Stochastic Point Process Research Articles

A geometric approach to scaling individual distributions to macroecological patterns

Understanding macroecological patterns across scales is a central goal of ecology and a key need for conservation biology. Much research has focused on quantifying and understanding macroecological patterns such as the species-area relationship (SAR), the endemic-area relationship (EAR) and relative species abundance curve (RSA). Understanding how these aggregate patterns emerge from underlying spatial pattern at individual level, and how they relate to each other, has both basic and applied relevance. To address this challenge, we develop a novel spatially explicit geometric framework to understand multiple macroecological patterns, including the SAR, EAR, RSA, and their relationships. First, we provide a general theory that can be used to derive the asymptotic slopes of the SAR and EAR, and demonstrates the dependency of RSAs on the shape of the sampling region. Second, assuming specific shapes of the sampling region, species geographic ranges, and individual distribution patterns therein based on theory of stochastic point processes, we demonstrate various well-documented macroecological patterns can be recovered, including the tri-phasic SAR and various RSAs (e.g., Fisher’s logseries and the Poisson lognormal distribution). We also demonstrate that a single equation unifies RSAs across scales, and provide a new prediction of the EAR. Finally, to demonstrate the applicability of the proposed model to ecological questions, we provide how beta diversity changes with spatial extent and its grain over multiple scales. Emergent macroecological patterns are often attributed to ecological and evolutionary mechanisms, but our geometric approach still can recover many previously observed patterns based on simple assumptions about species geographic ranges and the spatial distribution of individuals, emphasizing the importance of geometric considerations in macroecological studies.

A hard-core stochastic process with simultaneous births and deaths

We introduce and examine a class of stochastic spatial point processes with births and deaths, related to spatial loss networks introduced in Ferrari et al. In these processes, a point stays in the system until it is removed due to interaction with a conflicting new arrival. In particular, we consider an interaction scheme where two points are conflicting if closed balls of radius 1/2 around them overlap and a new arriving point “ kills” each conflicting point independently with probability ρ. The new point is accepted if all conflicting points are killed. We construct this process on the whole Euclidean space , . If ρ is large enough, we show existence of a stationary regime and exponential convergence to the stationary distribution. Such stochastic models have been studied earlier as models for populations of interacting individuals or as spatial queuing and resource sharing networks.

Min–Max User-Pair Association Criterion and Outage Performance of K-Tier Relay-Based Heterogeneous Networks

This paper focuses on the user-pair association for multi-tier relay-based dual-hop heterogeneous networks (HetNets) and proposes a novel min–max user-pair association (MM-UPA) criterion by integrating the bias factors, which is an evolution of conventional single-hop user association based on receive signal strength (RSS). The core idea of the proposed MM-UPA is that, in each tier the nearest relay to a typical user-pair is characterized by the maximum of the RSS reciprocals $${1 / {P_{SR}^{j} }}$$ and $${1/ {P_{RD}^{j} }}$$ by the source and destination from relay. Then, a typical user-pair is associated to the relay based on minimizing these maximums of each tier. This proposed MM-UPA criterion is fit especially for the relay-based HetNets due to effectively exploiting the source-relay and relay-destination links simultaneously. With the MM-UPA criterion, by using stochastic geometry and homogeneous Poisson point processes, we present the analytical expression of the probability that a typical user-pair is associated with a relay of the kth tier as well as the corresponding statistical description of the distances from the source and destination of a typical user-pair to its associated relay. Finally, the average outage probability of a typical user-pair relay communication link is derived. The presented simulations and numerical results validate the derivations.

Exploratory analysis of lightning-ignited wildfires in the Warren Region, Western Australia

An exploratory analysis of lightning-ignited wildfire data for the Warren Region of Western Australia was carried out for the period from April 1976 to December 2016. Temporal patterns in the series were examined in terms of characterizing the seasonal cycle, and detecting long-term trends and changes in seasonality over time. A generalized additive modelling approach was used to ensure that temporal features were determined by the data rather than a priori assumed mathematical forms (e.g. linear or low-order polynomial functions). The spatial organization of the data was evaluated using concepts from the theory of stochastic point processes. Results indicate a strong seasonality in the monthly lightning ignition series, the presence of a long-term trend and an interaction between trend and seasonality. There is also strong evidence of spatial variation in the number of ignitions per unit area in terms of location and distance from nearest ignition. Within the Warren Region, observation platforms for fire detection and reporting protocols have remained stable over the period of record, and changes in land use are unlikely to have altered the pattern of lightning ignition. Thus, the above results might reflect an interplay between: landscape attributes (e.g. vegetation classes, elevation, slope, aspect); changes in rainfall and fuel moisture; changes in fuel management practices; and, perhaps, an increase in the frequency of dry thunderstorms and fire weather conditions.

Optimal Monitoring Time Interval for Radio-Electronic Equipment Unsteady Failure Rate

The method for determining the optimal duration of monitoring interval of non-stationary Poisson failure rate of radio-electronic equipment (REE) for a wide range of stationary processes, modulating it by intensity. To know the optimal du-ration of monitoring interval, taking into account the flow double stochasticity is necessary to calculate and predict REE reliability factor, operating in near-Earth space as a part of artificial Earth satellites (AES). In the course of calculation REE is exposed to ionizing radiation of unsteady flows and electrostatic charges, leading to short duration or irreversible failures. Based on the known linear model of double stochastic point process a detailed mathematical analysis is made for the first time ever and analytical expressions were obtained for the optimal monitoring time interval with respect to minimum combined standard uncertainty. Since the true nature of unsteadiness is unknown, analysis is carried out for Gaussian and Gauss-Markov modulating processes, covering a wide range of random processes. The analysis results confirm practical similarity of optimal monitoring time interval values for a wide range of modulating processes.

Recurrent spatio-temporal modeling of check-ins in location-based social networks.

Social networks are getting closer to our real physical world. People share the exact location and time of their check-ins and are influenced by their friends. Modeling the spatio-temporal behavior of users in social networks is of great importance for predicting the future behavior of users, controlling the users’ movements, and finding the latent influence network. It is observed that users have periodic patterns in their movements. Also, they are influenced by the locations that their close friends recently visited. Leveraging these two observations, we propose a probabilistic model based on a doubly stochastic point process with a periodic-decaying kernel for the time of check-ins and a time-varying multinomial distribution for the location of check-ins of users in the location-based social networks. We learn the model parameters by using an efficient EM algorithm, which distributes over the users, and has a linear time complexity. Experiments on synthetic and real data gathered from Foursquare show that the proposed inference algorithm learns the parameters efficiently and our method models the real data better than other alternatives.

The spatial and temporal scales of local dengue virus transmission in natural settings: a retrospective analysis

BackgroundDengue is a vector-borne disease caused by the dengue virus (DENV). Despite the crucial role of Aedes mosquitoes in DENV transmission, pure vector indices poorly correlate with human infections. Therefore there is great need for a better understanding of the spatial and temporal scales of DENV transmission between mosquitoes and humans. Here, we have systematically monitored the circulation of DENV in individual Aedes spp. mosquitoes and human patients from Caratinga, a dengue endemic city in the state of Minas Gerais, in Southeast Brazil. From these data, we have developed a novel stochastic point process pattern algorithm to identify the spatial and temporal association between DENV infected mosquitoes and human patients.MethodsThe algorithm comprises of: (i) parameterization of the variogram for the incidence of each DENV serotype in mosquitoes; (ii) identification of the spatial and temporal ranges and variances of DENV incidence in mosquitoes in the proximity of humans infected with dengue; and (iii) analysis of the association between a set of environmental variables and DENV incidence in mosquitoes in the proximity of humans infected with dengue using a spatio-temporal additive, geostatistical linear model.ResultsDENV serotypes 1 and 3 were the most common virus serotypes detected in both mosquitoes and humans. Using the data on each virus serotype separately, our spatio-temporal analyses indicated that infected humans were located in areas with the highest DENV incidence in mosquitoes, when incidence is calculated within 2.5–3 km and 50 days (credible interval 30–70 days) before onset of symptoms in humans. These measurements are in agreement with expected distances covered by mosquitoes and humans and the time for virus incubation. Finally, DENV incidence in mosquitoes found in the vicinity of infected humans correlated well with the low wind speed, higher air temperature and northerly winds that were more likely to favor vector survival and dispersal in Caratinga.ConclusionsWe have proposed a new way of modeling bivariate point pattern on the transmission of arthropod-borne pathogens between vector and host when the location of infection in the latter is known. This strategy avoids some of the strong and unrealistic assumptions made by other point-process models. Regarding virus transmission in Caratinga, our model showed a strong and significant association between high DENV incidence in mosquitoes and the onset of symptoms in humans at specific spatial and temporal windows. Together, our results indicate that vector surveillance must be a priority for dengue control. Nevertheless, localized vector control at distances lower than 2.5 km around premises with infected vectors in densely populated areas are not likely to be effective.

A Stochastic Geometry Approach to Full‐Duplex MIMO Relay Network

Cellular networks are extensively modeled by placing the base stations on a grid, with relays and destinations being placed deterministically. These networks are idealized for not considering the interferences when evaluating the coverage/outage and capacity. Realistic models that can overcome such limitation are desirable. Specifically, in a cellular downlink environment, the full‐duplex (FD) relaying and destination are prone to interferences from unintended sources and relays. However, this paper considered two‐hop cellular network in which the mobile nodes aid the sources by relaying the signal to the dead zone. Further, we model the locations of the sources, relays, and destination nodes as a point process on the plane and analyze the performance of two different hops in the downlink. Then, we obtain the success probability and the ergodic capacity of the two‐hop MIMO relay scheme, accounting for the interference from all other adjacent cells. We deploy stochastic geometry and point process theory to rigorously analyze the two‐hop scheme with/without interference cancellation. These attained expressions are amenable to numerical evaluation and are corroborated by simulation results.

A restricted Linked Stress Release Model (LSRM) for the Corinth gulf (Greece)

The earthquake occurrence in the Corinth Gulf (Greece) is modeled by means of the Linked Stress Release Model, which combines the gradual increase of the strain energy due to continuous tectonic loading and its sudden release when an event occurs. The modeling is based on the theory of stochastic point processes and it is determined by the conditional intensity function. The modeled time–dependent seismicity incorporates stress transfer and interactions among the major fault segments that belong either to the western or to the eastern part of the study area. Static stress changes caused by the coseismic slip of eight characteristic events that occurred during the instrumental era were calculated. Their perturbations reveal that earthquakes occurring in the eastern part provoke excitation in the western part while the events that occurred in the western part do not affect the seismicity of the eastern part. This result was taken into account and the values of the model parameters were constrained accordingly. The analysis is also performed on synthetic earthquake catalogs developed by an earthquake simulator algorithm that can replicate the seismic activity in the study area and yields to comparable results.

Statistical Features of the 2010 Beni-Ilmane, Algeria, Aftershock Sequence

The aftershock sequence of the 2010 Beni-Ilmane (M W 5.5) earthquake is studied in depth to analyze the spatial and temporal variability of seismicity parameters of the relationships modeling the sequence. The b value of the frequency–magnitude distribution is examined rigorously. A threshold magnitude of completeness equal to 2.1, using the maximum curvature procedure or the changing point algorithm, and a b value equal to 0.96 ± 0.03 have been obtained for the entire sequence. Two clusters have been identified and characterized by their faulting type, exhibiting b values equal to 0.99 ± 0.05 and 1.04 ± 0.05. Additionally, the temporal decay of the aftershock sequence was examined using a stochastic point process. The analysis was done through the restricted epidemic-type aftershock sequence (RETAS) stochastic model, which allows the possibility to recognize the prevailing clustering pattern of the relaxation process in the examined area. The analysis selected the epidemic-type aftershock sequence (ETAS) model to offer the most appropriate description of the temporal distribution, which presumes that all events in the sequence can cause secondary aftershocks. Finally, the fractal dimensions are estimated using the integral correlation. The obtained D 2 values are 2.15 ± 0.01, 2.23 ± 0.01 and 2.17 ± 0.02 for the entire sequence, and for the first and second cluster, respectively. An analysis of the temporal evolution of the fractal dimensions D −2, D 0, D 2 and the spectral slope has been also performed to derive and characterize the different clusters included in the sequence.

A doubly stochastic rainfall model with exponentially decaying pulses

We develop a doubly stochastic point process model with exponentially decaying pulses to describe the statistical properties of the rainfall intensity process. Mathematical formulation of the point process model is described along with second-order moment characteristics of the rainfall depth and aggregated processes. The derived second-order properties of the accumulated rainfall at different aggregation levels are used in model assessment. A data analysis using 15 years of sub-hourly rainfall data from England is presented. Models with fixed and variable pulse lifetime are explored. The performance of the model is compared with that of a doubly stochastic rectangular pulse model. The proposed model fits most of the empirical rainfall properties well at sub-hourly, hourly and daily aggregation levels.

EARTHQUAKE FORECAST BY TIME-DEPENDENT HAZARD MODELS

This presentation outlines methodological aspects of earthquake forecasting. The recurring debates concerning predictability of earthquakes clearly show how this problem is centred on the difficulty of systematically testing the numerous methodologies that in the years have been proposed and sustained by the supporters of prediction. This difficulty starts, sometimes, from the lack of a quantitative and rigorous definition of the concerned precursor, and other times from the lack of continuous observations, upon which statistical analyses could be based. After an introduction concerning the definition of earthquake precursors, the way how to validate forecast hypotheses and the cost associated to their operational application, I give two examples of time-dependent hazard models, for long-term and short-term earthquake forecasts respectively. Considering the long-term forecast modelling, the effect of stress change due to previous historical earthquakes on the probability of occurrence of future earthquakes on neighbouring faults is taken into account. Following a standard methodology developed a couple of decades ago, the probability of occurrence in the next 50 years for a characteristic earthquake on known seismogenic structures can be estimated by a time-dependent renewal model. Then, a physical model for the Coulomb stress change caused by previous earthquakes on these structures is applied. The influence of this stress change on the occurrence rate of characteristic earthquakes is computed taking into account the permanent perturbation (clock advance). The method so developed is applied to the computation of earthquake hazard of the main seismogenic structures recognized in the Southern Apennines region, for which both historical and paleoseismological data are available. A popular short-term time dependent hazard forecast model is the epidemic model. In this model earthquakes are regarded as the realization of a stochastic point process, and their magnitude distribution is described by the Gutenberg-Richter law with a constant b-value. The occurrence rate density is computed by the sum of two terms, one representing the independent, or spontaneous activity, and the other representing the activity induced by previous earthquakes. While the first term depends only on space, the second one is factored into three terms that respectively include the magnitude, time and location of the past earthquakes. An example of application of the epidemic model to the 2009 L’Aquila seismic series is shown.

On a unified point process approach for the characterization of bioelectric discrete phenomena.

This paper discusses a unified method based on the theory of point processes to characterize various types of bioelectric discrete signals such as heart beat timing, myoelectric activity, discharge of primary sensory neurons or neurons in the central nervous systems. The doubly stochastic point processes, in which the discrete event occurring intensity is stochastic, forms the most general class to characterize the discrete phenomena. In this paper the self-exciting process has been shown to be useful to characterize wide range of discrete biosignals. The modeling of conditional intensity function is the essential part of the characterization. When the intensity has a parametric model, the maximum likelihood parameter estimation will be the useful way to characterize the phenomena. The effectiveness of the method is demonstrated by a specific modeling of the spontaneous neuronal burst discharges recorded from the brain thalamus during the neuro surgery. The first approximation model has four parameters obtained by the instantaneous nonlinearly transformed sinusoidal function. An extended model allows arbitrary periodic intensity with refractory period. Predicted interval histograms show good agreement with the observed ones indicating the validity of the proposed method.

Optimal Trading Strategies within the Hamilton-Jacobi-Bellman Equation - An Application to Statistical Arbitrage

We propose optimal trading strategies based on the Hamilton-Jacobi-Bellman equation when a trader is allowed to place only market orders with a particular focus on a statistical arbitrage. A process that a placed limit order is filled is often modeled as a stochastic point process. In case of a market order, however, it is filled with a hundred percent probability meaning that it is not a stochastic process and is rather a deterministic one. They are therefore represented using the Heaviside step function and their infinitesimal generators are just time derivative and give delta functions. An integral linear utility function with an inventory penalty and a discount enables us to obtain optimal number of market order with a simple calculus and those in a limit T → ∞, i.e. a stationary limit. We show that, at least within our formulation, we cannot generate a positive PnL by trading multiple set of assets whose noises are correlated but their drifts are zero which contradicts the common statement that Given correlated multiple assets, we can exploit those correlations by trading an appropriate linear combination of However, we show that if multiple assets are cointegrated, we can create a mean-reverting portfolio and exploit them. In that case, an optimal number of market order which takes current status of market as well as a trader's preference into account is given in a simple but intuitive and reasonable form and it does generate positive PnL while controlling risk of the portfolio.

General Point Sampling with Adaptive Density and Correlations

Analyzing and generating sampling patterns are fundamental problems for many applications in computer graphics. Ideally, point patterns should conform to the problem at hand with spatially adaptive density and correlations. Although there exist excellent algorithms that can generate point distributions with spatially adaptive density or anisotropy, the pair-wise correlation model, blue noise being the most common, is assumed to be constant throughout the space. Analogously, by relying on possibly modulated pair-wise difference vectors, the analysis methods are designed to study only such spatially constant correlations. In this paper, we present the first techniques to analyze and synthesize point patterns with adaptive density and correlations. This provides a comprehensive framework for understanding and utilizing general point sampling. Starting from fundamental measures from stochastic point processes, we propose an analysis framework for general distributions, and a novel synthesis algorithm that can generate point distributions with spatio-temporally adaptive density and correlations based on a locally stationary point process model. Our techniques also extend to general metric spaces. We illustrate the utility of the new techniques on the analysis and synthesis of real-world distributions, image reconstruction, spatio-temporal stippling, and geometry sampling.

An adaptive hawkes process formulation for estimating time-of-day zonal trip arrivals with location-based social networking check-in data

Location-Based Social Networking (LBSN) services, such as Foursquare, Facebook check-ins, and Geo-tagged Twitter tweets, have emerged as new secondary data sources for studying individual travel mobility patterns at a fine-grained level. However, the differences between human social behavioral and travel patterns can cause significant sampling bias for travel demand estimation. This paper presents a dynamic model to estimate time-of-day zonal trip arrival patterns. In the proposed model, the state propagation is formulated by the Hawkes process; the observation model implements LBSN sampling. The proposed model is applied to Foursquare check-in data collected from Austin, Texas in 2010 and calibrated with Origin-Destination (OD) data and time of day factor from Capital Area Metropolitan Planning Organization (CAMPO). The proposed model is compared with a simple aggregation model of trip purposes and time of day based on a prior daily OD estimation model using the LBSN data. The results illustrate the promising benefits of applying stochastic point process models and state-space modeling in time-of-day zonal arrival pattern estimation with the LBSN data. The proposed model can significantly reduce the number of parameters to calibrate in order to reduce the sampling bias of LBSN data for trip arrival estimation.

Application of a linked stress release model in Corinth Gulf and Central Ionian Islands (Greece)

Spatio-temporal stress changes and interactions between adjacent fault segments consist of the most important component in seismic hazard assessment, as they can alter the occurrence probability of strong earthquake onto these segments. The investigation of the interactions between adjacent areas by means of the linked stress release model is attempted for moderate earthquakes (M ≥ 5.2) in the Corinth Gulf and the Central Ionian Islands (Greece). The study areas were divided in two subareas, based on seismotectonic criteria. The seismicity of each subarea is investigated by means of a stochastic point process and its behavior is determined by the conditional intensity function, which usually gets an exponential form. A conditional intensity function of Weibull form is used for identifying the most appropriate among the models (simple, independent and linked stress release model) for the interpretation of the earthquake generation process. The appropriateness of the models was decided after evaluation via the Akaike information criterion. Despite the fact that the curves of the conditional intensity functions exhibit similar behavior, the use of the exponential-type conditional intensity function seems to fit better the data.

Quantifying inhomogeneity of spatial point patterns

This article compares measures for quantifying the level of inhomogeneity of a given spatial point pattern. Comparisons are based on two main metrics: first, we evaluate the performance of the measures on both certain stochastic point processes and on very specific point patterns designed to expose potential weaknesses of the inhomogeneity measures. Second, we evaluate the computational complexity of the measures. Results show that choosing a measure is a tradeoff between accurate assessment of inhomogeneity and computational complexity. The only exception is the proposed extended wrap-around measure, which is performing well in terms of both metrics.

A reference decision model of first responders' decision-making

This paper introduces a generic decision model that connects three basic components in the response decision process: the stochastic information flow consisting of help requests and resource availabilities; situation awareness based on a spatiotemporal risk picture and response decisions allocating resources for response actions. The formalism utilised is the stochastic marked point process. The paper is written to support 'black-box' modelling of situation awareness in virtual training simulation, where a trainee's performance is compared with, e.g., the performance of experienced commanders with respect to key performance indicators (KPIs) obtained from such simulations. The paper also briefly discusses the use of the introduced decision model in developing agent-based models and simulations. It is also noted that the decision model encompasses the recognition-primed decision (RPD) model of naturalistic decision-making.

A reference decision model of first responders' decision-making

This paper introduces a generic decision model that connects three basic components in the response decision process: the stochastic information flow consisting of help requests and resource availabilities; situation awareness based on a spatiotemporal risk picture and response decisions allocating resources for response actions. The formalism utilised is the stochastic marked point process. The paper is written to support 'black-box' modelling of situation awareness in virtual training simulation, where a trainee's performance is compared with, e.g., the performance of experienced commanders with respect to key performance indicators (KPIs) obtained from such simulations. The paper also briefly discusses the use of the introduced decision model in developing agent-based models and simulations. It is also noted that the decision model encompasses the recognition-primed decision (RPD) model of naturalistic decision-making.