Multi-mode industrial soft sensor method based on mixture Laplace variational auto-encoder

Tianming Zhang, Gaowei Yan*, Rong Li, Shuyi Xiao, Yusong Pang

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


The industrially collected process data usually exhibit non-Gaussian and multi-mode characteristics. Due to sensor failures, irregular disturbances, and transmission problems, there are unavoidable outliers that make the data exhibit heavy-tailed characteristics. To this end, a variational auto-encoder regression method based on the mixture Laplacian distribution (MLVAER) is proposed, by introducing a type-II multivariate Laplacian distribution in the latent variable space for robust modeling, and further extending it to the mixture form to accommodate multi-mode processes, the corresponding reparameterization trick is finally proposed for the mixture form of this distribution for neural network gradient descent training. The model based on this distribution assumption has higher degrees of freedom than the model based on the traditional multivariate Laplace distribution assumption when the network structure is the same. Numerical simulation and experiments on two industrial examples demonstrate that the proposed algorithm reduces the root mean square error by over 15% compared to other algorithms.

Original languageEnglish
Article number114435
JournalMeasurement: Journal of the International Measurement Confederation
Publication statusPublished - 2024

Bibliographical note

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Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


The authors are grateful for the financial support from the National Natural Science Foundation of China ( 61973226 , 62003233 ), the Gemeng Science and Technology Innovation Foundation Project ( 2022-05 ), the Shanxi Province Major Special Program of Science and Technology ”Unveiling and Commanding” Project ( 202201090301013 ). and Shanxi Province Science Foundation for Youths, China ( 202203021222101 ).


  • Heavy tail
  • Mixture Laplace
  • Multi-mode
  • Soft sensor
  • Variational auto-encoder


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