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• A new ICA algorithm by concurrent consideration of independence and slowness is developed. • SICA is no longer restricted by the non-Gaussian hypothesis. • SICA explores the temporal information of slow varying to sort the features. • The supervised SICA is also deduced for the classification problem. For the process data analytics, numerous statistical methods are designed to extract informative features to reveal latent characteristics and correlation patterns among the variables. As a classic statistical method, independent component analysis extracts mutually independent components by searching for the maximum direction of non-Gaussianity from the mixed signals. However, the algorithm does not work for the latent Gaussian components, and the disorder of extracted features also brings trouble to the practical applications. In the present work, a slow independent component analysis algorithm is proposed to tackle the problems by concurrently considering the high-order statistic and slowness. It constructs a dual-objective optimization criterion function, simultaneously incorporating independence and slow varying characteristics into the feature extraction procedure, which ties together the ideas of independent component analysis and slow feature analysis under the same mathematical umbrella. Through adjusting the controllable sub-optimization objective weight, the proposed slow independent component analysis adds insight into the different roles of independence and slow varying characteristics in calibration modeling, and, thus, provides possibilities to combine the advantages of independent component analysis and slow feature analysis. Compared with the traditional method, the new algorithm no longer relies solely on higher-order statistic but also utilizes the slow varying characteristic to extract latent Gaussian or non-Gaussian components from mixed signals. Besides the unsupervised paradigm, the proposed algorithm is extended to the supervised form for classification task by utilizing the temporal structure as well as the label information. The effectiveness of the proposed algorithm is illustrated by blind source separation task and industrial fault diagnosis task using real-world datasets including speech signals and multiphase flow facility. [ABSTRACT FROM AUTHOR]