Identifiability of discrete Input-Output hidden Markov models with external signals

Abstract

In this paper, we consider a bivariate process (Xt, Yt) t∈Z which, conditionally on a signal (Wt) t∈Z , is a hidden Markov model whose transition and emission kernels depend on (Wt) t∈Z. The resulting process (Xt, Yt, Wt) t∈Z is referred to as an input-output hidden Markov model or hidden Markov model with external signals. We prove that this model is identifiable and that the associated maximum likelihood estimator is consistent. Introducing an Expectation Maximization-based algorithm, we train and evaluate the performance of this model in several frameworks. In addition to learning dependencies between (Xt, Yt) t∈Z and (Wt) t∈Z , our approach based on hidden Markov models with external signals also outperforms state-of-the-art algorithms on real-world fashion sequences.

Identifiability of discrete input–output hidden Markov models with external signals

“…A first simulated time series is generated using a hidden Markov model with external signals and seasonal components as defined above (shmm-es). As external signal (W t ) t 1 , a signal of the fashion dataset introduced in [David et al, 2022] is used to provide a realistic setting. The external sequence is smoothed using a moving average with a sliding window of length 8 and divided by the mean of the first year to rescale the signal.…”

“…1 https://github.com/etidav/hmm_with_external_signals Figure 5: Time series "eu-female-top-325" from [David et al, 2022] representing an emerging fashion trend on social media with its linked influencers external signal. The influencers sequence is smoothed using a moving average with a sliding window of length 8.…”

“…A sequence 1 from the fashion dataset introduced in [David et al, 2022] was selected as it shows a sudden change of level, seasonality and noise intensity as illustrated in Figure 5. Several models are trained using this time series.…”

“…Despite all these results, recent state-of-the-art forecasting models are for a large part based on recurrent neural networks [Hochreiter andSchmidhuber, 1997, Salinas et al, 2020] or sequence to sequence deep learning architectures [Vaswani et al, 2017, Li et al, 2019. Intrinsically designed to deal with numerous heterogeneous data and include external signals, these new approaches overcome some limitations of HMMs and reach unprecedented accuracy levels in various frameworks and for numerous data sets, see for instance [Lim et al, 2021, Salinas et al, 2020, David et al, 2022. However, these results have a cost: i) most of the recent state-of-the-art models are black boxes as the final forecast usually cannot be explained ; ii) very few theoretical guarantees exist for such deep learning architecture-based algorithms.…”

“…Then, we implement the MLE using the EM algorithm and show on a synthetic data set that it can recover the true set of parameters and that the addition of external signals does not prevent an efficient training process. Finally, we evaluate the proposed method on real world retail times series using the fashion data set introduced in [David et al, 2022]. This data set gathers the evolution of thousands of fashion items on social media and provides for each of them an external signal representing the behaviour of influencers.…”