# Compressible Diagnosis of Membrane Fouling Based on Transfer Entropy
## 1 Introduction
proposed a multi-level interpretable logic tree to clarify the nonlinear relationship between root causes, intermediate causes, and faults [21]. In addition, several intelligent tools are also introduced to explain the nonlinear process of membrane fouling. For example, [PERSON] et al. developed an accident-relevance tree based on the analysis of the formation mechanism of quality accidents [22]. The method located the root causes of quality accidents utilizing the fuzzy mechanism and the vague nature of datasets. [PERSON] et al. combined the decision tree learner and ANN to diagnose power transformer faults of membrane fouling, which reduced the calculation cost and improved the accuracy of fault classification simultaneously [23]. Other similar nonlinear methods can be observed in [24; 25]. However, the methods mentioned in [21; 22; 23; 24; 25] only focus on the causality between local variables by constructing a tree causality diagram, rather than the interaction between all relevant variables. To provide the causal variables of faults comprehensively, [PERSON] et al. synthesized principal component analysis (PCA) and Bayesian network to capture the nonlinear dependence of high-dimensional process data [26]. Then, the root causal variables of faults were diagnosed with the discretization of continuous data. In [27], a Bayesian network was developed to describe the relationship between alarm variables and root causal variables in thermal power plants. The parameters of the network were updated in a recursive way, which promoted the accurate detection of the root causal variables. Furthermore, [PERSON] et al. proposed a recursive kernel PCA and Bayesian network to diagnose sludge bulking in the wastewater treatment process [28]. This method effectively captured the nonlinear and time-varying characteristics of sludge bulking to diagnose the root causal variables with high accuracy. However, once the diagnosis models in [26; 27; 28] were constructed with given datasets, they always held the invariant information transfer path due to their complexity. When the operating conditions of MBR are changed frequently or drastically, it may be difficult to maintain acceptable accuracy for those models. Additionally, the relationship between those variables of membrane fouling exhibits both time-varying and nonlinear characteristics primarily because membrane fouling is a dynamic and complex process that is influenced by multiple, interacting factors. These characteristics can make it challenging to diagnose membrane fouling effectively. Based on the above analysis, this paper proposes a compressible diagnosis model (CDM) based on transfer entropy. This model is used to depict the fault transfer topology (FTT) of membrane fouling and further explore the root cause following the operating conditions. The novelties of this work are as follows. 1. Based on the mechanism analysis associated with membrane fouling, the relationship between causal variables and membrane fouling is clarified with the feature extraction algorithm. Then, the related variables of membrane fouling are obtained under different operating conditions. Different from the data-driven diagnosis models in [15; 16; 17; 18; 19] with given causal variables, the feature extraction algorithm will enable the proposed CDM to transform raw data into informative representations that can be utilized for diagnosis. 2. Instead of using a mapping relationship with simple input-output representation such as decision trees [21; 22] and the Bayesian network [24; 25], a topology based on transfer entropy is constructed. This approach not only provides a qualitative evaluation of the causal relationships between variables by observing the dynamic transfer path, but it also offers a quantitative description of those variables. It helps uncover the path of fault occurrence and further obtain the fault cause priority that may change over time as the operating conditions change. 3.