Conditional Intensity Function Research Articles

A novel point process model for neuronal spike trains

Point process provides a mathematical framework for characterizing neuronal spiking activities. Classical point process methods often focus on the conditional intensity function, which describes the likelihood at any time point given its spiking history. However, these models do not describe the central tendency or importance of the spike train observations. Based on the recent development on the notion of center-outward rank for point process, we propose a new modeling framework on spike train data. The new likelihood of a spike train is a product of the marginal probability on the number of spikes and the probability of spike timings conditioned on the same number. In particular, the conditioned distribution is calculated by adopting the well-known Isometric Log-Ratio transformation. We systematically compare the new likelihood with the state-of-the-art point process likelihoods in terms of ranking, outlier detection, and classification using simulations and real spike train data. This new framework can effectively identify templates as well as outliers in spike train data. It also provides a reasonable model, and the parameters can be efficiently estimated with conventional maximum likelihood methods. It is found that the proposed likelihood provides an appropriate ranking on the spike train observations, effectively detects outliers, and accurately conducts classification tasks in the given data.

Entropy-Aware Time-Varying Graph Neural Networks with Generalized Temporal Hawkes Process: Dynamic Link Prediction in the Presence of Node Addition and Deletion

This paper addresses the problem of learning temporal graph representations, which capture the changing nature of complex evolving networks. Existing approaches mainly focus on adding new nodes and edges to capture dynamic graph structures. However, to achieve more accurate representation of graph evolution, we consider both the addition and deletion of nodes and edges as events. These events occur at irregular time scales and are modeled using temporal point processes. Our goal is to learn the conditional intensity function of the temporal point process to investigate the influence of deletion events on node representation learning for link-level prediction. We incorporate network entropy, a measure of node and edge significance, to capture the effect of node deletion and edge removal in our framework. Additionally, we leveraged the characteristics of a generalized temporal Hawkes process, which considers the inhibitory effects of events where past occurrences can reduce future intensity. This framework enables dynamic representation learning by effectively modeling both addition and deletion events in the temporal graph. To evaluate our approach, we utilize autonomous system graphs, a family of inhomogeneous sparse graphs with instances of node and edge additions and deletions, in a link prediction task. By integrating these enhancements into our framework, we improve the accuracy of dynamic link prediction and enable better understanding of the dynamic evolution of complex networks.

An objective approach to identifying individual atrial fibrillation triggers: A simulation study

Background and Objective:Growing evidence shows that certain acute exposures, especially alcohol, may trigger episodes of paroxysmal atrial fibrillation (AF). However, there is a lack of methods for assessing the relation between triggers and AF episodes in individual patients. The present paper proposes an approach to identifying AF triggers based on the assumption that the post-trigger AF burden is larger than the pre-trigger AF burden during the analysis time interval. Method:For the purpose of identification, a measure of relational strength between pre- and post-trigger burden is introduced, accounting for the cumulative effect of the triggers contained in the observation interval. The proposed approach is explored for different types of AF episode pattern, generated using the alternating, bivariate Hawkes model, whose conditional intensity function is designed to account for the effect of alcohol. In total, 7200 different AF patterns were generated for different numbers of AF triggers and alcohol units. Results:The simulation study demonstrates that, depending on the pattern type, the relational strength increases 3–6 times with alcohol consumption in comparison with no consumption. Conclusions:The proposed approach to identifying triggers in individual patients with paroxysmal AF should facilitate the implementation of longitudinal studies for the objective assessment of trigger effect on AF occurrence.

ATPP : A Mobile App Prediction System Based on Deep Marked Temporal Point Processes

Predicting the next application (app) a user will open is essential for improving the user experience, e.g., app pre-loading and app recommendation. Unlike previous solutions that only predict which app the user will open, this article predicts both the next app and the time to open it. Time prediction is essential to avoid loading the next app too early and consuming unnecessary resources on smartphones. To predict the next app and open time jointly, we model the app usage sequence as a marked temporal point process (MTPP), whose conditional intensity function can capture the probability of a new app usage event. We develop a novel data-driven MTPP-based app prediction system, named ATPP (App Temporal Point Process), which adopts a recurrent neural network architecture to learn the MTPP conditional intensity function for app prediction. ATPP adopts a set of techniques to incorporate the unique features of app prediction in our RNN architecture, including learning the correlated usage behavior of different apps by representation learning, the temporal dependency of app usage by an attention mechanism, and the location-related app usage behavior by feature extraction and fusion layer. We conduct extensive experiments on a large-scale anonymized app usage dataset to verify ATPP’s effectiveness.

Model ETAS Spatio-Temporal pada Analisis Pemetaan Intensitas Kegempaan di Wilayah Sumatera

High seismic activity forces Indonesia to mitigate natural disasters to minimize the impact of these disasters. One of the mitigation efforts that can be done is to know the possibility of an earthquake occurring in an area. Earthquakes, which are random phenomena, cannot be determined with certainty. Therefore, studies in seismological statistics can be used. One of the studies in seismological statistics that discusses the occurrence of earthquakes is the epidemic-type aftershock sequence (ETAS) model. This model describes the main seismic activity followed by aftershocks. The ETAS model, which only considers the parameters of time and event magnitude, was developed into a Spatio-temporal ETAS model, which also describes spatial parameters or locations. This study used the Spatio-temporal ETAS model to analyze the Sumatra region’s seismic activity from 2000-2022. Earthquake data sources are obtained from the United State Geological Survey (USGS). Based on data analysis, the Spatio-temporal ETAS model expressed in terms of the conditional intensity function shows that earthquakes that occur in the Sumatera region with a large magnitude tend to produce a large conditional intensity function as well. The spatial component of the spatio-temporal ETAS model can be explained by mapping the peak ground acceleration (PGA). The results of the PGA mapping show that seismic activity in the Sumatera region is prone to occur along the north and west coasts of the Sumatra region, with PGA values ranging from 0.2 g to 0.9 g. The larger the PGA value means, the higher the risk of an earthquake occurring in the area marked with a reddish-yellow dot or contour.

Multivariate conditional aging intensity functions and load-sharing models

The aging intensity functions analyze the aging property quantitatively, in the sense that the larger the aging intensity, the stronger the tendency of aging. They are useful tools to describe reliability properties of distributions. In the literature, the aging intensity functions have been studied in the univariate and bivariate case but without considering the possibility of observing a dynamic history. In this paper, the concept of aging intensity function is extended to the multivariate case by the use of the multivariate conditional hazard rate functions. Some properties of those functions are studied and a focus on the bivariate case is performed. Finally, the multivariate conditional aging intensity functions are studied for the order dependent version of the time-homogeneous load-sharing model and a study on the comparison among surviving components in a system is provided.

Identification and prediction of attacks to industrial control systems using temporal point processes

The task of identifying malicious activities in logs and predicting threats is crucial nowadays in industrial sector. In this paper, we focus on the identification of past malicious activities and in the prediction of future threats by proposing a novel technique based on the combination of Marked Temporal Point Processes (MTTP) and Neural Networks. Differently from the traditional formulation of Temporal Point Processes, our method does not make any prior assumptions on the functional form of the conditional intensity function and on the distribution of the events. Our approach is based the adoption of Neural Networks with the goal of improving the capabilities of learning arbitrary and unknown event distributions by taking advantage of the Deep Learning theory. We conduct a series of experiments using industrial data coming from gas pipelines, showing that our framework is able to represent in a convenient way the information gathered from the logs and predict future menaces in an unsupervised way, as well as classifying the past ones. The results of the experimental evaluation, showing outstanding values for precision and recall, confirm the effectiveness of our approach.

Motion Picture Editing as a Hawkes Process

In this article I analyse motion picture editing as a point process to explore the temporal structure in the timings of cuts in motion pictures, modelling the editing in 134 Hollywood films released between 1935 and 2005 as a Hawkes process with an exponential kernel. The results show that the editing in Hollywood films can be modelled as a Hawkes process and that the conditional intensity function provides a direct description of the instantaneous cutting rate of a film, revealing the structure of a film’s editing at a range of scales. The parameters of the exponential kernel show a clear trend over time to a more rapid editing style with an increase in the rate of exogenous events and small increase in the rate of endogenous events. This is consistent with the shift from a classical to an intensified continuity editing style. There are, however, few differences between genres indicating the consistency of editing practices in Hollywood cinema over time and different types of films.

Rank-Based Mixture Models for Temporal Point Processes

Temporal point process, an important area in stochastic process, has been extensively studied in both theory and applications. The classical theory on point process focuses on time-based framework, where a conditional intensity function at each given time can fully describe the process. However, such a framework cannot directly capture important overall features/patterns in the process, for example, characterizing a center-outward rank or identifying outliers in a given sample. In this article, we propose a new, data-driven model for regular point process. Our study provides a probabilistic model using two factors: (1) the number of events in the process, and (2) the conditional distribution of these events given the number. The second factor is the key challenge. Based on the equivalent inter-event representation, we propose two frameworks on the inter-event times (IETs) to capture large variability in a given process—One is to model the IETs directly by a Dirichlet mixture, and the other is to model the isometric logratio transformed IETs by a classical Gaussian mixture. Both mixture models can be properly estimated using a Dirichlet process (for the number of components) and Expectation-Maximization algorithm (for parameters in the models). In particular, we thoroughly examine the new models on the commonly used Poisson processes. We finally demonstrate the effectiveness of the new framework using two simulations and one real experimental dataset.

Graph Neural Point Process for Temporal Interaction Prediction

Temporal graphs are ubiquitous data structures in many scenarios, including social networks, user-item interaction networks, etc. In this paper, we focus on predicting the exact time of the next interaction, given a node pair on a temporal graph. This novel problem can support interesting applications, such as time-sensitive items recommendation, congestion prediction on road networks, and many others. We present Graph Neural Point Process (GNPP) to tackle this problem. GNPP relies on the graph neural message passing and the temporal point process framework. Most previous graph neural models only utilize the chronological order of observed events and ignore exact timestamps. In GNPP, we adapt a time encoding scheme to map real-valued timestamps to a high-dimensional vector space so that the temporal information can be precisely captured. Further, GNPP considers the structural information of graphs by conducting message-passing aggregation. The obtained representation defines a conditional intensity function that models events' generation mechanisms to predict future event times. We evaluate this model on synthetic and real-world datasets where it outperforms some recently proposed neural point process models and GNNs. We further conduct ablation comparisons and visualizations to shed some light on the learned model and understand the functionality of important components.

Continuous temporal network embedding by modeling neighborhood propagation process

Network embedding has become the focal point of increasing research interests in academic and industrial domains. Temporal networks are evolved by adding nodes and edges sequentially, which should be regarded as neighborhood propagation processes driven by chronological interactive events between nodes at arbitrary time granularity. In this scenario, both the occurrence times and connected nodes of interactive events are the key factors for embedding. Existing models of neighborhood propagation process focus on conventional temporal point process modeled by manual parametric forms of conditional intensity function, which merely measures the event occurrence probability at occurrence time. However, modeling event propagation without considering the connections of nodes in the network is inaccurate, requiring further research to capture both occurrence times and target nodes for temporal network embedding. In this paper, we propose a point process-based continuous temporal network embedding method by modeling neighborhood propagation process, named NPPCTNE, where the neighborhood propagation history is embedded into a hidden vector to jointly model the node and time of each interactive event. For propagation node prediction, the generating process of the next target node is simulated respectively by transient and steady-state responses to aggregate the long-term and short-term propagation effects of the continuous network system. For propagation time prediction, an intensity-free temporal point process is applied, which directly estimates the generative time probability distribution of the propagation sequence to learn the whole evolution process of the temporal network. Experiments on four real-world datasets show the efficacy of our model compared to state-of-the-arts.

Nostradamus: A novel event propagation prediction approach with spatio-temporal characteristics in non-Euclidean space

The prediction of event propagation has received extensive attention from the knowledge discovery community for applications such as virus spread analytics, social network analysis, earthquake location prediction, and typhoon tracking. The data describing these phenomena are multidimensional asynchronous event data that affect each other and show complex dynamic patterns in the continuous-time domain. Unlike the discrete characteristics formed by sampling at equal intervals of asynchronous time series, the timestamps of asynchronous events are in the continuous-time field. The study of these dynamic processes and the mining of their potential correlations provide a foundation for applying event propagation prediction, traceability, and causal inference at both the micro and macro levels.Most of the existing methods represent data as being embedded in the Euclidean space. However, when embedding a real-world graph with a tree-likeliness graph, Euclidean space cannot visually represent a graph. Inspired by the characteristics of hyperbolic space, we propose a model called Nostradamus to capture the propagation of the events of interest from historical events in a graph via the hyperbolic graph neural Hawkes process with Spatio-temporal characteristics. The Nostradamus’ core concept is to integrate the Hawkes process’s conditional intensity function with a hyperbolic graph convolutional recurrent neural network.

Astrologer: Exploiting graph neural Hawkes process for event propagation prediction with spatio-temporal characteristics

The prediction of event propagation has received extensive attention from the knowledge discovery community for applications such as social network analysis. The data describing these phenomena are multidimensional asynchronous event data that affect each other and show complex dynamic patterns in the continuous-time domain. The study of these dynamic processes and the mining of their potential correlations provide a foundation for the application of event propagation forecasting.However, conventional forecasting methods often make strong assumptions about the generative processes of the event data that may or may not reflect the reality, and the strong parametric assumptions also restrict the expressive power of the respective processes. Therefore, it is difficult to capture both the temporal and spatial effects of past event sequences.Most of the existing methods capture the intensity function of the Hawkes processes conditioned only on the historical events while ignoring the spatial information and the influences among different events.In this work, we propose the Astrologer, a graph neural Hawkes process that can capture the propagation of events from historical events on graph. The underlying idea of Astrologer is to incorporate the conditional intensity function of the Hawkes processes with a graph convolutional recurrent neural network. Using both synthetic and real-world datasets, we show that, Astrologer can learn the dynamics of event propagation without knowing the actual parametric forms. Astrologer can also learn the dynamics and achieve better predictive performance than other parametric alternatives based on particular prior assumptions.

A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach

Abstract In this paper, we investigate the problem of the local linear estimation of the conditional ageing intensity function, when the variable of interest is subject to random right-censored. We establish under appropriate conditions the asymptotic normality of this estimator.

Conditional intensity: A powerful tool for modelling and analysing point process data

SummaryThe conditional intensity function of a spatial point process describes how the probability that a point of the process occurs ‘at’ a particular point in its carrier space depends on the realisation of the process in the remainder of the carrier space. Provided that the point process is simple, the conditional intensity determines all of the properties of the process, in particular its likelihood function. In this paper, we review the use of the conditional intensity function in the formulation of point process models and in making inferences from point process data, giving separate consideration to temporal, spatial and spatiotemporal settings. We argue that the conditional intensity function should take centre‐stage in spatiotemporal point process modelling and analysis.

Robust point-process Granger causality analysis in presence of exogenous temporal modulations and trial-by-trial variability in spike trains.

Assessing directional influences between neurons is instrumental to understand how brain circuits process information. To this end, Granger causality, a technique originally developed for time-continuous signals, has been extended to discrete spike trains. A fundamental assumption of this technique is that the temporal evolution of neuronal responses must be due only to endogenous interactions between recorded units, including self-interactions. This assumption is however rarely met in neurophysiological studies, where the response of each neuron is modulated by other exogenous causes such as, for example, other unobserved units or slow adaptation processes. Here, we propose a novel point-process Granger causality technique that is robust with respect to the two most common exogenous modulations observed in real neuronal responses: within-trial temporal variations in spiking rate and between-trial variability in their magnitudes. This novel method works by explicitly including both types of modulations into the generalized linear model of the neuronal conditional intensity function (CIF). We then assess the causal influence of neuron i onto neuron j by measuring the relative reduction of neuron j's point process likelihood obtained considering or removing neuron i. CIF's hyper-parameters are set on a per-neuron basis by minimizing Akaike's information criterion. In synthetic data sets, generated by means of random processes or networks of integrate-and-fire units, the proposed method recovered with high accuracy, sensitivity and robustness the underlying ground-truth connectivity pattern. Application of presently available point-process Granger causality techniques produced instead a significant number of false positive connections. In real spiking responses recorded from neurons in the monkey pre-motor cortex (area F5), our method revealed many causal relationships between neurons as well as the temporal structure of their interactions. Given its robustness our method can be effectively applied to real neuronal data. Furthermore, its explicit estimate of the effects of unobserved causes on the recorded neuronal firing patterns can help decomposing their temporal variations into endogenous and exogenous components.

A constrained-memory stress release model (CM-SRM) for the earthquake occurrence in the Corinth Gulf (Greece)

The complexity of seismogenesis requires the development of stochastic models, the application of which aims to improve our understanding on the seismic process and the associated underlying mechanisms. Seismogenesis in the Corinth Gulf (Greece) is modeled through a Constrained-Memory Stress Release Model (CM-SRM), which combines the gradual increase of the strain energy due to continuous slow tectonic strain loading and its sudden release during an earthquake occurrence. The data are treated as a point process, which is uniquely defined by the associated conditional intensity function. In the original form of the Simple Stress Release Model (SSRM), the conditional intensity function depends on the entire history of the process. In an attempt to identify the most appropriate parameterization that better fits the data and describes the earthquake generation process, we introduce a constrained “ $$m$$ -memory” point process, implying that only the $$m$$ most recent arrival times are taken into account in the conditional intensity function, for some suitable $$m \in N$$ . Modeling of this process is performed for moderate earthquakes (M ≥ 5.2) occurring in the Corinth Gulf since 1911, by considering in each investigation different number of steps backward in time. The derived model versions are compared with the SSRM in its original form and evaluated in terms of information criteria and residual analysis.

Outcome-Oriented Deep Temporal Phenotyping of Disease Progression.

Chronic diseases evolve slowly throughout a patient's lifetime creating heterogeneous progression patterns that make clinical outcomes remarkably varied across individual patients. A tool capable of identifying temporal phenotypes based on the patients different progression patterns and clinical outcomes would allow clinicians to better forecast disease progression by recognizing a group of similar past patients, and to better design treatment guidelines that are tailored to specific phenotypes. To build such a tool, we propose a deep learning approach, which we refer to as outcome-oriented deep temporal phenotyping (ODTP), to identify temporal phenotypes of disease progression considering what type of clinical outcomes will occur and when based on the longitudinal observations. More specifically, we model clinical outcomes throughout a patient's longitudinal observations via time-to-event (TTE) processes whose conditional intensity functions are estimated as non-linear functions using a recurrent neural network. Temporal phenotyping of disease progression is carried out by our novel loss function that is specifically designed to learn discrete latent representations that best characterize the underlying TTE processes. The key insight here is that learning such discrete representations groups progression patterns considering the similarity in expected clinical outcomes, and thus naturally provides outcome-oriented temporal phenotypes. We demonstrate the power of ODTPby applying it to a real-world heterogeneous cohort of 11779 stage III breast cancer patients from the U.K. National Cancer Registration and Analysis Service. The experiments show that ODTPidentifies temporal phenotypes that are strongly associated with the future clinical outcomes and achieves significant gain on the homogeneity and heterogeneity measures over existing methods. Furthermore, we are able to identify the key driving factors that lead to transitions between phenotypes which can be translated into actionable information to support better clinical decision-making.

Survival analysis of failures based on Hawkes process with Weibull base intensity

In this paper, we construct a Hawkes process with time-varying base intensity to model the sequence of failure, i.e., failure events of the compressor station, and we combine survival analysis and point process model on various failure events of the compressor station based on Hawkes process. To our best knowledge, until now, nearly all relevant literature of the Hawkes point processes assumes that the base intensity of the conditional intensity function is time-invariant. This assumption is apparently too harsh to be verified. For example, in the practical application, including financial analysis, reliability analysis, survival analysis and social network analysis, the truth variation of the base intensity of the failure occurrence over time is not constant. The constant base intensity will not reflect the base intensity trend of the failure occurring over time. Thus, in order to solve this problem, in this paper, we propose a new time-varying base intensity, e.g. which is treated as obeying Weibull distribution. First, we introduce the base intensity into a Hawkes process that obeys the Weibull distribution, and then we propose an effective learning algorithm based on the maximum likelihood estimator. Experiments on the constant base intensity synthetic data, time-varying base intensity synthetic data, and real-world data show that our method can learn the triggering patterns of the Hawkes processes and the time-varying base intensity simultaneously and robustly. Experiments on real-world data also reveal the Granger causality of different types of failures and the base probability of failure varying over time. We put forward some suggestions for practical production based on the experimental results.

Ageing Intensity Function for Conditionally Specified Models

Jiang et al. have introduced a quantitative measure known as the ageing intensity function for evaluating the ageing properties of a component/system. The present study extends this ageing intensity function to the conditionally specified and conditional survival models. In a two component system, these two conditional ageing intensity functions provide the ageing patterns of one component when the other component has either failed or survived a specified period of time. The proposed conditionally ageing intensity function models provide simple expressions for some commonly used bivariate lifetime models. Characterization relationships are established for bivariate exponential, bivariate Weibull, conditional proportional hazards and bivariate weighted models.