# Soil Classification from Piezoone Penetration Test Using Fuzzy Clustering and Neuro-Fuzzy Theory
## 3 Fuzzy Clustering and Neuro-Fuzzy Modeling
### FCM (Fuzzy C-Means) Clustering Algorithm  
Before developing the neuro-fuzzy model for PCPT-based soil classification, a grouping procedure was carried out to establish the unique structure between soil behavior type and PCPT input parameters, as well as to determine the appropriate number of soil types in the database and the input variables. Generally, some techniques exist to find structures in the database and divide them into small groups. An unsupervised learning strategy, clustering, can be used for that purpose, with the FCM (fuzzy C-means) algorithm being most widely used. This algorithm searches for fuzzy divisions \(\overline{F}=\left\{\overline{F_{1}},\overline{F_{2}},\cdot\cdot\cdot,\overline {F_{c}}\right\}\) to minimize the function, as expressed in Equation (6), when a dataset composed of \(n\) items is divided into \(c\) clusters. \[J_{m}(U,V:X)=\sum_{i=1}^{c}\sum_{k=1}^{n}\left(\mu_{ik}\right)^{m}\left\|X_{k} -V_{i}\right\|^{2}, \tag{6}\]  
where \(V=\left\{V_{1},V_{2},\cdot\cdot\cdot,V_{c}\right\}\) is the set of \(c\) central vectors, and \(\left\|X_{k}-V_{i}\right\|\) is the geometric distance between data \(X_{k}\) and the center of the \(i\)th cluster. In addition, \(\mu_{ik}\) is the grade of \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Sites** & **Nos.** & **Soil Type by USCS** \\ \cline{3-4}  & & Pyeongtaek & 2 & CL, SP, SW \\ \cline{2-4} Gyeonggi & Siheung & 3 & CL, SM, SP \\ \cline{2-4}  & Ilsan & 1 & SM, SP \\ \cline{2-4}  & Incheon & 3 & CL, ML \\ \hline \multirow{2}{*}{Chungnam} & Seocheon & 4 & CL, SM, SP \\ \cline{2-4}  & Asan & 1 & CL, SM \\ \hline \multirow{2}{*}{Jeonnam} & Yeongam & 1 & CH, SP, SW \\ \cline{2-4}  & Gwangyang & 1 & CL, CH, ML, SP \\ \hline \multirow{2}{*}{Jeonbuk} & Kunjang & 9 & CH, CL, ML \\ \hline \multirow{6}{*}{Gyeongnam} & Yangsan & 4 & CL, SM \\ \cline{2-4}  & Tongyeong & 2 & CH \\ \cline{2-4}  & Hadong & 2 & CL, CH, MH, SP \\ \cline{2-4}  & Ulsan & 2 & SM, SP-SC \\ \cline{2-4}  & Yongwon & 4 & CH \\ \cline{2-4}  & Cheonseong & 4 & CH \\ \cline{2-4}  & Gaduk & 4 & CH \\ \hline Kangwon & Naegok & 2 & CL, MH, SP, SW \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Included number of PCPTs and USCS classes for each site. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**USCS** & **Nos.** & \(q_{t}\) (MPa)** & \(f_{s}\) (MPa) & \(u_{bt}\) (MPa) \\ \hline CH & 2746 & 0.108 to 1.250 & 0.0001 to 0.030 & 0.051 to 0.683 \\ \hline CL & 1861 & 0.028 to 6.520 & 0.0001 to 0.052 & 0.002 to 1.088 \\ \hline MH & 36 & 0.608 to 1.640 & 0.003 to 0.028 & 0.018 to 0.255 \\ \hline ML & 284 & 0.436 to 6.818 & 0.004 to 0.096 & \(-\)0.092 to 0.562 \\ \hline SM, SP-SC & 148 & 0.217 to 160.328 & 0.006 to 5.740 & \(-\)0.960 to 3.256 \\ \hline SP, SW & 98 & 1.232 to 36.263 & 0.005 to 0.857 & \(-\)0.168 to 0.528 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of soil data points and corresponding PCPT data range. cluster \(\overline{F}_{i}\) including data \(X_{k}\) and satisfies Equation (7) in the element of the fuzzy partition matrix \(U=[\mu_{ij}]\) with the size of \((c\times n)\). \[\mu_{ik}\in[0,1],\sum\limits_{i=1}^{c}\mu_{ik}=1 \tag{7}\]  
The procedure of the FCM clustering algorithm is summarized below. 1. Assume partition number \(c\) (\(2\leq c\leq n\)) and fuzziness of partition \(m\). 2. Select initial values of fuzzy partition matrix, \(U(t)\). Random values are assumed for satisfying Equation (6). 3. Calculate the center of cluster \(V\) using Equation (8). \[V_{i}^{(t+1)}=\frac{\sum\limits_{i=1}^{c}\left({\mu_{ik}}^{(t)}\right)^{m}X_{k }}{\sum\limits_{i=1}^{c}\mu_{ik}^{(t)}},m>1,i=1,\cdot\cdot\cdot,c\] (8)
4. Recompose fuzzy partition matrix using Equation (9). \[\mu_{ik}=\frac{1}{\sum\limits_{j=1}^{c}\left(\frac{\left|X_{k}-V_{i}\right|^{2 }}{\left|X_{k}-V_{j}\right|^{2}}\right)^{1/(m-1)}},i=1,\cdot\cdot\cdot,c,k=1, \cdot\cdot\cdot,n\] (9)
5. Complete the procedure if \(\left|U(t+1)-U(t)\right|<\delta\). Otherwise, repeat phase 4. Here, \(\delta\) is assumed to be \(10^{-3}\). 6.