## 4 Results and Discussion  
### Variation of Membership of Different Stage  
To prove the accuracy of the membership-fusing model, characterized by Equations (2), (4) and (5) with the determined parameters, a group of 18 sludge samples collected in the first day in September, 2019 was chosen to measure the mcrA gene sequences to verify whether the sequencing data satisfy the membership functions. Figure 6a exhibits the distribution of the relative abundance of different genera methanogens from all samples of the three stages, from which the value of quantity ratio of _Methanosaeta_ to _Methanobacterium_ could be obtained according to Equation (1). The six paralleling \(x\) values of stage I were calculated to be 7.15, 7.09, 7.01, 7.26, 7.08 and 6.98, between 2.01 and 7.45, within the range of \(x\) where the membership function of \(S_{\text{I}}\) predominates. This result demonstrates the processing stage affiliation of the tested paralleling samples being Stage I, which in turn reveals the applicability of \(B(S_{\text{I}})\) and \(A(S_{\text{I}})\). The six paralleling \(x\) values obtained from samples of Stage II are 1.63, 1.58, 1.62, 1.57, 1.71 and 1.69, ranging from 0 to 2.01, indicating these samples are most likely from  
Figure 4: Basic membership functions with the change of parameters \(\tau_{1}\) and \(\tau_{2}\). Figure 5: Normalized membership functions with the determined parameters. Stage II and the feasibility of \(B(S_{\text{II}})\) and \(A(S_{\text{II}})\). The six paralleling \(x\) values of Stage III are 15.69, 15.61, 15.73, 15.72, 15.70 and 15.65, all above 15.45, reflecting these samples most probably belong to Stage III and the fitness of \(B(S_{\text{III}})\) and \(A(S_{\text{III}})\). All \(x\) values generated from the 18 sludge samples distribute outside the range, from 7.45 to 15.45, showing the rationality of \(B(S_{\text{un}})\) and \(A(S_{\text{un}})\). These three results indicate the accuracy of the membership functions was verified in the aspect of the rational distribution of \(x\) values and the unique methanogen structure of each processing stage could also be properly distinguished by the model. For each processing stage, the four integrated memberships generated from the six paralleling samples are shown in Figure 6b, from which the judgment on the stage affiliation of a group of paralleling sludge samples could be made. The determined stage could then be checked to test if it corresponds to the stage where this group of paralleling samples were collected. For Stage I, the integrated memberships of \(S_{\text{I}}\), \(S_{\text{II}}\), \(S_{\text{III}}\), and \(S_{\text{un}}\) are 0.99, 0.00, 0.00 and 0.01, respectively, whose total value is 1.00 and \(C(S_{\text{I}})\) is the maximum. This consequence confirms that this group of paralleling samples most likely belong to Stage I, in consistent with the stage where the six paralleling samples were collected. The 99% possibility of this group of paralleling samples collected from Stage I reveals the vague boundary of the methanogen community structure owing to the relatively steady change of the inlet quality and running operation condition of different sampling time. The integrated memberships of \(S_{\text{I}}\), \(S_{\text{II}}\), \(S_{\text{III}}\), and \(S_{\text{un}}\) for Stage II are 0.02, 0.98, 0.00 and 0.00, whose sum is 1.00 and \(C(S_{\text{II}})\) is the largest. This result indicates that the judgment on the processing stage affiliation of these six paralleling sludge samples being Stage II, corresponding with where these samples were gathered. The possibility of 98% belonging to Stage II rather than 100% uncovers the relatively small fluctuation of the wastewater quality treated by thermal hydrolysis of different sampling time. The 2% possibility being Stage I elucidates the higher similarity of the methanogen community structure between Stage I and II compared with that between Stage II and III, or Stage I and III.