3.2. Statistical Analysis
To determine the regression equation using multiple regression analysis, HRC and THR were selected as the independent variables with heating rate and sample mass being the dependent variables. The descriptive statistics of the input data are listed in Table 2
. The analysis of variance showing the influence of HRC and THR on heating rate and sample mass in this regression analysis is presented in Table 3
and Table 4
. It is clearly seen that the dependent variables have a greater significance in the estimation of HRC than THR. The test statistic of HRC has an F-value of 53.85, which is larger than the critical value
. This analysis signifies that there is a significant statistical difference in the means of the variables. However, the F-value for THR is quite smaller than the critical value; hence, the null hypothesis for equal population means cannot be rejected.
In Table 5
, the multiple linear regression model summarized for HRC and THR are presented. It can be seen that the adjusted R-square for HRC is higher than THR. HRC has a linear relationship with sample mass and heating rate, while THR is almost constant throughout the range of heating rates applied. It should be noted that to get a very accurate prediction of these flammability parameters, especially for THR, a method that can handle non-linear modeling could be used. Hence, the next section applies ANFIS networks in the prediction of HRC and THR.
3.3. ANFIS Network Prediction Results
The present study employed ANFIS networks to model the relationship between sample mass, heating rate, heat release capacity and total heat release rate measured from the MCC experiment. To develop the ANFIS model, the hold-out data splitting technique was adopted. Twenty-four randomly selected data-points out of the 28 experimental data were used for training, while the remaining 4 represented the test data for the model. For improved accuracy, the test data covered the entire range of the available dataset. Table 6
shows the datasets used for developing the models [14
The membership function for the model was selected by trial and error, and the hybrid learning algorithm was adopted for the training process. The model structure for HRC and THR, as illustrated in Figure 5
, consists of two inputs, three membership functions for each input and one output.
Three logical operators—and, or and not—are adopted in ANFIS applications. However, depending on the fuzzy logic rules extracted, any of the operators can be used to suit the structure of input data. In this research, only the ‘and’ logical operator was utilized.
The neurons in Layer 3 consist of fuzzy rules, the conditions of each rule and the consequences. The fuzzy IF-THEN rules generated for the membership functions from the input data of the developed models are detailed from 1–10. These conditional statements describe how the outputs were formulated according to the three membership functions applied.
If (input1 is in1mf1) and (input2 is in2mf1), then (output is out1mf1) (1).
If (input1 is in1mf1) and (input2 is in2mf2), then (output is out1mf2) (1).
If (input1 is in1mf1) and (input2 is in2mf3), then (output is out1mf3) (1).
If (input1 is in1mf2) and (input2 is in2mf1), then (output is out1mf4) (1).
If (input1 is in1mf2) and (input2 is in2mf2), then (output is out1mf5) (1).
If (input1 is in1mf2) and (input2 is in2mf3), then (output is out1mf6) (1).
If (input1 is in1mf3) and (input2 is in2mf1), then (output is out1mf7) (1).
If (input1 is in1mf3) and (input2 is in2mf2), then (output is out1mf8) (1).
If (input1 is in1mf3) and (input2 is in2mf3), then (output is out1mf9) (1).
The models were trained using 100 iterations. The modeling parameters for the developed ANFIS network after the training process are as listed in Table 7
Plots of experimental data against predicted data from the HRC ANFIS model during training and testing are illustrated in Figure 6
, Figure 7
and Figure 8
. From the simulation, the minimal training Root Mean Squared Error (RMSE) was 0.0224, while the average testing error obtained was 0.625. It is quite clearly seen that the predicted data show a close proximity to the experimental data. A surface plot demonstrating the relationship between the predicted HRC, sample mass and heating rate is presented in Figure 8
. The shape of the curve is similar to the one illustrated in Figure 3
; hence, the plots from the ANFIS model show that the model has a high predictive ability.
Similarly, plots of experimental and predicted THR datasets were obtained from the Neuro-Fuzzy Designer app. The minimum average training and testing RMSE for THR were 0.00781 and 0.9395, respectively.
indicates the performance of ANFIS models in estimating THR and HRC from the MCC experiment. The basic attributes considered are the adjusted R-squared and the root mean squared errors. It is quite obvious from Table 8
that the RMSEs in all the predictions are less than one, indicating an excellent performance. Although, the training of THR outperformed the other models in terms of prediction errors, no obvious differences can be observed. The learning ability of the developed THR model was more accurate than the generalization one, as shown in the training and testing plots (Figure 9
, Figure 10
and Figure 11
). Furthermore, the training of the THR model was better than the HRC model, while the test results of HRC outperformed THR. In general, the predictions were in good agreement and fitted the experimental data accurately. Considering the R2
values obtained, one notable conclusion can be made: the model predicted HRC better than THR since both training and testing of HRC had the best results. This is due to the fact that HRC has a direct and significant statistical relationship with the input parameters, whereas THR is almost constant at any given heating rate and sample mass, thus presenting an uneven statistical distribution. It should also be noted that the test results are an indication of the excellent ability of the developed models to predict data beyond the limits of the training range.
The average training and testing errors in the present study have been compared with the results obtained from prediction of HRC and THR with the feed-forward back propagation neural network (FFBPNN) by Mensah et al. [14
]. Table 9
shows the RMSE obtained from both the models. Similarly, Figure 12
, Figure 13
, Figure 14
and Figure 15
give a visual representation of the variations in the predicted data from the ANFIS and FFBPNN models.
Although the results from both ANFIS and FFBPNN models are seemingly good, the comparison in Table 9
indicates the presence of significant differences in the attainable prediction errors as well as the training time. From the table, the ANFIS models attained very low average errors in all cases (both training and testing). The high errors presented in the ANN models could be attributed to the limited amount of data used for the simulation. The results further affirm the superiority and accuracy in the application of ANFIS over ANN.
The combination of fuzzy reasoning and artificial neural networks optimizes and improves the learning and generalization capabilities of models. The ability of the system to tackle non-linearities in datasets is also greatly enhanced. This improvement can be observed in the application of ANFIS in flammability studies covered in the present study. The insignificant RMSE values obtained show that ANFIS is suitable for predicting HRC and THR from MCC experiments. With sufficient training, testing data and the right selection of input parameters, this modeling method can be accurately extended to a double scale analysis, such as the prediction of cone calorimeter test data from MCC test results.