## 2 System Detectability Based on Bayesian Networks
The parameterization in Equation (1) has also been applied to path modeling by [PERSON] [28]. The graphical model is developed to portray linear causal relationships based on sample correlations. Path modeling is preferable because it concisely illustrates the causal features of BNs. Let \(\Omega_{k}\) be an active path between \(F_{i}\) and \(F_{j}\), where \(k\) is the number of active paths between \(F_{i}\) and \(F_{j}\). Such an active path does not go against the direction of an arc after having gone forward. Next, the path coefficient, \(\delta(\cdot,\cdot)\), can be expressed as follows:  
\[\delta\big{(}F_{i},F_{j}\big{)}=\left\{ \begin{array}{ll}\rho_{i\to j}& if\big{\{}F_{i}\to F_{j}\big{\}}=\Omega_{k}\\ \sum\limits_{k}f_{k}^{-1}\big{(}\rho_{q\to r}\big{)}&for\quad other \big{\{}F_{q}\to F_{r}\big{\}}\in\Omega_{k}\end{array} \right., \tag{2}\]  
where \(\rho_{q\to r}=\prod_{i}\delta\big{(}F_{q},F_{r}\big{)}\) is the Pearson correlation along the incoming paths from \(F_{q}\) to \(F_{r}\), and \(\
u\) is the number of directed arcs in an active path from \(F_{q}\) to \(F_{r}\). The causal influences represented by a path coefficient can easily be read from the graph illustrating that model. Under the linear Gaussian parameterization of the BN in Equation (1), the causal impacts between two physical variables, \(F_{i}\) and \(F_{j}\), can be determined. Specifically, if there is one directed path from \(F_{i}\) to \(F_{j}\), it is easy to derive that the causal influence is equal to the product of all path coefficients on the corresponding directed path. Otherwise, the causal effect of \(F_{i}\) on \(F_{j}\) is the sum of all directed paths from \(F_{i}\) to \(F_{j}\). That is, the causal influences of \(F_{i}\) on \(F_{j}\), denoted by \(\pi_{i\to j}\), can be expressed as follows:  
\[\pi_{i\to j}=\left\{ \begin{array}{ll}\prod_{\lambda}\delta\big{(}F_{i},F_{j }\big{)}&,k=1\\ \sum\limits_{k}\{\prod_{\lambda}\delta\big{(}F_{i},F_{j}\big{)}\}&,k\geq 2\\ 1&,i=j\\ 0&,otherwise\end{array} \right. \tag{3}\]  
where \(k\) is the number of active paths, \(\Omega_{k}\); \(\lambda\) is the number of directed arcs in an active path from \(F_{i}\) to \(F_{j}\). To achieve the target product quality, the wet grinding process is always kept under control. An effective method for this is to build monitoring control charts on sensor outputs to detect abnormalities, known as assignable variation, as abnormalities are often relatively high in magnitude. Letting \(\mathcal{S}=\)[\(S_{1},\,S_{2},\,\ldots,\,S_{m}\)] be the sensor set to detect the assignable variation in the series of physical variables, \(\mathcal{F}=\{F_{1},\,F_{2},\,\ldots,\,F_{n}\}\) (\(m\leq n\)), then control charts can be built for the output of each sensor to flag and identify the occurrence points of system abnormalities. The single mean shift of physical parameters sensed is conducted to improve the wet-grinding system's detection accuracy. Assuming a single mean shift in \(F_{i}\) is \(\varepsilon_{i}\), the causal effect of an ancestor, \(F_{j}\), on this variable, \(F_{i}\), can be obtained as Equation (4); here, \(\varepsilon_{i}\) is the single mean shift in \(F_{j}\) induced by \(\varepsilon_{i}\). \[\varepsilon_{j}=\pi_{i\to j}\varepsilon_{i} \tag{4}\]  
Equation (4) shows how the mean shift \(\varepsilon_{i}\) in a variable \(F_{i}\) propagates to any variable \(F_{j}\) in the BN.