1998年，前南斯拉夫的教授奥普里科维奇(S. Opricovic) 在其论文《Multicriteria optimization of civil engineering systems》摆事实讲道理的喷了一把TOPSIS方法。这篇文章指出按照TOPSIS方法整出来的最优法未必是最优解。并提出了多属性妥协解排序法( VlseKriterijumska Optimizacija I Kompromisno Resenje, VIKOR)
TOPSIS方法是多目标决策分析中一种常用的有效方法, 是黄庆来为解决单个决策者的多属性决策问题所提出来的一种接近于线性加权平均方法的排序法, 其基本思想是: 最终方案尽可能接近正理想解又尽可能 远离负理想解. TOPSIS方法比较合理且易于理解, 计算也相对简单并能够用简单的算术形式来描述备选方案的极小值, 而且在比较过程中也可以使用客观权重.
VIKOR方法就是一种魔改的方法，然后在这个套路上有了一堆的名词，啥前景理论等等，各种名词超多，找几篇文章看一下概念就行。
事实上，对TOPSIS魔改最多的地方就是求距离的地方。而且魔改起来非常容易。
相似度的魔改方向:
相似度的算法太多了，相关的理论也多。这些不但是topsis的基础，也是很多神经网络，机器学习等的基本问题。能用到topsis上当然也可以用到神经网络上。
比较是科学研究的一种基本方法。在统计学术语中，概率函数之间的比较用于推断样本之间的联系、相关性或关系。现在推荐一个有46种用于比较概率函数的优化距离和相似性度量的工具包。
它是基于R语言实现的，一个基础性的求相似度的框架。非常好用，可用于比较单变量或多变量概率函数的聚类、分类、统计推断、拟合优度、非参数统计、信息理论和机器学习任务等等。是一个非常好的框架。
权重的魔改方向
权重有主观赋权法跟客观赋权法。然后就是两者结合的方法，作为偷懒，懒得被人喷说你这个是拍脑袋的，所以尽量是基于客观的赋权法。
客观赋权法有如下几种：
★熵权法
★因子分析权数法(FAM)
★主成分分析权数法(PCA)
★独立性权系数法
上面几种当中，后面三种相对简单，拿出去不好吹，一般都是用熵权法魔改一下。
主观赋权法主要有两种：
★AHP法
★ANP法
★灰度相关
AHP用烂了，所以这个相对少点人用了，而且ANP已经涵盖了AHP法，所以即便是用的AHP目前一般都说成了ANP。
在概率论或信息论中，KL散度( Kullback–Leibler divergence)，又称相对熵（relative entropy)，是描述两个概率分布P和Q差异的一种方法。它是非对称的，这意味着$D(P||Q) ≠ D(Q||P)$。特别的，在信息论中，$D(P||Q)$表示当用概率分布Q来拟合真实分布P时，产生的信息损耗，其中P表示真实分布，Q表示P的拟合分布。有人将KL散度称为KL距离，但事实上，KL散度并不满足距离的概念因为:
1）KL散度不是对称的；
2）KL散度不满足三角不等式。
上面的概念理解不理解先不管，记住下面的散度距离公式：
$$ d = \sum_{i = 1}^N P_i \cdot log(\frac{P_i}{Q_i})$$
如果相似度用到了KL散度也就是包含有熵的概念，求权重如果不有那么点熵权的东西就说不过去了。
原始数据如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline X1 &7 &5.9 &1.9 &636 &0.22 &0.032 &0.762 &0.248\\ \hline X2 &2.4 &2.6 &0.9 &567 &0.04 &0.059 &1.14 &0.185\\ \hline X3 &2.8 &2.2 &0.4 &584 &0.04 &0.037 &1.06 &0.033\\ \hline X4 &2.5 &2.3 &0.5 &715 &0.072 &0.039 &1.31 &0.023\\ \hline X5 &4 &2.6 &0.1 &719 &0.04 &0.086 &1.19 &0.09\\ \hline X6 &3.6 &2.7 &0.8 &695 &0.24 &0.033 &0.962 &0.521\\ \hline X7 &7.5 &5.4 &1.3 &718 &0.104 &0.033 &0.829 &0.014\\ \hline X8 &135 &48.8 &1.7 &1030 &0.52 &0.06 &2.56 &3.51\\ \hline X9 &152 &42.4 &0.9 &1160 &0.22 &0.316 &2.48 &0.032\\ \hline X10 &83.2 &53.8 &17.5 &890 &1.6 &0.034 &2.15 &0.142\\ \hline X11 &79 &252 &4.3 &1170 &0.36 &0.055 &0.422 &0.165\\ \hline X12 &77.2 &318 &1.4 &1310 &0.24 &0.075 &0.637 &0.014\\ \hline X13 &37.2 &7.8 &0.6 &599 &0.088 &0.041 &0.47 &0.025\\ \hline X14 &4.1 &3.7 &2.5 &397 &0.26 &0.241 &0.175 &0.015\\ \hline X15 &5.5 &4.7 &1.6 &425 &0.088 &0.258 &0.181 &0.013\\ \hline X16 &9.9 &26.2 &106 &518 &0.104 &0.35 &0.005 &0.205\\ \hline B1 &50 &50 &2 &300 &0.1 &0.05 &0.05 &0.01\\ \hline B2 &150 &150 &5 &500 &0.2 &0.5 &0.05 &0.1\\ \hline B3 &250 &250 &20 &1000 &0.3 &1 &0.1 &1\\ \hline B4 &350 &350 &30 &2000 &2 &5 &1.5 &4.8\\ \hline \end{array} $$数据说明：上面为检测的地下水水质的一组样品。其中B1到B4为国家标准的虚拟样品
规一化的公式很多，但是采用KL散度的方式求解，最忌讳用极差法，因为极差法的极值就是0跟1，导致后来算的时候出现分母为零，不好算。：
$$极差法的一般公式为： X_{new}=\left| \frac{{X-X_{min}}}{{X_{max}-X_{min}}} \right|$$
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline X1 &0.0132 &0.0106 &0.017 &0.1976 &0.0918 &0 &0.2963 &0.0497\\ \hline X2 &0 &0.0012 &0.0076 &0.1571 &0 &0.0054 &0.4442 &0.0365\\ \hline X3 &0.0012 &0 &0.0028 &0.1671 &0 &0.001 &0.4129 &0.0048\\ \hline X4 &0.0003 &0.0003 &0.0038 &0.2441 &0.0163 &0.0014 &0.5108 &0.0027\\ \hline X5 &0.0046 &0.0012 &0 &0.2465 &0 &0.0109 &0.4638 &0.0167\\ \hline X6 &0.0035 &0.0014 &0.0066 &0.2324 &0.102 &0.0002 &0.3746 &0.1067\\ \hline X7 &0.0147 &0.0092 &0.0113 &0.2459 &0.0327 &0.0002 &0.3225 &0.0008\\ \hline X8 &0.3815 &0.134 &0.0151 &0.4294 &0.2449 &0.0056 &1 &0.7307\\ \hline X9 &0.4304 &0.1156 &0.0076 &0.5059 &0.0918 &0.0572 &0.9687 &0.0046\\ \hline X10 &0.2325 &0.1484 &0.1643 &0.3471 &0.7959 &0.0004 &0.8395 &0.0276\\ \hline X11 &0.2204 &0.7182 &0.0397 &0.5118 &0.1633 &0.0046 &0.1632 &0.0324\\ \hline X12 &0.2152 &0.908 &0.0123 &0.5941 &0.102 &0.0087 &0.2474 &0.0008\\ \hline X13 &0.1001 &0.0161 &0.0047 &0.1759 &0.0245 &0.0018 &0.182 &0.0031\\ \hline X14 &0.0049 &0.0043 &0.0227 &0.0571 &0.1122 &0.0421 &0.0665 &0.001\\ \hline X15 &0.0089 &0.0072 &0.0142 &0.0735 &0.0245 &0.0455 &0.0689 &0.0006\\ \hline X16 &0.0216 &0.069 &1 &0.1282 &0.0327 &0.064 &0 &0.0407\\ \hline B1 &0.1369 &0.1374 &0.0179 &0 &0.0306 &0.0036 &0.0176 &0\\ \hline B2 &0.4246 &0.425 &0.0463 &0.1176 &0.0816 &0.0942 &0.0176 &0.0188\\ \hline B3 &0.7123 &0.7125 &0.1879 &0.4118 &0.1327 &0.1948 &0.0372 &0.2067\\ \hline B4 &1 &1 &0.2823 &1 &1 &1 &0.5851 &1\\ \hline \end{array} $$$$欧式距离的公式为： y_{ij} = \frac {x_{ij}} { \sqrt{ {\sum \limits_{i=1}^{n}{x_{ij}^2}} } }= \frac {x_{ij}} { ( {\sum \limits_{i=1}^{n}{x_{ij}^2}} )^{\frac 1 2} }$$
结果：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline X1 &0.0134185565 &0.009538742 &0.0167209603 &0.1606002488 &0.08072808 &0.0062008568 &0.1434715527 &0.0408599787\\ \hline X2 &0.0046006479 &0.0042035134 &0.0079204549 &0.1431766369 &0.0146778327 &0.0114328297 &0.2146424805 &0.030480226\\ \hline X3 &0.0053674226 &0.003556819 &0.0035202022 &0.1474694108 &0.0146778327 &0.0071697406 &0.1995798503 &0.0054370133\\ \hline X4 &0.0047923416 &0.0037184926 &0.0044002527 &0.1805490218 &0.0264200989 &0.0075572942 &0.2466505697 &0.0037894335\\ \hline X5 &0.0076677466 &0.0042035134 &0.0008800505 &0.1815590863 &0.0146778327 &0.0166648025 &0.2240566244 &0.0148282181\\ \hline X6 &0.0069009719 &0.004365187 &0.0070404043 &0.1754986995 &0.0880669963 &0.0063946335 &0.1811281283 &0.0858389069\\ \hline X7 &0.0143770248 &0.008730374 &0.011440657 &0.1813065701 &0.0381623651 &0.0063946335 &0.1560865055 &0.0023066117\\ \hline X8 &0.258786447 &0.0788967134 &0.0149608592 &0.2600915978 &0.1908118254 &0.0116266064 &0.4820041667 &0.5783005049\\ \hline X9 &0.29137437 &0.0685496034 &0.0079204549 &0.2929186927 &0.08072808 &0.0612334605 &0.4669415365 &0.0052722553\\ \hline X10 &0.1594891288 &0.086980393 &0.1540088446 &0.2247393418 &0.5871133088 &0.0065884103 &0.4048081869 &0.023395633\\ \hline X11 &0.1514379949 &0.4074174542 &0.0378421732 &0.2954438539 &0.1321004945 &0.0106577226 &0.0794553744 &0.0271850665\\ \hline X12 &0.147987509 &0.5141220256 &0.0123207076 &0.3307961099 &0.0880669963 &0.014533258 &0.119936193 &0.0023066117\\ \hline X13 &0.0713100432 &0.0126105402 &0.0052803032 &0.1512571525 &0.032291232 &0.0079448477 &0.0884929525 &0.0041189495\\ \hline X14 &0.0078594402 &0.0059819229 &0.0220012635 &0.1002488974 &0.0954059127 &0.0467002025 &0.0329495036 &0.0024713697\\ \hline X15 &0.0105431515 &0.0075986589 &0.0140808086 &0.1073193486 &0.032291232 &0.0499944076 &0.0340792008 &0.0021418537\\ \hline X16 &0.0189776728 &0.0423584814 &0.9328535727 &0.1308033473 &0.0381623651 &0.0678218708 &0.0009414144 &0.0337753856\\ \hline B1 &0.0958468322 &0.0808367965 &0.0176010108 &0.0757548343 &0.0366945818 &0.0096888387 &0.0094141439 &0.0016475798\\ \hline B2 &0.2875404967 &0.2425103894 &0.044002527 &0.1262580572 &0.0733891636 &0.0968883869 &0.0094141439 &0.0164757979\\ \hline B3 &0.4792341612 &0.4041839824 &0.1760101081 &0.2525161144 &0.1100837454 &0.1937767738 &0.0188282878 &0.1647579786\\ \hline B4 &0.6709278256 &0.5658575753 &0.2640151621 &0.5050322288 &0.733891636 &0.9688838689 &0.2824243164 &0.7908382973\\ \hline \end{array} $$
公式：
$1、 \rho _{ij}=\frac {x_{ij}} {\sum \limits_{i=1}^{n}{x_{ij}}}$
$2、 e_{j}=-k {\sum \limits_{i=1}^{n}{\rho _{ij}\times ln({\rho _{ij}}) } },(其中有 \quad k= \frac{1}{ln(n)}, 且 \quad 当 \rho _{ij}=0 时,令 \quad ln({\rho _{ij}})=-1000 )$
$3、 \quad d_{j}=1-e_{j} $
$4、 \quad \omega_{j}=\frac {d_{j}} {\sum \limits_{j=1}^{m}{d_{j}}} $
结果：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{2 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline 权重 &0.1075 &0.1276 &0.1934 &0.0151 &0.0986 &0.2029 &0.0557 &0.1992\\ \hline 权重顺序 &5 &4 &3 &8 &6 &1 &7 &2\\ \hline \end{array} $$
公式：
$$ \mathbf{S^+}= ( z^+_1 ,z^+_2 ,z^+_3,…,z^+_m )\quad \quad 有： z^+_j =\omega_{j} \cdot Max( n_{ij}) \quad 且 1≤i≤n ; \quad j=1,2,… ,m $$
$$ \mathbf{S^-}= ( z^-_1 ,z^-_2 ,z^-_3,…,z^-_m )\quad \quad 有： z^-_j =\omega_{j} \cdot Min( n_{ij}) \quad 且 1≤i≤n ; \quad j=1,2,… ,m $$
结果：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline \mathbf{S^+} &0.07213462 &0.07221371 &0.18043193 &0.00760945 &0.07233498 &0.19659094 &0.0268402 &0.15755697\\ \hline \end{array} $$$$ \begin{array}{c|c|c|c|c|c|c}{M_{1 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline \mathbf{S^-} &0.00049464 &0.00045391 &0.00017022 &0.00114142 &0.0014467 &0.00125818 &5.242E-5 &0.00032824\\ \hline \end{array} $$
公式说明由于分析的内容具有物理属性性质，非文本之类分析，故熵非信息熵，采用的属性熵，取对数为$ln$：
$$ \mathbf{K^+}= ( k^+_{ij})_{n \times m} \quad \quad 有： k^+_{ij} = z_j^+ ln{\frac {z_j^+} {\omega_{j} \cdot n_{ij}} } + (1-z^+_j)ln{\frac {1-z_j^+} {1- \omega_{j} \cdot n_{ij}} } \quad 其中 \quad n_{ij} \in N \quad z^+_j \in \mathbf{S^+} $$
$$ \mathbf{K^-}= ( k^-_{ij})_{n \times m} \quad \quad 有： k^-_{ij} = ln{\frac {z_j^-} {\omega_{j} \cdot n_{ij}} } + (1-z^-_j)ln{\frac {1-z_j^-} {1-\omega_{j} \cdot n_{ij}} } \quad 其中 \quad n_{ij} \in N \quad z^-_j \in \mathbf{S^-} $$
特殊情况处理：比如极差法进行归一化处理，会导致分母为零，或者出现$ln0$的情况，这需要近似处理，如把1变成0.99999999； 0变成0.0000001等计算机好处理的方式
对抗矩阵$ \mathbf{K^+} $如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline X1 &0.2141 &0.2264 &0.5652 &0.0035 &0.0974 &0.8182 &0.0139 &0.3293\\ \hline X2 &0.2904 &0.285 &0.6986 &0.0042 &0.2147 &0.6988 &0.0069 &0.3737\\ \hline X3 &0.2794 &0.297 &0.8442 &0.004 &0.2147 &0.7898 &0.0081 &0.6411\\ \hline X4 &0.2875 &0.2938 &0.8041 &0.003 &0.1732 &0.7796 &0.005 &0.6977\\ \hline X5 &0.2539 &0.285 &1.094 &0.0029 &0.2147 &0.6256 &0.0063 &0.4846\\ \hline X6 &0.2614 &0.2823 &0.7197 &0.0031 &0.0918 &0.8122 &0.0097 &0.22\\ \hline X7 &0.2092 &0.2327 &0.6328 &0.0029 &0.1477 &0.8122 &0.0123 &0.7757\\ \hline X8 &0.0254 &0.0821 &0.585 &0.0014 &0.0454 &0.6955 &0 &0.008\\ \hline X9 &0.0202 &0.091 &0.6986 &0.001 &0.0974 &0.3771 &0 &0.6459\\ \hline X10 &0.0502 &0.076 &0.1867 &0.0019 &0.0018 &0.8064 &0.0004 &0.4142\\ \hline X11 &0.0531 &0.0037 &0.4212 &0.0009 &0.0665 &0.7125 &0.0262 &0.3912\\ \hline X12 &0.0544 &0.0003 &0.6196 &0.0006 &0.0918 &0.6521 &0.0174 &0.7757\\ \hline X13 &0.0994 &0.2066 &0.7714 &0.0039 &0.1592 &0.7698 &0.0238 &0.6846\\ \hline X14 &0.2521 &0.2597 &0.5165 &0.0062 &0.0867 &0.4279 &0.0473 &0.7648\\ \hline X15 &0.2312 &0.2426 &0.5958 &0.0058 &0.1592 &0.4151 &0.0465 &0.7873\\ \hline X16 &0.1896 &0.1227 &0 &0.0047 &0.1477 &0.3581 &0.141 &0.3581\\ \hline B1 &0.0805 &0.0806 &0.5561 &0.008 &0.1504 &0.7311 &0.0797 &0.8286\\ \hline B2 &0.0208 &0.0208 &0.395 &0.0049 &0.1036 &0.2928 &0.0797 &0.4683\\ \hline B3 &0.0039 &0.0039 &0.1662 &0.0015 &0.0777 &0.1728 &0.0616 &0.1308\\ \hline B4 &0 &0 &0.1076 &0 &0 &0 &0.0033 &0\\ \hline \end{array} $$对抗矩阵$ \mathbf{K^-} $如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &Cl^- &SO_4^{2-} &NO_3^- &TDS &Fe^{3+} &Zn &Mn &NO_2^-\\ \hline X1 &0.0004 &0.0003 &0.0026 &0.0004 &0.0041 &0 &0.0077 &0.0068\\ \hline X2 &0 &0 &0.001 &0.0003 &0 &0.0003 &0.0117 &0.0048\\ \hline X3 &0 &0 &0.0003 &0.0003 &0 &0 &0.0108 &0.0004\\ \hline X4 &0 &0 &0.0004 &0.0006 &0.0003 &0 &0.0135 &0.0002\\ \hline X5 &0.0001 &0 &0 &0.0006 &0 &0.0009 &0.0122 &0.0019\\ \hline X6 &0 &0 &0.0008 &0.0005 &0.0047 &0 &0.0098 &0.0156\\ \hline X7 &0.0005 &0.0003 &0.0016 &0.0006 &0.0009 &0 &0.0084 &0\\ \hline X8 &0.0257 &0.0083 &0.0022 &0.0014 &0.0138 &0.0003 &0.0268 &0.1201\\ \hline X9 &0.0293 &0.007 &0.001 &0.0017 &0.0041 &0.0083 &0.026 &0.0003\\ \hline X10 &0.015 &0.0093 &0.0292 &0.001 &0.0527 &0 &0.0224 &0.0035\\ \hline X11 &0.0142 &0.0508 &0.0065 &0.0018 &0.0085 &0.0002 &0.0041 &0.0042\\ \hline X12 &0.0138 &0.0651 &0.0018 &0.0022 &0.0047 &0.0006 &0.0064 &0\\ \hline X13 &0.0058 &0.0006 &0.0005 &0.0003 &0.0006 &0 &0.0046 &0.0002\\ \hline X14 &0.0001 &0.0001 &0.0035 &0 &0.0053 &0.0057 &0.0016 &0\\ \hline X15 &0.0002 &0.0002 &0.0021 &0.0001 &0.0006 &0.0063 &0.0017 &0\\ \hline X16 &0.0008 &0.0038 &0.1976 &0.0002 &0.0009 &0.0096 &0 &0.0054\\ \hline B1 &0.0084 &0.0085 &0.0027 &0 &0.0008 &0.0001 &0.0004 &0\\ \hline B2 &0.0288 &0.0291 &0.0077 &0.0002 &0.0035 &0.0151 &0.0004 &0.0022\\ \hline B3 &0.0501 &0.0503 &0.0336 &0.0013 &0.0065 &0.0345 &0.0008 &0.0315\\ \hline B4 &0.0719 &0.0722 &0.0513 &0.0043 &0.0679 &0.211 &0.0155 &0.169\\ \hline \end{array} $$
青龙、东面、雅蠛蝶$K_{d}^+ $ 如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &Zn &NO_2^- &NO_3^- &SO_4^{2-} &Cl^- &Fe^{3+} &Mn &TDS\\ \hline X1 &0.8182 &0.3293 &0.5652 &0.2264 &0.2141 &0.0974 &0.0139 &0.0035\\ \hline X2 &0.6988 &0.3737 &0.6986 &0.285 &0.2904 &0.2147 &0.0069 &0.0042\\ \hline X3 &0.7898 &0.6411 &0.8442 &0.297 &0.2794 &0.2147 &0.0081 &0.004\\ \hline X4 &0.7796 &0.6977 &0.8041 &0.2938 &0.2875 &0.1732 &0.005 &0.003\\ \hline X5 &0.6256 &0.4846 &1.094 &0.285 &0.2539 &0.2147 &0.0063 &0.0029\\ \hline X6 &0.8122 &0.22 &0.7197 &0.2823 &0.2614 &0.0918 &0.0097 &0.0031\\ \hline X7 &0.8122 &0.7757 &0.6328 &0.2327 &0.2092 &0.1477 &0.0123 &0.0029\\ \hline X8 &0.6955 &0.008 &0.585 &0.0821 &0.0254 &0.0454 &0 &0.0014\\ \hline X9 &0.3771 &0.6459 &0.6986 &0.091 &0.0202 &0.0974 &0 &0.001\\ \hline X10 &0.8064 &0.4142 &0.1867 &0.076 &0.0502 &0.0018 &0.0004 &0.0019\\ \hline X11 &0.7125 &0.3912 &0.4212 &0.0037 &0.0531 &0.0665 &0.0262 &0.0009\\ \hline X12 &0.6521 &0.7757 &0.6196 &0.0003 &0.0544 &0.0918 &0.0174 &0.0006\\ \hline X13 &0.7698 &0.6846 &0.7714 &0.2066 &0.0994 &0.1592 &0.0238 &0.0039\\ \hline X14 &0.4279 &0.7648 &0.5165 &0.2597 &0.2521 &0.0867 &0.0473 &0.0062\\ \hline X15 &0.4151 &0.7873 &0.5958 &0.2426 &0.2312 &0.1592 &0.0465 &0.0058\\ \hline X16 &0.3581 &0.3581 &0 &0.1227 &0.1896 &0.1477 &0.141 &0.0047\\ \hline B1 &0.7311 &0.8286 &0.5561 &0.0806 &0.0805 &0.1504 &0.0797 &0.008\\ \hline B2 &0.2928 &0.4683 &0.395 &0.0208 &0.0208 &0.1036 &0.0797 &0.0049\\ \hline B3 &0.1728 &0.1308 &0.1662 &0.0039 &0.0039 &0.0777 &0.0616 &0.0015\\ \hline B4 &0 &0 &0.1076 &0 &0 &0 &0.0033 &0\\ \hline \end{array} $$白虎、西面、法克鱿 $ K_{d}^- $ 如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &Zn &NO_2^- &NO_3^- &SO_4^{2-} &Cl^- &Fe^{3+} &Mn &TDS\\ \hline X1 &0 &0.0068 &0.0026 &0.0003 &0.0004 &0.0041 &0.0077 &0.0004\\ \hline X2 &0.0003 &0.0048 &0.001 &0 &0 &0 &0.0117 &0.0003\\ \hline X3 &0 &0.0004 &0.0003 &0 &0 &0 &0.0108 &0.0003\\ \hline X4 &0 &0.0002 &0.0004 &0 &0 &0.0003 &0.0135 &0.0006\\ \hline X5 &0.0009 &0.0019 &0 &0 &0.0001 &0 &0.0122 &0.0006\\ \hline X6 &0 &0.0156 &0.0008 &0 &0 &0.0047 &0.0098 &0.0005\\ \hline X7 &0 &0 &0.0016 &0.0003 &0.0005 &0.0009 &0.0084 &0.0006\\ \hline X8 &0.0003 &0.1201 &0.0022 &0.0083 &0.0257 &0.0138 &0.0268 &0.0014\\ \hline X9 &0.0083 &0.0003 &0.001 &0.007 &0.0293 &0.0041 &0.026 &0.0017\\ \hline X10 &0 &0.0035 &0.0292 &0.0093 &0.015 &0.0527 &0.0224 &0.001\\ \hline X11 &0.0002 &0.0042 &0.0065 &0.0508 &0.0142 &0.0085 &0.0041 &0.0018\\ \hline X12 &0.0006 &0 &0.0018 &0.0651 &0.0138 &0.0047 &0.0064 &0.0022\\ \hline X13 &0 &0.0002 &0.0005 &0.0006 &0.0058 &0.0006 &0.0046 &0.0003\\ \hline X14 &0.0057 &0 &0.0035 &0.0001 &0.0001 &0.0053 &0.0016 &0\\ \hline X15 &0.0063 &0 &0.0021 &0.0002 &0.0002 &0.0006 &0.0017 &0.0001\\ \hline X16 &0.0096 &0.0054 &0.1976 &0.0038 &0.0008 &0.0009 &0 &0.0002\\ \hline B1 &0.0001 &0 &0.0027 &0.0085 &0.0084 &0.0008 &0.0004 &0\\ \hline B2 &0.0151 &0.0022 &0.0077 &0.0291 &0.0288 &0.0035 &0.0004 &0.0002\\ \hline B3 &0.0345 &0.0315 &0.0336 &0.0503 &0.0501 &0.0065 &0.0008 &0.0013\\ \hline B4 &0.211 &0.169 &0.0513 &0.0722 &0.0719 &0.0679 &0.0155 &0.0043\\ \hline \end{array} $$按照权重由小到大顺序得到$K_{a}^+ , K_{a}^-$ 下标a 表示的是权重增加的方向的意思，即ASC的意思
朱雀、南面、草泥马$K_{a}^+ $ 如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &TDS &Mn &Fe^{3+} &Cl^- &SO_4^{2-} &NO_3^- &NO_2^- &Zn\\ \hline X1 &0.0035 &0.0139 &0.0974 &0.2141 &0.2264 &0.5652 &0.3293 &0.8182\\ \hline X2 &0.0042 &0.0069 &0.2147 &0.2904 &0.285 &0.6986 &0.3737 &0.6988\\ \hline X3 &0.004 &0.0081 &0.2147 &0.2794 &0.297 &0.8442 &0.6411 &0.7898\\ \hline X4 &0.003 &0.005 &0.1732 &0.2875 &0.2938 &0.8041 &0.6977 &0.7796\\ \hline X5 &0.0029 &0.0063 &0.2147 &0.2539 &0.285 &1.094 &0.4846 &0.6256\\ \hline X6 &0.0031 &0.0097 &0.0918 &0.2614 &0.2823 &0.7197 &0.22 &0.8122\\ \hline X7 &0.0029 &0.0123 &0.1477 &0.2092 &0.2327 &0.6328 &0.7757 &0.8122\\ \hline X8 &0.0014 &0 &0.0454 &0.0254 &0.0821 &0.585 &0.008 &0.6955\\ \hline X9 &0.001 &0 &0.0974 &0.0202 &0.091 &0.6986 &0.6459 &0.3771\\ \hline X10 &0.0019 &0.0004 &0.0018 &0.0502 &0.076 &0.1867 &0.4142 &0.8064\\ \hline X11 &0.0009 &0.0262 &0.0665 &0.0531 &0.0037 &0.4212 &0.3912 &0.7125\\ \hline X12 &0.0006 &0.0174 &0.0918 &0.0544 &0.0003 &0.6196 &0.7757 &0.6521\\ \hline X13 &0.0039 &0.0238 &0.1592 &0.0994 &0.2066 &0.7714 &0.6846 &0.7698\\ \hline X14 &0.0062 &0.0473 &0.0867 &0.2521 &0.2597 &0.5165 &0.7648 &0.4279\\ \hline X15 &0.0058 &0.0465 &0.1592 &0.2312 &0.2426 &0.5958 &0.7873 &0.4151\\ \hline X16 &0.0047 &0.141 &0.1477 &0.1896 &0.1227 &0 &0.3581 &0.3581\\ \hline B1 &0.008 &0.0797 &0.1504 &0.0805 &0.0806 &0.5561 &0.8286 &0.7311\\ \hline B2 &0.0049 &0.0797 &0.1036 &0.0208 &0.0208 &0.395 &0.4683 &0.2928\\ \hline B3 &0.0015 &0.0616 &0.0777 &0.0039 &0.0039 &0.1662 &0.1308 &0.1728\\ \hline B4 &0 &0.0033 &0 &0 &0 &0.1076 &0 &0\\ \hline \end{array} $$玄武、北面、菊花熊 $ K_{a}^- $ 如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &TDS &Mn &Fe^{3+} &Cl^- &SO_4^{2-} &NO_3^- &NO_2^- &Zn\\ \hline X1 &0.0004 &0.0077 &0.0041 &0.0004 &0.0003 &0.0026 &0.0068 &0\\ \hline X2 &0.0003 &0.0117 &0 &0 &0 &0.001 &0.0048 &0.0003\\ \hline X3 &0.0003 &0.0108 &0 &0 &0 &0.0003 &0.0004 &0\\ \hline X4 &0.0006 &0.0135 &0.0003 &0 &0 &0.0004 &0.0002 &0\\ \hline X5 &0.0006 &0.0122 &0 &0.0001 &0 &0 &0.0019 &0.0009\\ \hline X6 &0.0005 &0.0098 &0.0047 &0 &0 &0.0008 &0.0156 &0\\ \hline X7 &0.0006 &0.0084 &0.0009 &0.0005 &0.0003 &0.0016 &0 &0\\ \hline X8 &0.0014 &0.0268 &0.0138 &0.0257 &0.0083 &0.0022 &0.1201 &0.0003\\ \hline X9 &0.0017 &0.026 &0.0041 &0.0293 &0.007 &0.001 &0.0003 &0.0083\\ \hline X10 &0.001 &0.0224 &0.0527 &0.015 &0.0093 &0.0292 &0.0035 &0\\ \hline X11 &0.0018 &0.0041 &0.0085 &0.0142 &0.0508 &0.0065 &0.0042 &0.0002\\ \hline X12 &0.0022 &0.0064 &0.0047 &0.0138 &0.0651 &0.0018 &0 &0.0006\\ \hline X13 &0.0003 &0.0046 &0.0006 &0.0058 &0.0006 &0.0005 &0.0002 &0\\ \hline X14 &0 &0.0016 &0.0053 &0.0001 &0.0001 &0.0035 &0 &0.0057\\ \hline X15 &0.0001 &0.0017 &0.0006 &0.0002 &0.0002 &0.0021 &0 &0.0063\\ \hline X16 &0.0002 &0 &0.0009 &0.0008 &0.0038 &0.1976 &0.0054 &0.0096\\ \hline B1 &0 &0.0004 &0.0008 &0.0084 &0.0085 &0.0027 &0 &0.0001\\ \hline B2 &0.0002 &0.0004 &0.0035 &0.0288 &0.0291 &0.0077 &0.0022 &0.0151\\ \hline B3 &0.0013 &0.0008 &0.0065 &0.0501 &0.0503 &0.0336 &0.0315 &0.0345\\ \hline B4 &0.0043 &0.0155 &0.0679 &0.0719 &0.0722 &0.0513 &0.169 &0.211\\ \hline \end{array} $$
$K_{d}^+ , K_{d}^-$ 分别累加得到一组对抗矩阵$P_{d} , Q_{d}$
其中$K_{d}^+ \longrightarrow P_{d} $ 可以看为青龙东方位的雅蠛蝶累加
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &雅蠛蝶1 &雅蠛蝶2 &雅蠛蝶3 &雅蠛蝶4 &雅蠛蝶5 &雅蠛蝶6 &雅蠛蝶7 &雅蠛蝶8\\ \hline X1 &0.818 &1.148 &1.713 &1.939 &2.153 &2.251 &2.264 &2.268\\ \hline X2 &0.699 &1.073 &1.771 &2.056 &2.347 &2.561 &2.568 &2.572\\ \hline X3 &0.79 &1.431 &2.275 &2.572 &2.851 &3.066 &3.074 &3.078\\ \hline X4 &0.78 &1.477 &2.281 &2.575 &2.863 &3.036 &3.041 &3.044\\ \hline X5 &0.626 &1.11 &2.204 &2.489 &2.743 &2.958 &2.964 &2.967\\ \hline X6 &0.812 &1.032 &1.752 &2.034 &2.296 &2.387 &2.397 &2.4\\ \hline X7 &0.812 &1.588 &2.221 &2.453 &2.663 &2.81 &2.823 &2.826\\ \hline X8 &0.696 &0.704 &1.289 &1.371 &1.396 &1.441 &1.441 &1.443\\ \hline X9 &0.377 &1.023 &1.722 &1.813 &1.833 &1.93 &1.93 &1.931\\ \hline X10 &0.806 &1.221 &1.407 &1.483 &1.534 &1.535 &1.536 &1.538\\ \hline X11 &0.712 &1.104 &1.525 &1.529 &1.582 &1.648 &1.674 &1.675\\ \hline X12 &0.652 &1.428 &2.047 &2.048 &2.102 &2.194 &2.211 &2.212\\ \hline X13 &0.77 &1.454 &2.226 &2.432 &2.532 &2.691 &2.715 &2.719\\ \hline X14 &0.428 &1.193 &1.709 &1.969 &2.221 &2.308 &2.355 &2.361\\ \hline X15 &0.415 &1.202 &1.798 &2.041 &2.272 &2.431 &2.478 &2.484\\ \hline X16 &0.358 &0.716 &0.716 &0.839 &1.028 &1.176 &1.317 &1.322\\ \hline B1 &0.731 &1.56 &2.116 &2.196 &2.277 &2.427 &2.507 &2.515\\ \hline B2 &0.293 &0.761 &1.156 &1.177 &1.198 &1.301 &1.381 &1.386\\ \hline B3 &0.173 &0.304 &0.47 &0.474 &0.478 &0.555 &0.617 &0.618\\ \hline B4 &0 &0 &0.108 &0.108 &0.108 &0.108 &0.111 &0.111\\ \hline \end{array} $$其中$K_{d}^- \longrightarrow Q_{d} $ 可以看为白虎西方位的法克鱿累加
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &法克鱿1 &法克鱿2 &法克鱿3 &法克鱿4 &法克鱿5 &法克鱿6 &法克鱿7 &法克鱿8\\ \hline X1 &0 &0.007 &0.009 &0.01 &0.01 &0.014 &0.022 &0.022\\ \hline X2 &0 &0.005 &0.006 &0.006 &0.006 &0.006 &0.018 &0.018\\ \hline X3 &0 &0 &0.001 &0.001 &0.001 &0.001 &0.012 &0.012\\ \hline X4 &0 &0 &0.001 &0.001 &0.001 &0.001 &0.014 &0.015\\ \hline X5 &0.001 &0.003 &0.003 &0.003 &0.003 &0.003 &0.015 &0.016\\ \hline X6 &0 &0.016 &0.016 &0.016 &0.017 &0.021 &0.031 &0.032\\ \hline X7 &0 &0 &0.002 &0.002 &0.002 &0.003 &0.012 &0.012\\ \hline X8 &0 &0.12 &0.123 &0.131 &0.157 &0.17 &0.197 &0.199\\ \hline X9 &0.008 &0.009 &0.01 &0.017 &0.046 &0.05 &0.076 &0.078\\ \hline X10 &0 &0.003 &0.033 &0.042 &0.057 &0.11 &0.132 &0.133\\ \hline X11 &0 &0.004 &0.011 &0.062 &0.076 &0.084 &0.088 &0.09\\ \hline X12 &0.001 &0.001 &0.002 &0.068 &0.081 &0.086 &0.092 &0.095\\ \hline X13 &0 &0 &0.001 &0.001 &0.007 &0.008 &0.012 &0.013\\ \hline X14 &0.006 &0.006 &0.009 &0.009 &0.009 &0.015 &0.016 &0.016\\ \hline X15 &0.006 &0.006 &0.008 &0.009 &0.009 &0.009 &0.011 &0.011\\ \hline X16 &0.01 &0.015 &0.213 &0.216 &0.217 &0.218 &0.218 &0.218\\ \hline B1 &0 &0 &0.003 &0.011 &0.02 &0.021 &0.021 &0.021\\ \hline B2 &0.015 &0.017 &0.025 &0.054 &0.083 &0.086 &0.087 &0.087\\ \hline B3 &0.034 &0.066 &0.1 &0.15 &0.2 &0.207 &0.207 &0.209\\ \hline B4 &0.211 &0.38 &0.431 &0.503 &0.575 &0.643 &0.659 &0.663\\ \hline \end{array} $$$K_{a}^+ , K_{a}^-$ 分别累加得到一组对抗矩阵$P_{a} , Q_{a}$
其中$K_{a}^+ \longrightarrow P_{a} $ 可以看为朱雀南方位的草泥马累加
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &草泥马1 &草泥马2 &草泥马3 &草泥马4 &草泥马5 &草泥马6 &草泥马7 &草泥马8\\ \hline X1 &0.004 &0.017 &0.115 &0.329 &0.555 &1.121 &1.45 &2.268\\ \hline X2 &0.004 &0.011 &0.226 &0.516 &0.801 &1.5 &1.873 &2.572\\ \hline X3 &0.004 &0.012 &0.227 &0.506 &0.803 &1.647 &2.288 &3.078\\ \hline X4 &0.003 &0.008 &0.181 &0.469 &0.762 &1.567 &2.264 &3.044\\ \hline X5 &0.003 &0.009 &0.224 &0.478 &0.763 &1.857 &2.341 &2.967\\ \hline X6 &0.003 &0.013 &0.105 &0.366 &0.648 &1.368 &1.588 &2.4\\ \hline X7 &0.003 &0.015 &0.163 &0.372 &0.605 &1.238 &2.013 &2.826\\ \hline X8 &0.001 &0.001 &0.047 &0.072 &0.154 &0.739 &0.747 &1.443\\ \hline X9 &0.001 &0.001 &0.098 &0.119 &0.21 &0.908 &1.554 &1.931\\ \hline X10 &0.002 &0.002 &0.004 &0.054 &0.13 &0.317 &0.731 &1.538\\ \hline X11 &0.001 &0.027 &0.094 &0.147 &0.151 &0.572 &0.963 &1.675\\ \hline X12 &0.001 &0.018 &0.11 &0.164 &0.165 &0.784 &1.56 &2.212\\ \hline X13 &0.004 &0.028 &0.187 &0.286 &0.493 &1.264 &1.949 &2.719\\ \hline X14 &0.006 &0.054 &0.14 &0.392 &0.652 &1.169 &1.933 &2.361\\ \hline X15 &0.006 &0.052 &0.212 &0.443 &0.685 &1.281 &2.068 &2.484\\ \hline X16 &0.005 &0.146 &0.293 &0.483 &0.606 &0.606 &0.964 &1.322\\ \hline B1 &0.008 &0.088 &0.238 &0.319 &0.399 &0.955 &1.784 &2.515\\ \hline B2 &0.005 &0.085 &0.188 &0.209 &0.23 &0.625 &1.093 &1.386\\ \hline B3 &0.001 &0.063 &0.141 &0.145 &0.149 &0.315 &0.446 &0.618\\ \hline B4 &0 &0.003 &0.003 &0.003 &0.003 &0.111 &0.111 &0.111\\ \hline \end{array} $$其中$K_{a}^- \longrightarrow Q_{a} $ 可以看为玄武北方位的菊花熊累加
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times8}} &菊花熊1 &菊花熊2 &菊花熊3 &菊花熊4 &菊花熊5 &菊花熊6 &菊花熊7 &菊花熊8\\ \hline X1 &0 &0.008 &0.012 &0.013 &0.013 &0.015 &0.022 &0.022\\ \hline X2 &0 &0.012 &0.012 &0.012 &0.012 &0.013 &0.018 &0.018\\ \hline X3 &0 &0.011 &0.011 &0.011 &0.011 &0.011 &0.012 &0.012\\ \hline X4 &0.001 &0.014 &0.014 &0.014 &0.014 &0.015 &0.015 &0.015\\ \hline X5 &0.001 &0.013 &0.013 &0.013 &0.013 &0.013 &0.015 &0.016\\ \hline X6 &0.001 &0.01 &0.015 &0.015 &0.015 &0.016 &0.032 &0.032\\ \hline X7 &0.001 &0.009 &0.01 &0.01 &0.011 &0.012 &0.012 &0.012\\ \hline X8 &0.001 &0.028 &0.042 &0.068 &0.076 &0.078 &0.198 &0.199\\ \hline X9 &0.002 &0.028 &0.032 &0.061 &0.068 &0.069 &0.069 &0.078\\ \hline X10 &0.001 &0.023 &0.076 &0.091 &0.1 &0.13 &0.133 &0.133\\ \hline X11 &0.002 &0.006 &0.014 &0.029 &0.079 &0.086 &0.09 &0.09\\ \hline X12 &0.002 &0.009 &0.013 &0.027 &0.092 &0.094 &0.094 &0.095\\ \hline X13 &0 &0.005 &0.006 &0.011 &0.012 &0.013 &0.013 &0.013\\ \hline X14 &0 &0.002 &0.007 &0.007 &0.007 &0.011 &0.011 &0.016\\ \hline X15 &0 &0.002 &0.002 &0.003 &0.003 &0.005 &0.005 &0.011\\ \hline X16 &0 &0 &0.001 &0.002 &0.006 &0.203 &0.209 &0.218\\ \hline B1 &0 &0 &0.001 &0.01 &0.018 &0.021 &0.021 &0.021\\ \hline B2 &0 &0.001 &0.004 &0.033 &0.062 &0.07 &0.072 &0.087\\ \hline B3 &0.001 &0.002 &0.009 &0.059 &0.109 &0.143 &0.174 &0.209\\ \hline B4 &0.004 &0.02 &0.088 &0.16 &0.232 &0.283 &0.452 &0.663\\ \hline \end{array} $$
偏序的核心规则为对称性
设：在$ \mathbf{P}, \mathbf{Q}$矩阵中任意两样本(行)$x,y$
$$ 对于 \mathbf{P}矩阵: p_{x1} \geqslant p_{y1} 且p_{x2} \geqslant p_{y2} 且 p_{x3} \geqslant p_{y3} {\cdots}且p_{xm} \geqslant p_{ym} 同时对于 \mathbf{Q}矩阵: q_{x1} \leqslant q_{y1} 且 q_{x2} \leqslant q_{y2} 且q_{x3} \leqslant q_{y3}{\cdots} 且 q_{xm} \leqslant q_{ym}\\ $$
记作：$\quad \quad PS_{(x)}\geqslant PS_{(y)}$
关系矩阵值有：$$a_{xy}= \begin{cases} 1, PS_{(x)} {\geqslant} PS_{(y)} \\ 0, 其它 \end{cases} $$
由$ \mathbf{P_d} , \mathbf{Q_d}$求出关系矩阵，哈斯矩阵与上蹿哈斯图
由$ \mathbf{P_a} , \mathbf{Q_a}$求出关系矩阵，哈斯矩阵与上蹿哈斯图
TOPSIS中关键一步是：计算各评价方案（样本）到正、负理想解的距离。
KL散度的计算公式如下：
$$ d_i^+=\sum \limits_{j=1}^{m} \left\{ z_j^+ ln{\frac {z_j^+} {\omega_{j} \cdot n_{ij}} } + (1-z^+_j)ln{\frac {1-z_j^+} {1- \omega_{j} \cdot n_{ij}} } \right\} \quad \quad i表示行,j表示列\quad 其中 \quad n_{ij} \in N \quad z^+_j \in \mathbf{S^+}$$
$$ d_i^-=\sum \limits_{j=1}^{m} \left\{ z_j^- ln{\frac {z_j^-} {\omega_{j} \cdot n_{ij}} } + (1-z^-_j)ln{\frac {1-z_j^-} {1- \omega_{j} \cdot n_{ij}} } \right\} \quad \quad i表示行,j表示列\quad 其中 \quad n_{ij} \in N \quad z^-_j \in \mathbf{S^-}$$
结果如下：
$$ \begin{array}{c|c|c|c|c|c|c}{M_{20 \times2}} &d^+ &d^-\\ \hline X1 &2.268 &0.0223\\ \hline X2 &2.5723 &0.0181\\ \hline X3 &3.0782 &0.0118\\ \hline X4 &3.0438 &0.015\\ \hline X5 &2.9669 &0.0157\\ \hline X6 &2.4001 &0.0315\\ \hline X7 &2.8255 &0.0123\\ \hline X8 &1.4428 &0.1986\\ \hline X9 &1.9313 &0.0777\\ \hline X10 &1.5377 &0.1331\\ \hline X11 &1.6754 &0.0903\\ \hline X12 &2.212 &0.0946\\ \hline X13 &2.7187 &0.0128\\ \hline X14 &2.3613 &0.0164\\ \hline X15 &2.4835 &0.0111\\ \hline X16 &1.3218 &0.2184\\ \hline B1 &2.5149 &0.0209\\ \hline B2 &1.3858 &0.0869\\ \hline B3 &0.6183 &0.2086\\ \hline B4 &0.1109 &0.663\\ \hline \end{array} $$偏序规则略:
夹逼的思路是一种非常重要的思路，求极限中经常用到。本算例中只是把信息论中的KL散度的相似度概念代替了线性的欧式距离的相似度的概念而已。
不论是topsis 还是VIKOR方法都非常好魔改，只要把现在火的一塌糊涂的ML（不是 make love），深度学习，等的一些基本概念弄进去，就不算组合也能整个100来种方法。
本文更侧重于夹逼的思路，即从多个角度，构建不同的取偏序的规矩。通过可视化的层级图的变化，理解神经网络的学习规则。
这种样品多的哈斯图很占页面，就不打字，光是计算的主要的几个矩阵，一下子就弄上了1、2万字。目前的只弄了6个哈斯图。事实上这种可以弄出非常多的哈斯图出来。