Reshaping an adaptive semi-supervised multi-output soft sensor for wastewater treatment process monitoring with stationary wavelet time difference and co-training algorithm

In wastewater treatment processes, effluent variables often exhibit a strong multiple variable nexus, high nonlinearity, and time-varying characteristics. The scarcity of labeled data usually adds further complexities for soft sensor modeling. To address these issues, this paper proposed an adaptive semi-supervised multi-output soft sensor with the help of the time difference technique and the co-training algorithm (ST-Co MGPR-MBiLSTM). Firstly, a semi-supervised temporal network is constructed by co-training the multi-output Gaussian process regression (MGPR) and the multi-output Bi-directional long and short-term memory network (MBiLSTM), which can extract temporal features from an unlabeled dataset and establish complex relationships between input variables and multi-output variables for a simultaneous accurate multivariate prediction. During the semi-supervised training process, pseudo-label selection guided by uncertainty and reliability is employed to control pseudo-label quality. Secondly, we apply the stationary wavelet transform (SWT) to the inputs and outputs, and retain only the level-J approximation as a denoised representation; subsequently, we perform first-order time differencing (TD) on the inputs and outputs. This method can preserve temporal alignment, stabilize the data distribution, and capture local dynamics, thereby enabling robust dynamic behavior modeling. Furthermore, the model parameters are optimized by using Kalman gain coefficients, which further strengthen the model's robustness in dynamic environments. The effectiveness of the proposed soft sensor is validated using real-world and simulated wastewater treatment data.