Generally, these days people live longer but often with increased impairment and disabilities; therefore, they can benefit from assistive technologies. In this paper, we focus on the completion of activities of daily living (ADLs) by such patients, using so-called Smart Homes and Sensor Technology to collect data, and provide a suitable analysis to support the management of these conditions. The activities here are cast as states of a Markov-type process, while changes of state are indicated by sensor activations. This facilitates the extraction of key performance indicators (KPIs) in Smart Homes, e.g., the duration of an important activity, as well as the identification of anomalies in such transitions and durations. The use of semi-Markov models for such a scenario is described, where the state durations are represented by mixed gamma models. This approach is illustrated and evaluated using a publicly available Smart Home dataset comprising an event log of sensor activations, together with an annotated record of the actual activities. Results indicate that the methodology is well-suited to such scenarios.

Within the realm of business context, process duration signifies time spent by customers between successive activities. This temporal perspective offers important insight to customer behavior, highlighting potential bottlenecks, and influencing business management decisions. The distribution of these process duration often changes over time due to factors such as seasonality, emerging legislation, changes to supply chains, and customer demand. Referred to as concept drift, these variations pose challenges for robust process modeling, understanding, and refinement. Subsequently, gamma mixture models are widely employed to model durations. These source data can, however, become left-truncated and right-censored within any specific observation window thereby necessitating a (well-known) modification to the likelihood function. The approach reported in this article leveraged this adapted likelihood across a series of observation windows, applying the likelihood ratio test to identify duration changes/concept drift. Due to its flexibility in modelling any duration distribution, the gamma mixture model was used with Nelder–Mead optimized likelihood for the left-truncated and right-censored data. The number of gamma components was determined by the Bayesian information criterion. The proposed framework underwent validation through simulated exponential samples, leading to recommendations for its practical application. Subsequently, we applied the methodology to three real-life event logs exhibiting diverse characteristics. Experimental results showcase the effectiveness of our approach in terms of data fitting, as compared to Kaplan–Meier curves, and in detecting instances of drift. This comprehensive validation underscores the practical utility and reliability of our framework for dynamic business scenarios.

Within a business context, anomalies can be viewed as indicators for inefficiencies or fraud, which impact upon product quality and customer satisfaction. The development of approaches to monitor, detect and predict anomalous business processes remains an important research topic. In this paper, we propose a method, combining Discrete-time Markov chains (DTMCs) and hitting probabilities (HP), for detecting anomalies occurring in the execution of business processes. Our method extends standard DTMCs to be able to estimate the probability of occurring for a process instance even though it is partially recorded (i.e., the initial executions are missing). The proposed method, denoted as HPDTMC, does not rely on prior knowledge about anomalies and the business process and can be trained on datasets already consisting of anomalies. A Šidák correction is applied to balance the probability of instances of varying length since naturally, process instances with more executions have lower sequence probability and more likely to be detected as anomalies by using DTMCs. We demonstrate the effectiveness of the method by evaluating it on two artificial datasets and one real-life dataset against seven classic anomaly detection methods. In the experiments, our approach reached an F 1 score of 0.904 on average. Moreover, the proposed method outperforms competitors under noisy conditions. The main contribution of this paper is the proposed noise-robust method which is able to detect fully or partially recorded process instances of varying lengths.

Concept drift, which refers to changes in the underlying process structure or customer behaviour over time, is inevitable in business processes, causing challenges in ensuring that the learned model is a proper representation of the new data. Due to factors such as seasonal effects and policy updates, concept drifts can occur in customer transitions and time spent throughout the process, either suddenly or gradually. In a concept drift context, we can discard the old data and retrain the model using new observations (sudden drift) or combine the old data with the new data to update the model (gradual drift) or maintain the model as unchanged (no drift). In this paper, we model a response to concept drift as a sequential decision making problem by combing a hierarchical Markov model and a Markov decision process (MDP). The approach can detect concept drift, retrain the model and update customer profiles automatically. We validate the proposed approach on 68 artificial datasets and a real-world hospital billing dataset, with experimental results showing promising performance.