## 4 Results and Discussion
### Variation of Membership with Different Number of Interference Data
This result reflects that the calculation of integrated memberships would depend more on \(x\) values of \(S_{\text{I}}\) than \(x\) values of \(S_{\text{un}}\), indicating the stage affiliation judgment would more rely on the valid information rather than the uncertain one. Condition (6) explains this situation. When intersecting with any other fuzzy set, \(S_{\text{un}}\) does not work, so that the influence brought by the interference information would be eliminated, and the interference data could be reasonably fused to make a reliable stage affiliation judgment. It could be concluded that the smaller the number of interference data belonging to \(S_{\text{I}}\), the higher the integrated membership of \(S_{\text{I}}\), the more credible this group of data, and the more reliable the judgment on the processing stage affiliation being stage I. When \(x\) values belonging to \(S_{\text{un}}\) appear, the determination of the processing stage would be more dependent on valid \(x\) values belonging to \(S_{\text{I}}\) than invalid \(x\) values pertaining to \(S_{\text{un}}\). Consequently, the model could authentically reflect the appearance of imprecise and conflict data, fusing all the information reasonably and make a reasonable judgment on the processing stage affiliation for stage I. The reliability of the model in addressing different numbers of interference data is guaranteed. The five groups of quantity ratios of _Methanosaeta_ to _Methanobacterium_ by introducing one to five interference data together with the non-interference quantity ratios for Stage II and III are shown in Figures 8a and 9a, and the corresponding integrated memberships of the four fuzzy sets are exhibited in Figures 8b and 9b. All the interfering \(x\) values belong to \(S_{\text{un}}\). For Stage II, \(C(S_{\text{II}})\) descends, \(C(S_{\text{un}})\) rises and \(C(S_{\text{II}})\) is the largest as the number of the interfering \(x\) increases, in consistent with the shifts of \(C(S_{\text{I}})\) and \(C(S_{\text{un}})\) of Stage I. Owing to the initial value of \(C(S_{\text{un}})\) obtained from the non-interference data group of Stage II being too small, the increase of \(C(S_{\text{un}})\) does not appear such apparently, but is still in an order of magnitude of growth. The range of the fluctuation of \(C(S_{\text{III}})\) is negligible, corresponding to that of Stage I. The apparent increase of \(C(S_{\text{I}})\) could be attributed to the relatively high initial value of \(C(S_{\text{I}})\), but is still below \(C(S_{\text{II}})\). For Stage III, \(C(S_{\text{III}})\) declines, \(C(S_{\text{un}})\) raises up and \(C(S_{\text{III}})\) is the  
Figure 7: Performance of the model fusing different numbers of interference data of Stage I: (**a**) Quantity ratios of _Methanosaeta_ to _Methanobacterium_ of Stage I with different numbers of interference data (For the x-coordinate, letters a to \(t\) represent the six paralleling data, numbers 0 to 5 represent the number of interference data.) (**b**) Integrated memberships of Stage I with different numbers of interference data. (Stage I stands for the state of sludge samples after treated by centrifugal thickening and pre-dewatering, Stage II stands for the state of sludge samples after treated by thermal hydrolysis, Stage III stands for the state of sludge samples after experiencing anaerobic digestion, and stage uncertain stands for the uncertainty of the processing stage affiliation of a sludge sample.) maximal as the number of interference data increases, which is the same with the trends of \(C(S_{\text{I}})\) and \(C(S_{\text{un}})\) of Stage I. \(C(S_{\text{II}})\) and \(C(S_{\text{I}})\) are in a relatively small range of change around 0, the same shifts as \(C(S_{\text{II}})\) and \(C(S_{\text{III}})\) of Stage I. For the data group with five interfering \(x\) values of Stage II or III, the integrated memberships generated are close to the normalized memberships obtained from the last non-interfering \(x\) value in this data group of Stage II or III, being 1.69 or 15.65.