## 5 Conclusions  
This study proposes a soft sensor construction method named MOSE-MLV-VSPLS in an ensemble learning framework to accurately measure industrial processes' key parameters under complex data relationships. As an ensemble learning-based approach, at the base model generation stage, the proposed base model generation method first uses the MIC algorithm on each subsample space obtained by bootstrap random resampling to optimize the variable space. Then it uses the adaptive multi-kernel functions to obtain the latent variable space describing the data relationships from different perspectives, followed by the construction of the diversified base model. Compared with the EPLS algorithm, which does not use variable selection, and the LV-VSEPLS algorithm, which is based on a single kernel function, the proposed algorithm has the advantages of effectively removing redundant variables and significantly increasing the level of diversity of the underlying model's strong nonlinear data relationships.  
Since both the accuracy and diversity affect the performance of an ensemble learning-based soft sensor model, in the base model ensemble phase, a comprehensive diversity quantification indicator is designed, and the base model ensemble pruning, which considers the accuracy and diversity quantification metrics simultaneously, is transformed into a multi-objective optimization problem. The solution is performed using a multi-objective gray wolf optimization algorithm, which is an average ensemble of a selected set of high-performing and complementary base models. Compared with the full ensemble approach, the complexity of the ensemble modeling is significantly reduced.  
The comparative experimental results for two application examples show that the ensemble soft sensor model with the MOSE-MLV-VSPLS method achieves satisfactory performance by an effective diversity enhancement mechanism and the selective mean ensemble of smaller base models. The MOSE-MLV-VSPLS method has strong generality in homogeneous integration modeling. The ensemble strategy can be updated, and other base models can be used according to different application requirements.  
Figure 8: Prediction results of different methods on the debutanizer column test set. (**a**) Predictions of MLV-VSEPLS. (**b**) Predictions of MOSE-MLV-VSEPLS.  
It is worth noting that in this study, for the generation of the multi-kernel LVS in the base model construction, the kind of kernel function for each LVS is single, and in the future, hybrid kernel functions can be used for a better description of the data space. On the other hand, after converting the ensemble pruning problem into a multi-objective optimization, the success of the proposed method depends to a large extent on the design of a suitable diversity of definitions of the objective functions, which is open to question. As new process data continue to be added to the data space, the model update mechanism of this algorithm still needs further research and strengthening.  
Methodology, L.P.; software, L.P. and L.H.; validation, L.P.; formal analysis, L.H.; investigation, L.P. and Y.S.; resources, L.G.; writing--original draft preparation, L.P.; writing--review and editing, L.H.; supervision, L.G.; project administration, L.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the key research and development program of Shaanxi Province, China (Grant no. 2022 GY-134); the Science and Technology Foundation of Xi'an University of Architecture and Technology, China (Grant no. ZR19059); and the Special Scientific Research Project of the Education Department of Shaanxi Provincial Government of China under Grant no. 21 JK0732.  
Not applicable.  
Data Availability Statement: Data sharing is not applicable to this article, as no datasets were generated during the current study.  
The authors declare no conflict of interest.