Abstract Soft sensors are the most commonly used tools to estimate the hard-to-measure variables in the chemical processes and other industries, mainly due to unknown mechanism, significant measurement delay and highly unacceptable costs. However, a small number of labeled data and a large number of unlabeled data are not fully investigated and coordination of them to train the models and to improve the prediction accuracy is even rare. Multiple tasks learning or multiple outputs learning adds more complexity to this problem. In this light, this paper proposed semi-supervised multiple-output learning soft sensor models with co-training MPLS (Multiple-output partial least squares), co-training MRVM (Multiple-output relevance vector machines), tri-training MPLS and tri-training MRVM. Co-training MPLS model is developed by extending the traditional co-training PLS model. Co-training MRVM is promoted by replacing MPLS with MRVM. Tri-training MPLS and tri-training MRVM are built by combining tri-training algorithm with MPLS and MRVM. The proposed four models can make full use of appropriate unlabeled data to optimize the regression model, and then to directly strengthen multiple-output variables prediction. These models were firstly demonstrated by a numerical example, then accessed by a wastewater plant (WWTP) simulated with well-established WWTP validation platform, Benchmark Simulation Model No. 1 (BSM1). The results proved that the proposed models were able to significantly improve the prediction performance and efficiency.

中文翻译：

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开发具有协同训练和三重训练 MPLS 和 MRVM 的半监督多输出软传感器

摘要 软传感器是化学过程和其他行业中最常用的估计难以测量的变量的工具，主要是由于机制未知、测量延迟大和成本高昂。然而，少量标记数据和大量未标记数据没有得到充分研究，协调它们来训练模型和提高预测精度甚至很少见。多任务学习或多输出学习增加了这个问题的复杂性。有鉴于此，本文提出了具有协同训练 MPLS（多输出偏最小二乘法）、协同训练 MRVM（多输出相关向量机）、三重训练 MPLS 和三重训练的半监督多输出学习软传感器模型。 - 训练 MRVM。Co-training MPLS 模型是在传统的 Co-training PLS 模型的基础上发展起来的。Co-training MRVM 是通过用 MRVM 代替 MPLS 来推广的。Tri-training MPLS和tri-training MRVM是将tri-training算法与MPLS和MRVM相结合构建的。所提出的四种模型可以充分利用适当的未标记数据来优化回归模型，进而直接加强多输出变量预测。这些模型首先通过数值示例进行演示，然后通过使用完善的污水处理厂验证平台、基准模拟模型 1 (BSM1) 模拟的污水处理厂 (WWTP) 进行访问。结果证明所提出的模型能够显着提高预测性能和效率。Tri-training MPLS和tri-training MRVM是将tri-training算法与MPLS和MRVM相结合构建的。提出的四种模型可以充分利用合适的未标记数据优化回归模型，进而直接加强多输出变量预测。这些模型首先通过数值示例进行演示，然后通过使用完善的污水处理厂验证平台、基准模拟模型 1 (BSM1) 模拟的污水处理厂 (WWTP) 进行访问。结果证明所提出的模型能够显着提高预测性能和效率。Tri-training MPLS和tri-training MRVM是将tri-training算法与MPLS和MRVM相结合构建的。所提出的四种模型可以充分利用适当的未标记数据来优化回归模型，进而直接加强多输出变量预测。这些模型首先通过数值示例进行演示，然后通过使用完善的污水处理厂验证平台、基准模拟模型 1 (BSM1) 模拟的污水处理厂 (WWTP) 进行访问。结果证明所提出的模型能够显着提高预测性能和效率。然后直接加强多输出变量预测。这些模型首先通过数值示例进行演示，然后通过使用完善的污水处理厂验证平台、基准模拟模型 1 (BSM1) 模拟的污水处理厂 (WWTP) 进行访问。结果证明所提出的模型能够显着提高预测性能和效率。然后直接加强多输出变量预测。这些模型首先通过数值示例进行演示，然后通过使用完善的污水处理厂验证平台、基准模拟模型 1 (BSM1) 模拟的污水处理厂 (WWTP) 进行访问。结果证明所提出的模型能够显着提高预测性能和效率。