\u5b66\u8fc7\u7a7a\u95f4\u63d2\u503c\u7684\u4eba\u90fd\u77e5\u9053\u514b\u91cc\u91d1\u63d2\u503c\uff0c\u4f46\u662f\u5b83\u7684\u53d8\u79cd\u7e41\u591a\u3001\u516c\u5f0f\u590d\u6742\uff0c\u8fd8\u6709\u4e2a\u534a\u65b9\u5dee\u51fd\u6570\u8ba9\u4eba\u4e0d\u77e5\u6240\u4e91<\/p>\n
\u672c\u6587\u8bb2\u7b80\u5355\u4ecb\u7ecd\u57fa\u672c\u514b\u91cc\u91d1\u63d2\u503c\u7684\u539f\u7406\uff0c\u53ca\u5176\u63a8\u7406\u8fc7\u7a0b\uff0c\u5168\u6587\u5206\u4e3a\u4e5d\u4e2a\u90e8\u5206\uff1a<\/p>\n
0.\u5f15\u8a00\uff0d\u4ece\u53cd\u8ddd\u79bb\u63d2\u503c\u8bf4\u8d77
\n1.\u514b\u91cc\u91d1\u63d2\u503c\u7684\u5b9a\u4e49
\n2.\u5047\u8bbe\u6761\u4ef6
\n3.\u65e0\u504f\u7ea6\u675f\u6761\u4ef6
\n4.\u4f18\u5316\u76ee\u6807\uff0f\u4ee3\u4ef7\u51fd\u6570
\n5.\u4ee3\u4ef7\u51fd\u6570\u7684\u6700\u4f18\u89e3
\n6.\u534a\u65b9\u5dee\u51fd\u6570
\n7.\u666e\u901a\u514b\u91cc\u91d1\u4e0e\u7b80\u5355\u514b\u91cc\u91d1
\n8.\u5c0f\u7ed3<\/p>\n
<\/p>\n
\u7a7a\u95f4\u63d2\u503c\u95ee\u9898\uff0c\u5c31\u662f\u5728\u5df2\u77e5\u7a7a\u95f4\u4e0a\u82e5\u5e72\u79bb\u6563\u70b9 \\((x_i,y_i)\\) \u7684\u67d0\u4e00\u5c5e\u6027(\u5982\u6c14\u6e29\uff0c\u6d77\u62d4)\u7684\u89c2\u6d4b\u503c\\(z_i=z(x_i,y_i)\\)\u7684\u6761\u4ef6\u4e0b\uff0c\u4f30\u8ba1\u7a7a\u95f4\u4e0a\u4efb\u610f\u4e00\u70b9\\((x,y)\\)\u7684\u5c5e\u6027\u503c\u7684\u95ee\u9898\u3002<\/p>\n
\u76f4\u89c2\u6765\u8bb2\uff0c\u6839\u636e\u5730\u7406\u5b66\u7b2c\u4e00\u5b9a\u5f8b\uff0c<\/p>\n
All attribute values on a geographic surface are related to each other, but closer values are more strongly related than are more distant ones.<\/p><\/blockquote>\n
\u5927\u610f\u5c31\u662f\uff0c\u5730\u7406\u5c5e\u6027\u6709\u7a7a\u95f4\u76f8\u5173\u6027\uff0c\u76f8\u8fd1\u7684\u4e8b\u7269\u4f1a\u66f4\u76f8\u4f3c\u3002\u7531\u6b64\u4eba\u4eec\u53d1\u660e\u4e86\u53cd\u8ddd\u79bb\u63d2\u503c\uff0c\u5bf9\u4e8e\u7a7a\u95f4\u4e0a\u4efb\u610f\u4e00\u70b9\\((x,y)\\)\u7684\u5c5e\u6027\\(z=z(x,y)\\)\uff0c\u5b9a\u4e49\u53cd\u8ddd\u79bb\u63d2\u503c\u516c\u5f0f\u4f30\u8ba1\u91cf<\/p>\n
\\[\\hat{z} = \\sum^{n}_{i=1}{\\frac{1}{d^\\alpha}z_i}\\]<\/p>\n
\u5176\u4e2d\\(\\alpha\\)\u901a\u5e38\u53d61\u6216\u80052\u3002<\/p>\n
\u5373\uff0c\u7528\u7a7a\u95f4\u4e0a\u6240\u6709\u5df2\u77e5\u70b9\u7684\u6570\u636e\u52a0\u6743\u6c42\u548c\u6765\u4f30\u8ba1\u672a\u77e5\u70b9\u7684\u503c\uff0c\u6743\u91cd\u53d6\u51b3\u4e8e\u8ddd\u79bb\u7684\u5012\u6570\uff08\u6216\u8005\u5012\u6570\u7684\u5e73\u65b9\uff09\u3002\u90a3\u4e48\uff0c\u8ddd\u79bb\u8fd1\u7684\u70b9\uff0c\u6743\u91cd\u5c31\u5927\uff1b\u8ddd\u79bb\u8fdc\u7684\u70b9\uff0c\u6743\u91cd\u5c31\u5c0f\u3002<\/p>\n
\u53cd\u8ddd\u79bb\u63d2\u503c\u53ef\u4ee5\u6709\u6548\u7684\u57fa\u4e8e\u5730\u7406\u5b66\u7b2c\u4e00\u5b9a\u5f8b\u4f30\u8ba1\u5c5e\u6027\u503c\u7a7a\u95f4\u5206\u5e03\uff0c\u4f46\u4ecd\u7136\u5b58\u5728\u5f88\u591a\u95ee\u9898\uff1a<\/p>\n
\n
- \\(\\alpha\\)\u7684\u503c\u4e0d\u786e\u5b9a<\/li>\n
- \u7528\u5012\u6570\u51fd\u6570\u6765\u63cf\u8ff0\u7a7a\u95f4\u5173\u8054\u7a0b\u5ea6\u4e0d\u591f\u51c6\u786e<\/li>\n<\/ul>\n
\u56e0\u6b64\u66f4\u52a0\u51c6\u786e\u7684\u514b\u91cc\u91d1\u63d2\u503c\u65b9\u6cd5\u88ab\u63d0\u51fa\u6765\u4e86<\/p>\n
1.\u514b\u91cc\u91d1\u63d2\u503c\u7684\u5b9a\u4e49<\/h1>\n
\u76f8\u6bd4\u53cd\u8ddd\u79bb\u63d2\u503c\uff0c\u514b\u91cc\u91d1\u63d2\u503c\u516c\u5f0f\u66f4\u52a0\u62bd\u8c61<\/p>\n
\\[\\hat{z_o} = \\sum^{n}_{i=1}{\\lambda_iz_i}\\]<\/p>\n
\u5176\u4e2d\\(\\hat{z_o}\\)\u662f\u70b9\\((x_o,y_o)\\)\u5904\u7684\u4f30\u8ba1\u503c\uff0c\u5373\\(z_o=z(x_o,y_o)\\) \u3002<\/p>\n
\u8fd9\u91cc\u7684\\(\\lambda_i\\)\u662f\u6743\u91cd\u7cfb\u6570\u3002\u5b83\u540c\u6837\u662f\u7528\u7a7a\u95f4\u4e0a\u6240\u6709\u5df2\u77e5\u70b9\u7684\u6570\u636e\u52a0\u6743\u6c42\u548c\u6765\u4f30\u8ba1\u672a\u77e5\u70b9\u7684\u503c\u3002\u4f46\u6743\u91cd\u7cfb\u6570\u5e76\u975e\u8ddd\u79bb\u7684\u5012\u6570\uff0c\u800c\u662f\u80fd\u591f\u6ee1\u8db3\u70b9\\((x_o,y_o)\\)\u5904\u7684\u4f30\u8ba1\u503c\\(\\hat{z_o}\\)\u4e0e\u771f\u5b9e\u503c\\(z_o\\)\u7684\u5dee\u6700\u5c0f\u7684\u4e00\u5957\u6700\u4f18\u7cfb\u6570\uff0c\u5373<\/p>\n
\\[\\min_{\\lambda_i} Var(\\hat{z_o}-z_o)\\]<\/p>\n
\u540c\u65f6\u6ee1\u8db3\u65e0\u504f\u4f30\u8ba1\u7684\u6761\u4ef6<\/p>\n
\\[E(\\hat{z_o}-z_o)=0\\]<\/p>\n
2.\u5047\u8bbe\u6761\u4ef6<\/h1>\n
\u4e0d\u540c\u7684\u514b\u91cc\u91d1\u63d2\u503c\u65b9\u6cd5\u7684\u4e3b\u8981\u5dee\u5f02\u5c31\u662f\u5047\u8bbe\u6761\u4ef6\u4e0d\u540c\u3002\u672c\u6587\u4ec5\u4ecb\u7ecd\u666e\u901a\u514b\u91cc\u91d1\u63d2\u503c\u7684\u5047\u8bbe\u6761\u4ef6\u4e0e\u5e94\u7528\u3002<\/p>\n
\u666e\u901a\u514b\u91cc\u91d1\u63d2\u503c\u7684\u5047\u8bbe\u6761\u4ef6\u4e3a\uff0c\u7a7a\u95f4\u5c5e\u6027\\(z\\)\u662f\u5747\u4e00\u7684\u3002\u5bf9\u4e8e\u7a7a\u95f4\u4efb\u610f\u4e00\u70b9\\((x,y)\\)\uff0c\u90fd\u6709\u540c\u6837\u7684\u671f\u671bc\u4e0e\u65b9\u5dee\\(\\sigma^2\\)\u3002<\/p>\n
\u5373\u5bf9\u4efb\u610f\u70b9\\((x,y)\\)\u90fd\u6709<\/p>\n
\\[E[z(x,y)] = E[z] = c\\]<\/p>\n
\\[Var[z(x,y)] = \\sigma^2\\]<\/p>\n
\u6362\u4e00\u79cd\u8bf4\u6cd5\uff1a\u4efb\u610f\u4e00\u70b9\u5904\u7684\u503c\\(z(x,y)\\)\uff0c\u90fd\u7531\u533a\u57df\u5e73\u5747\u503c\\(c\\)\u548c\u8be5\u70b9\u7684\u968f\u673a\u504f\u5dee\\(R(x,y)\\)\u7ec4\u6210\uff0c\u5373<\/p>\n
\\[z(x,y)=E[z(x,y)] + R(x,y)] = c + R(x,y)\\]<\/p>\n
\u5176\u4e2d\\(R(x,y)\\)\u8868\u793a\u70b9\\((x,y)\\)\u5904\u7684\u504f\u5dee\uff0c\u5176\u65b9\u5dee\u5747\u4e3a\u5e38\u6570<\/p>\n
\\[Var[R(x,y)] = \\sigma^2\\]<\/p>\n
3.\u65e0\u504f\u7ea6\u675f\u6761\u4ef6<\/h1>\n
\u5148\u5206\u6790\u65e0\u504f\u4f30\u8ba1\u6761\u4ef6\\(E(\\hat{z_o}-z_o)=0\\)\uff0c\u5c06\\(\\hat{z_o} = \\sum^{n}_{i=1}{\\lambda_iz_i}\\)\u5e26\u5165\u5219\u6709<\/p>\n
\\[E(\\sum^{n}_{i=1}{\\lambda_iz_i}- z_o)=0\\]<\/p>\n
\u53c8\u56e0\u4e3a\u5bf9\u4efb\u610f\u7684z\u90fd\u6709\\(E[z] = c\\)\uff0c\u5219<\/p>\n
\\[c \\sum^{n}_{i=1}{\\lambda_i}- c=0\\]<\/p>\n
\u5373<\/p>\n
\\[\\sum^{n}_{i=1}{\\lambda_i} = 1\\]<\/p>\n
\u8fd9\u662f\\(\\lambda_i\\)\u7684\u7ea6\u675f\u6761\u4ef6\u4e4b\u4e00\u3002<\/p>\n
4.\u4f18\u5316\u76ee\u6807\/\u4ee3\u4ef7\u51fd\u6570J<\/h1>\n
\u518d\u5206\u6790\u4f30\u8ba1\u8bef\u5dee\\(Var(\\hat{z_o}-z_o)\\)\u3002\u4e3a\u65b9\u4fbf\u516c\u5f0f\u63a8\u7406\uff0c\u7528\u7b26\u53f7\\(J\\)\u8868\u793a\uff0c\u5373<\/p>\n
\\[J = Var(\\hat{z_o}-z_o)\\]<\/p>\n
\u5219\u6709<\/p>\n
\\[\\begin{array}{r@{\\;=\\;}l} J &= Var(\\sum^{n}_{i=1}{\\lambda_iz_i} – z_o) \\\\&= Var(\\sum^{n}_{i=1}{\\lambda_iz_i}) – 2 Cov(\\sum^{n}_{i=1}{\\lambda_iz_i}, z_o) + Cov(z_o, z_o) \\\\&= \\sum^{n}_{i=1}\\sum^{n}_{j=0}{\\lambda_i\\lambda_jCov( z_i, z_j)} – 2 \\sum^{n}_{i=1}{\\lambda_iCov(z_i, z_o)} + Cov(z_o, z_o) \\end{array} \\]<\/p>\n
\u4e3a\u7b80\u5316\u63cf\u8ff0\uff0c\u5b9a\u4e49\u7b26\u53f7 \\(C_{ij} = Cov(z_i,z_j) = Cov(R_i,R_j)\\)\uff0c\u8fd9\u91cc\\(R_i = z_i – c\\)\uff0c\u5373\u70b9\\((x_i,y_i)\\)\u5904\u7684\u5c5e\u6027\u503c\u76f8\u5bf9\u4e8e\u533a\u57df\u5e73\u5747\u5c5e\u6027\u503c\u7684\u504f\u5dee\u3002<\/p>\n
\u5219\u6709<\/p>\n
\\[J = \\sum^{n}_{i=1}\\sum^{n}_{j}{\\lambda_i\\lambda_jC_{ij}} – 2 \\sum^{n}_{i=1}{\\lambda_iC_{io}} + C_{oo} \\]<\/p>\n
5.\u4ee3\u4ef7\u51fd\u6570\u7684\u6700\u4f18\u89e3<\/h1>\n
\u518d\u5b9a\u4e49\u534a\u65b9\u5dee\u51fd\u6570 \\(r_{ij} = \\sigma^2 -C_{ij}\\)\uff0c\u5e26\u5165J\u4e2d\uff0c\u6709<\/p>\n
\\[\\begin{array}{r@{\\;=\\;}l}J & = \\sum^{n}_{i=1}\\sum^{n}_{j=0}{\\lambda_i\\lambda_j(\\sigma^2 – r_{ij})} – 2 \\sum^{n}_{i=1}{\\lambda_i(\\sigma^2 – r_{io})} + \\sigma^2 – r_{oo} \\\\ &=\\sum^{n}_{i=1}\\sum^{n}_{j=0}{\\lambda_i\\lambda_j(\\sigma^2)} -\\sum^{n}_{i=1}\\sum^{n}_{j=0}{\\lambda_i\\lambda_j( r_{ij})}-2\\sum^{n}_{i=1}{\\lambda_i(\\sigma^2)}+2 \\sum^{n}_{i=1}{\\lambda_i(r_{io})}+\\sigma^2 – r_{oo} \\end{array} \\]<\/p>\n
\u8003\u8651\u5230\\(\\sum^{n}_{i=1}{\\lambda_i} = 1\\)<\/p>\n
\\[\\begin{array}{r@{\\;=\\;}l}J &= \\sigma^2-\\sum^{n}_{i=1}\\sum^{n}_{j}{\\lambda_i\\lambda_j(r_{ij})}-2 \\sigma^2 +2 \\sum^{n}_{i=1}{\\lambda_i(r_{io})}+ \\sigma^2 – r_{oo}\\\\&=2 \\sum^{n}_{i=1}{\\lambda_i(r_{io})} -\\sum^{n}_{i=1}\\sum^{n}_{j=0}{\\lambda_i\\lambda_j(r_{ij})} – r_{oo} \\end{array} \\]<\/p>\n
\u6211\u4eec\u7684\u76ee\u6807\u662f\u5bfb\u627e\u4f7fJ\u6700\u5c0f\u7684\u4e00\u7ec4 \\(\\lambda_i\\)\uff0c\u4e14J\u662f\\(\\lambda_i\\)\u7684\u51fd\u6570\uff0c\u56e0\u6b64\u76f4\u63a5\u5c06J\u5bf9\\(\\lambda_i\\)\u6c42\u504f\u5bfc\u6570\u4ee4\u5176\u4e3a0\u5373\u53ef\u3002\u5373<\/p>\n
\\[\\frac{\\partial J}{\\partial \\lambda_i}= 0;i=1,2,\\cdots,n\\]<\/p>\n
\u4f46\u662f\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6211\u4eec\u8981\u4fdd\u8bc1\u6c42\u89e3\u51fa\u6765\u7684\u6700\u4f18 \\(\\lambda_i\\) \u6ee1\u8db3\u516c\u5f0f\\(\\sum^{n}_{i=1}{\\lambda_i} = 1\\)\uff0c\u8fd9\u662f\u4e00\u4e2a\u5e26\u7ea6\u675f\u6761\u4ef6\u7684\u6700\u4f18\u5316\u95ee\u9898\u3002\u4f7f\u7528\u62c9\u683c\u6717\u65e5\u4e58\u6570\u6cd5\u6c42\u89e3\uff0c\u6c42\u89e3\u65b9\u6cd5\u4e3a\u6784\u9020\u4e00\u4e2a\u65b0\u7684\u76ee\u6807\u51fd\u6570<\/p>\n
\\[J + 2\\phi(\\sum^{n}_{i=1}{\\lambda_i}-1)\\]<\/p>\n
\u5176\u4e2d\\(\\phi\\)\u662f\u62c9\u683c\u6717\u65e5\u4e58\u6570\u3002\u6c42\u89e3\u4f7f\u8fd9\u4e2a\u4ee3\u4ef7\u51fd\u6570\u6700\u5c0f\u7684\u53c2\u6570\u96c6\\({\\phi,\\lambda_1,\\lambda_2,\\cdots,\\lambda_n}\\)\uff0c\u5219\u80fd\u6ee1\u8db3\u5176\u5728\\(\\sum^{n}_{i=1}{\\lambda_i} = 1\\)\u7ea6\u675f\u4e0b\u6700\u5c0f\u5316\\(J\\)\u3002\u5373<\/p>\n
\\[\\left\\{\\begin{array}{r@{\\;=\\;}l}\\frac{\\partial(J + 2\\phi(\\sum^{n}_{i=1}{\\lambda_i}-1))}{\\partial \\lambda_k} &= 0;k=1,2,\\cdots,n\\\\ \\frac{\\partial(J + 2\\phi(\\sum^{n}_{i=1}{\\lambda_i}-1))}{\\partial \\phi} &= 0 \\end{array} \\right.\\]<\/p>\n
\\[\\left\\{\\begin{array}{r@{\\;=\\;}l} \\frac{\\partial (2 \\sum^{n}_{i=1}{\\lambda_i(r_{io})} – \\sum^{n}_{i=1}\\sum^{n}_{j}{\\lambda_i\\lambda_j(r_{ij})} – r_{oo}+ 2\\phi(\\sum^{n}_{i=1}{\\lambda_i}-1))}{\\partial \\lambda_k} & = 0;k=1,2,\\cdots,n\\\\ \\frac{\\partial ( 2 \\sum^{n}_{i=1}{\\lambda_i(r_{io})} – \\sum^{n}_{i=1}\\sum^{n}_{j}{\\lambda_i\\lambda_j(r_{ij})} – r_{oo}+ 2\\phi(\\sum^{n}_{i=1}{\\lambda_i}-1))}{\\partial \\phi} &= 0 \\end{array} \\right.\\]<\/p>\n
\\[\\left\\{\\begin{array}{r@{\\;=\\;}l} 2r_{ko} – \\sum^{n}_{j=1}{(r_{kj}+r_{jk})\\lambda_j}+2\\phi&=0;k=1,2,\\cdots,n\\\\ \\sum^{n}_{i=1}{\\lambda_i} &= 1 \\end{array} \\right.\\]<\/p>\n
\u7531\u4e8e\\(C_{ij}=Cov(z_i,z_j)=C_{ji}\\)\uff0c\u56e0\u6b64\u540c\u6837\u5730\\(r_{ij}=r_{ji}\\)\uff0c\u90a3\u4e48\u6709<\/p>\n
\\[\\left\\{\\begin{array}{r@{\\;=\\;}l} r_{ko} – \\sum^{n}_{j=1}{r_{kj}\\lambda_j}+\\phi&= 0;k=1,2,\\cdots,n\\\\ \\sum^{n}_{i=1}{\\lambda_i} &= 1 \\end{array} \\right.\\]<\/p>\n
\u5f0f\u5b50\u4e2d\u534a\u65b9\u5dee\u51fd\u6570\\(r_{ij}\\)\u5341\u5206\u91cd\u8981\uff0c\u6700\u540e\u4f1a\u8be6\u7ec6\u89e3\u91ca\u5176\u8ba1\u7b97\u4e0e\u5b9a\u4e49<\/p>\n
\u5728\u4ee5\u4e0a\u8ba1\u7b97\u4e2d\u6211\u4eec\u5f97\u5230\u4e86\u5bf9\u4e8e\u6c42\u89e3\u6743\u91cd\u7cfb\u6570\\(\\lambda_j\\)\u7684\u65b9\u7a0b\u7ec4\u3002\u5199\u6210\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u5f62\u5f0f\u5c31\u662f\uff1a<\/p>\n
\\begin{equation}\\left\\{\\begin{array}{r@{\\;=\\;}l} r_{11}\\lambda_1 + r_{12}\\lambda_2 + \\cdots + r_{1n}\\lambda_n – \\phi&= r_{1o}\\\\r_{21}\\lambda_1 + r_{22}\\lambda_2 + \\cdots + r_{2n}\\lambda_n – \\phi&= r_{2o}\\\\&\\cdots\\\\ r_{n1}\\lambda_1 + r_{n2}\\lambda_2 + \\cdots + r_{nn}\\lambda_n – \\phi&= r_{no}\\\\ \\lambda_1 + \\lambda_2 + \\cdots + \\lambda_n &= 1\\\\ \\end{array} \\right.\\end{equation}<\/p>\n
\u5199\u6210\u77e9\u9635\u5f62\u5f0f\u5373\u4e3a<\/p>\n
\\[\\begin{bmatrix}r_{11}&r_{12}&\\cdots&r_{1n}&1\\\\ r_{21}&r_{22}&\\cdots&r_{2n}&1\\\\\\cdots&\\cdots&\\cdots&\\cdots&\\cdots\\\\r_{n1}&r_{n2}&\\cdots&r_{nn}&1\\\\1&1&\\cdots&1&0\\end{bmatrix}\\begin{bmatrix} \\lambda_1\\\\ \\lambda_2\\\\\\cdots\\\\\\lambda_n\\\\-\\phi\\end{bmatrix}=\\begin{bmatrix} r_{1o}\\\\ r_{2o}\\\\\\cdots\\\\r_{no}\\\\1\\end{bmatrix}\\]<\/p>\n
\u5bf9\u77e9\u9635\u6c42\u9006\u5373\u53ef\u6c42\u89e3\u3002<\/p>\n
\u552f\u4e00\u672a\u77e5\u7684\u5c31\u662f\u4e0a\u6587\u4e2d\u5b9a\u4e49\u7684\u534a\u65b9\u5dee\u51fd\u6570\\(r_{ij}\\)\uff0c\u63a5\u4e0b\u6765\u5c06\u8be6\u7ec6\u8ba8\u8bba<\/p>\n
6.\u534a\u65b9\u5dee\u51fd\u6570<\/h1>\n
\u4e0a\u6587\u4e2d\u5bf9\u534a\u65b9\u5dee\u51fd\u6570\u7684\u5b9a\u4e49\u4e3a<\/p>\n
\\[r_{ij} = \\sigma^2 -C_{ij}\\]<\/p>\n
\u5176\u7b49\u4ef7\u5f62\u5f0f\u4e3a<\/p>\n
\\[r_{ij} = \\frac{1}{2}E[(z_i-z_j)^2]\\]<\/p>\n
\u8fd9\u4e5f\u662f\u534a\u65b9\u5dee\u51fd\u6570\u540d\u79f0\u7684\u6765\u7531\uff0c\u63a5\u4e0b\u6765\u8bc1\u660e\u8fd9\u4e8c\u8005\u662f\u7b49\u4ef7\u7684\uff1a<\/p>\n
\u6839\u636e\u4e0a\u6587\u5b9a\u4e49 \\(R_i = z_i – c\\)\uff0c\u6709\\(z_i-z_j = R_i – R_j\\)\uff0c\u5219<\/p>\n
\\[\\begin{array}{r@{\\;=\\;}l} r_{ij} &= \\frac{1}{2}E[(R_i-R_j)^2]\\\\&= \\frac{1}{2}E[R_i^2-2R_iR_j+R_j^2]\\\\&= \\frac{1}{2}E[R_i^2]+\\frac{1}{2}E[R_j^2]-E[R_iR_j] \\end{array} \\]<\/p>\n
\u53c8\u56e0\u4e3a\uff1a<\/p>\n
\\[E[R_i^2] =E[R_j^2] = E[(z_i – c)^2] = Var(z_i) = \\sigma^2 \\]<\/p>\n
\\[E[R_iR_j] = E[(z_i – c)(z_j-c)] = Cov(z_i,z_j) = C_{ij}\\]<\/p>\n
\u4e8e\u662f\u6709<\/p>\n
\\[\\begin{array}{r@{\\;=\\;}l} r_{ij} &= \\frac{1}{2}E[(z_i-z_j)^2]\\\\&= \\frac{1}{2}E[R_i^2]+\\frac{1}{2}E[R_j^2]-E[R_iR_j]\\\\&= \\frac{1}{2}\\sigma^2+\\frac{1}{2}\\sigma^2- C_{ij}\\\\&=\\sigma^2 -C_{ij}\\end{array}\\]<\/p>\n
\\( \\sigma^2 -C_{ij} = \\frac{1}{2}E[(z_i-z_j)^2]\\)\u5f97\u8bc1\uff0c\u73b0\u5728\u7684\u95ee\u9898\u5c31\u662f\u5982\u4f55\u8ba1\u7b97<\/p>\n
\\[r_{ij} = \\frac{1}{2}E[(z_i-z_j)^2]\\]<\/p>\n
\u8fd9\u65f6\u9700\u8981\u7528\u5230\u5730\u7406\u5b66\u7b2c\u4e00\u5b9a\u5f8b\uff0c\u7a7a\u95f4\u4e0a\u76f8\u8fd1\u7684\u5c5e\u6027\u76f8\u8fd1\u3002\\(r_{ij} = \\frac{1}{2}(z_i-z_j)^2\\)\u8868\u8fbe\u4e86\u5c5e\u6027\u7684\u76f8\u4f3c\u5ea6\uff1b\u7a7a\u95f4\u7684\u76f8\u4f3c\u5ea6\u5c31\u7528\u8ddd\u79bb\u6765\u8868\u8fbe\uff0c\u5b9a\u4e49i\u4e0ej\u4e4b\u95f4\u7684\u51e0\u4f55\u8ddd\u79bb<\/p>\n
\\[d_{ij} = d(z_i,z_j) = d( (x_i,y_i), (x_j,y_j)) = \\sqrt{(x_i-x_j)^2 + (y_i – y_j)^2}\\]<\/p>\n
\u514b\u91cc\u91d1\u63d2\u503c\u5047\u8bbe\\(r_{ij}\\)\u4e0e\\(d_{ij}\\)\u5b58\u5728\u7740\u51fd\u6570\u5173\u7cfb\uff0c\u8fd9\u79cd\u51fd\u6570\u5173\u7cfb\u53ef\u4ee5\u662f\u7ebf\u6027\u3001\u4e8c\u6b21\u51fd\u6570\u3001\u6307\u6570\u3001\u5bf9\u6570\u5173\u7cfb\u3002\u4e3a\u4e86\u786e\u8ba4\u8fd9\u79cd\u5173\u7cfb\uff0c\u6211\u4eec\u9700\u8981\u9996\u5148\u5bf9\u89c2\u6d4b\u6570\u636e\u96c6<\/p>\n
\\[\\{z(x_1,y_1),z(x_2,y_2),z(x_3,y_3),\\cdots,z(x_{n-1},y_{n-1}),z(x_n,y_n)\\}\\]<\/p>\n
\u8ba1\u7b97\u4efb\u610f\u4e24\u4e2a\u70b9\u7684 \u8ddd\u79bb\\(d_{ij}= \\sqrt{(x_i-x_j)^2 + (y_i – y_j)^2}\\)\u548c \u534a\u65b9\u5dee \\(\\sigma^2 -C_{ij} =\\frac{1}{2}E[(z_i-z_j)^2]\\)\uff0c\u8fd9\u65f6\u4f1a\u5f97\u5230\\(n^2\\)\u4e2a\\((d_{ij}, r_{ij})\\)\u7684\u6570\u636e\u5bf9\u3002<\/p>\n
\u5c06\u6240\u6709\u7684\\(d\\)\u548c\\(r\\)\u7ed8\u5236\u6210\u6563\u70b9\u56fe\uff0c\u5bfb\u627e\u4e00\u4e2a\u6700\u4f18\u7684\u62df\u5408\u66f2\u7ebf\u62df\u5408\\(d\\)\u4e0e\\(r\\)\u7684\u5173\u7cfb\uff0c\u5f97\u5230\u51fd\u6570\u5173\u7cfb\u5f0f<\/p>\n
\\[r = r(d)\\]<\/p>\n
\u90a3\u4e48\u5bf9\u4e8e\u4efb\u610f\u4e24\u70b9\\(\u001c(x_i,y_i), (x_j,y_j)\\)\uff0c\u5148\u8ba1\u7b97\u5176\u8ddd\u79bb\\(d_{ij}\\)\uff0c\u7136\u540e\u6839\u636e\u5f97\u5230\u7684\u51fd\u6570\u5173\u7cfb\u5c31\u53ef\u4ee5\u5f97\u5230\u8fd9\u4e24\u70b9\u7684\u534a\u65b9\u5dee\\(r_{ij}\\)<\/p>\n
7. \u7b80\u5355\u514b\u91cc\u91d1\uff08simple kriging\uff09\u4e0e\u666e\u901a\u514b\u91cc\u91d1\uff08ordinary kriging\uff09\u7684\u533a\u522b<\/h1>\n
\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u5747\u4e3a\u666e\u901a\u514b\u91cc\u91d1\uff08ordinary kriging\uff09\u7684\u516c\u5f0f\u4e0e\u63a8\u7406\u3002<\/p>\n
\u4e8b\u5b9e\u4e0a\u666e\u901a\u514b\u91cc\u91d1\u63d2\u503c\u8fd8\u6709\u7b80\u5316\u7248\uff0c\u5373\u7b80\u5355\u514b\u91cc\u91d1\uff08simple kriging\uff09\u63d2\u503c\u3002\u4e8c\u8005\u7684\u5dee\u5f02\u5c31\u5728\u4e8e\u5982\u4f55\u5b9a\u4e49\u63d2\u503c\u5f62\u5f0f\uff1a<\/p>\n
\u4e0a\u6587\u8bb2\u5230\uff0c\u666e\u901a\u514b\u91cc\u91d1\u63d2\u503c\u5f62\u5f0f\u4e3a<\/p>\n
\\[\\hat{z_o} = \\sum^{n}_{i=1}{\\lambda_iz_i}\\]<\/p>\n
\u800c\u7b80\u5355\u514b\u91cc\u91d1\u7684\u5f62\u5f0f\u5219\u4e3a<\/p>\n
\\[\\hat{z_o} – c= \\sum^{n}_{i=1}{\\lambda_i(z_i-c)}\\]<\/p>\n
\u8fd9\u91cc\u7684\u7b26\u53f7\\(c\\)\u5728\u4e0a\u6587\u4ecb\u7ecd\u8fc7\u4e86\uff0c\u662f\u5c5e\u6027\u503c\u7684\u6570\u5b66\u671f\u671b\uff0c\u5373\\(E[z] = c\\)\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u666e\u901a\u514b\u91cc\u91d1\u63d2\u503c\u4e2d\uff0c\u8ba4\u4e3a\u672a\u77e5\u70b9\u7684\u5c5e\u6027\u503c\u662f\u5df2\u77e5\u70b9\u7684\u5c5e\u6027\u503c\u7684\u52a0\u6743\u6c42\u548c\uff1b\u800c\u5728\u7b80\u5355\u514b\u91cc\u91d1\u63d2\u503c\u4e2d\uff0c\u5047\u8bbe\u672a\u77e5\u70b9\u7684\u5c5e\u6027\u503c\u76f8\u5bf9\u4e8e\u5e73\u5747\u503c\u7684\u504f\u5dee\u662f\u5df2\u77e5\u70b9\u7684\u5c5e\u6027\u503c\u76f8\u5bf9\u4e8e\u5e73\u5747\u503c\u7684\u504f\u5dee\u7684\u52a0\u6743\u6c42\u548c\uff0c\u7528\u516c\u5f0f\u8868\u8fbe\u5373\u4e3a\uff1a<\/p>\n
\\[\\hat{R_o} = \\sum^{n}_{i=1}{\\lambda_iR_i}\\]<\/p>\n
\u8fd9\u91cc\u7684\\(R_i\\)\u5728\u4e0a\u6587\u5b9a\u4e49\u8fc7\u4e86\uff1a\\(R_i = z_i – c\\)\u3002<\/p>\n
\u4f46\u662f\u4e3a\u4ec0\u4e48\u8fd9\u6837\u7684\u514b\u91cc\u91d1\u63d2\u503c\u79f0\u4e3a\u201c\u7b80\u5355\u514b\u91cc\u91d1\u201d\u5462\uff1f\u7531\u4e8e\u6709\u5047\u8bbe\\(E[z] = c\\)\uff0c\u4e5f\u5c31\u662f\u8bf4\\(E(R_i + c) = c\\)\uff0c\u5373\\(E(R_i) = 0\\)\u3002\u90a3\u4e48\u4e0a\u9762\u7684\u516c\u5f0f\\(\\hat{R_o} = \\sum^{n}_{i=1}{\\lambda_iR_i}\\)\u4e24\u8fb9\u7684\u671f\u671b\u4e00\u5b9a\u76f8\u540c\uff0c\u90a3\u4e48\u5728\u6c42\u89e3\u672a\u77e5\u53c2\u6570\\(\\lambda_i\\)\u5c31\u4e0d\u9700\u8981\u6709\u65e0\u504f\u7ea6\u675f\u6761\u4ef6\\(\\sum^{n}_{i=1}{\\lambda_i} = 1\\)\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8fd9\u6837\u7684\u4f30\u8ba1\u516c\u5f0f\u5929\u751f\u5c31\u80fd\u6ee1\u8db3\u65e0\u504f\u6761\u4ef6\u3002\u56e0\u6b64\u5b83\u88ab\u79f0\u4e3a\u7b80\u5355\u514b\u91cc\u91d1\u3002<\/p>\n
\u4ece\u5728\u4e0a\u6587\uff08\u7b2c4\u8282\u4f18\u5316\u76ee\u6807\/\u4ee3\u4ef7\u51fd\u6570J\uff09\u4e2d\u53ef\u4ee5\u77e5\u9053\uff0c\u4f18\u5316\u76ee\u6807\u7684\u63a8\u7406\u548c\u6c42\u89e3\u8fc7\u7a0b\u662f\u901a\u8fc7\u5bf9\u5c5e\u6027\u503c\u76f8\u5bf9\u4e8e\u671f\u671b\u7684\u504f\u5dee\u91cf\\(R_i\\)\u8fdb\u884c\u6570\u5b66\u8ba1\u7b97\u800c\u8fdb\u884c\u7684\u3002\u4e5f\u5c31\u662f\u8bf4\u8fd9\u4e24\u79cd\u514b\u91cc\u91d1\u63d2\u503c\u65b9\u6cd5\u867d\u7136\u63d2\u503c\u5f62\u5f0f\u4e0d\u4e00\u6837\uff0c\u6c42\u89e3\u65b9\u6cd5\u662f\u4e00\u6837\u7684\uff0c\u91cd\u8981\u7684\u533a\u522b\u662f\u7b80\u5355\u514b\u91cc\u91d1\u63d2\u503c\u4e0d\u9700\u8981\u7ea6\u675f\u6761\u4ef6\\(\\sum^{n}_{i=1}{\\lambda_i} = 1\\)\uff0c\u6c42\u89e3\u65b9\u7a0b\u7ec4\u4e3a\uff1a<\/p>\n
\\begin{equation}\\left\\{\\begin{array}{r@{\\;=\\;}l} r_{11}\\lambda_1 + r_{12}\\lambda_2 + \\cdots + r_{1n}\\lambda_n + \\phi&= r_{1o}\\\\r_{21}\\lambda_1 + r_{22}\\lambda_2 + \\cdots + r_{2n}\\lambda_n + \\phi&= r_{2o}\\\\&\\cdots\\\\ r_{n1}\\lambda_1 + r_{n2}\\lambda_2 + \\cdots + r_{nn}\\lambda_n + \\phi&= r_{no}\\\\ \\end{array} \\right.\\end{equation}<\/p>\n
\u8fd8\u6709\u66f4\u91cd\u8981\u7684\u4e00\u70b9\uff0c\u7b80\u5355\u514b\u91cc\u91d1\u7684\u63d2\u503c\u516c\u5f0f\u4e3a\uff1a<\/p>\n
\\[\\hat{z_o} = \\sum^{n}_{i=1}{\\lambda_i(z_i-c)}+c\\]<\/p>\n
\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u8ba1\u7b97\u672a\u77e5\u70b9\u5c5e\u6027\u503c\\(\\hat{z_o}\\)\u524d\uff0c\u9700\u8981\u77e5\u9053\u8be5\u5730\u533a\u7684\u5c5e\u6027\u503c\u671f\u671b\\(c\\)\u3002\u4e8b\u5b9e\u4e0a\u6211\u4eec\u5728\u8fdb\u884c\u63d2\u503c\u524d\u5f88\u96be\u77e5\u9053\u8fd9\u4e2a\u5730\u533a\u7684\u771f\u5b9e\u5c5e\u6027\u503c\u671f\u671b\u3002\u6709\u4e9b\u7814\u7a76\u8005\u53ef\u80fd\u4f1a\u91c7\u7528\u5bf9\u89c2\u6d4b\u6570\u636e\u7b80\u5355\u6c42\u5e73\u5747\u7684\u65b9\u6cd5\u8ba1\u7b97\u671f\u671b\u503c\\(c\\)\uff0c\u800c\u8003\u8651\u5230\u7a7a\u95f4\u91c7\u6837\u70b9\u4f4d\u7f6e\u4ee3\u8868\u6027\u53ef\u80fd\u6709\u504f\u5dee\uff08\u6bd4\u5982\u91c7\u6837\u70b9\u805a\u96c6\u5728\u67d0\u4e00\u5c0f\u7247\u5730\u533a\uff0c\u6ca1\u6709\u4ee3\u8868\u6027\uff09\uff0c\u7b80\u5355\u5e73\u5747\u4f30\u8ba1\u7684\u671f\u671b\u4e5f\u53ef\u80fd\u662f\u6709\u504f\u5dee\u7684\u3002\u8fd9\u662f\u7b80\u5355\u514b\u91cc\u91d1\u65b9\u6cd5\u7684\u5c40\u9650\u6027\u3002<\/p>\n
8.\u5c0f\u7ed3<\/h1>\n
\u603b\u7684\u6765\u8bf4\uff0c\u8fdb\u884c\u514b\u91cc\u91d1\u63d2\u503c\u5206\u4e3a\u8fd9\u51e0\u4e2a\u6b65\u9aa4\uff1a<\/p>\n
\n
- \u5bf9\u4e8e\u89c2\u6d4b\u6570\u636e\uff0c\u4e24\u4e24\u8ba1\u7b97\u8ddd\u79bb\u4e0e\u534a\u65b9\u5dee<\/li>\n
- \u5bfb\u627e\u4e00\u4e2a\u62df\u5408\u66f2\u7ebf\uff08\u6216\u8005\u5176\u4ed6\u6a21\u578b\uff09\u6a21\u62df\u8ddd\u79bb\u4e0e\u534a\u65b9\u5dee\u7684\u5173\u7cfb\uff0c\u4ece\u800c\u80fd\u6839\u636e\u4efb\u610f\u8ddd\u79bb\u8ba1\u7b97\u51fa\u76f8\u5e94\u7684\u534a\u65b9\u5dee<\/li>\n
- \u8ba1\u7b97\u51fa\u6240\u6709\u5df2\u77e5\u70b9\u4e4b\u95f4\u7684\u534a\u65b9\u5dee\\(r_{ij}\\)\uff0c\u76f4\u63a5\u4f7f\u7528\u516c\u5f0f\\(r_{ij} = \\frac{1}{2}(z_i-z_j)^2\\)<\/li>\n
- \u5bf9\u4e8e\u672a\u77e5\u70b9\\(z_o\\)\uff0c\u8ba1\u7b97\u5b83\u5230\u6240\u6709\u5df2\u77e5\u70b9\\(z_i\\)\u7684\u8ddd\u79bb\\(d_{io}\\)\uff0c\u5e76\u901a\u8fc7\u7b2c2\u6b65\u7684\u62df\u5408\u66f2\u7ebf\uff0c\u4f30\u8ba1\u534a\u65b9\u5dee\\(r_{io}\\)<\/li>\n
- \u6c42\u89e3\u7b2c\u56db\u8282\u4e2d\u7684\u65b9\u7a0b\u7ec4\uff0c\u5f97\u5230\u6700\u4f18\u7cfb\u6570\\(\\lambda_i\\)<\/li>\n
- \u4f7f\u7528\u6700\u4f18\u7cfb\u6570\u5bf9\u5df2\u77e5\u70b9\u7684\u5c5e\u6027\u503c\u8fdb\u884c\u52a0\u6743\u6c42\u548c\uff0c\u5f97\u5230\u672a\u77e5\u70b9\\(z_o\\)\u7684\u4f30\u8ba1\u503c\uff0c\u5373\u4e3a\\(\\hat{z_o} = \\sum^{n}_{i=1}{\\lambda_iz_i}\\)<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"