# Soil Classification from Piezoone Penetration Test Using Fuzzy Clustering and Neuro-Fuzzy Theory
## 3 Fuzzy Clustering and Neuro-Fuzzy Modeling
### Neuro-Fuzzy Algorithm  
After the FCM clustering described in Section 3.2, a neuro-fuzzy model was developed on the basis of the optimum number of clusters (three, i.e., clay, silt, and sand) and input variables (\(q_{t}\), \(R_{f}\), \(B_{q}\)). The fuzzy technique has the advantage of presenting data or events which cannot be numerically expressed, whereas the selection of the fuzzy membership function for inference is not always objective and precise. The decisions of fuzzy membership functions by [PERSON] [6] and [PERSON] and [PERSON] [7] were dependent upon subjective trial-and-error methods or experimental methods. Thus, any revision or supplement considering local characteristics may not be easily considered even though it is necessary. However, the neuro-fuzzy method combining a neutral network and the fuzzy method has the merits of both techniques and is expected to overcome the previously mentioned defects. The neural network has the advantage of facing variations of data, but the input data are in numeric form. On the other hand, the fuzzy technique allows presenting numeric data using a membership function. Thus, the techniques are complementary. Moreover, the neuro-fuzzy technique can objectively decide membership functions and can be easily updated if needed through determining the optimized membership functions using the neural network algorithm. Figure 6 shows a schematic diagram of the ANFIS neuro-fuzzy model with two input parameters. The general neuro-fuzzy model uses membership functions in the input and output phases. In this case, more time is required to complete calculation and convergence. Thus, the ANFIS (adaptive network-based fuzzy inference system) adopts a first-order function in the output instead of a membership function. The relevant procedure of the ANFIS model is summarized below. Figure 5: Change in success rate with respect to m value. Figure 6: Procedure of ANFIS neuro-fuzzy model. 1. The neuro-fuzzy output shown in Figure 6 is defined as a first-order function as shown in Equation (10) if two input parameters are assumed. Here, \(p\), \(q\), and \(r\) are constants to be decided after neural network training, while \(x\) and \(y\) are input parameters, which are PCPT indices. \[f_{1}=p_{1}x+q_{1}y+r_{1},\] \[f_{2}=p_{2}x+q_{2}y+r_{2},\] (10) \[f=f_{1}+f_{2}\]
2. Total outputs in the system considering weighting factors \(w_{1}\) and \(w_{2}\) are given by Equation (11). Weighting factors are calculated using Equation (12) after evaluating each fuzzy membership function for given input parameters. Here, \(\mu_{A1}\) corresponds to the finalized fuzzy membership function of \(x\), while \(\mu_{B1}\) corresponds to the finalized fuzzy membership function \(y\). \[f=\frac{w_{1}f_{1}+w_{2}f_{2}}{w_{1}+w_{2}}\.\] (11) \[w_{1}=\mu_{A1}\mu_{B1},w_{2}=\mu_{A2}\mu_{B2}.\] (12)
3. The training procedure is completed after optimizing the parameters (\(p\), \(q\), and \(r\)) of the first-order function and of membership functions to minimize output error, \(e\), defined by output \(f_{k}\) and estimation \(T_{k}\). \[e=\frac{1}{2}{\sum{\left({f_{k}-T_{k}}\right)^{2}}}.\] (13) Commercial soft computing package, Matlab was used to complete the training procedure when the least square error in Equation (13) was within the target error value, \(\varepsilon=0.01\) or when the maximum training loop attained 200. If any of the predetermined conditions were not satisfied, is the outcome was regarded as \"bad\". Figure 7 shows the commonly used candidate membership functions for the input process--i.e., triangular, Gaussian, bell-shapes, and sigmoidal (S-shaped) membership functions. Tables 4-7 show the various neuro-fuzzy analysis results to find the best combination of fuzzy membership functions for input variables. The success rate for each class, which is defined by the match with the soil type in the database, and the averaged success rates per each combination are presented. As shown in the tables, the success rate generally ranged from 70% to 79%.