初始概率

$$\begin{array}{c|c|c|c|c|c|c}{M_{10 \times1}} &初始概率 P\\ \hline A &0.5\\ \hline B &0.3\\ \hline C &0.6\\ \hline D &0.5\\ \hline E &0.4\\ \hline F &0.3\\ \hline G &0.6\\ \hline H &0.2\\ \hline I &0.1\\ \hline J &0.6\\ \hline \end{array} $$

概率关系矩阵

$$R=\begin{array}{c|c|c|c|c|c|c}{M_{10 \times10}} &A &B &C &D &E &F &G &H &I &J\\ \hline A &0 &0.45 &0 &0.4 &0.45 &0.75 &0 &0.45 &0.45 &0\\ \hline B &0.25 &0 &0.28 &0.35 &0 &0.2 &0.38 &0.35 &0.35 &0.34\\ \hline C &0.55 &0.65 &0 &0.65 &0.7 &0.5 &0 &0.65 &0.65 &0.66\\ \hline D &0.4 &0.6 &0.51 &0 &0.55 &0.3 &0.53 &0.55 &0.55 &0.53\\ \hline E &0.3 &0.5 &0.39 &0.5 &0 &0.35 &0.47 &0 &0 &0.43\\ \hline F &0.4 &0.25 &0 &0.1 &0.25 &0 &0.27 &0.25 &0.25 &0.27\\ \hline G &0.55 &0.75 &0 &0.7 &0.75 &0.55 &0 &0 &0 &0.66\\ \hline H &0.1 &0.15 &0 &0.25 &0.25 &0.1 &0 &0 &0.3 &0.24\\ \hline I &0.05 &0 &0 &0.15 &0 &0.05 &0.15 &0.2 &0 &0.15\\ \hline J &0.55 &0.75 &0.59 &0.7 &0.75 &0.5 &0.66 &0 &0 &0\\ \hline \end{array} $$

交叉影响矩阵的求解

$$ C_{ij}= \frac {1}{1-P_j}[ln( \frac {R_{ij}}{1-R_{ij}} ) - ln(\frac {P_i}{1-P_i} )] $$

$$CIA=\begin{array}{c|c|c|c|c|c|c}{M_{10 \times10}} &A &B &C &D &E &F &G &H &I &J\\ \hline A &0 &-0.287 &0 &-0.811 &-0.334 &1.569 &0 &-0.251 &-0.223 &0\\ \hline B &-0.503 &0 &-0.243 &0.457 &0 &-0.77 &0.894 &0.285 &0.254 &0.46\\ \hline C &-0.41 &0.305 &0 &0.427 &0.736 &-0.579 &0 &0.267 &0.237 &0.645\\ \hline D &-0.811 &0.579 &0.1 &0 &0.334 &-1.21 &0.3 &0.251 &0.223 &0.3\\ \hline E &-0.884 &0.579 &-0.105 &0.811 &0 &-0.305 &0.713 &0 &0 &0.309\\ \hline F &0.884 &-0.359 &0 &-2.7 &-0.419 &0 &-0.368 &-0.314 &-0.279 &-0.368\\ \hline G &-0.41 &0.99 &0 &0.884 &1.155 &-0.293 &0 &0 &0 &0.645\\ \hline H &-1.622 &-0.498 &0 &0.575 &0.479 &-1.158 &0 &0 &0.599 &0.584\\ \hline I &-1.494 &0 &0 &0.925 &0 &-1.067 &1.157 &1.014 &0 &1.157\\ \hline J &-0.41 &0.99 &-0.104 &0.884 &1.155 &-0.579 &0.645 &0 &0 &0\\ \hline \end{array} $$

交叉影响矩阵转置

$$Ori=\begin{array}{c|c|c|c|c|c|c}{M_{10 \times10}} &A &B &C &D &E &F &G &H &I &J\\ \hline A &0 &-0.50263 &-0.40959 &-0.81093 &-0.88367 &0.88367 &-0.40959 &-1.62186 &-1.49443 &-0.40959\\ \hline B &-0.28667 &0 &0.30511 &0.57924 &0.57924 &-0.35902 &0.99021 &-0.49758 &0 &0.99021\\ \hline C &0 &-0.24291 &0 &0.10001 &-0.10462 &0 &0 &0 &0 &-0.10375\\ \hline D &-0.81093 &0.45652 &0.42715 &0 &0.81093 &-2.69985 &0.88367 &0.57536 &0.92525 &0.88367\\ \hline E &-0.33445 &0 &0.73639 &0.33445 &0 &-0.41886 &1.15525 &0.47947 &0 &1.15525\\ \hline F &1.56945 &-0.77 &-0.57924 &-1.21043 &-0.30511 &0 &-0.29256 &-1.15847 &-1.06745 &-0.57924\\ \hline G &0 &0.89437 &0 &0.30036 &0.7133 &-0.36831 &0 &0 &1.15656 &0.64457\\ \hline H &-0.25084 &0.28532 &0.26697 &0.25084 &0 &-0.31414 &0 &0 &1.01366 &0\\ \hline I &-0.22297 &0.25362 &0.2373 &0.22297 &0 &-0.27924 &0 &0.59889 &0 &0\\ \hline J &0 &0.46001 &0.64457 &0.30036 &0.30903 &-0.36831 &0.64457 &0.58404 &1.15656 &0\\ \hline \end{array} $$

对称化矩阵，平移与更改符号得到手性对称矩阵

$$\begin{array} {c|cccccccccc|cccccccccc}{M_{20 \times20}} &+A &+B &+C &+D &+E &+F &+G &+H &+I &+J &-A &-B &-C &-D &-E &-F &-G &-H &-I &-J\\ \hline +A &0 &0 &0 &0 &0 &\color{blue}{0.884} &0 &0 &0 &0 &0 &\color{red}{0.503} &\color{red}{0.41} &\color{red}{0.811} &\color{red}{0.884} &0 &\color{red}{0.41} &\color{red}{1.622} &\color{red}{1.494} &\color{red}{0.41}\\ +B &0 &0 &\color{blue}{0.305} &\color{blue}{0.579} &\color{blue}{0.579} &0 &\color{blue}{0.99} &0 &0 &\color{blue}{0.99} &\color{red}{0.287} &0 &0 &0 &0 &\color{red}{0.359} &0 &\color{red}{0.498} &0 &0\\ +C &0 &0 &0 &\color{blue}{0.1} &0 &0 &0 &0 &0 &0 &0 &\color{red}{0.243} &0 &0 &\color{red}{0.105} &0 &0 &0 &0 &\color{red}{0.104}\\ +D &0 &\color{blue}{0.457} &\color{blue}{0.427} &0 &\color{blue}{0.811} &0 &\color{blue}{0.884} &\color{blue}{0.575} &\color{blue}{0.925} &\color{blue}{0.884} &\color{red}{0.811} &0 &0 &0 &0 &\color{red}{2.7} &0 &0 &0 &0\\ +E &0 &0 &\color{blue}{0.736} &\color{blue}{0.334} &0 &0 &\color{blue}{1.155} &\color{blue}{0.479} &0 &\color{blue}{1.155} &\color{red}{0.334} &0 &0 &0 &0 &\color{red}{0.419} &0 &0 &0 &0\\ +F &\color{blue}{1.569} &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &\color{red}{0.77} &\color{red}{0.579} &\color{red}{1.21} &\color{red}{0.305} &0 &\color{red}{0.293} &\color{red}{1.158} &\color{red}{1.067} &\color{red}{0.579}\\ +G &0 &\color{blue}{0.894} &0 &\color{blue}{0.3} &\color{blue}{0.713} &0 &0 &0 &\color{blue}{1.157} &\color{blue}{0.645} &0 &0 &0 &0 &0 &\color{red}{0.368} &0 &0 &0 &0\\ +H &0 &\color{blue}{0.285} &\color{blue}{0.267} &\color{blue}{0.251} &0 &0 &0 &0 &\color{blue}{1.014} &0 &\color{red}{0.251} &0 &0 &0 &0 &\color{red}{0.314} &0 &0 &0 &0\\ +I &0 &\color{blue}{0.254} &\color{blue}{0.237} &\color{blue}{0.223} &0 &0 &0 &\color{blue}{0.599} &0 &0 &\color{red}{0.223} &0 &0 &0 &0 &\color{red}{0.279} &0 &0 &0 &0\\ +J &0 &\color{blue}{0.46} &\color{blue}{0.645} &\color{blue}{0.3} &\color{blue}{0.309} &0 &\color{blue}{0.645} &\color{blue}{0.584} &\color{blue}{1.157} &0 &0 &0 &0 &0 &0 &\color{red}{0.368} &0 &0 &0 &0\\ \hline -A &0 &\color{red}{0.503} &\color{red}{0.41} &\color{red}{0.811} &\color{red}{0.884} &0 &\color{red}{0.41} &\color{red}{1.622} &\color{red}{1.494} &\color{red}{0.41} &0 &0 &0 &0 &0 &\color{blue}{0.884} &0 &0 &0 &0\\ -B &\color{red}{0.287} &0 &0 &0 &0 &\color{red}{0.359} &0 &\color{red}{0.498} &0 &0 &0 &0 &\color{blue}{0.305} &\color{blue}{0.579} &\color{blue}{0.579} &0 &\color{blue}{0.99} &0 &0 &\color{blue}{0.99}\\ -C &0 &\color{red}{0.243} &0 &0 &\color{red}{0.105} &0 &0 &0 &0 &\color{red}{0.104} &0 &0 &0 &\color{blue}{0.1} &0 &0 &0 &0 &0 &0\\ -D &\color{red}{0.811} &0 &0 &0 &0 &\color{red}{2.7} &0 &0 &0 &0 &0 &\color{blue}{0.457} &\color{blue}{0.427} &0 &\color{blue}{0.811} &0 &\color{blue}{0.884} &\color{blue}{0.575} &\color{blue}{0.925} &\color{blue}{0.884}\\ -E &\color{red}{0.334} &0 &0 &0 &0 &\color{red}{0.419} &0 &0 &0 &0 &0 &0 &\color{blue}{0.736} &\color{blue}{0.334} &0 &0 &\color{blue}{1.155} &\color{blue}{0.479} &0 &\color{blue}{1.155}\\ -F &0 &\color{red}{0.77} &\color{red}{0.579} &\color{red}{1.21} &\color{red}{0.305} &0 &\color{red}{0.293} &\color{red}{1.158} &\color{red}{1.067} &\color{red}{0.579} &\color{blue}{1.569} &0 &0 &0 &0 &0 &0 &0 &0 &0\\ -G &0 &0 &0 &0 &0 &\color{red}{0.368} &0 &0 &0 &0 &0 &\color{blue}{0.894} &0 &\color{blue}{0.3} &\color{blue}{0.713} &0 &0 &0 &\color{blue}{1.157} &\color{blue}{0.645}\\ -H &\color{red}{0.251} &0 &0 &0 &0 &\color{red}{0.314} &0 &0 &0 &0 &0 &\color{blue}{0.285} &\color{blue}{0.267} &\color{blue}{0.251} &0 &0 &0 &0 &\color{blue}{1.014} &0\\ -I &\color{red}{0.223} &0 &0 &0 &0 &\color{red}{0.279} &0 &0 &0 &0 &0 &\color{blue}{0.254} &\color{blue}{0.237} &\color{blue}{0.223} &0 &0 &0 &\color{blue}{0.599} &0 &0\\ -J &0 &0 &0 &0 &0 &\color{red}{0.368} &0 &0 &0 &0 &0 &\color{blue}{0.46} &\color{blue}{0.645} &\color{blue}{0.3} &\color{blue}{0.309} &0 &\color{blue}{0.645} &\color{blue}{0.584} &\color{blue}{1.157} &0\\ \hline \end{array} $$

基于截距的聚类分析

手性对称矩阵的阈值集合$\ddot \Delta $　得出对应 19个结构。

序号 阈值集合中——特征阈值 聚类特征-对应截距$\lambda $数值区段 ISM运算过程
10.100010<$\lambda$<0.1
20.250840.1<$\lambda$<0.2508
30.334450.2508<$\lambda$<0.3345
40.502630.3345<$\lambda$<0.5026
50.584040.5026<$\lambda$<0.584
60.644570.584<$\lambda$<0.6446
70.71330.6446<$\lambda$<0.7133
80.736390.7133<$\lambda$<0.7364
90.770.7364<$\lambda$<0.77
100.894370.77<$\lambda$<0.8944
110.925250.8944<$\lambda$<0.9252
121.013660.9252<$\lambda$<1.0137
131.067451.0137<$\lambda$<1.0674
141.156561.0674<$\lambda$<1.1566
151.158471.1566<$\lambda$<1.1585
161.494431.1585<$\lambda$<1.4944
171.569451.4944<$\lambda$<1.5694
181.621861.5694<$\lambda$<1.6219
192.699851.6219<$\lambda$<2.6999

取绝对值，不进行平移对称化矩阵如下：

$$ABS-ORI=\begin{array}{c|c|c|c|c|c|c}{M_{10 \times10}} &A &B &C &D &E &F &G &H &I &J\\ \hline A &0 &0.50263 &0.40959 &0.81093 &0.88367 &0.88367 &0.40959 &1.62186 &1.49443 &0.40959\\ \hline B &0.28667 &0 &0.30511 &0.57924 &0.57924 &0.35902 &0.99021 &0.49758 &0 &0.99021\\ \hline C &0 &0.24291 &0 &0.10001 &0.10462 &0 &0 &0 &0 &0.10375\\ \hline D &0.81093 &0.45652 &0.42715 &0 &0.81093 &2.69985 &0.88367 &0.57536 &0.92525 &0.88367\\ \hline E &0.33445 &0 &0.73639 &0.33445 &0 &0.41886 &1.15525 &0.47947 &0 &1.15525\\ \hline F &1.56945 &0.77 &0.57924 &1.21043 &0.30511 &0 &0.29256 &1.15847 &1.06745 &0.57924\\ \hline G &0 &0.89437 &0 &0.30036 &0.7133 &0.36831 &0 &0 &1.15656 &0.64457\\ \hline H &0.25084 &0.28532 &0.26697 &0.25084 &0 &0.31414 &0 &0 &1.01366 &0\\ \hline I &0.22297 &0.25362 &0.2373 &0.22297 &0 &0.27924 &0 &0.59889 &0 &0\\ \hline J &0 &0.46001 &0.64457 &0.30036 &0.30903 &0.36831 &0.64457 &0.58404 &1.15656 &0\\ \hline \end{array} $$

基于截距的聚类分析

手性对称矩阵的阈值集合$\ddot \Delta $　得出对应 18个结构。

序号 阈值集合中——特征阈值 聚类特征-对应截距$\lambda $数值区段 ISM运算过程
100<$\lambda$<0
20.250840<$\lambda$<0.2508
30.334450.2508<$\lambda$<0.3345
40.584040.3345<$\lambda$<0.584
50.644570.584<$\lambda$<0.6446
60.71330.6446<$\lambda$<0.7133
70.736390.7133<$\lambda$<0.7364
80.770.7364<$\lambda$<0.77
90.894370.77<$\lambda$<0.8944
100.925250.8944<$\lambda$<0.9252
111.013660.9252<$\lambda$<1.0137
121.067451.0137<$\lambda$<1.0674
131.156561.0674<$\lambda$<1.1566
141.158471.1566<$\lambda$<1.1585
151.494431.1585<$\lambda$<1.4944
161.569451.4944<$\lambda$<1.5694
171.621861.5694<$\lambda$<1.6219
182.699851.6219<$\lambda$<2.6999