Abstract
In this paper, a novel generic clustering-based algorithm for approximating the topology and the parameters of discrete state space Input/Output Hidden Markov Models (IOHMMs) with continuous observation spaces is introduced. The algorithm can accommodate any continuous space clustering method, whether incremental or not; it can easily be extended to Input/Output Hidden Semi-Markov Models (IOHSMMs) as well as standard HMMs and HSMMs. In this paper, the proposed algorithm is implemented with the Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN*) clustering algorithm. This algorithm brings numerous benefits such as the capability to learn the topology and the parameters of the model with more or less conservatism and the capability to define distributions from several frameworks of the uncertainty theory such as the probabilities, the possibilities or the imprecise probabilities. The algorithm is validated on synthetic and real-world datasets.
Highlights
Input/Output Hidden Markov Models (IOHMMs) [1] constitute a powerful stochastic modelling framework for time series relying on strong mathematical foundations
We address the problem of learning first-order discrete state space IOHMMs topology and parameters λ from continuous observation sequences
Depending on the usage one expect to have with the IOHMM model learned, Gaussian mixture models (GMMs)-based learning might lead to better IOHMM models for classification problems while HDBSCAN*based learning might lead IOHMM models better for representing/extracting dynamical systems from partial knowledge or knowledge distilled over the long term
Summary
Input/Output Hidden Markov Models (IOHMMs) [1] constitute a powerful stochastic modelling framework for time series relying on strong mathematical foundations. B. CONTRIBUTIONS In this paper, we propose a generic clustering-based algorithm for learning the topology and the parameters λ of discrete state space IOHMMs with continuous observation spaces. As an extension of the IOHMMs, Input/Output Hidden Semi-Markov Models (IOHSMMs) [27] allow the underlying stochastic process X to be a semi-Markov chain: each state is given a duration constraint as the number of observations allowed to be produced while being in this state With this model, elements aij, i = j, are defined by aij(u, δ) = p x(k) = j|x(k−1) = i, d(x(k−1)) = δ, u(k) = u , denoting. The hidden state space being known (line 3 in Algorithm.1), elements of the credal sets can be complemented with distribution parameters obtained from an Expectation-Maximization algorithm (e.g., Baum-Welch).
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