Soft sensor based on localized semi-supervised relevance vector machine for penicillin fermentation process with asymmetric data

•An LSL algorithm is proposed to estimate the label for unlabeled data.•An LSRVM soft sensor is proposed for batch processes with asymmetric data.•A penicillin fermentation process is used to verify the proposed soft sensor.•The proposed soft sensor has better performance than other soft sensors. Due to operating conditions and on-site measurement instrument performance, the obtained batch process data present asymmetric characteristics. Compared with the symmetric data, it is more than a challenge to establish an effective soft sensor model for batch process with asymmetric data due to the following issues: (1) weak model performance due to the small amount of labeled data and (2) lack of an effective similarity evaluation between labeled data and unlabeled data with multiphase characteristics. To address these issues, a localized semi-supervised relevance vector machine-based soft sensor is proposed for multiphase batch processes with asymmetric data. First, a sequence-constrained fuzzy c-means algorithm is used to divide asymmetric data into phases. Then, a localized semi-supervised-based algorithm is proposed to estimate the label for unlabeled data of each phase. This algorithm consists of similar dataset construction based on a designed comprehensive similarity and label estimation based on just-in-time learning. On this basis, the soft sensor model for each phase is constructed based on relevance vector machine. Finally, an experiment of the penicillin fermentation process illustrates the effectiveness of the proposed method.