Thepresented equations can be efficiently solved using well-known numerical methods.

Hidden semi-Markov Models (HSMM's) - while broadly in use - are restricted to a discrete and uniform time grid. They are thus not well suited to explain often irregularly spaced discrete event data from continuous-time phenomena. We show that non-sampling-based latent state inference used in HSMM's can be generalized to latent Continuous-Time semi-Markov Chains (CTSMC's). We formulate integro-differential forward and backward equations adjusted to the observation likelihood and introduce an exact integral equation for the Bayesian posterior marginals and a scalable Viterbi-type algorithm for posterior path estimates. The presented equations can be efficiently solved using well-known numerical methods. As a practical tool, variable-step HSMM's are introduced. We evaluate our approaches in latent state inference scenarios in comparison to classical HSMM's.

Hidden semi-Markov Models (HSMM's) - while broadly in use - are restricted to

Recommender systems are widely used for suggesting books, education materials, and products to users by exploring their behaviors. In reality, users' preferences often change over time, leading to studies on time-dependent recommender systems. However, most existing approaches that deal with time information remain primitive. In this paper, we extend existing methods and propose a hidden semi-Markov model to track the change of users' interests. Particularly, this model allows for capturing the different durations of user stays in a (latent) interest state, which can better model the heterogeneity of user interests and focuses. We derive an expectation maximization algorithm to estimate the parameters of the framework and predict users' actions. Experiments on three real-world datasets show that our model significantly outperforms the state-of-the-art time-dependent and static benchmark methods. Further analyses of the experiment results indicate that the performance improvement is related to the heterogeneity of state durations and the drift of user interests in the dataset.

Hidden semi-Markov Models (HSMM's) - while broadly in use - are restricted to a discrete and uniform time grid. They are thus not well suited to explain often irregularly spaced discrete event data from continuous-time phenomena. We show that non-sampling-based latent state inference used in HSMM's can be generalized to latent Continuous-Time semi-Markov Chains (CTSMC's). We formulate integro-differential forward and backward equations adjusted to the observation likelihood and introduce an exact integral equation for the Bayesian posterior marginals and a scalable Viterbi-type algorithm for posterior path estimates. The presented equations can be efficiently solved using well-known numerical methods.

Evolving consumer demands and market trends have led to businesses increasingly embracing a production approach that prioritizes flexibility and customization. Consequently, factory workers must engage in tasks that are more complex than before. Thus, productivity depends on each worker's skills in assembling products. Therefore, analyzing the behavior of a worker is crucial for work improvement. However, manual analysis is time consuming and does not provide quick and accurate feedback. Machine learning have been attempted to automate the analyses; however, most of these methods need several labels for training. To this end, we extend the Gaussian process hidden semi-Markov model (GP-HSMM), to enable the rapid and automated analysis of worker behavior without pre-training. The model does not require labeled data and can automatically and accurately segment continuous motions into motion classes. The proposed model is a probabilistic model that hierarchically connects GP-HSMM and HSMM, enabling the extraction of behavioral patterns with different granularities. Furthermore, it mutually infers the parameters between the GP-HSMM and HSMM, resulting in accurate motion pattern extraction. We applied the proposed method to motion data in which workers assembled products at an actual production site. The accuracy of behavior pattern extraction was evaluated using normalized Levenshtein distance (NLD). The smaller the value of NLD, the more accurate is the pattern extraction. The NLD of motion patterns captured by GP-HSMM and HSMM layers in our proposed method was 0.50 and 0.33, respectively, which are the smallest compared to that of the baseline methods.