Causal discovery in multi-rate time series encounters greater challenges compared to regular time series. This stems from a potential problem that has not been noticed and explored in existing studies:information granularity heterogeneity, which refers to the natural difference in information granularity between fast sampling rate data (high information granularity) and slow sampling rate data (low information granularity). Such an imbalance in information granularity can hinder forecasting relationships modeling and induce biased causal learning. Therefore, we propose aMutual Information-iNspired causalDiscovery framework (MIND), aiming to derive rate-agnostic features with consistent information granularity to alleviate information granularity heterogeneity problem. Technically, MIND comprises Stage 1 (pre-training) and Stage 2 (fine-tuning and causal discovery). In Stage 1, empowered by pseudo-slow sampling rate data (generated through the interleaved down sampling strategy) and mutual information, we can eliminate the influence of sampling rates and drive rate-aware encoders (RAEs) to sense key information (i.e., rate-agnostic) that remains unchanged across varying sampling rates. In Stage 2, the well-trained RAEs can extract rate-agnostic features from real multi-rate time series, thus facilitating effective forecasting relationships modeling and yield accurate causal discovery. Empirically, MIND realizes superior performance on various multi-rate scenarios, including four simulation datasets and one real-world dataset.

Data-driven methods have been extensively utilized in establishing predictive models from historical data for process monitoring and prediction of quality variables. However, most data-driven approaches assume that training data and testing data come from steady-state operating regions and follow the same distribution, which may not be the case when it comes to complex industrial processes. To avoid these restrictive assumptions and account for practical implementation, a novel online transfer learning technique is proposed to dynamically learn cross-domain features based on the variational Bayesian inference in this work. Stemming from the probabilistic slow feature analysis, a transfer slow feature analysis (TSFA) technique is presented to transfer dynamic models learned from different source processes to enhance prediction performance in the target process. In particular, two weighting functions associated with transition and emission equations are introduced and updated dynamically to quantify the transferability from source domains to the target domain at each time instant. Instead of point estimation, a variational Bayesian inference scheme is designed to learn the parameters under probability distributions accounting for corresponding uncertainties. The effectiveness of the proposed technique with applications to soft sensor modelling is demonstrated by a simulation example, a public dataset and an industrial case study.

Higher-order cumulants analysis (HCA) is an up-to-date method that utilizes higher-order cumulants rather than lower-order statistics (e.g., variances) to achieve the state monitoring purpose. Although HCA has a strong capability for state monitoring, it still exhibits many inadequacies for monitoring dynamic processes. Currently, there are various approaches (e.g., dynamic principle component analysis and dynamic independent component analysis) that are applicable to dynamic features. However, the key step of dynamic state monitoring methods is determination of the time lags or the lag structure. Almost all the reported dynamic methods select a single number of time lags for all variables. This simple selection method may not be appropriate since it is generally not possible that all variables have the same lag structure. In order to address this issue, a new lag selection method for each individual variable is proposed in this study. Hence, two dynamic higher-order cumulants analysis (DHCA) approaches are proposed for state monitoring, among which one is based on the conventional lag selection method and another is based on the new lag selection method proposed in this study. The two kinds of DHCA approaches are tested on the Tennessee Eastman process, and are demonstrated to be superior to all the compared methods.