Math: 客观赋权方法列表

熵权法：

\[{w_j} = \frac{{1 - k\sum\limits_{i = 1}^m {{f_{ij}}\ln {f_{ij}}} }}{{n - \sum\limits_{j = 1}^n {\left( {k\sum\limits_{i = 1}^m {{f_{ij}}\ln {f_{ij}}} } \right)} }},{f_{ij}} = \frac{{{r_{ij}}}}{{\sum\limits_{i = 1}^m {{r_{ij}}} }},k = \frac{1}{{\ln m}}\]

变异系数法：

\[{w_j} = \frac{{{\raise0.7ex\hbox{${\sqrt {\frac{1}{m}\sum\limits_{i = 1}^m {{{\left( {{r_{ij}} - \overline {{r_{ij}}} } \right)}^{\left( 2 \right)}}} } }$} \!\mathord{\left/ {\vphantom {{\sqrt {\frac{1}{m}\sum\limits_{i = 1}^m {{{\left( {{r_{ij}} - \overline {{r_{ij}}} } \right)}^{\left( 2 \right)}}} } } {\left| {\overline {{r_j}} } \right|}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\left| {\overline {{r_j}} } \right|}$}}}}{{\sum\limits_{j = 1}^n {\left( {{\raise0.7ex\hbox{${\sum\limits_{i = 1}^m {{{\left( {{r_{ij}} - \overline {{r_j}} } \right)}^{\left( 2 \right)}}} }$} \!\mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^m {{{\left( {{r_{ij}} - \overline {{r_j}} } \right)}^{\left( 2 \right)}}} } {\left| {\overline {{r_j}} } \right|}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\left| {\overline {{r_j}} } \right|}$}}} \right)} }},\overline {{r_j}} = \frac{1}{m}\sum\limits_{i = 1}^m {{r_{ij}}} \]

均方差法

\[{w_j} = \frac{{\sqrt {\frac{1}{m}\sum\limits_{i = 1}^m {{{\left( {{r_{ij}} - \overline {{r_j}} } \right)}^2}} } }}{{\sum\limits_{j = 1}^n {\sqrt {\frac{1}{m}\sum\limits_{i = 1}^m {{{\left( {{r_{ij}} - \overline {{r_j}} } \right)}^2}} } } }},\overline {{r_j}} = \frac{1}{m}\sum\limits_{i = 1}^m {{r_{ij}}} \]

离差最大化法

\[{w_j} = {\textstyle{{\sum\limits_{i = 1}^m {\sum\limits_{k = 1}^m {\left| {{r_{ij}} - {r_{kj}}} \right|} } } \over {\sum\limits_{j = 1}^m {\sum\limits_{i = 1}^n {\sum\limits_{k = 1}^n {\left| {{r_{ij}} - {r_{kj}}} \right|} } } }}}\]

复相关系数法

\[{w_j} = \frac{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{p_j}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{p_j}}$}}}}{{\sum\limits_{j = 1}^n {\left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{p_j}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{p_j}}$}}} \right)} }}\]

式中：p_j为x_j对x₁……x_n的复相关系数。

References

[1] 倪广亚, 刘学录, 李沁汶, 等. 基于数据信息特征的土地资源评价客观赋权方法的研究. 中国农学通报, 2014, 30(20): 255~262.