Continuous-time Event Research Articles

Modeling Continuous Time Sequences with Intermittent Observations using Marked Temporal Point Processes

A large fraction of data generated via human activities such as online purchases, health records, spatial mobility, etc. can be represented as a sequence of events over a continuous-time. Learning deep learning models over these continuous-time event sequences is a non-trivial task as it involves modeling the ever-increasing event timestamps, inter-event time gaps, event types, and the influences between different events within and across different sequences. In recent years, neural enhancements to marked temporal point processes (MTPP) have emerged as a powerful framework to model the underlying generative mechanism of asynchronous events localized in continuous time. However, most existing models and inference methods in the MTPP framework consider only the complete observation scenario i.e., the event sequence being modeled is completely observed with no missing events – an ideal setting that is rarely applicable in real-world applications. A recent line of work which considers missing events while training MTPP utilizes supervised learning techniques that require additional knowledge of missing or observed label for each event in a sequence, which further restricts its practicability as in several scenarios the details of missing events is not known a priori . In this work, we provide a novel unsupervised model and inference method for learning MTPP in presence of event sequences with missing events. Specifically, we first model the generative processes of observed events and missing events using two MTPP, where the missing events are represented as latent random variables. Then, we devise an unsupervised training method that jointly learns both the MTPP by means of variational inference. Such a formulation can effectively impute the missing data among the observed events, which in turn enhances its predictive prowess, and can identify the optimal position of missing events in a sequence. Experiments with eight real-world datasets show that IMTPP outperforms the state-of-the-art MTPP frameworks for event prediction and missing data imputation, and provides stable optimization.

Personalized Dynamic Treatment Regimes in Continuous Time: A Bayesian Approach for Optimizing Clinical Decisions with Timing

Accurate models of clinical actions and their impacts on disease progression are critical for estimating personalized optimal dynamic treatment regimes (DTRs) in medical/health research, especially in managing chronic conditions. Traditional statistical methods for DTRs usually focus on estimating the optimal treatment or dosage at each given medical intervention, but overlook the important question of “when this intervention should happen.” We fill this gap by developing a two-step Bayesian approach to optimize clinical decisions with timing. In the first step, we build a generative model for a sequence of medical interventions—which are discrete events in continuous time—with a marked temporal point process (MTPP) where the mark is the assigned treatment or dosage. Then this clinical action model is embedded into a Bayesian joint framework where the other components model clinical observations including longitudinal medical measurements and time-to-event data conditional on treatment histories. In the second step, we propose a policy gradient method to learn the personalized optimal clinical decision that maximizes the patient survival by interacting the MTPP with the model on clinical observations while accounting for uncertainties in clinical observations learned from the posterior inference of the Bayesian joint model in the first step. A signature application of the proposed approach is to schedule follow-up visitations and assign a dosage at each visitation for patients after kidney transplantation. We evaluate our approach with comparison to alternative methods on both simulated and real-world datasets. In our experiments, the personalized decisions made by the proposed method are clinically useful: they are interpretable and successfully help improve patient survival.

Learning Temporal Point Processes for Efficient Retrieval of Continuous Time Event Sequences

Recent developments in predictive modeling using marked temporal point processes (MTPPs) have enabled an accurate characterization of several real-world applications involving continuous-time event sequences (CTESs). However, the retrieval problem of such sequences remains largely unaddressed in literature. To tackle this, we propose NEUROSEQRET which learns to retrieve and rank a relevant set of continuous-time event sequences for a given query sequence, from a large corpus of sequences. More specifically, NEUROSEQRET first applies a trainable unwarping function on the query sequence, which makes it comparable with corpus sequences, especially when a relevant query-corpus pair has individually different attributes. Next, it feeds the unwarped query sequence and the corpus sequence into MTPP guided neural relevance models. We develop two variants of the relevance model which offer a tradeoff between accuracy and efficiency. We also propose an optimization framework to learn binary sequence embeddings from the relevance scores, suitable for the locality-sensitive hashing leading to a significant speedup in returning top-K results for a given query sequence. Our experiments with several datasets show the significant accuracy boost of NEUROSEQRET beyond several baselines, as well as the efficacy of our hashing mechanism.

A Comparative Study of Some Point Process Models for Dynamic Networks

Modeling dynamic networks has attracted much interest in recent years, which helps understand networks’ behavior. Many works have been dedicated to modeling discrete‐time networks, but less work is done for continuous‐time networks. Point processes as powerful tools for modeling discrete events in continuous time have been widely used for modeling events over networks and their dynamics. These models have solid mathematical assumptions, making them interpretable but decreasing their generalizability for different datasets. Hence, neural point processes were introduced that don’t have strong assumptions on generative functions. However, these models can be impractical in the case of a large number of event types. This research presents a comparative study of different point process (Hawkes) models for continuous‐time networks. Furthermore, a previously introduced neural point process (neural Hawkes) model is applied for modeling network interactions. In this work, network clustering is used for specifying interaction types. These methods are compared using different synthetic and real‐world datasets, and their efficiency is evaluated on these datasets. The experiments represent that each model is appropriate for a group of datasets. In addition, the effect of clustering on results is discussed, and experiments for different clusters are presented.

Unbiased Simulation of Rare Events in Continuous Time

For rare events described in terms of Markov processes, truly unbiased estimation of the rare event probability generally requires the avoidance of numerical approximations of the Markov process. Recent work in the exact and varepsilon-strong simulation of diffusions, which can be used to almost surely constrain sample paths to a given tolerance, suggests one way to do this. We specify how such algorithms can be combined with the classical multilevel splitting method for rare event simulation. This provides unbiased estimations of the probability in question. We discuss the practical feasibility of the algorithm with reference to existing varepsilon-strong methods and provide proof-of-concept numerical examples.

Extraction of rapid kinetics from smFRET measurements using integrative detectors

SUMMARYHidden Markov models (HMMs) are used to learn single-molecule kinetics across a range of experimental techniques. By their construction, HMMs assume that single-molecule events occur on slower timescales than those of data acquisition. To move beyond that HMM limitation and allow for single-molecule events to occur on any timescale, we must treat single-molecule events in continuous time as they occur in nature. We propose a method to learn kinetic rates from single-molecule Förster resonance energy transfer (smFRET) data collected by integrative detectors, even if those rates exceed data acquisition rates. To achieve that, we exploit our recently proposed “hidden Markov jump process” (HMJP), with which we learn transition kinetics from parallel measurements in donor and acceptor channels. HMJPs generalize the HMM paradigm in two critical ways: (1) they deal with physical smFRET systems as they switch between conformational states in continuous time, and (2) they estimate transition rates between conformational states directly without having recourse to transition probabilities or assuming slow dynamics. Our continuous-time treatment learns the transition kinetics and photon emission rates for dynamic regimes that are inaccessible to HMMs, which treat system kinetics in discrete time. We validate our framework’s robustness on simulated data and demonstrate its performance on experimental data from FRET-labeled Holliday junctions.

Estimating Transfer Entropy in Continuous Time Between Neural Spike Trains or Other Event-Based Data.

Transfer entropy (TE) is a widely used measure of directed information flows in a number of domains including neuroscience. Many real-world time series for which we are interested in information flows come in the form of (near) instantaneous events occurring over time. Examples include the spiking of biological neurons, trades on stock markets and posts to social media, amongst myriad other systems involving events in continuous time throughout the natural and social sciences. However, there exist severe limitations to the current approach to TE estimation on such event-based data via discretising the time series into time bins: it is not consistent, has high bias, converges slowly and cannot simultaneously capture relationships that occur with very fine time precision as well as those that occur over long time intervals. Building on recent work which derived a theoretical framework for TE in continuous time, we present an estimation framework for TE on event-based data and develop a k-nearest-neighbours estimator within this framework. This estimator is provably consistent, has favourable bias properties and converges orders of magnitude more quickly than the current state-of-the-art in discrete-time estimation on synthetic examples. We demonstrate failures of the traditionally-used source-time-shift method for null surrogate generation. In order to overcome these failures, we develop a local permutation scheme for generating surrogate time series conforming to the appropriate null hypothesis in order to test for the statistical significance of the TE and, as such, test for the conditional independence between the history of one point process and the updates of another. Our approach is shown to be capable of correctly rejecting or accepting the null hypothesis of conditional independence even in the presence of strong pairwise time-directed correlations. This capacity to accurately test for conditional independence is further demonstrated on models of a spiking neural circuit inspired by the pyloric circuit of the crustacean stomatogastric ganglion, succeeding where previous related estimators have failed.

Recurrent Poisson Process Unit for Speech Recognition

Over the past few years, there has been a resurgence of interest in using recurrent neural network-hidden Markov model (RNN-HMM) for automatic speech recognition (ASR). Some modern recurrent network models, such as long shortterm memory (LSTM) and simple recurrent unit (SRU), have demonstrated promising results on this task. Recently, several scientific perspectives in the fields of neuroethology and speech production suggest that human speech signals may be represented in discrete point patterns involving acoustic events in the speech signal. Based on this hypothesis, it may pose some challenges for RNN-HMM acoustic modeling: firstly, it arbitrarily discretizes the continuous input into the interval features at a fixed frame rate, which may introduce discretization errors; secondly, the occurrences of such acoustic events are unknown. Furthermore, the training targets of RNN-HMM are obtained from other (inferior) models, giving rise to misalignments. In this paper, we propose a recurrent Poisson process (RPP) which can be seen as a collection of Poisson processes at a series of time intervals in which the intensity evolves according to the RNN hidden states that encode the history of the acoustic signal. It aims at allocating the latent acoustic events in continuous time. Such events are efficiently drawn from the RPP using a sampling-free solution in an analytic form. The speech signal containing latent acoustic events is reconstructed/sampled dynamically from the discretized acoustic features using linear interpolation, in which the weight parameters are estimated from the onset of these events. The above processes are further integrated into an SRU, forming our final model, called recurrent Poisson process unit (RPPU). Experimental evaluations on ASR tasks including ChiME-2, WSJ0 and WSJ0&1 demonstrate the effectiveness and benefits of the RPPU. For example, it achieves a relative WER reduction of 10.7% over state-of-the-art models on WSJ0.

Nonparametric inference for continuous-time event counting and link-based dynamic network models

A flexible approach for modeling both dynamic event counting and dynamic link-based networks based on counting processes is proposed, and estimation in these models is studied. We consider nonparametric likelihood based estimation of parameter functions via kernel smoothing. The asymptotic behavior of these estimators is rigorously analyzed by allowing the number of nodes to tend to infinity. The finite sample performance of the estimators is illustrated through an empirical analysis of bike share data.

A semiparametric extension of the stochastic block model for longitudinal networks

We propose an extension of the stochastic block model for recurrent interaction events in continuous time, where every individual belongs to a latent group and conditional interactions between two individuals follow an inhomogeneous Poisson process with intensity driven by the individuals’ latent groups. We show that the model is identifiable and estimate it with a semiparametric variational expectation-maximization algorithm. We develop two versions of the method, one using a nonparametric histogram approach with an adaptive choice of the partition size, and the other using kernel intensity estimators. We select the number of latent groups by an integrated classification likelihood criterion. We demonstrate the performance of our procedure on synthetic experiments, analyse two datasets to illustrate the utility of our approach, and comment on competing methods.

Inferring social structure from continuous-time interaction data.

Relational event data, which consist of events involving pairs of actors over time, are now commonly available at the finest of temporal resolutions. Existing continuous-time methods for modeling such data are based on point processes and directly model interaction "contagion," whereby one interaction increases the propensity of future interactions among actors, often as dictated by some latent variable structure. In this article, we present an alternative approach to using temporal-relational point process models for continuous-time event data. We characterize interactions between a pair of actors as either spurious or as resulting from an underlying, persistent connection in a latent social network. We argue that consistent deviations from expected behavior, rather than solely high frequency counts, are crucial for identifying well-established underlying social relationships. This study aims to explore these latent network structures in two contexts: one comprising of college students and another involving barn swallows.

Sampled-data output feedback control based on a new event-triggered control scheme

In this paper, a new event-triggered control (ETC) scheme is proposed to investigate the observer-based output feedback control problems. To reduce the usage of communication resources, the control input is designed as a specific time-varying signal between two consecutive event-triggering instants, which is different from the general ETC methods. For the continuous-time event detection case, it is shown that the obtained lower bound of the minimum inter-event interval is not smaller than the one using the existing method. In addition, a novel discrete-time event detection scheme and the corresponding ETC policy based on the sampling output is proposed. Employing the stability analysis method for the time-delay system, a sufficient condition on the existence of an observer-based controller is presented. Finally, the simulation results are given to show that the proposed ETC scheme is beneficial to further reduce the frequency of the communication between sensors and the control unit.

Inference of causality in epidemics on temporal contact networks.

Investigating into the past history of an epidemic outbreak is a paramount problem in epidemiology. Based on observations about the state of individuals, on the knowledge of the network of contacts and on a mathematical model for the epidemic process, the problem consists in describing some features of the posterior distribution of unobserved past events, such as the source, potential transmissions, and undetected positive cases. Several methods have been proposed for the study of these inference problems on discrete-time, synchronous epidemic models on networks, including naive Bayes, centrality measures, accelerated Monte-Carlo approaches and Belief Propagation. However, most traced real networks consist of short-time contacts on continuous time. A possibility that has been adopted is to discretize time line into identical intervals, a method that becomes more and more precise as the length of the intervals vanishes. Unfortunately, the computational time of the inference methods increase with the number of intervals, turning a sufficiently precise inference procedure often impractical. We show here an extension of the Belief Propagation method that is able to deal with a model of continuous-time events, without resorting to time discretization. We also investigate the effect of time discretization on the quality of the inference.

Hybrid Modeling of Strategic Loading of a Marine Hybrid Power Plant With Experimental Validation

Recent developments in marine power systems, energy storage devices (ESDs) technology, and modification to rules and regulations increase the opportunities to improve efficiency and reduce emissions. One particular application is the strategic loading, where the ESD is charged and discharged cyclically, altering the instantaneous fuel consumption, thus aiming to reduce the average fuel consumption. Due to the ESD switching behavior, a hybrid simulation framework is an appropriate dynamic modeling tool. The hybrid simulation model is important in proper design and verification of control strategies for hybrid power plants. A hybrid model was derived, modeling transients as continuous-time events and modeling instantaneous behavior changes as discrete events. Due to the complexity of the system and its hybrid nature (continuous and discrete times), it is important to validate the derived model, such that are known its accuracy and limitations. The developed hybrid model was validated using experiments at the Hybrid Machinery Laboratory, Norwegian University of Science and Technology. The analyzed effects are the steady state, transient behavior, and losses. The transient behaviors include Generator-set (genset) dynamics and load ramps. The losses include the production losses, transmission losses, and ESD losses. The non-modeled effects include the load fluctuation, genset speed variation about the given set-point, and thermal effects on the genset and on the ESD. The results show good correlation between the hybrid model and the experiments. The fuel consumption estimation error stayed below 3% for all 15 analyzed cases, as well as having less than 9% deviation for the NO x gas emissions estimation. The model is considered as a good approximation for the real operation, enabling its use for design and research purposes.

Path to Purchase: A Mutually Exciting Point Process Model for Online Advertising and Conversion

This paper studies the effects of various types of online advertisements on purchase conversion by capturing the dynamic interactions among advertisement clicks themselves. It is motivated by the observation that certain advertisement clicks may not result in immediate purchases, but they stimulate subsequent clicks on other advertisements, which then lead to purchases. We develop a novel model based on mutually exciting point processes, which consider advertisement clicks and purchases as dependent random events in continuous time. We incorporate individual random effects to account for consumer heterogeneity and cast the model in the Bayesian hierarchical framework. We construct conversion probability to properly evaluate the conversion effects of online advertisements. We develop simulation algorithms for mutually exciting point processes to compute the conversion probability and for out-of-sample prediction. Model comparison results show the proposed model outperforms the benchmark models that ignore exciting effects among advertisement clicks. Using a proprietary data set, we find that display advertisements have relatively low direct effect on purchase conversion, but they are more likely to stimulate subsequent visits through other advertisement formats. We show that the commonly used measure of conversion rate is biased in favor of search advertisements and underestimates the conversion effect of display advertisements the most. Our model also furnishes a useful tool to predict future purchases and advertisement clicks for the purpose of targeted marketing and customer relationship management.This paper was accepted by Eric Bradlow, special issue on business analytics.

Comparative analysis of Streptococcus pneumoniaetransmission in Portuguese and Finnish day-care centres

BackgroundDay-care centre (DCC) attendees play a central role in maintaining the circulation of Streptococcus pneumoniae (pneumococcus) in the population. The prevalence of pneumococcal carriage is highest in DCC attendees but varies across countries and is found to be consistently lower in Finland than in Portugal. We compared key parameters underlying pneumococcal transmission in DCCs to understand which of these contributed to the observed differences in carriage prevalence.MethodsLongitudinal data about serotype-specific carriage in DCC attendees in Portugal (47 children in three rooms; mean age 2 years; range 1–3 years) and Finland (91 children in seven rooms; mean age 4 years; range 1–7 years) were analysed with a continuous-time event history model in a Bayesian framework. The monthly rates of within-room transmission, community acquisition and clearing carriage were estimated.ResultsThe posterior mean of within-room transmission rate was 1.05 per month (Portugal) vs. 0.63 per month (Finland). The smaller rate of clearance in Portugal (0.57 vs. 0.73 per month) is in accordance with the children being younger. The overall community rate of acquisition was larger in the Portuguese setting (0.25 vs. 0.11 per month), in agreement with that the groups belonged to a larger DCC. The model adequately predicted the observed levels of carriage prevalence and longitudinal patterns in carriage acquisition and clearance.ConclusionsThe difference in prevalence of carriage (61% in Portuguese vs. 26% among Finnish DCC attendees) was assigned to the longer duration of carriage in younger attendees and a significantly higher rate of within-room transmission and community acquisition in the Portuguese setting.

Path to Purchase: A Mutually Exciting Point Process Model for Online Advertising and Conversion

This paper studies the effects of various types of online advertisements on purchase conversion by capturing the dynamic interactions among advertisement clicks themselves. It is motivated by the observation that certain advertisement clicks may not result in immediate purchases, but they stimulate subsequent clicks on other advertisements which then lead to purchases. We develop a novel model based on mutually exciting point processes, which consider advertisement clicks and purchases as dependent random events in continuous time. We incorporate individual random effects to account for consumer heterogeneity and cast the model in the Bayesian hierarchical framework. We construct conversion probability to properly evaluate the conversion effects of online advertisements. Simulation algorithms for mutually exciting point processes are developed to compute the conversion probability and for out-of-sample prediction. Model comparison results show the proposed model outperforms the benchmark models that ignore exciting effects among advertisement clicks. We find that display advertisements have relatively low direct effect on purchase conversion, but they are more likely to stimulate subsequent visits through other advertisement formats. We show that the commonly used measure of conversion rate is biased in favor of search advertisements and underestimates the conversion effect of display advertisements the most. Our model also furnishes a useful tool to predict future purchases and advertisement clicks for the purpose of targeted marketing and customer relationship management.

Modelling security market events in continuous time: Intensity based, multivariate point process models

A continuous time econometric modelling framework for multivariate financial market event (or ‘transactions’) data is developed in which the model is specified via the vector conditional intensity. Generalised Hawkes models are introduced that incorporate inhibitory events and dependence between trading days. Novel omnibus specification tests based on a multivariate random time change theorem are proposed. A bivariate point process model of the timing of trades and mid-quote changes is then presented for a New York Stock Exchange stock and related to the market microstructure literature. The two-way interaction of trades and quote changes in continuous time is found to be important empirically.

Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models

A continuous time econometric modelling framework for multivariate financial market event (or 'transactions') data is developed in which the model is specified via the vector conditional intensity. Generalised Hawkes models are introduced that incorporate inhibitory events and dependence between trading days. Sufficient conditions for identification of linear Hawkes models and for stationarity of some new non-linear Hawkes models are derived. Novel omnibus specification tests based on a multivariate random time change theorem are proposed. A computationally efficient thinning algorithm for simulation of generalised Hawkes processes is also developed. A bivariate point process model of the timing of trades and mid-quote changes is presented for a New York Stock Exchange stock and related to the market microstructure literature. The two-way interaction of trades and quote changes is found to be important empirically.

A LIKELIHOOD FRAMEWORK FOR INFERRING THE EVOLUTION OF GEOGRAPHIC RANGE ON PHYLOGENETIC TREES

At a time when historical biogeography appears to be again expanding its scope after a period of focusing primarily on discerning area relationships using cladograms, new inference methods are needed to bring more kinds of data to bear on questions about the geographic history of lineages. Here we describe a likelihood framework for inferring the evolution of geographic range on phylogenies that models lineage dispersal and local extinction in a set of discrete areas as stochastic events in continuous time. Unlike existing methods for estimating ancestral areas, such as dispersal-vicariance analysis, this approach incorporates information on the timing of both lineage divergences and the availability of connections between areas (dispersal routes). Monte Carlo methods are used to estimate branch-specific transition probabilities for geographic ranges, enabling the likelihood of the data (observed species distributions) to be evaluated for a given phylogeny and parameterized paleogeographic model. We demonstrate how the method can be used to address two biogeographic questions: What were the ancestral geographic ranges on a phylogenetic tree? How were those ancestral ranges affected by speciation and inherited by the daughter lineages at cladogenesis events? For illustration we use hypothetical examples and an analysis of a Northern Hemisphere plant clade (Cercis), comparing and contrasting inferences to those obtained from dispersal-vicariance analysis. Although the particular model we implement is somewhat simplistic, the framework itself is flexible and could readily be modified to incorporate additional sources of information and also be extended to address other aspects of historical biogeography.