# Compressible Diagnosis of Membrane Fouling Based on Transfer Entropy
## 4 Results and Discussion
### Results of Membrane Fouling Diagnosis Model  
According to the results of feature variable selection, COD, influent NH\({}_{3}\)-N, influent TP, ORP in the anaerobic zone, DO in the aerobic zone, sludge concentration in the aerobic zone, effluent flow, aeration, effluent turbidity, water temperature, pH, and TMP are determined as the related variables for the diagnosis of membrane fouling and replaced by numbers 1 to 12, respectively. Once a membrane fouling incident occurs, we can trace the causal variables from them. They are used as the inputs of the CDM. In total, 2000 groups of data are selected as samples to construct the initial FTT as shown in Figure 8. The adjacency matrix between variables is shown in **A\({}_{\textbf{1}}\)**. It can be found from Figure 8 that the FTT has high complexity, so it is necessary to extract stronger causality. The FTT can be simplified by setting a threshold value. The adjacency matrix between variables is shown in **A\({}_{\textbf{2}}\)**. As shown in Figure 9, the simplified FTT is obtained, which can represent the relationship between fault variables.  
As shown in Figure 10, the information compressible strategy is used to further simplify the fault transfer topology. The adjacency matrix between the variables is shown in **A\({}_{\textbf{3}}\)**. In this experiment, the root causal variables of membrane fouling are diagnosed according to the simplified fault transfer topology, and the effectiveness of the CDM is compared with other diagnosis methods.  
Figure 6: The RMSE during the training process.  
Figure 7: The RMSE during the detection process.  
Figure 8: Initial fault transfer topology.  
Figure 9: Fault transfer topology after setting the threshold.  
\[\mathbf{A_{3}}=\left(\begin{array}{cccccccccccc}0&0&0&0.810403&0&0&0.942594&1.24 7714&0&0&0&1.058295\\ 0&0&0&0.730733&0&0&1.000398&1.228595&0&0&0&1.100775\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0.822288&0&0&0&0&0.958113\\ 0&0&0&0.756785&0&0&0&1.065932&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1.267596&0&0&0&1.146267\\ 0&0&0&0.846681&0&0&1.018529&1.201856&0&0&0&1.119884\\ 0&0&0&0&0&0&0&0&0&0&0\end{array}\right) \tag{19}\]  
To evaluate the diagnostic efficiency of the fault transfer topology (FTT) with an information compressible strategy (ICS) and a threshold, the results were compared with some other methods: FTT with threshold, initial FTT, Bayesian network (BN), ANN, and fuzzy logic (FL). The comparison results of FTT with ICS and a threshold to other methods are related to diagnosis time, number of connections, and accuracy. In Table 2, it can be seen that the proposed FTT with ICS and a threshold achieves the lowest number of connections compared to FTT without ICS or a threshold. It means that the designed ICS and threshold can simplify the failover topology which also contributes to speeding the diagnosis of membrane fouling. In addition, the accuracy of FTT with ICS and a threshold is also best compared to other methods, which indicates that the proposed FFT in the CDM is in favor of exploring the root causal variables of membrane fouling.  
Figure 10: Fault transfer topology based on information compressible strategy.  
**Funding:** This work was supported by the National Natural Science Foundation of China under Grants 61890930-5 and 61622301, the National Key Research and Development Project under Grants 2018 YFC1900800-5, and the Beijing Natural Science Foundation under Grant 4172005.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The datasets generated during and/or analyzed during the current study are not publicly available due to corporate privacy but are available from the corresponding author on reasonable request.  
**Conflicts of Interest:** The authors declare no conflict of interest.