RSR——AISM联合求解过程

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原始数据如下

$$ \begin{array}{c|c|c|c|c|c|c}{M_{16 \times17}} &X1 &-X2 &X3 &X4 &-X5 &-X6 &X7 &-X8 &-X9 &X10 &-X11 &-X12 &X13 &X14 &X15 &X16 &X17\\ \hline 合肥市 &78.33 &377 &262.8130871 &187195.5463 &1.434479554 &716 &76.33 &43.84288954 &8.27 &115623 &32.8 &25.7 &60.6 &4.178814383 &6149 &43.08 &97.36\\ \hline 淮北市 &16.67 &185.9 &211.450237 &175695.4612 &0.879166667 &822 &65.88 &41.40900474 &7.22 &47654 &39.2 &31 &50.5 &4.265402844 &2180 &45.38 &98.84\\ \hline 毫州市 &47.46 &202.4 &279.1789671 &174279.2739 &0.724982982 &628 &42.22 &33.25821596 &7.9 &33313 &60.9 &48 &50.8 &3.755868545 &6100 &37.32 &97.96\\ \hline 宿州市 &58.47 &178.4 &312.7701082 &182180.4434 &0.570386988 &582 &43.96 &38.24110128 &6.61 &34773 &51.4 &46.7 &49.6 &2.949852507 &5222 &38.57 &95.14\\ \hline 蚌埠市 &36.97 &417.1 &311.8723823 &179972.9364 &1.337406015 &573 &58.58 &41.62206606 &6.74 &60469 &69.2 &31.1 &47.5 &3.162333099 &1519 &42.73 &98\\ \hline 阜阳市 &54.39 &210.7 &207.0672414 &174936.0614 &1.01754386 &816 &44.62 &33.42137931 &8.67 &32855 &64.3 &37.9 &48.8 &3.908045977 &3566 &40.58 &96.31\\ \hline 淮南市 &31.36 &596 &222.3538462 &137345.1327 &2.680412371 &631 &65.04 &43.26730769 &4.42 &37140 &160.5 &149.2 &49.3 &2.932692308 &998 &44.64 &98\\ \hline 滁州市 &81.53 &526.2 &414.3041247 &128228.0937 &1.27008149 &310 &54.54 &42.44326306 &5.7 &70429 &75 &25.1 &42.3 &3.895508708 &6576 &45.91 &97.23\\ \hline 六安市 &136.63 &474.4 &1081.763322 &341185.607 &0.438543247 &315 &47.09 &38.57577855 &6.05 &33370 &142.7 &52.6 &50.5 &1.470588235 &2561 &42.59 &98.46\\ \hline 马鞍山市 &30.23 &1217.9 &443.1037913 &258782.7808 &2.748565966 &583 &69.12 &54.22302609 &5.4 &89867 &136.2 &213.5 &46.6 &1.460869565 &1085 &45.28 &97.6\\ \hline 芜湖市 &58.35 &798.1 &640.0682587 &403875.0626 &1.246898263 &627 &66.41 &45.53008292 &5.51 &96154 &83.3 &121.1 &47.4 &1.558872305 &2522 &43.06 &97.05\\ \hline 宣城市 &188.9 &529.9 &3643.156454 &785575.3647 &0.145450794 &216 &56.33 &50.73510638 &3.11 &58819 &90.3 &36.1 &43.1 &3.262411348 &1389 &41.68 &96.04\\ \hline 铜陵市 &32.04 &765.1 &849.163535 &463430.8511 &0.901004304 &546 &57.16 &43.25007962 &3.15 &58726 &130.8 &202.2 &48.2 &2.707006369 &384 &45.15 &94.81\\ \hline 池州市 &116.37 &687.8 &3600.676787 &646155.706 &0.191019645 &180 &54.92 &40.41919687 &3.3 &56217 &122.8 &143.3 &45.7 &4.015670911 &1880 &44.86 &96.59\\ \hline 安庆市 &157 &513.6 &1474.112448 &514710.2306 &0.34841304 &348 &49.98 &40.64765045 &6.24 &50574 &101.9 &88.2 &46 &1.813685078 &8227 &40.44 &96.45\\ \hline 黄山市 &180.54 &310.5 &7535.785714 &1091363.312 &0.041203401 &145 &52.49 &43.65306122 &4.74 &57853 &53.9 &27.4 &57.8 &2.267573696 &906 &47.91 &95.28\\ \hline \end{array} $$

编秩后的秩次矩阵如下

$$Rank=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times17}} &X1 &X2 &X3 &X4 &X5 &X6 &X7 &X8 &X9 &X10 &X11 &X12 &X13 &X14 &X15 &X16 &X17\\ \hline 合肥市 &10 &11 &4 &8 &3 &3 &16 &4 &2 &16 &16 &15 &16 &15 &14 &9 &10\\ \hline 淮北市 &1 &15 &2 &5 &9 &1 &13 &10 &4 &6 &15 &13 &12.5 &16 &8 &14 &16\\ \hline 毫州市 &6 &14 &5 &3 &10 &5 &1 &16 &3 &2 &12 &8 &14 &11 &13 &1 &12\\ \hline 宿州市 &9 &16 &7 &7 &11 &8 &2 &14 &6 &4 &14 &9 &11 &8 &12 &2 &2\\ \hline 蚌埠市 &5 &10 &6 &6 &4 &9 &11 &9 &5 &12 &10 &12 &7 &9 &6 &7 &13.5\\ \hline 阜阳市 &7 &13 &1 &4 &7 &2 &3 &15 &1 &1 &11 &10 &9 &13 &11 &4 &5\\ \hline 淮南市 &3 &5 &3 &2 &2 &4 &12 &6 &13 &5 &1 &3 &10 &7 &3 &10 &13.5\\ \hline 滁州市 &11 &7 &8 &1 &5 &13 &7 &8 &9 &13 &9 &16 &1 &12 &15 &15 &9\\ \hline 六安市 &13 &9 &12 &10 &12 &12 &4 &13 &8 &3 &2 &7 &12.5 &2 &10 &6 &15\\ \hline 马鞍山市 &2 &1 &9 &9 &1 &7 &15 &1 &11 &14 &3 &1 &5 &1 &4 &13 &11\\ \hline 芜湖市 &8 &2 &10 &11 &6 &6 &14 &3 &10 &15 &8 &5 &6 &3 &9 &8 &8\\ \hline 宣城市 &16 &6 &15 &15 &15 &14 &9 &2 &16 &11 &7 &11 &2 &10 &5 &5 &4\\ \hline 铜陵市 &4 &3 &11 &12 &8 &10 &10 &7 &15 &10 &4 &2 &8 &6 &1 &12 &1\\ \hline 池州市 &12 &4 &14 &14 &14 &15 &8 &12 &14 &8 &5 &4 &3 &14 &7 &11 &7\\ \hline 安庆市 &14 &8 &13 &13 &13 &11 &5 &11 &7 &7 &6 &6 &4 &4 &16 &3 &6\\ \hline 黄山市 &15 &12 &16 &16 &16 &16 &6 &5 &12 &9 &13 &14 &15 &5 &2 &16 &3\\ \hline \end{array} $$

权重求解过程——以熵权法为例

先把原始矩阵归一化

$$N=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times17}} &X1 &X2 &X3 &X4 &X5 &X6 &X7 &X8 &X9 &X10 &X11 &X12 &X13 &X14 &X15 &X16 &X17\\ \hline 合肥市 &0.358 &0.8089 &0.0076 &0.0612 &0.4854 &0.1566 &1 &0.4951 &0.0719 &1 &1 &0.9968 &1 &0.9691 &0.7351 &0.5439 &0.6328\\ \hline 淮北市 &0 &0.9928 &0.0006 &0.0493 &0.6905 &0 &0.6936 &0.6112 &0.2608 &0.1788 &0.9499 &0.9687 &0.4481 &1 &0.229 &0.7611 &1\\ \hline 毫州市 &0.1788 &0.9769 &0.0098 &0.0478 &0.7474 &0.2866 &0 &1 &0.1385 &0.0055 &0.78 &0.8785 &0.4645 &0.8183 &0.7288 &0 &0.7816\\ \hline 宿州市 &0.2427 &1 &0.0144 &0.056 &0.8045 &0.3545 &0.051 &0.7623 &0.3705 &0.0232 &0.8543 &0.8854 &0.3989 &0.5309 &0.6169 &0.118 &0.0819\\ \hline 蚌埠市 &0.1179 &0.7704 &0.0143 &0.0537 &0.5212 &0.3678 &0.4796 &0.6011 &0.3471 &0.3336 &0.715 &0.9682 &0.2842 &0.6067 &0.1447 &0.5109 &0.7916\\ \hline 阜阳市 &0.219 &0.9689 &0 &0.0485 &0.6394 &0.0089 &0.0704 &0.9922 &0 &0 &0.7533 &0.9321 &0.3552 &0.8726 &0.4057 &0.3078 &0.3722\\ \hline 淮南市 &0.0853 &0.5983 &0.0021 &0.0095 &0.0252 &0.2821 &0.669 &0.5226 &0.7644 &0.0518 &0 &0.3413 &0.3825 &0.5248 &0.0783 &0.6912 &0.7916\\ \hline 滁州市 &0.3766 &0.6654 &0.0283 &0 &0.5461 &0.7563 &0.3612 &0.5619 &0.5342 &0.454 &0.6695 &1 &0 &0.8681 &0.7895 &0.8111 &0.6005\\ \hline 六安市 &0.6965 &0.7152 &0.1194 &0.2211 &0.8532 &0.7489 &0.1428 &0.7464 &0.4712 &0.0062 &0.1394 &0.854 &0.4481 &0.0035 &0.2776 &0.4976 &0.9057\\ \hline 马鞍山市 &0.0787 &0 &0.0322 &0.1356 &0 &0.353 &0.7886 &0 &0.5881 &0.6888 &0.1903 &0 &0.235 &0 &0.0894 &0.7517 &0.6923\\ \hline 芜湖市 &0.242 &0.4038 &0.0591 &0.2862 &0.5547 &0.288 &0.7092 &0.4146 &0.5683 &0.7648 &0.6045 &0.4904 &0.2787 &0.0349 &0.2726 &0.542 &0.5558\\ \hline 宣城市 &1 &0.6619 &0.4689 &0.6825 &0.9615 &0.8951 &0.4137 &0.1664 &1 &0.3137 &0.5497 &0.9416 &0.0437 &0.6424 &0.1281 &0.4117 &0.3052\\ \hline 铜陵市 &0.0892 &0.4356 &0.0876 &0.348 &0.6824 &0.4077 &0.438 &0.5234 &0.9928 &0.3126 &0.2326 &0.06 &0.3224 &0.4443 &0 &0.7394 &0\\ \hline 池州市 &0.5789 &0.51 &0.4631 &0.5378 &0.9447 &0.9483 &0.3723 &0.6584 &0.9658 &0.2823 &0.2952 &0.3726 &0.1858 &0.911 &0.1907 &0.712 &0.4417\\ \hline 安庆市 &0.8148 &0.6775 &0.1729 &0.4013 &0.8865 &0.7001 &0.2275 &0.6475 &0.4371 &0.2141 &0.4589 &0.6651 &0.2022 &0.1258 &1 &0.2946 &0.4069\\ \hline 黄山市 &0.9515 &0.8729 &1 &1 &1 &1 &0.3011 &0.5042 &0.7068 &0.302 &0.8348 &0.9878 &0.847 &0.2876 &0.0666 &1 &0.1166\\ \hline \end{array} $$

对规范化矩阵运用对应的赋权法求得权重——本处以熵权法为例

$$w=\begin{array}{c|c|c|c|c|c|c}{M_{2 \times17}} &X1 &X2 &X3 &X4 &X5 &X6 &X7 &X8 &X9 &X10 &X11 &X12 &X13 &X14 &X15 &X16 &X17\\ \hline 权值 &0.0848 &0.0542 &0.2802 &0.0906 &0.1167 &0.0391 &0.0056 &0.0029 &0.0176 &0.0302 &0.0372 &0.1103 &0.0016 &0.0225 &0.1056 &0.0008 &0\\ \hline 权值排序 &6 &7 &1 &5 &2 &8 &13 &14 &12 &10 &9 &3 &15 &11 &4 &16 &17\\ \hline \end{array} $$

加权秩和比矩阵如下

$$WRSR=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times1}} &WRSR\\ \hline 合肥市 &0.5298\\ \hline 淮北市 &0.4115\\ \hline 毫州市 &0.4651\\ \hline 宿州市 &0.5676\\ \hline 蚌埠市 &0.4436\\ \hline 阜阳市 &0.3706\\ \hline 淮南市 &0.2043\\ \hline 滁州市 &0.5823\\ \hline 六安市 &0.6246\\ \hline 马鞍山市 &0.3338\\ \hline 芜湖市 &0.5075\\ \hline 宣城市 &0.7778\\ \hline 铜陵市 &0.4544\\ \hline 池州市 &0.6818\\ \hline 安庆市 &0.716\\ \hline 黄山市 &0.8293\\ \hline \end{array} $$

确定RSR(WRSR)的分布

　　1、将WRSR值按照从小到大的顺序排列

　　2、列出各组频数

　　3、计算各组累计频数

　　4、确定各组RSR的秩次R及平均秩次

　　5、计算向下累计频率最后一项用 $ \left( 1-1/4n \right) \times 100\% $ 修正

　　6、根据累计频率，根据“百分数与概率单位对照表”，求其所对应概率单位 Probit 值

$$统计=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times7}} &RSR &f &∑f &R & \bar {R} &R' \% &Probit\\ \hline 1 &0.2043 &1 &1 &1 &1 &0.0625 &3.4618\\ \hline 2 &0.3338 &1 &2 &2 &2 &0.125 &3.8497\\ \hline 3 &0.3706 &1 &3 &3 &3 &0.1875 &4.111\\ \hline 4 &0.4115 &1 &4 &4 &4 &0.25 &4.3255\\ \hline 5 &0.4436 &1 &5 &5 &5 &0.3125 &4.5098\\ \hline 6 &0.4544 &1 &6 &6 &6 &0.375 &4.6814\\ \hline 7 &0.4651 &1 &7 &7 &7 &0.4375 &4.8414\\ \hline 8 &0.5075 &1 &8 &8 &8 &0.5 &5\\ \hline 9 &0.5298 &1 &9 &9 &9 &0.5625 &5.156\\ \hline 10 &0.5676 &1 &10 &10 &10 &0.625 &5.3186\\ \hline 11 &0.5823 &1 &11 &11 &11 &0.6875 &5.4874\\ \hline 12 &0.6246 &1 &12 &12 &12 &0.75 &5.6745\\ \hline 13 &0.6818 &1 &13 &13 &13 &0.8125 &5.8853\\ \hline 14 &0.716 &1 &14 &14 &14 &0.875 &6.1503\\ \hline 15 &0.7778 &1 &15 &15 &15 &0.9375 &6.5301\\ \hline 16 &0.8293 &1 &16 &16 &16 &0.9844 &7.1444\\ \hline \end{array} $$

基于最小二乘法的数据拟合求出回归系数

　　$ RSR= a + b * Probit $ 对应 $ Y= a + b * X $ 即为一条直线。

　　最小二乘法的原理是，所有点距离该直线的垂直距离之和最小。b 与a 分别为其回归系数。

　　$b= \frac { \sum \limits_{i=1}^{n}(x_iy_i- \bar {x} \bar {y}) } {\sum \limits_{i=1}^{n} (x_i^2- \bar{x}^2 )}$

　　$a= \bar {y} -b \bar {x} $

$$回归的数据DATA=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times2}} &RSR(Y) &Probit(X)\\ \hline 1 &0.2043 &3.4618\\ \hline 2 &0.3338 &3.8497\\ \hline 3 &0.3706 &4.111\\ \hline 4 &0.4115 &4.3255\\ \hline 5 &0.4436 &4.5098\\ \hline 6 &0.4544 &4.6814\\ \hline 7 &0.4651 &4.8414\\ \hline 8 &0.5075 &5\\ \hline 9 &0.5298 &5.156\\ \hline 10 &0.5676 &5.3186\\ \hline 11 &0.5823 &5.4874\\ \hline 12 &0.6246 &5.6745\\ \hline 13 &0.6818 &5.8853\\ \hline 14 &0.716 &6.1503\\ \hline 15 &0.7778 &6.5301\\ \hline 16 &0.8293 &7.1444\\ \hline \end{array} $$$$\begin{array}{c|c|c|c|c|c|c}{M_{1 \times2}} &a &b\\ \hline 回归值 &-0.3285 &0.1675\\ \hline \end{array} $$

绘制回归图表

分档经验法则

　　常见的分成三挡，其根据主要是概率单位的统计学中的正态分布原则。$ Probit <4 $ 为差（下，劣）； $ Probit >6 $　为优；$ 6> Probit >4$为中。

$$RSR=\begin{array}{c|c|c|c|c|c|c}{M_{16 \times1}} &RSR\\ \hline 合肥市 &0.5298\\ \hline 淮北市 &0.4115\\ \hline 毫州市 &0.4651\\ \hline 宿州市 &0.5676\\ \hline 蚌埠市 &0.4436\\ \hline 阜阳市 &0.3706\\ \hline 淮南市 &0.2043\\ \hline 滁州市 &0.5823\\ \hline 六安市 &0.6246\\ \hline 马鞍山市 &0.3338\\ \hline 芜湖市 &0.5075\\ \hline 宣城市 &0.7778\\ \hline 铜陵市 &0.4544\\ \hline 池州市 &0.6818\\ \hline 安庆市 &0.716\\ \hline 黄山市 &0.8293\\ \hline \end{array} $$

　　把RSR经过计算转化成 Probit 。并加入基准分档线得到如下矩阵

$$Probit=\begin{array}{c|c|c|c|c|c|c}{M_{18 \times1}} &Probit\\ \hline 合肥市 &5.1242\\ \hline 淮北市 &4.418\\ \hline 毫州市 &4.7382\\ \hline 宿州市 &5.35\\ \hline 蚌埠市 &4.6095\\ \hline 阜阳市 &4.1737\\ \hline 淮南市 &3.181\\ \hline 滁州市 &5.4378\\ \hline 六安市 &5.69\\ \hline 马鞍山市 &3.954\\ \hline 芜湖市 &4.991\\ \hline 宣城市 &6.6052\\ \hline 铜陵市 &4.6744\\ \hline 池州市 &6.0319\\ \hline 安庆市 &6.2361\\ \hline 黄山市 &6.9121\\ \hline Poor-A &4\\ \hline A-Good &6\\ \hline \end{array} $$