Markov chains are simple yet powerful mathematical structures to model temporally dependent processes. They generally assume stationary data, i.e., fixed transition probabilities between observations/states. However, live, real-world processes, like in the context of activity tracking, biological time series, or industrial monitoring, often switch behavior over time. Such behavior switches can be modeled as transitions between higher-level modes (e.g., running, walking, etc.). Yet all modes are usually not previously known, often exhibit vastly differing transition probabilities, and can switch unpredictably. Thus, to track behavior changes of live, real-world processes, this study proposes an online and efficient method to construct Evolving Markov chains (EMCs). EMCs adaptively track transition probabilities, automatically discover modes, and detect mode switches in an online manner. In contrast to previous work, EMCs are of arbitrary order, the proposed update scheme does not rely on tracking windows, only updates the relevant region of the probability tensor, and enjoys geometric convergence of the expected estimates. Our evaluation of synthetic data and real-world applications on human activity recognition, electric motor condition monitoring, and eye-state recognition from electroencephalography (EEG) measurements illustrates the versatility of the approach and points to the potential of EMCs to efficiently track, model, and understand live, real-world processes.

Machine learning (ML) is increasingly employed in real-world applications like medicine or economics, thus, potentially affecting large populations. However, ML models often do not perform homogeneously, leading to underperformance or, conversely, unusually high performance in certain subgroups (e.g., sex=female ∧ marital_status=married). Identifying such subgroups can support practical decisions on which subpopulation a model is safe to deploy or where more training data is required. However, an efficient and coherent framework for effective search is missing. Consequently, we introduce SubROC, an open-source, easy-to-use framework based on Exceptional Model Mining for reliably and efficiently finding strengths and weaknesses of classification models in the form of interpretable population subgroups. SubROC incorporates common evaluation measures (ROC and PR AUC), efficient search space pruning for fast exhaustive subgroup search, control for class imbalance, adjustment for redundant patterns, and significance testing. We illustrate the practical benefits of SubROC in case studies as well as in comparative analyses across multiple datasets.