Causal Inference in Disease Spread across a Heterogeneous Social System

Diffusion processes are governed by external triggers and internal dynamics in complex systems. Timely and cost-effective control of infectious disease spread critically relies on uncovering the underlying diffusion mechanisms, which is challenging due to invisible causality between events and their time-evolving intensity. We infer causal relationships between infections and quantify the reflexivity of a meta-population, the level of feedback on event occurrences by its internal dynamics (likelihood of a regional outbreak triggered by previous cases). These are enabled by our new proposed model, the Latent Influence Point Process (LIPP) which models disease spread by incorporating macro-level internal dynamics of meta-populations based on human mobility. We analyse 15-year dengue cases in Queensland, Australia. From our causal inference, outbreaks are more likely driven by statewide global diffusion over time, leading to complex behavior of disease spread. In terms of reflexivity, precursory growth and symmetric decline in populous regions is attributed to slow but persistent feedback on preceding outbreaks via inter-group dynamics, while abrupt growth but sharp decline in peripheral areas is led by rapid but inconstant feedback via intra-group dynamics. Our proposed model reveals probabilistic causal relationships between discrete events based on intra- and inter-group dynamics and also covers direct and indirect diffusion processes (contact-based and vector-borne disease transmissions).Comment: arXiv admin note: substantial text overlap with arXiv:1711.0635

Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes

We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the "shock") as a function of the current time $t$. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays $\sim 1/t^{1-(m+1)\theta}$ of the rate of triggered events as a function of the distance $m$ of the events to the initial shock in the type space, where $0 < \theta <1$ for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via a kind of inter-breeding genealogy.Comment: 40 pages, 8 figure

Multivariate Hawkes Processes for Large-scale Inference

In this paper, we present a framework for fitting multivariate Hawkes processes for large-scale problems both in the number of events in the observed history $n$ and the number of event types $d$ (i.e. dimensions). The proposed Low-Rank Hawkes Process (LRHP) framework introduces a low-rank approximation of the kernel matrix that allows to perform the nonparametric learning of the $d^2$ triggering kernels using at most $O(ndr^2)$ operations, where $r$ is the rank of the approximation ($r \ll d,n$). This comes as a major improvement to the existing state-of-the-art inference algorithms that are in $O(nd^2)$. Furthermore, the low-rank approximation allows LRHP to learn representative patterns of interaction between event types, which may be valuable for the analysis of such complex processes in real world datasets. The efficiency and scalability of our approach is illustrated with numerical experiments on simulated as well as real datasets.Comment: 16 pages, 5 figure

Scalable Bayesian Learning for State Space Models using Variational Inference with SMC Samplers

We present a scalable approach to performing approximate fully Bayesian inference in generic state space models. The proposed method is an alternative to particle MCMC that provides fully Bayesian inference of both the dynamic latent states and the static parameters of the model. We build up on recent advances in computational statistics that combine variational methods with sequential Monte Carlo sampling and we demonstrate the advantages of performing full Bayesian inference over the static parameters rather than just performing variational EM approximations. We illustrate how our approach enables scalable inference in multivariate stochastic volatility models and self-exciting point process models that allow for flexible dynamics in the latent intensity function.Comment: To appear in AISTATS 201