Table 1.
Average values for observed (OBS) growth rate (Gr, h−1) and average values estimated by ANN [10] and by ANFIS models of Leuconostoc mesenteroides growth under aerobic conditions during model development with the training data set [10].
Table 2.
Average values for observed (OBS) growth rate (Gr, h−1) and average values estimated by ANN [10] and by ANFIS models of Leuconostoc mesenteroides growth under anaerobic conditions during model development with the training data set [10].
Figure 1.
Framework of adaptive network-based fuzzy inference system.
Layer 1: fuzzification layer; Layer 2: production layer; Layer 3: normalization layer; Layer 4: de-fuzzification layer; Layer 5: total output layer.
Figure 2.
Flowchart of ANFIS for predicting the growth rate of Leuconostoc mesenteroides.
In the training process, the number of membership functions is set for each input parameter until prediction performance is satisfactory. After the training procedure obtains the training results, the testing data are input into the trained ANFIS model to obtain the testing results.
Table 3.
Average values for observed (OBS) growth rate (Gr, h−1) and average values estimated by ANN [10] and by ANFIS models of Leuconostoc mesenteroides growth under aerobic conditions using testing data set [10].
Table 4.
Average for observed (OBS) growth rate (Gr, h−1) and average values estimated by ANN [10] and by ANFIS models of Leuconostoc mesenteroides growth under anaerobic conditions during model development with the testing data set [10].
Table 5.
Comparison of performance indices between ANN and ANFIS models under aerobic conditions.
Table 6.
Comparison of performance indices between ANN and ANFIS models under anaerobic conditions.
Table 7.
Five bootstrap data sets derived from the original data set under aerobic conditions.
Table 8.
Five bootstrap data sets derived from original data set under anaerobic conditions.
Figure 3.
Fuzzy rule architecture of the Gaussian membership function.
The four input (temperature, pH, NaCl, NaNO2) and one output (growth rate) parameters for the adaptive network-based fuzzy inference system model were used to the predict growth rate of Leuconostoc mesenteroides under aerobic and anaerobic conditions. Each input parameter divides three Gaussian membership functions (i.e. small, medium and large areas). The number of fuzzy rules is 81.
Figure 4.
Final Gaussian membership functions of the four input parameters derived by training under aerobic and anaerobic conditions.
The adaptive network-based fuzzy inference system was trained using 30 sets of experimental data in 100 learning cycles under aerobic and anaerobic conditions. The final Gaussian membership functions were obtained for the four inputs under aerobic conditions (A) temperature, (B) pH, (C) NaCl and (D) NaNO2 and under anaerobic conditions (E) temperature, (F) pH, (G) NaCl and (H) NaNO2. Each input divides three Gaussian membership functions (i.e., small, medium and large areas).
Figure 5.
Comparison of actual growth rates for Leuconostoc mesenteroides and growth rates predicted by ANFIS model and by ANN model under aerobic conditions.
Under aerobic conditions, all specific growth rates predicted by the ANFIS models when using the (A) training data set and the (B) testing data set were closer to the 45° line compared to the rates predicted by the ANN models, which confirmed the superior prediction accuracy of the ANFIS models.
Figure 6.
Comparison of residual values (predicted–observed growth rate) of Leuconostoc mesenteroides obtained by ANFIS model and by ANN model under aerobic conditions.
The spread of residual values was narrower for the ANFIS models for (A) the training data set and for (B) the testing data set, which indicated their better prediction performance under aerobic conditions.
Figure 7.
Actual values for growth rate of Leuconostoc mesenteroides compared with values predicted by ANFIS model and by ANN model under anaerobic conditions.
Under anaerobic conditions, the growth rates predicted by ANFIS models in (A) the training data set and in (B) the testing data set were closer to the 45° line compared to those predicted by the ANN models. In other words, the predictive accuracy of the ANFIS models was higher than that of ANN models under anaerobic conditions.
Figure 8.
Comparison of residual values (predicted–observed growth rate) of Leuconostoc mesenteroides obtained by ANFIS model and by ANN model under anaerobic conditions.
The better prediction performance of the ANFIS models under anaerobic conditions was confirmed by their narrower spread of residual values for (A) the training data set and for (B) the testing data set.
Figure 9.
The input layer includes nodes for four inputs (temperature, pH, NaCl, and NaNO2). In the hidden layer, three nodes transfer data to the output layer via a separate weighted logistic transfer function. In the output layer, one node (growth rate) transfers data via a separate weighted identity transfer function to obtain the predictive output.
Table 9.
Comparison of original data set and five bootstrap data sets in terms of mean values for four inputs and one output under aerobic conditions (ANOVA test).
Table 10.
Comparison of original data set and five bootstrap data sets in terms of mean values for four inputs and one output under anaerobic conditions (ANOVA test).
Table 11.
Comparison of performance indices between ANN and ANFIS models using 30-bootstrap data set with 10-fold cross-validation under aerobic conditions.
Table 12.
Comparison of performance indices between ANN and ANFIS models using 30-bootstrap data set with 10-fold cross-validation under anaerobic conditions.
Figure 10.
Sensitivity analysis of four input variables under aerobic conditions using ANFIS model.
Under aerobic conditions, the sensitivity values for temperature, NaCl, pH and NaCO2 were 0.88, 0.13, 0.07 and 0.04, respectively. The most influential (sensitive) parameter affecting the growth rate of Leuconostoc mesenteroides under aerobic conditions was temperature and, to a lesser extent, NaCl.
Figure 11.
Sensitivity analysis of four input variables under anaerobic conditions using ANFIS model.
Under anaerobic conditions, the sensitivity values for temperature, NaCl, NaCO2 and pH were 0.46, 0.32, 0.25 and 0.19. The most influential (sensitive) parameters affecting the growth rate of Leuconostoc mesenteroides under anaerobic conditions were temperature and, to a lesser extent, NaCl.
Figure 12.
Sensitivity analysis of four input variables under aerobic conditions using ANN model.
Under aerobic conditions, the sensitivity values for temperature, NaCl, pH and NaCO2 were 0.20, 0.06, 0.04 and 0.02, respectively. The most influential (sensitive) parameter affecting the growth rate of Leuconostoc mesenteroides under aerobic conditions was temperature and, to a lesser extent, NaCl.
Figure 13.
Sensitivity analysis of four input variables under anaerobic conditions using ANN model.
Under anaerobic conditions, the sensitivity values for temperature, NaCl, NaCO2 and pH were 0.52, 0.16, 0.09 and 0.16. The most influential (sensitive) parameters affecting the growth rate of Leuconostoc mesenteroides under anaerobic conditions were temperature and, to a lesser extent, NaCl.