Point Process Model Research Articles

Earthquake Predictability and Forecast Evaluation Using Likelihood-Based Marginal and Conditional Scores

Abstract Earthquake probability forecasts are typically based on simulations of seismicity generated by statistical (point process) models or direct calculation when feasible. To systematically assess various aspects of such forecasts, the Collaborative Studies on Earthquake Predictability testing center has utilized N- (number), M- (magnitude), S- (space), conditional likelihood-, and T- (Student’s t) tests to evaluate earthquake forecasts in a gridded space–time range. This article demonstrates the correct use of point process likelihood to evaluate forecast performance covering marginal and conditional scores, such as numbers, occurrence times, locations, magnitudes, and correlations among space–time–magnitude cells. The results suggest that for models that only rely on the internal history but not on external observation to do simulation, such as the epidemic-type aftershock sequence model, test and scoring can be rigorously implemented via the likelihood function. Specifically, gridding the space domain unnecessarily complicates testing, and evaluating spatial forecasting directly via marginal likelihood might be more promising.

Algorithmic Identification of the Precursory Scale Increase Phenomenon in Earthquake Catalogs

Abstract The precursory scale increase (Ψ) phenomenon describes the sudden increase in rate and magnitude in a precursory area AP, at precursor time TP, and with precursor magnitude MP prior to the upcoming large earthquake with magnitude Mm. Scaling relations between the Ψ variables form the basis of the “Every Earthquake a Precursor According to Scale” (EEPAS) earthquake forecasting model. EEPAS is a well-established space–time point process model that forecasts large earthquakes in the medium term, that is, the coming months to decades, depending on Mm. In Aotearoa New Zealand, EEPAS contributes to hybrid models for public earthquake forecasting and to the source model of time-varying seismic hazard models, including the latest revision of the National Seismic Hazard Model. The Ψ phenomenon was recently shown not to be unique for a given earthquake, with smaller precursory areas AP associated with larger precursor times TP and vice versa. This trade-off between AP and TP has also been found for the spatial and temporal distributions of the EEPAS models. Detailed analysis of the Ψ phenomenon has so far been limited by the manual and labor-intensive procedure of identifying Ψ in earthquake catalogs. Here, we introduce two algorithms to automatically detect Ψ and apply them to real and simulated earthquake catalog data. By randomizing the catalog and removing aftershocks, we confirm that the Ψ phenomenon is a feature of space–time earthquake clustering prior to major earthquakes. Multiple Ψ identifications confirm the trade-off between AP and TP, and the scaling relations for both real and simulated catalogs are consistent with the original scaling relations on which EEPAS is based. We identify opportunities for future work to refine the algorithms and apply them to physics-based simulated catalogs to enhance the understanding of Ψ. A better understanding of Ψ has the potential to improve forecasting of large upcoming earthquakes.

Predicting the Probability of Abrupt Changes to Wave‐Generated Seafloor Sand Ripples

AbstractA new, non‐dimensional ripple reset parameter and a stochastic point process model is used to estimate the likelihood of propagating ocean waves to form ripples on sandy seabeds. The ripple reset parameter is a function only of water depth, significant wave height, and mean grain size. Ripple formation is estimated by the magnitude of an intensity function based on a time series of the ripple reset parameter. The point process model is trained with a time series of observed waves and ripple change, and is then applied to predict the probability that a ripple field with a different geometry will form within a given time interval from another time series of wave data. The model is trained and tested with four field deployments at three field sites to determine its skill in predicting the ripple formation (a) at one field site over one time period after being trained with observations from the same site over a different time period, and (b) at one field site after being trained with observations from another field site. Results show that while the model is sufficient at predicting ripple formation in both scenarios, it is sensitive to the quality and quantity of the training data. Increasing the amount of training data greatly improves model performance. Employing a stochastic model based on a simple ripple reset parameter reduces tunable model parameters and provides a prediction of the probability for ripple formation given only a water depth, grain size, and time series of wave heights.

A non-homogeneous alternating renewal process model for interval censoring

Abstract Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on a non-homogeneous alternating renewal process in which interval generation is assumed to be time-dependent, and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to the homogeneous approach through a number of illustrative examples.

Nonparametric isotropy test for spatial point processes using random rotations

In spatial statistics, point processes are often assumed to be isotropic meaning that their distribution is invariant under rotations. Statistical tests for the null hypothesis of isotropy found in the literature are based either on asymptotics or on Monte Carlo simulation of a parametric null model. Here, we present a nonparametric test based on resampling the Fry points of the observed point pattern. Empirical levels and powers of the test are investigated in a simulation study for four point process models with anisotropy induced by different mechanisms. Finally, a real data set is tested for isotropy.

Landscape positioning of Neolithic mustatil stone structures along the margins of the Nefud Desert, Saudi Arabia

Monumental rectangular stone structures called mustatils are an important emerging feature of the Holocene archaeological record of northwestern Arabia. To date, few have been excavated, with available radiocarbon dates suggesting an age range of ca. 5400–4200 BC. Here we present a rigorous spatial analysis to identify the patterning and landscape context of 169 mustatils in the southern and western margins of the Nefud Desert. This included: (1) a systematic survey of satellite imagery to identify mustatils; (2) viewshed analysis to examine location and landscape visibility; (3) a point process model to understand how diverse environmental and landscape variables affect mustatil locations; (4) mark correlation function to assess spatial patterning of mustatils based on their size. Results indicate that mustatil locations are determined most by proximity to water (likely locations of enhanced surface water occurrence under the enhanced humidity of the Mid-Holocene), on east facing slopes, close to rocky areas, at elevations between 880 and 950 masl, and on or near topographic ridges (positive topographic position index). Viewshed analysis showed that mustatils are preferentially located in areas that have good views, but not the best that are available, indicating complex landscape positioning that balances a range of topographic and behavioural factors. Using a rank permutation method with size (length) of mustatils as a proxy for labour mobilization we show that mustatils within clusters are not arranged hierarchically based on size, and were likely built by non-stratified groups of people. Our analyses show that people were choosing multiple factors when deciding where to build mustatils, and that the distribution of mustatils may relate to different groups of people resulting in the construction of complex ritual landscapes. As one of the earliest examples of large-scale monumental stone structure construction in global prehistory, understanding mustatils can enlighten us on human-environment interaction during the Neolithic.

Impacts of innovation school system in Korea: a latent space item response model with Neyman–Scott point process

Abstract South Korea’s educational system has faced criticism for its lack of focus on critical thinking and creativity, resulting in high levels of stress and anxiety among students. As part of the government’s effort to improve the educational system, the innovation school system was introduced in 2009, which aims to develop students’ creativity as well as their non-cognitive skills. To better understand the differences between innovation and regular school systems in South Korea, we propose a novel method that combines the latent space item response model with the Neyman–Scott point process model. Our method accounts for the heterogeneity of items and students, captures relationships between respondents and items, and identifies item and student clusters that can provide a comprehensive understanding of students’ behaviours/perceptions on non-cognitive outcomes. Our analysis reveals that students in the innovation school system show a higher sense of citizenship, while those in the regular school system tend to associate confidence in appearance with social ability. A comparison with exploratory item factor analysis highlights our method’s advantages in terms of uncertainty quantification of the clustering process and more detailed and nuanced clustering results. Our method is made available to an existing R package, lsirm12pl.

SB-ETAS: using simulation based inference for scalable, likelihood-free inference for the ETAS model of earthquake occurrences

The rapid growth of earthquake catalogs, driven by machine learning-based phase picking and denser seismic networks, calls for the application of a broader range of models to determine whether the new data enhances earthquake forecasting capabilities. Additionally, this growth demands that existing forecasting models efficiently scale to handle the increased data volume. Approximate inference methods such as inlabru, which is based on the Integrated nested Laplace approximation, offer improved computational efficiencies and the ability to perform inference on more complex point-process models compared to traditional MCMC approaches. We present SB-ETAS: a simulation based inference procedure for the epidemic-type aftershock sequence (ETAS) model. This approximate Bayesian method uses sequential neural posterior estimation (SNPE) to learn posterior distributions from simulations, rather than typical MCMC sampling using the likelihood. On synthetic earthquake catalogs, SB-ETAS provides better coverage of ETAS posterior distributions compared with inlabru. Furthermore, we demonstrate that using a simulation based procedure for inference improves the scalability from O(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n^2)$$\\end{document} to O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n\\log n)$$\\end{document}. This makes it feasible to fit to very large earthquake catalogs, such as one for Southern California dating back to 1981. SB-ETAS can find Bayesian estimates of ETAS parameters for this catalog in less than 10 h on a standard laptop, a task that would have taken over 2 weeks using MCMC. Beyond the standard ETAS model, this simulation based framework allows earthquake modellers to define and infer parameters for much more complex models by removing the need to define a likelihood function.

Estimating the scale-dependent influence of natural terrestrial corridors on the positioning of settlements: A multi-scale study of Roman forts in Wales

Natural terrestrial corridors have been shown to have influenced the positioning of past settlements. The scale at which this pattern-process relationship operates is often un-estimated and thus remains unclear. This paper proposes the comparison of multiple point process models as an approach for estimating the optimal scale at which this relationship is strongest. With this approach, it is revealed that the positioning of Roman forts used during the conquest of Wales was most influenced by natural terrestrial corridors at a scale of 1,100m. At this scale, the Roman army stationed at these forts could control natural corridors – both via on-the-ground response as well as through overseeing movements by the native communities of Wales. Comparing multiple scenarios, it is also shown that the control of river-systems did not influence the positioning of Roman forts at the expense of controlling these natural terrestrial corridors used by those on foot. With archaeological interpretations susceptible to change as a result of the scale at which this pattern-process relationship is measured, the estimation of the optimal scale is pivotal for non-biased inferences on the processes thought to have influenced the positioning of settlements in the past.

Predicting Question Popularity for Community Question Answering

In this paper, we study the problem of predicting popularities of questions in Community Question Answering (CQA). To address this problem, we propose a Posterior Attention Recurrent Point Process Model (PARPP) to take both the interaction of users and the Matthew effect into account for question popularity prediction. Our PARPP uses long short-term memory (LSTM) to encode the observed history and another LSTM network to record each step of decoding information. At each decoding step, it uses prior attention to capture answers that have a greater impact on the problem. When a new answer is observed, it uses Bayes’ rule to modify prior attention and obtain posterior attention. Then, the posterior attention is used to update the decoding status. We further introduce a convergence strategy to capture the Matthew effect in CQA. We conduct experiments on a Zhihu dataset crawled from a famous Chinese CQA forum. The experimental results show that our model outperforms several state-of-the-art methods. We further analyze the attention mechanism in our model. Our analysis shows that the proposed attention mechanism can better capture the impact of each answer on the future popularity of the question, which makes our model more interpretable. Our study would shed light on other similar studies such as answer ranking in response to the question and finding experts who have expertise on the topics of the questions.

Detecting rhythmic spiking through the power spectra of point process model residuals.

Objective. Oscillations figure prominently as neurological disease hallmarks and neuromodulation targets. To detect oscillations in a neuron's spiking, one might attempt to seek peaks in the spike train's power spectral density (PSD) which exceed a flat baseline. Yet for a non-oscillating neuron, the PSD is not flat: The recovery period ('RP', the post-spike drop in spike probability, starting with the refractory period) introduces global spectral distortion. An established 'shuffling' procedure corrects for RP distortion by removing the spectral component explained by the inter-spike interval (ISI) distribution. However, this procedure sacrifices oscillation-related information present in the ISIs, and therefore in the PSD. We asked whether point process models (PPMs) might achieve more selective RP distortion removal, thereby enabling improved oscillation detection.Approach. In a novel 'residuals' method, we first estimate the RP duration (nr) from the ISI distribution. We then fit the spike train with a PPM that predicts spike likelihood based on the time elapsed since the most recent of any spikes falling within the precedingnrmilliseconds. Finally, we compute the PSD of the model's residuals.Main results. We compared the residuals and shuffling methods' ability to enable accurate oscillation detection with flat baseline-assuming tests. Over synthetic data, the residuals method generally outperformed the shuffling method in classification of true- versus false-positive oscillatory power, principally due to enhanced sensitivity in sparse spike trains. In single-unit data from the internal globus pallidus (GPi) and ventrolateral anterior thalamus (VLa) of a parkinsonian monkey-in which alpha-beta oscillations (8-30 Hz) were anticipated-the residuals method reported the greatest incidence of significant alpha-beta power, with low firing rates predicting residuals-selective oscillation detection.Significance. These results encourage continued development of the residuals approach, to support more accurate oscillation detection. Improved identification of oscillations could promote improved disease models and therapeutic technologies.

An unbounded intensity model for point processes

We develop a model for point processes on the real line, where the intensity can be locally unbounded without inducing an explosion. In contrast to an orderly point process, for which the probability of observing more than one event over a short time interval is negligible, the bursting intensity causes an extreme clustering of events around the singularity. We propose a nonparametric approach to detect such bursts in the intensity. It relies on a heavy traffic condition, which admits inference for point processes over a finite time interval. With Monte Carlo evidence, we show that our testing procedure exhibits size control under the null, whereas it has high rejection rates under the alternative. We implement our approach on high-frequency data for the EUR/USD spot exchange rate, where the test statistic captures abnormal surges in trading activity. We detect a nontrivial amount of intensity bursts in these data and describe their basic properties. Trading activity during an intensity burst is positively related to volatility, illiquidity, and the probability of observing a drift burst. The latter effect is reinforced if the order flow is imbalanced or the price elasticity of the limit order book is large.

Optimal decision rules for marked point process models

We study a Markov decision problem in which the state space is the set of finite marked point patterns in the plane, the actions represent thinnings, the reward is proportional to the mark sum which is discounted over time, and the transitions are governed by a birth-death-growth process. We show that thinning points with large marks maximises the discounted total expected reward when births follow a Poisson process and marks grow logistically. Explicit values for the thinning threshold and the discounted total expected reward over finite and infinite horizons are also provided. When the points are required to respect a hard core distance, upper and lower bounds on the discounted total expected reward are derived.

Predicting urban signal-controlled intersection congestion events using spatio-temporal neural point process

ABSTRACT The urban traffic signal-controlled intersections are of great significance for solving the problem of urban road congestion. Previous research on congestion prediction mainly aggregated data at the level of road segments or traffic flow at a coarse regulated time interval. Fine-grained prediction of congestion events at the lane-level and cycle-level enables detailed a understanding of spatio-temporal dependencies, leading to congestion reduction, improved efficiency. This paper presents a Spatio-Temporal Neural Point Process (STNPP) model that combines Graph Neural Networks and Neural Temporal Point Process to predict congestion events at urban intersections. The proposed model allows for complete prediction of congestion events, including their occurrence, development, dissipation. In the process of spatial correlation modeling, graph neural networks are used to model the spatial relationships between both region and intersections. The current intersection and its upstream/downstream areas are modeled separately. To model the temporal correlations at individual intersections, we focus on a specific lane and capture the evolution of congestion events using the Neural Point Process Gated Recurrent Unit (NPPGRU), which captures the temporal granularity changes of signal-controlled cycles in congestion events. Using actual traffic speed and signal-controlled data from Hangzhou city, we validate that the proposed method achieves stable predictive performance.

Unveiling the web of interactions: Analyzing dynamic customer engagements across multiple websites

Online customers exhibit unique dynamic patterns during their journey. In the early phases, customers gather different information from multiple websites. Collectively, the information serves as references, helping customers to make informed decisions. Further, inferior options are eliminated until the journey converges on a single website for conversion. In this study, we examine the dynamic interactions among online customers and websites. To capture the dynamics, we propose a metric, momentum of information-building and incorporate the measure into a dynamically and mutually exciting marked point process model. We apply the proposed model to clickstream data for hotel bookings and find strong evidence of dynamic behavior. Also, we show that ignoring the dynamics may lead to bias in estimating the carryover and spillover effects and diminish the prediction accuracy of website visits and conversions. Furthermore, we offer recommendations for effective advertising strategies focusing on key aspects such as advertising density, timing, and targeting.

A self-excitation point process model for quantifying nanoparticle agglomeration

ABSTRACT Nanoparticle agglomeration refers to the spontaneous clustering of nanoparticles caused by the attractive forces. Agglomeration impacts the physical and mechanical properties of nanoparticles and their composites. Understanding and controlling nanoparticle agglomeration is crucial for optimizing various applications, from nanomedicine to nanoelectronics. In this work, we formulated a self-exciting point process model to analyse nanoparticle agglomeration based on microstructural input. The model is utilised to categorise a specific set of points within the microstructure as either independent (dispersed) or dependent (agglomerated), along with their respective probabilities. We employed this approach to study two distinct scenarios: (a) Agglomeration in experimentally generated microstructures of titanium nanoparticles, and (b) Analysing agglomeration patterns in simulated microstructures of carbon nanotube networks, generated using a stochastic microstructure model. We obtain a quantitative understanding of the connection between the sonication duration and the level of agglomeration in titanium nanoparticles. As the sonication period increases, both the agglomeration percentage and the size of the agglomerates decrease. Furthermore, our analysis of simulated carbon nanotube microstructures, including equiaxed and rope-like agglomeration, shows a close alignment between results obtained from the point process model and those generated by the stochastic microstructure model.

Convolutional Non-Homogeneous Poisson Process and its Application to Wildfire Ignition Risk Quantification for Power Delivery Networks

To quantify wildfire ignition risks on power delivery networks, the current practice predominantly relies on the empirically calculated fire danger indices, which may not well capture the effects of dynamically changing environmental factors. This article proposes a spatio-temporal point process model, known as the Convolutional Non-homogeneous Poisson Process (cNHPP), and applies the model to quantify wildfire ignition risks for power delivery networks. The proposed model captures both the current (i.e., instantaneous) and cumulative (i.e., historical) effects of key environmental processes (i.e., covariates) on wildfire risks, as well as the spatio-temporal dependency among different segments of the power delivery network. The computation and interpretation of the intensity function are thoroughly investigated. We apply the proposed approach to estimate wildfire ignition risks on major transmission lines in California, using historical fire data, meteorological and vegetation data obtained from the National Oceanic and Atmospheric Administration and National Aeronautics and Space Administration. A comprehensive comparison study is performed to show the applicability and predictive capability of the proposed approach.

Validation of point process predictions with proper scoring rules

AbstractWe introduce a class of proper scoring rules for evaluating spatial point process forecasts based on summary statistics. These scoring rules rely on Monte Carlo approximations of expectations and can therefore easily be evaluated for any point process model that can be simulated. In this regard, they are more flexible than the commonly used logarithmic score and other existing proper scores for point process predictions. The scoring rules allow for evaluating the calibration of a model to specific aspects of a point process, such as its spatial distribution or tendency toward clustering. Using simulations, we analyze the sensitivity of our scoring rules to different aspects of the forecasts and compare it to the logarithmic score. Applications to earthquake occurrences in northern California, United States and the spatial distribution of Pacific silver firs in Findley Lake Reserve in Washington highlight the usefulness of our scores for scientific model selection.

Unveiling Multi-Agent Strategies: A Data-Driven Approach for Extracting and Evaluating Team Tactics from Football Event and Freeze-Frame Data

Studying collective behavior in opposing multi-agent teams is crucial across game theory, robotics, and sports analytics. In sports, especially football, team tactics involve intricate strategic spatial and action behaviors displayed as event sequences during possession. Understanding and analyzing these tactics is essential for successful training, strategic planning, and on-field success. While traditional approaches, such as notational and statistical analyses, offer valuable insights into team tactics, they often lack a comprehensive consideration of contextual information, thereby limiting the holistic evaluation of teams’ performances. To bridge this gap and capture the nuanced intricacies of team tactics, we employed advanced methodologies. The sequential pattern mining algorithm PrefixSpan was utilized to extract tactical patterns from possession sequences, enabling a deeper understanding of how teams strategize and adapt during play. Additionally, the neural marked spatio temporal point process (NMSTPP) model was leveraged to model and predict team behaviors, facilitating a fair comparison among teams. The evaluation of team possessions was further enhanced through the innovative holistic possession utilization score metrics, providing a more nuanced assessment of performance. In our experimental exploration, we identified and classified five distinct team tactics, validated the efficacy of the NMSTPP model when integrating StatsBomb 360 data, and conducted a comprehensive analysis of English Premier League teams during the 2022/2023 season. The results were visualized using radar plots and scatter plots with mean shift clustering. Lastly, the potential applications to RoboCup were discussed.

Standard form of master equations for general non-Markovian jump processes: The Laplace-space embedding framework and asymptotic solution

We present a standard form of master equations (MEs) for general one-dimensional non-Markovian (history-dependent) jump processes, complemented by an asymptotic solution derived from an expanded system-size approach. The ME is obtained by developing a general Markovian embedding using a suitable set of auxiliary field variables. This Markovian embedding uses a Laplace-convolution operation applied to the velocity trajectory. We introduce an asymptotic method tailored for this ME standard, generalizing the system-size expansion for these jump processes. Under specific stability conditions tied to a single noise source, upon coarse graining, the generalized Langevin equation (GLE) emerges as a universal approximate model for point processes in the weak-coupling limit. This methodology offers a unified analytical tool set for general non-Markovian processes, reinforcing the universal applicability of the GLE founded in microdynamics and the principles of statistical physics. Published by the American Physical Society 2024