D_4\\cup\\cdots\\cup D_k\\\\
T=D_3
\\end{cases}\\\\
\\cdots\\\\
\\text{\u7ec4\\(k-1\\)}:&
\\begin{cases}
S=D_1\\cup D_2\\cup\\cdots\\cup D_{k-2}\\cup D_k\\\\
T=D_{k-1}
\\end{cases}\\\\
\\text{\u7ec4\\(k\\)}:&
\\begin{cases}
S=D_1\\cup D_2\\cup\\cdots\\cup D_{k-1}\\\\
T=D_k
\\end{cases}\\\\
\\end{cases}
\\]
\u8fdb\u884c\\(k\\)\u6b21\u6d4b\u8bd5\u9a8c\u8bc1\u540e\u5bf9\u6d4b\u8bd5\u7ed3\u679c\u53d6\u5e73\u5747\u4f5c\u4e3a\u8bc4\u4f30\u7ed3\u679c\u3002\u5bf9\u4e8e\u4efb\u4e00\u7ec4\uff0c\u8bad\u7ec3\u96c6\u548c\u6d4b\u8bd5\u96c6\u540c\u6837\u6ee1\u8db3\\(S\\cup T=D,S\\cap T=\\emptyset\\)\uff0c\u901a\u5e38\u5c06\u4ea4\u53c9\u9a8c\u8bc1\u6cd5\u79f0\u4e3a\\(k\\)\u6298\u4ea4\u53c9\u9a8c\u8bc1\uff08\u5e38\u53d6\\(k=10\\)\uff09.<\/p>\n\n\n\n
\u8fd9\u79cd\u65b9\u6cd5\u662f\u4ee5\u81ea\u4e3b\u91c7\u6837\u6cd5\uff08bootstrap sampling\uff09\u4e3a\u57fa\u7840\u7684\uff1a
\u5bf9\u4e8e\u4e00\u4e2a\u5305\u542b\\(m\\)\u4e2a\u6837\u672c\u7684\u6570\u636e\u96c6\\(D\\)\uff0c\u5bf9\u5b83\u91c7\u6837\u4ea7\u751f\u6570\u636e\u96c6\\(D’\\)\uff08\u91c7\u6837\u65b9\u6cd5\u4e3a\\(m\\)\u6b21\u6709\u653e\u56de\u968f\u673a\u91c7\u6837\uff09\uff0c\u53ef\u4ee5\u77e5\u9053\u4efb\u610f\u4e00\u4e2a\u6837\u672c\\(s\\in D\\)\u59cb\u7ec8\u6ca1\u6709\u88ab\u91c7\u6837\u5230\u7684\u6982\u7387\u4e3a\uff1a
\\[
\\left(1-\\frac{1}{m}\\right)^m
\\]
\u6709\uff1a
\\[
\\lim_{m\\rightarrow\\infty}\\left(1-\\frac{1}{m}\\right)^m=\\frac{1}{e}\\approx0.368
\\]
\u53ef\u4ee5\u77e5\u9053\u6570\u636e\u96c6\\(D\\)\u4e2d\u6709\u5927\u7ea6\\(36.8\\%\\)\u7684\u6837\u672c\u4e0d\u4f1a\u88ab\u91c7\u6837\u5230\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u7528\\(D’\\)\u4f5c\u4e3a\u8bad\u7ec3\u96c6\uff0c\\(D-D’\\)\u4f5c\u4e3a\u6d4b\u8bd5\u96c6\u8fdb\u884c\u8bad\u7ec3\u548c\u6d4b\u8bd5.<\/p>\n\n\n\n
\n\u81ea\u52a9\u6cd5\u9002\u5408\u5728\u6570\u636e\u96c6\u8f83\u5c0f\u3001\u96be\u4ee5\u6709\u6548\u5212\u5206\u8bad\u7ec3\u96c6\u548c\u6d4b\u8bd5\u96c6\u7684\u65f6\u5019\u6bd4\u8f83\u6709\u7528\uff0c\u5728\u6570\u636e\u91cf\u8db3\u591f\u5927\u7684\u65f6\u5019\u6700\u597d\u4f7f\u7528\u7559\u51fa\u6cd5\u6216\u4ea4\u53c9\u9a8c\u8bc1\u6cd5<\/p>\n<\/blockquote>\n\n\n\n
\u7ebf\u6027\u6a21\u578b<\/h2>\n\n\n\n
\u7ed9\u5b9a\u7531\\(d\\)\u4e2a\u5c5e\u6027\u63cf\u8ff0\u7684\u793a\u4f8b\\(\\pmb{x}=(x_1,x_2,\\cdots,x_d)^T\\)\uff0c\u5176\u4e2d\\(x_i\\)\u4e3a\\(\\pmb{x}\\)\u5728\u7b2c\\(i\\)\u4e2a\u5c5e\u6027\u4e0a\u7684\u53d6\u503c\uff0c\u7ebf\u6027\u6a21\u578b\u8bd5\u56fe\u5b66\u5f97\u4e00\u4e2a\u5173\u4e8e\\(\\pmb{x}\\)\u7684\u7ebf\u6027\u51fd\u6570\u6765\u8fdb\u884c\u9884\u6d4b\u7684\u51fd\u6570\uff1a
\\[
f(\\pmb{x})=w_1x_1+w_2x_2+\\cdots+w_dx_d+b
\\]
\u5411\u91cf\u8868\u793a\u4e3a\uff1a
\\[
f(\\pmb{x})=\\pmb{w}^{T}\\pmb{x}+b
\\]
\u5176\u4e2d\\(\\pmb{w}=(w_1,w_2,\\cdots,w_d)^T\\)\uff0c\u5728\\(\\pmb{w}\\)\u548c\\(b\\)\u5b66\u5f97\u4e4b\u540e\uff0c\u6a21\u578b\u5c31\u53ef\u4ee5\u786e\u5b9a.<\/p>\n\n\n\n\u7ebf\u6027\u56de\u5f52<\/h3>\n\n\n\n
\u7ed9\u5b9a\u6570\u636e\u96c6\\(D=\\{(\\pmb{x}_i,y_i)|i=1,2,\\cdots,m\\}\\)\uff0c\u5176\u4e2d\\(\\pmb{x}_i=(x_{i1},x_{i2},\\cdots,x_{id})^T,y_i\\in\\mathbb{R}\\)\uff0c\u7ebf\u6027\u56de\u5f52\u8bd5\u56fe\u5b66\u5f97\u4e00\u4e2a\u7ebf\u6027\u6a21\u578b\u4ee5\u5c3d\u53ef\u80fd\u51c6\u786e\u5730\u9884\u6d4b\u5b9e\u503c\u8f93\u51fa\u6807\u8bb0\uff0c\u53ef\u91c7\u7528\u591a\u5143\u7ebf\u6027\u56de\u5f52\u7b97\u6cd5\u6765\u8bd5\u56fe\u5b66\u5f97\u51fd\u6570\\(f(\\pmb{x})\\)\u4f7f\u5f97\uff1a
\\[
f(\\pmb{x}_i)=\\pmb{w}^{T}\\pmb{x}_i+b\\simeq y_i
\\]
\u591a\u5143\u7ebf\u6027\u56de\u5f52\u91c7\u7528\u6700\u5c0f\u4e8c\u4e58\u6cd5\u6765\u5bf9\\(\\pmb{w}\\)\u4ee5\u53ca\\(b\\)\u8fdb\u884c\u4f30\u8ba1\uff0c\u7b97\u6cd5\u5982\u4e0b\uff1a
\u9996\u5148\u5c06\u6570\u636e\u96c6\\(D\\)\u8868\u793a\u4e3a\u4e00\u4e2a\\(m\\times(d+1)\\)\u7684\u77e9\u9635\\(\\pmb{X}\\)\uff1a
\\[
\\pmb{X}=\\left(\\begin{matrix}
x_{11}&x_{12}&\\cdots&x_{1d}&1\\\\
x_{21}&x_{22}&\\cdots&x_{2d}&1\\\\
\\vdots&\\vdots&\\ddots&\\vdots&\\vdots\\\\
x_{m1}&x_{m2}&\\cdots&x_{md}&1
\\end{matrix}\\right)=\\left(\\begin{matrix}
\\pmb{x}_1^T&1\\\\
\\pmb{x}_2^T&1\\\\
\\vdots&\\vdots\\\\
\\pmb{x}_m^T&1
\\end{matrix}\\right)
\\]
\u5e76\u5c06\u6807\u8bb0\\(y_1,y_2,\\cdots,y_{m}\\)\u5199\u6210\u5411\u91cf\u5f62\u5f0f\\(\\pmb{y}=(y_1,y_2,\\cdots,y_{m})^{T}\\)\uff0c\u6709\\(\\pmb{w}\\)\u7684\u8fd1\u4f3c\u5411\u91cf\uff1a
\\[
\\hat{\\pmb{w}}^{*}=(\\pmb{X}^{T}\\pmb{X})^{-1}\\pmb{X}^T\\pmb{y}
\\]
\u4ee4\\(\\hat{\\pmb{x}}_i=(\\pmb{x}_i^{T},1)^T\\)\uff08\u5373\u5728\\(\\pmb{x}_i\\)\u7684\u6700\u540e\u62fc\u4e0a\u4e00\u4e2a\\(1\\)\uff09\uff0c\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u591a\u5143\u7ebf\u6027\u56de\u5f52\u6a21\u578b\uff1a
\\[
f(\\hat{\\pmb{x}}_i)=\\hat{\\pmb{x}}_i^T\\hat{\\pmb{w}}^{*}=\\hat{\\pmb{x}}_i^T(\\pmb{X}^{T}\\pmb{X})^{-1}\\pmb{X}^T\\pmb{y}
\\]
\u8fd9\u91cc\u4ee5\u52a0\u5229\u798f\u5c3c\u4e9a\u623f\u5c4b\u4ef7\u683c\u6570\u636e\u96c6\u4e3a\u4f8b\uff0c\u8be5\u6570\u636e\u96c6\u63cf\u8ff0\u5982\u4e0b\uff1a<\/p>\n\n\n\n