# Compressible Diagnosis of Membrane Fouling Based on Transfer Entropy
## 1 Introduction  
Membrane fouling is an important factor affecting wastewater treatment processes with membrane bioreactors (MBRs), which can lead to the loss of membrane flux, deterioration of effluent quality, etc. [1, 2, 3]. To prevent this phenomenon, it is necessary to accurately diagnose future cases of membrane fouling before implementing efficient strategies [4, 5]. However, in MBRs, the biochemical process that triggers membrane fouling is complicated and involves many causal variables, such as aeration, reflux, and dosing. The relationship between these variables is time-varying and nonlinear [6, 7]. Therefore, it is challenging to establish an accurate diagnosis model for membrane fouling. To establish the causal relationship between the variables and membrane fouling, some scholars have studied the mechanism of membrane fouling to diagnose its occurrence directly [8, 9, 10, 11]. [PERSON] et al. analyzed the growth of filter cake in the process of low-pressure cross-flow microfiltration in MBR with fluid dynamics gauging [12]. Then, the diagnosis of membrane fouling was realized by quantifying the significance of membrane pore-level fouling phenomena at the early stage of filtration. In [13], a mechanism model was built for diagnosing membrane fouling by combining adenosine triphosphate and total cell count. The results displayed that this model was suitable for biological fouling diagnosis. In addition, MBR is affected by the flow distribution and hydraulic conditions in the reactor. A residence time distribution technique was developed to determine the impact of membrane geometry, orientation, and mixing efficiency on MBR performance [14]. [PERSON] et al. employed Raman chemical images to identify the morphology of membrane fouling [15]. This strategy could diagnose different types of membrane fouling with its observed appearance under specific static conditions. However, the above mechanisms are difficult to adapt to different working conditions. Since the correction of a large number of parameters is complicated, they are time-consuming for maintaining model accuracy. To resolve this problem, some data-driven diagnosis models, based on support vector regression (SVR), kernel function (KF), and artificial neural network (ANN) with strong adaptability, were established for membrane fouling [16; 17; 18; 19]. For example, [PERSON] et al. designed an SVR model with a LibSVM package to diagnose cases of membrane fouling in MBRs by mapping the relationship between extracellular polymer substances, organic loading rate, transmembrane pressure difference, and total membrane resistance [17]. The results showed that the relevant influencing factors of membrane fouling could be uncovered effectively. [PERSON] et al. proposed a multi-category diagnosis method based on KF for the detection and early warning of membrane fouling [18]. This method combined multiple binary classifiers to identify the causal variables of membrane fouling. [PERSON] et al. employed ANN to identify membrane fouling to minimize the risk of its occurrence [19]. The parameters of this model were updated based on the genetic algorithm, which was able to adapt to different operating conditions. Data-driven diagnosis models have the capability to sharpen the nonlinear relationship between input variables and output variables so that causal variables of membrane fouling can be distinguished. However, the existing models lack interpretability and struggle to determine interactions among different variables. Then, abundant variables with overlap and collinearity increase the complexity and confusion of diagnosis. To achieve a diagnosis process with interpretability, [PERSON] et al. simplified the causality diagram through Granger causality and maximum spanning trees to diagnose the root causal variables of process abnormalities [20]. However, Granger causality analysis is only applicable to the causality analysis of linear processes, which cannot explain the nonlinearity in membrane fouling. To conquer this challenge, [PERSON] et al.