## 3 Experimental Settings and Results Analysis  
### Experimental Settings  
The data used in the experiment are sourced from the medium and heavy plate production line of a steel plant, with a total of 19,616 data points collected. These include  
Figure 10: Bayesian neural network flow chart based on Adam variational inference. (**a**) Training part. (**b**) Prediction part.  
the energy consumption per ton of steel \(Y\) for each batch, along with corresponding energy consumption values \(y_{1},y_{2},y_{3}\). The anomalies in the data, as well as their relationships with specific variables and factors, have been labeled by on-site experts. All the following experiments were completed in Python.  
Experiment 1: Anomaly detection based on box plot  
According to Algorithm 1, the anomalies in the energy consumption per ton of steel \(Y\) for each batch are first identified. Then, the anomalies in the sub-energy consumption values \(y_{i}\) are investigated based on these results. Finally, the identified anomalies are compared with the labeled anomalies to verify the effectiveness of the algorithm. The threshold for anomaly detection was set based on the IQR method, as mentioned above. The specific parameters, such as factor 1.5 in the IQR calculation, were selected based on the nature of the data, ensuring that the threshold was sensitive enough to detect significant deviations without flagging too many false positives.  
Experiment 2: Root cause tracing based on Adam variational inference Bayesian neural network  
The time series data of each process variable \(x(t-l)\) and \(x(t)\) are used as inputs and outputs for the LSTM model, with RMSE (Root Mean Square Error) as the evaluation metric to measure the accuracy of the model's predictions. Based on the obtained Granger coefficients \(a_{ij}\) a causal relationship diagram is drawn and the weighted coefficients \(\mathcal{W}_{ij}\) for each path are calculated to identify the optimal causal path. The performance is compared with traditional Granger causality analysis (GC), neural network-based Granger causality analysis (NN-GC), LSTM-based Granger causality analysis (LSTM-GC), and the proposed LSTM-based weighted Granger causality analysis (LSTM-WGC). The relevant parameter settings are shown in Table 7.  
The correspondence between process variables and factors in the actual production process, derived from historical data, is converted into vectors according to (26) and used as the input and output for the Bayesian network. ELBO is used as the evaluation metric, and Adam is employed to accelerate the minimization of ELBO, thereby approximating \(\Pr(w|X,Z)\) with \(q(w)\). The model's performance is compared with three optimization algorithms: Stochastic Gradient Descent (SGD), Stochastic Gradient Descent with Momentum (SGD with momentum), and Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithms. The relevant parameter settings are shown in Table 8.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Input layer node & 9 \\ Hidden layer node & 25 \\ Output layer node & 9 \\ LSTM layers & 2 \\ Activation function & Sigmoid function \\ Error back propagation & Derivative of sigmoid function \\ Threshold & 0 \\ Loss function & Mean square error function \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameters of the experiment causality analysis.