## 2 Theory and Methodology
### Improved Particle Filter Online Recognition Method  
The conventional particle filter (PF) suffers from limitations including suboptimal importance density selection, particle degeneracy, and high computational burden. Furthermore, direct processing of anomalous measurements without proper treatment introducesbias in model predictions [27]. To address these challenges while mitigating the impact of abnormal data on estimation accuracy, this study proposes an improved particle filtering framework that integrates statistical modeling of multi-type anomalies [30]. The state vector (particle) \(M\) is randomly generated by the probability density function \(f(y_{0})\) of the initial state vector: \(y_{0}\)(\({}^{0}\)), \(i=1,2,\ldots,M\); \(y_{0}\) is the seepage pressure state vector. Taking into account the deployment of \(N_{Z}\) during seepage pressure monitoring for a sensor, the measurement equation can be expressed as in Equation (1):  
\[A_{k}\ =\ h(y_{k})+r_{k} \tag{1}\]  
where \(y_{k}\in R^{N_{Z}}\) is the state vector of the system at \(k\) time; \(A_{k}\in R^{N_{Z}}\) is the measurement vector at \(k\) time; \(h(\bullet)\) is the measurement function; \(r_{k}\in R^{N_{Z}}\) is the measurement noise vector at \(k\) time, which can be simulated as a normal distribution with zero mean and \(R_{k}\) covariance matrix. After measuring the quantity \(A_{k}\) at \(k\) time, the likelihood function of \(p\Big{(}A_{k}\Big{|}y_{k|k-1}^{(i)}\Big{)}\), each particle \(y_{k|k-1}^{(i)}\) can be calculated in Equation (2):  
\[ \begin{array}{l}\rho\Big{(}A_{k}\Big{|}y_{k|k-1}^{(i)}\Big{)}\propto\frac{ 1}{(2\pi)^{\frac{N_{Z}}{2}}\ |R_{k}|^{1/2}}\exp\\ \bigg{\{}-\frac{1}{2}\Big{(}\Big{(}A_{k}-h\Big{(}y_{k|k-1}^{(i)}\Big{)}\ \ R_{k}^{-1}\Big{(}A_{k}-h\Big{(}y_{k|k-1}^{(i)}\Big{)}\ \ ^{T}\Big{\}}\ \ \ \ \ i=1,2,\cdots,M\end{array}  \tag{2}\]where \(m_{k}^{0l}(l=1,\cdots,N_{L})\) is the \(l\) component of the vector \(m_{k}^{0}\), and \(m_{k}^{0l}\) is simulated as a uniform distribution on the interval \(\left[a^{l},b^{l}\right]\). \(l_{k,n}\in R^{N_{L}}\), \(n\in\{0,1,2\}\) is a multi-Bernoulli random vector, which can detect the occurrence of multiple types of outliers on different measurement channels. It is denoted as in Equation (8):  
\[l_{k,n}\ =\ \left[\delta(l_{k}(1)-n),\cdots,\delta(l_{k}(N_{L})-n)\right]^{T} \quad n\in\{0,1,2\} \tag{8}\]  
where \(\delta\) is the Dirac function; vectors \(l_{k,n}\) are used to represent the occurrence, location, and timing of different types of outliers in multiple measurement channels. For example, \(l_{k,0}\ =\ \left[1,1,\cdots,1\right]^{T}\) represents no outliers in the monitored data at a \(k\) moment, \(l_{k,1}\ =\ \left[1,0,\cdots,0\right]^{T}\) represents the first measurement channel showing random errors at a \(k\) moment, and \(l_{k,2}\ =\ \left[0,1,\cdots,0\right]^{T}\) represents coarse errors appearing in the second measurement channel at a \(k\) moment. The elaboration of the probability of outliers in monitoring data based on Bernoulli probability theory is grounded in the fact that outlier identification is essentially a type of binary classification problem, to which the Bernoulli probability model is well suited for probabilistic description. Thus, the binary nature of outlier detection aligns closely with the Bernoulli probability statistical method, and the model architecture is interpretable and easy to understand. Given the assumption of independence in Bernoulli experiments--where the outcome of each trial does not affect the others--the anomaly status of each data point can be analyzed independently during outlier detection without considering the influence of other data points. This significantly reduces the computational complexity of the detection process and improves processing efficiency.