对抗择优抽取算法,双权重求出两列一个正向指标,一个负向指标的综合评价值
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流程图
原始矩阵如下:
$$ \begin{array}{c|c|c|c|c|c|c}{M_{11 \times15}} & -PN1 & -PN2 & -PE1 & -PS1 & -PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
2011 &0.654302103 &278.5577711 &0.104246565 &426.623418 &0.484969128 &0.518335321 &0.734225621 &0.050682504 &8.714200492 &8.950944846 &0.850812256 &10.3 &7380.279843 &35115.65191 &176.2\\
\hline
2012 &0.64018787 &230.2072698 &0.100574919 &430.0655108 &0.498760764 &0.524293616 &0.720190933 &0.053800771 &8.311141515 &8.547470721 &0.868804512 &11.1 &8725.395288 &39690.62085 &170.1\\
\hline
2013 &0.634839501 &232.7438152 &0.095228796 &433.091812 &0.510793577 &0.534174192 &0.725746269 &0.053359371 &8.162741259 &8.838568347 &0.889545989 &12.82 &10872.14779 &43857.04467 &144.4\\
\hline
2014 &0.621034465 &198.6498147 &0.094446274 &435.3089733 &0.524130018 &0.545060211 &0.730176133 &0.05345935 &8.324123649 &10.36566358 &0.912120266 &8.67 &12075.052 &47967.53527 &159.4\\
\hline
2015 &0.617366121 &179.1637029 &0.093725645 &438.6801925 &0.540016156 &0.547408526 &0.727055177 &0.052997358 &8.456326593 &10.10966409 &0.932741508 &50.29 &13182.83334 &51652.87565 &157.2\\
\hline
2016 &0.614656439 &134.5724311 &0.093564023 &442.1742592 &0.558923488 &0.547604836 &0.732802092 &0.052527861 &8.333882334 &10.73475734 &0.951164027 &50.54 &14328.14133 &55939.46599 &138.94\\
\hline
2017 &0.605388917 &93.61972547 &0.086820781 &445.1131496 &0.577785857 &0.548797995 &0.737336045 &0.052199574 &8.053402438 &11.01950072 &0.958207523 &62.2 &15557.73277 &61583.50207 &252.86\\
\hline
牛逼 &0.4 &100 &0.05 &400 &0.3 &0.7 &0.8 &0.1 &10 &20 &0.95 &12 &30000 &50000 &150\\
\hline
很好 &0.6 &200 &0.1 &600 &0.5 &0.5 &0.7 &0.08 &8 &15 &0.9 &9 &20000 &40000 &100\\
\hline
良 &0.7 &300 &0.2 &800 &0.6 &0.4 &0.6 &0.05 &6 &10 &0.8 &6 &15000 &30000 &50\\
\hline
很垃圾 &0.8 &400 &0.3 &1000 &0.7 &0.3 &0.5 &0.02 &4 &5 &0.7 &3 &10000 &20000 &20\\
\hline
\end{array} $$
采用的归一方法如下
极差法
正向指标公式:$$ n_{ij} = \frac{{o_{ij}-min(o_{j})}}{{max(o_{j})-min(o_{j})}} $$
负向指标公式:$$ n_{ij} = \frac{max(o_{j})-{o_{ij}}}{{max(o_{j})-min(o_{j})}} $$
归一化矩阵如下
$$ \begin{array}{c|c|c|c|c|c|c}{M_{11 \times15}} & -PN1 & -PN2 & -PE1 & -PS1 & -PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
2011 &0.364 &0.396 &0.783 &0.956 &0.538 &0.546 &0.781 &0.384 &0.786 &0.263 &0.584 &0.123 &0 &0.364 &0.671\\
\hline
2012 &0.4 &0.554 &0.798 &0.95 &0.503 &0.561 &0.734 &0.423 &0.719 &0.236 &0.654 &0.137 &0.059 &0.474 &0.645\\
\hline
2013 &0.413 &0.546 &0.819 &0.945 &0.473 &0.585 &0.752 &0.417 &0.694 &0.256 &0.734 &0.166 &0.154 &0.574 &0.534\\
\hline
2014 &0.447 &0.657 &0.822 &0.941 &0.44 &0.613 &0.767 &0.418 &0.721 &0.358 &0.822 &0.096 &0.208 &0.673 &0.599\\
\hline
2015 &0.457 &0.721 &0.825 &0.936 &0.4 &0.619 &0.757 &0.412 &0.743 &0.341 &0.901 &0.799 &0.257 &0.761 &0.589\\
\hline
2016 &0.463 &0.866 &0.826 &0.93 &0.353 &0.619 &0.776 &0.407 &0.722 &0.382 &0.973 &0.803 &0.307 &0.864 &0.511\\
\hline
2017 &0.487 &1 &0.853 &0.925 &0.306 &0.622 &0.791 &0.402 &0.676 &0.401 &1 &1 &0.362 &1 &1\\
\hline
牛逼 &1 &0.979 &1 &1 &1 &1 &1 &1 &1 &1 &0.968 &0.152 &1 &0.721 &0.558\\
\hline
很好 &0.5 &0.653 &0.8 &0.667 &0.5 &0.5 &0.667 &0.75 &0.667 &0.667 &0.775 &0.101 &0.558 &0.481 &0.344\\
\hline
良 &0.25 &0.326 &0.4 &0.333 &0.25 &0.25 &0.333 &0.375 &0.333 &0.333 &0.387 &0.051 &0.337 &0.24 &0.129\\
\hline
很垃圾 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0.116 &0 &0\\
\hline
\end{array} $$正极值点构成
$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times15}} & -PN1 & -PN2 & -PE1 & -PS1 & -PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
\mathbf{Zone^+} &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1 &1\\
\hline
\end{array} $$负极值点构成
$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times15}} & -PN1 & -PN2 & -PE1 & -PS1 & -PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
\mathbf{Zone^-} &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0\\
\hline
\end{array} $$ 采用的是熵权法(EWM)求权重W1
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times15}} & -PN1 & -PN2 & -PE1 & -PS1 & -PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
EWM所得权重 &0.056255 &0.052243 &0.040037 &0.04453 &0.05917 &0.048601 &0.042516 &0.056495 &0.042611 &0.074201 &0.044361 &0.196456 &0.123409 &0.057263 &0.061853\\
\hline
权重大小顺序 &8 &9 &15 &11 &5 &10 &14 &7 &13 &3 &12 &1 &2 &6 &4\\
\hline
\end{array} $$ 采用的是CRITIC方法求权重W2
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times15}} & -PN1 & -PN2 & -PE1 & -PS1 & -PS2 &SN1 &SN2 &SN3 &SE1 &SS1 &RN1 &RN2 &RE1 &RE2 &RS1\\
\hline
CRITIC方法所得权重 &0.053176 &0.066843 &0.066333 &0.077907 &0.058671 &0.060044 &0.065755 &0.060084 &0.063687 &0.062617 &0.072847 &0.088362 &0.067664 &0.070149 &0.06586\\
\hline
权重大小顺序 &15 &6 &7 &2 &14 &13 &9 &12 &10 &11 &3 &1 &5 &4 &8\\
\hline
\end{array} $$
由两种权重方法,针对归一化矩阵分别求得评价值
欧几里得距离、欧式距离公式
$$ d = \sqrt {\sum_\limits{j=1}^m { \omega_{j}^2 \left({n_{ij}-Min(n_j) } \right)} ^2} $$
$$matrix=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &a(正向指标) &-b(负向指标)\\
\hline 2011 &0.1088 &0.1494\\
\hline 2012 &0.1119 &0.1421\\
\hline 2013 &0.1152 &0.1364\\
\hline 2014 &0.1212 &0.1312\\
\hline 2015 &0.2001 &0.1024\\
\hline 2016 &0.2046 &0.0998\\
\hline 2017 &0.2462 &0.0907\\
\hline 牛逼 &0.22 &0.0828\\
\hline 很好 &0.1368 &0.1269\\
\hline 良 &0.0719 &0.1895\\
\hline 很垃圾 &0.0143 &0.2582\\
\hline \end{array} $$
由妥协解公式求出基础决策矩阵(边界决策矩阵)
$$ Q_i = \left( 1-k \right) {rank(a_i)/n} + k rank(b_i)/n $$
上述妥协解中,需要把负向指标转化为正向指标,原则即两个指标同方向。
$$base=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times2}} &Q(k=0) &Q(k=1)\\
\hline 2011 &0.1818 &0.1818\\
\hline 2012 &0.2727 &0.2727\\
\hline 2013 &0.3636 &0.3636\\
\hline 2014 &0.4545 &0.4545\\
\hline 2015 &0.6364 &0.6364\\
\hline 2016 &0.7273 &0.7273\\
\hline 2017 &0.9091 &0.8182\\
\hline 牛逼 &0.8182 &0.9091\\
\hline 很好 &0.5455 &0.5455\\
\hline 良 &0.0909 &0.0909\\
\hline 很垃圾 &0 &0\\
\hline \end{array} $$
AECM运算之一,获得交点(拐点)
所谓拐点,就是上述线段中的交点
所谓排序分析,即每个决策系数k对应的Q值的优劣排序,数值越低越优。两个拐点之间要素的排序是稳定一致的
拐点处(交点),存在着至少一次,某两个要素的排序是一致的。
交点坐标位置接近,以至于观测不到交点,下面会变换坐标,使得拐点等距,这样方便观测拐点具体的值。
由上图得到交点加上k=0,k=1即得到所有拐点,结果如下。
$$\begin{array}{c|c|c|c|c|c|c}{M_{3 \times1}} &拐点对应的k值\\
\hline 0 &0\\
\hline 1 &0.5\\
\hline 2 &1\\
\hline \end{array} $$
AECM运算之二,排序聚类分析
$$Qk_{matrix}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times3}} &k=0 &k=0.5 &k=1\\
\hline 2011 &0.182 &0.182 &0.182\\
\hline 2012 &0.273 &0.273 &0.273\\
\hline 2013 &0.364 &0.364 &0.364\\
\hline 2014 &0.455 &0.455 &0.455\\
\hline 2015 &0.636 &0.636 &0.636\\
\hline 2016 &0.727 &0.727 &0.727\\
\hline 2017 &0.909 &0.864 &0.818\\
\hline 牛逼 &0.818 &0.864 &0.909\\
\hline 很好 &0.545 &0.545 &0.545\\
\hline 良 &0.091 &0.091 &0.091\\
\hline 很垃圾 &0 &0 &0\\
\hline \end{array} $$ 上述两列都是正向指标,数值越大越好。因此排序情况如下:
$$Q_{rank}=\begin{array}{c|c|c|c|c|c|c}{M_{11 \times3}} &k=0 &k=0.5 &k=1\\
\hline 2011 &9 &9 &9\\
\hline 2012 &8 &8 &8\\
\hline 2013 &7 &7 &7\\
\hline 2014 &6 &6 &6\\
\hline 2015 &4 &4 &4\\
\hline 2016 &3 &3 &3\\
\hline 2017 &1 &2 &2\\
\hline 牛逼 &2 &2 &1\\
\hline 很好 &5 &5 &5\\
\hline 良 &10 &10 &10\\
\hline 很垃圾 &11 &11 &11\\
\hline \end{array} $$
拐点与区段的排序如下:其中拐点中交点的位置有相等的情况出现。
序号 | 性质与对应k值 | 区段大小 | Q值排序 |
---|
1 | 0 | 0 | $2017\succ 牛逼\succ 2016\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
---|
2 | 0<$k$<0.5 | 0.5 | $2017\succ 牛逼\succ 2016\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
---|
3 | 0.5 | 0 | $牛逼\succ 2017 = 2016\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
---|
4 | 0.5<$k$<1 | 0.5 | $牛逼\succ 2017\succ 2016\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
---|
5 | 1 | 0 | $牛逼\succ 2017\succ 2016\succ 2015\succ 很好\succ 2014\succ 2013\succ 2012\succ 2011\succ 良\succ 很垃圾$ |
提取区段的位置
序号 |
聚类特征-对应k值区段 |
区段大小 |
Q值排序 |
1 | 0<$k$< 0.5 | 0.5 | $2017 \succ 牛逼 \succ 2016 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
---|
2 | 0.5<$k$< 1 | 0.5 | $牛逼 \succ 2017 \succ 2016 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
AECM运算之三,层级要素所占区段统计,统计矩阵的获得
层级,序号越小越优 |
要素所占区段,该层级要素的的占比 |
0 | 2017=0.5 牛逼=0.5 |
---|
1 | 牛逼=0.5 2017=0.5 |
---|
2 | 2016=1 |
---|
3 | 2015=1 |
---|
4 | 很好=1 |
---|
5 | 2014=1 |
---|
6 | 2013=1 |
---|
7 | 2012=1 |
---|
8 | 2011=1 |
---|
9 | 良=1 |
---|
10 | 很垃圾=1 |
AECM运算之四,优胜与劣汰两种情境最终排序结果
情境 |
最优妥协解 |
优胜情境 | $2017 \succ 牛逼 \succ 2016 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
---|
劣汰情境 | $2017 \succ 牛逼 \succ 2016 \succ 2015 \succ 很好 \succ 2014 \succ 2013 \succ 2012 \succ 2011 \succ 良 \succ 很垃圾$ |
扯蛋模型