Negative-Binomial Randomized Gamma Markov Processes for Heterogeneous Overdispersed Count Time Series

TL;DR

A negative-binomial-randomized gamma Markov process is proposed, which not only significantly improves the predictive performance of the proposed dynamical system, but also facilitates the fast convergence of the inference algorithm.

Abstract

Modeling count-valued time series has been receiving increasing attention since count time series naturally arise in physical and social domains. Poisson gamma dynamical systems (PGDSs) are newly-developed methods, which can well capture the expressive latent transition structure and bursty dynamics behind count sequences. In particular, PGDSs demonstrate superior performance in terms of data imputation and prediction, compared with canonical linear dynamical system (LDS) based methods. Despite these advantages, PGDS cannot capture the heterogeneous overdispersed behaviours of the underlying dynamic processes. To mitigate this defect, we propose a negative-binomial-randomized gamma Markov process, which not only significantly improves the predictive performance of the proposed dynamical system, but also facilitates the fast convergence of the inference algorithm. Moreover, we develop methods to estimate both factor-structured and graph-structured transition dynamics, which enable us to infer more explainable latent structure, compared with PGDSs. Finally, we demonstrate the explainable latent structure learned by the proposed method, and show its superior performance in imputing missing data and forecasting future observations, compared with the related models.

Figures (6)

• Figure 1: The realizations of the negative-binomial-randomized gamma Markov processes defined in Eq.\ref{['eq:13']}. Here $\epsilon_0^{(\theta)}$ and $\tau$ were set to 0 and 1, respectively. The initial values of the NBRGMPs in (a) and (b), were set to 1, (c) and (d) were set to 1000, and the chains were simulated until $t=50$. Each subplot contains ten independent realizations.
• Figure 2: The hierarchical structure of the NBRGMP. The red arrows indicate intractable dependencies that require data augmentation schemes for posterior inference.
• Figure 3: The graph structure of the latent dimensions of the transition kernel behind sequential count observations.
• Figure 4: Negative-binomial-randomized gamma dynamical systems (NBRGDSs) demonstrate strong ability in capturing heterogeneous overdispersion effects, and thus achieves faster convergence (a), lowest mean absolute error (b) and mean relative error (c), compared with the other related baselines. The stationary and non-stationary generative process i.e. $\delta^{(t)} = \delta$ and $\delta^{(t)}$, denoted as solid line and dotted line, respectively.
• Figure 5: The proposed NBRGDS consistently achieves lower mean absolute and relative errors, when we vary the overdispersed magnitude (the ratio of variance to expection) of the synthetic count sequences, compared with the other closely-related models.