**3. Results**

#### **3.1 Performances of the models**

This section discusses the results obtained from the modeling of the study. The ANN, ANFIS and MLR models were also compared. **Table 2** showed the performance of the models.

#### **3.2 Performance of the multilinear regression (MLR) model**

In the MLR model, the data was split into two subclasses of training and testing. The ratios of the training and testing phases were characterized based on the fact that the common configuration of the model was built with respect to training data set. Hence, the quantity of data in the training category plays an important function. The total number of data was 63 in which 70% (44) and 30% (18) were randomly selected for training and testing, respectively. **Figure 5** shows the scatter plot of the relationship between actual and predicted specific wear rate (SWR) of the PTFE composites.

As it was shown in **Figure 5**, the determination coefficient (DC) of the training and testing phases were determined as 0.5674 and 0.5267, respectively. In addition, the RMSE in training was found to be 0.1275 but the testing stage RMSE was computed as 0.2306. As per the prediction analysis the DC and RMSE in the testing phase were considered. Therefore, MLR model with a DC of 0.5267 and RMSE of 0.2306 did not indicate higher prediction accuracy of the specific wear rate of the PTFE reinforced composites. This is attributed to the nonlinearity and complex nature of specific wear rate of the composites and MLR model is commonly good at finding linear and non-complex relationship between predictor and response variables [25].

#### **3.3 Performance of the feed forward neural network (FFNN) model**


Various learning algorithms were tried in order to find the optimum FFNN architecture and among all of them, Levenberg–Marquardt was found to be the

**Table 2.** *Performance results of the models.* *Specific Wear Rate Modeling of Polytetraflouroethylene Composites via Artificial Neural… DOI: http://dx.doi.org/10.5772/intechopen.95242*

**Figure 5.** *Scatter plot of MLR model in (a) training and (b) testing stages.*

most effective. In the FFNN model, the data is categorized into three subsets of training, validation and testing. The ratios of training, validation and testing are characterized based on the fact that fundamental architecture of the FFNN model is built based on the training data set. The whole data of the specific wear rate measurement was 63 in which 44 (70%) was chosen for training, 9 (15%) was selected for validation and 9 (15%) was chosen for testing. Besides, the sigmoid tangent was selected as the transfer function. The ANN model was trained with a single hidden layer. In addition, the number of neurons in the hidden layer was approximated using (Eq. (6)). Khademi et al. [26] instead of performing trial and error approach.

$$\mathbf{N\_h} \le \mathbf{2x\_i} + \mathbf{1} \tag{14}$$

where Nh stands for the maximum number of neurons in the hidden layer and xi equals the number of predictors. Therefore, in this research, based on the predictors which were five (5), the maximum number of neurons in the hidden layer was computed as eleven (11). The optimized ANN architecture with a single layer was thus expressed as [5–11-1]1.

**Figure 6(a), (b),** and **(c)** shows the scatter plot of FFNN model in training, validation and testing stages, respectively. As seen in **Figure 6(a)** and **(b)** the FFNN model exhibited desirable results in both training and validation phases. Additionally, to estimate the prediction performance of the FFNN model, the DC was evaluated for the testing step as shown in the scatter plot of **Figure 6(c)**. As indicated in **Figure 6(c)**, the DC for testing of the FFNN model was determined as 0.9749 with a RMSE of 0.0559. This means that an FFNN model is more efficient in predicting the wear behavior of the composites, as compared to MLR model. This result was similar but higher than the previous study [27]. To round off, ANN model was found to be efficient in predicting the specific wear rate of the composites. This tallies with past studies of [28–29].

#### **3.4 Performance of the adaptive neuro fuzzy inference system (ANFIS) model**

In this study, ANFIS that used the hybrid learning algorithm was employed. The proportions training, validation and testing were chosen the same as the ones in FFNN modeling. To determine the best membership function, trial and error

**Figure 6.** *Scatter plot of FFNN model in (a) training (b) validation and (c) testing phases.*

approach was used and it was found that Gaussian membership function gave the best results at 50 epochs and 0.05 tolerance errors. **Figure 7** shows the scatter plot of the relationship between the actual and predicted specific wear rate of the composites training, validation and testing stages. **Figure 7** shows perfect coincidence of the target and the output data which demonstrated the capability of the ANFIS model. As it was indicated in the figure, the DC of the ANFIS in the testing stage was computed as 0.9971. More so, the RMSE was computed as 0.0225. To wrap up, ANFIS model was found to be capable of approximating the specific wear rate of the composites with satisfactory performance. This excellent performance of the ANFIS model agrees with the research by [30–31].

#### **3.5 Comparing the results of FFNN, ANFIS and MLR models**

In this article, the performance of FFNN, ANFIS and MLR models on predicting the specific wear rate of PTFE composites based on determination coefficient (DC) and root mean square error (RMSE) was investigated. The higher values of DC and lower values of RMSE indicate better and accurate prediction capability of model. For the purpose of the comparison, the data was split into 65% (40) and 35% (22) in training and testing, respectively for all the models. The comparative results of the models were shown in **Table 3** above. As seen in **Table 3**, the performances of the FFNN and ANFIS models were better than that of the MLR model. FFNN and ANFIS models outperformed the performance of the MLR model by 43.14% and 43.12% and 48.23% and 50.02% in training and testing phases, respectively. In other words, the prediction quality of MLR model was ineffective compared to the high prediction quality of ANN and ANFIS of 0.9783 and 0.9961, respectively. Their capabilities to predict the specific wear rate with minimum errors of 4% and 2%

*Specific Wear Rate Modeling of Polytetraflouroethylene Composites via Artificial Neural… DOI: http://dx.doi.org/10.5772/intechopen.95242*

#### **Figure 7.**

*Scatter plot of ANFIS model in (a) training (b) validation and (c) testing.*


#### **Table 3.**

*Comparative performance results of the models.*

(within acceptable level) as compared to the high error of MLR model of 21% is associated with their abilities to deal with nonlinear, noisy, complex relationship and to learn from outside environment and generalize. More so, the prediction performance of the ANFIS model was slightly higher than that of ANN model by 2%. This is because ANFIS model combines the attributes of both learning algorithm and fuzzy logic structure. **Figures 8** and **9** show the scatter plot of the models prediction quality and the simulated prediction results, respectively. It can be seen that ANFIS and FFNN models indicated perfect match with the actual SWR of composites while MLR model exhibited imperfect consistency with respect to the observed SWR of the composites.

#### **3.6 Sensitivity analysis**

Identification of most influential parameter in the study of wear is a significant step in achieving optimum results. In the light of this, a nonlinear FFNN sensitivity

**Figure 8.** *Scatter plot of (a) MLR, (b) ANFIS and (c) FFNN models in testing stages.*

#### **Figure 9.**

*Comparing the performance of the models: Testing stage.*

of the specific wear rate of the composites was applied in this study to establish the dominant parameters in place of using traditional linear methods. The five specific wear rate were evaluated and ranked based on the mean value of the DC of the single modeling obtained in training and testing phases of the FFNN modeling. The results of the ranking based on the sensitivity analysis of the specific wear rate was presented in **Table 4**.

As seen from **Table 4**, in terms of the experimental conditions sliding distance is the most influential parameter, then sliding speed and the least was the applied load. On the contrary [27] reported that the sliding speed had the greatest effect on *Specific Wear Rate Modeling of Polytetraflouroethylene Composites via Artificial Neural… DOI: http://dx.doi.org/10.5772/intechopen.95242*


**Table 4.**

*Sensitivity analysis results of each input parameter.*


#### **Table 5.**

*Results of t-test at 5% significance level.*

the volume loss of the polymer composites. This means that the various sliding distances can lead to different specific wear rate of the composites. The higher the applied load the more the composites will spend in the elastic deformation phases. With respect to the composites constitutions, volume fraction of the reinforcements had the greatest effect on the specific wear rate followed by density. This implies that as the volume fraction of the reinforcing phases was increased hardness with a corresponding increase in density that minimizes the specific wear rate of the composites. This agrees with the work of [13]. However, when all the parameters are compared it was found that volume fraction was the most influential and applied load presented the least effect on the specific wear rate of the composites.

The model's goodness of fit versus the actual values for the ANN and ANFIS models was tested using t-test at 5% level of significance and the outcomes revealed that there was no significance difference between the predicted and the actual values of the SWR. This was as shown in **Figures 6** and **7** and the t-test result was presented in **Table 5**.

### **4. Conclusions**

In this study, three various data driven models namely: feed forward neural network (FFNN), adaptive neuro fuzzy inference system (ANFIS) and multi linear regression (MLR) were applied in modeling and prediction of the specific wear rate (SWR) of polytetraflouroethylene (PFTE) composites. MLR model with DC of 0.5266 and RMSE of 0.2306 was found to be inefficient enough to predict the SWR of the composites. This is due to the complex and nonlinear relationship between the investigated variables and MLR model is usually good at establishing linear relationship between predictors and responses. FFNN model having DC equals 0.9802 and RMSE as 0.0471 was found to be capable in predicting the SWR of the PTFE reinforced composites. ANFIS model DC equal to 0.9967 was found to be talented in approximating the SWR of the composites. FFNN and ANFIS models were found to be highly qualitative in predicting the SWR of the composites, yet MLR model was found to be incapable in the same prediction scenario. The high prediction performance of the FFNN and ANFIS models is owing to their capability to deal with nonlinear, noisy and complex relationship which is typical of SWR of the polymer composites. Although, both ANFIS and FFNN models were capable of predicting the SWR of the composites, ANFIS was found to be more efficient in predicting the SWR of the composites than FFNN model. The sensitivity analysis of the built FFNN model indicated that sliding distance was the dominant parameter on the SWR of the composites in terms of the experimental conditions while volume fraction of the reinforcing phases was also influential parameter on the SWR with respect to the composites compositions. However, considering all the input parameters volume fraction of the reinforcements was the most dominant parameter and applied load was the least parameter influencing the SWR of the PTFE composites. The goodness of fit was rechecked using t-test at 5% significance level and the results affirmed the superiority of the FFNN and ANFIS models as powerful and efficient tools of modeling and prediction of SWR of the PTFE composites.
