Point Process Approach Research Articles

Extremes of independent stochastic processes: a point process approach

For each n ≥ 1, let \(\{ X_{in}, \quad i \geqslant 1 \}\) be independent copies of a nonnegative continuous stochastic process Xn = (Xn(s))s∈S indexed by a compact metric space S. We are interested in the process of partial maxima \(\tilde M_{n}(t,s) =\max \{ X_{in}(s), 1 \leqslant i\leqslant [nt] \},\quad t\geq 0, s\in S,\) where the brackets [ ⋅ ] denote the integer part. Under a regular variation condition on the sequence of processes Xn, we prove that the partial maxima process \(\tilde M_{n}\) weakly converges to a superextremal process \(\tilde M\) as \(n\to \infty \). We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.

Classification of Earthquake Distribution in the Banda Sea Collision Zone with Point Process Approach

Classification of Earthquake Distribution in the Banda Sea Collision Zone with Point Process Approach

Modelling the location decisions of manufacturing firms with a spatial point process approach

The paper is devoted to explore how the increasing availability of spatial micro-data, jointly with the diffusion of GIS software, allows to exploit micro-econometric methods based on stochastic spatial point processes in order to understand the factors that may influence the location decisions of new firms. By using the knowledge of the geographical coordinates of the newborn firms, their spatial distribution is treated as a realization of an inhomogeneous marked point process in the continuous space and the effect of spatial-varying factors on the location decisions is evaluated by parametrically modelling the intensity of the process. The study is motivated by the real issue of analysing the birth process of small and medium manufacturing firms in Tuscany, an Italian region, and it shows that the location choices of the new Tuscan firms is influenced on the one hand by the availability of infrastructures and the level of accessibility, and on the other by the presence and the characteristics of the existing firms. Moreover, the effect of these factors varies with the size and the level of technology of the new firms. Besides the specific Tuscan result, the study shows the potentiality of the described micro-econometric approach for the analysis of the spatial dynamics of firms.

A two-step approach to model precipitation extremes in California based on max-stable and marginal point processes

In modeling spatial extremes, the dependence structure is classically inferred by assuming that block maxima derive from max-stable processes. Weather stations provide daily records rather than just block maxima. The point process approach for univariate extreme value analysis, which uses more historical data and is preferred by some practitioners, does not adapt easily to the spatial setting. We propose a two-step approach with a composite likelihood that utilizes site-wise daily records in addition to block maxima. The procedure separates the estimation of marginal parameters and dependence parameters into two steps. The first step estimates the marginal parameters with an independence likelihood from the point process approach using daily records. Given the marginal parameter estimates, the second step estimates the dependence parameters with a pairwise likelihood using block maxima. In a simulation study, the two-step approach was found to be more efficient than the pairwise likelihood approach using only block maxima. The method was applied to study the effect of El Niño-Southern Oscillation on extreme precipitation in California with maximum daily winter precipitation from 35 sites over 55 years. Using site-specific generalized extreme value models, the two-step approach led to more sites detected with the El Niño effect, narrower confidence intervals for return levels and tighter confidence regions for risk measures of jointly defined events.

Investigating teleconnection drivers of bivariate heat waves in Florida using extreme value analysis

Maximum and minimum daily temperatures from the second half of the twentieth century are examined using a high resolution dataset of 833 grid cells across the state of Florida. A bivariate extreme value analysis point process approach is used to model characteristics including the frequency, magnitude, duration, and timing of periods or heat waves during which both daily maximum and minimum temperatures exceed their respective 90th percentile thresholds. The temperature dataset is combined with indices of the El Nino-Southern Oscillation (ENSO) and the Atlantic multi-decadal oscillation (AMO) to explore the influence of these oscillations on heat wave characteristics in Florida. In order to investigate the influence of a time varying signal (ENSO and AMO) on heat waves the signals are introduced into non-stationary models as covariates in the location and log-transformed scale parameters. The improvements to the model obtained by introducing covariates are examined using the deviance statistic whereby the difference in negative log-likelihood values between two models is tested for significance using a Chi squared distribution. Significant improvements in the non-stationary models with ENSO and AMO covariates indicate spatially varying impacts in the frequency, magnitude, and duration of heat waves. In particular, the warm phase of the AMO brings heat waves earlier in the summertime while also increasing their magnitude, frequency, and duration.

Forecasting Online Auctions via Self‐Exciting Point Processes

ABSTRACTModeling online auction prices is a popular research topic among statisticians and marketing analysts. Recent research mainly focuses on two directions: one is the functional data analysis (FDA) approach, in which the price–time relationship is modeled by a smooth curve, and the other is the point process approach, which directly models the arrival process of bidders and bids. In this paper, a novel model for the bid arrival process using a self‐exciting point process (SEPP) is proposed and applied to forecast auction prices. The FDA and point process approaches are linked together by using functional data analysis technique to describe the intensity of the bid arrival point process. Using the SEPP to model the bid arrival process, many stylized facts in online auction data can be captured. We also develop a simulation‐based forecasting procedure using the estimated SEPP intensity and historical bidding increment. In particular, prediction interval for the terminal price of merchandise can be constructed. Applications to eBay auction data of Harry Potter books and Microsoft Xbox show that the SEPP model provides more accurate and more informative forecasting results than traditional methods. Copyright © 2014 John Wiley & Sons, Ltd.

Statistical modeling and CMIP5 simulations of hot spell changes in China

A hot spell is an extreme weather event with one or more consecutive days with daily maximum temperature exceeding a certain threshold of high temperature. Statistical modeling of summer hot spells in China during 1960–2005 and their simulations in the historical experiment of the Coupled Model Intercomparison Project Phase 5 (CMIP5) are investigated in this study. A technique called the hot spell model (HSM), introduced by Furrer et al. (Clim Res 43:191–205, 2010) for modeling hot spells by extending the point process approach to extreme value theory, is applied. Specifically, the frequency of summer hot spells is modeled by a Poisson distribution, their intensity is modeled by a generalized Pareto distribution, and their duration is modeled by a geometric distribution. Results show that the HSM permits realistic modeling of summer hot spells in China. Trends in the frequency, duration, and intensity of hot spells were estimated based on the HSM for the observed period from 1960 to 2005. Furthermore, the performance in simulating hot spell characteristics and trends from the CMIP5 historical run were assessed based on the HSM. Climate models with good performance were selected to conduct an ensemble projection of hot spell intensity, frequency, and duration and their trends in future decades.

Point process analysis in brain networks of patients with diabetes

Noise and individual differences arise from disturbances in the effective use of resting-state functional magnetic resonance image (fMRI) datasets. In this study, the point process is used to treat fMRI datasets of healthy controls and patients with diabetes; then, functional brain networks of subjects are established using two sets of BOLD signals. The results illustrate that differences between the healthy controls and the patients were more obvious in point process signals than nonpoint process signals. Our results also suggest that there is a higher recognition accuracy of the signals by preprocessing with the point process. These findings may suggest that the point process approach can reduce BOLD signals noise, providing a new method for functional magnetic resonance data preprocessing, and may provide a promising method for early data preprocessing in computer-aided disease diagnostics.

Marked point process hotspot maps for homicide and gun crime prediction in Chicago

Crime hotspot maps are a widely used and successful method of displaying spatial crime patterns and allocating police resources. However, hotspot maps are often created over a single timescale using only one crime type. In the case of short-term hotspot maps that utilize several weeks of crime data, risk estimates suffer from a high variance, especially for low frequency crimes such as homicide. Long-term hotspot maps that utilize several years of data fail to take into account near-repeat effects and emerging hotspot trends. In this paper we show how point process models of crime can be extended to include leading indicator crime types, while capturing both short-term and long-term patterns of risk, through a marked point process approach. Several years of data and many different crime types are systematically combined to yield accurate hotspot maps that can be used for the purpose of predictive policing of gun-related crime. We apply the methodology to a large, open source data set which has been made available to the general public online by the Chicago Police Department.

Increased risk of heat waves in Florida: Characterizing changes in bivariate heat wave risk using extreme value analysis

Maximum and minimum daily temperatures from the second half of the 20th century are examined using a high resolution dataset of 833 grid cells across the state of Florida. A bivariate Extreme Value Analysis Point Process approach is used to model characteristics including the frequency, magnitude, duration, and timing of periods or heat waves during which both daily maximum and minimum temperatures exceed their respective 90th percentile thresholds. Variability in heat wave characteristics is examined across the state to give an indication of those areas where heat waves with certain characteristics may be more likely to occur. Changes in heat wave characteristics through time are examined by halving the temperature record and determining changes to heat wave characteristics between the two periods. This exploration of changes in heat wave risk through time gives a possible suggestion of trends in future heat wave risk. Findings indicate that there is considerable spatial variability in heat wave characteristics although heat waves have become increasingly frequent and intense throughout much of the state.

A point process approach to identifying and tracking transitions in neural spiking dynamics in the subthalamic nucleus of Parkinson's patients

Understanding the role of rhythmic dynamics in normal and diseased brain function is an important area of research in neural electrophysiology. Identifying and tracking changes in rhythms associated with spike trains present an additional challenge, because standard approaches for continuous-valued neural recordings--such as local field potential, magnetoencephalography, and electroencephalography data--require assumptions that do not typically hold for point process data. Additionally, subtle changes in the history dependent structure of a spike train have been shown to lead to robust changes in rhythmic firing patterns. Here, we propose a point process modeling framework to characterize the rhythmic spiking dynamics in spike trains, test for statistically significant changes to those dynamics, and track the temporal evolution of such changes. We first construct a two-state point process model incorporating spiking history and develop a likelihood ratio test to detect changes in the firing structure. We then apply adaptive state-space filters and smoothers to track these changes through time. We illustrate our approach with a simulation study as well as with experimental data recorded in the subthalamic nucleus of Parkinson's patients performing an arm movement task. Our analyses show that during the arm movement task, neurons underwent a complex pattern of modulation of spiking intensity characterized initially by a release of inhibitory control at 20-40 ms after a spike, followed by a decrease in excitatory influence at 40-60 ms after a spike.

Estimating animal resource selection from telemetry data using point process models

1. Analyses of animal resource selection functions (RSF) using data collected from relocations of individuals via remote telemetry devices have become commonplace. Increasing technological advances, however, have produced statistical challenges in analysing such highly autocorrelated data. Weighted distribution methods have been proposed for analysing RSFs with telemetry data. However, they can be computationally challenging due to an intractable normalizing constant and cannot be aggregated (i.e. collapsed) over time to make space-only inference. 2. In this study, we take a conceptually different approach to modelling animal telemetry data for making RSF inference. We consider the telemetry data to be a realization of a space-time point process. Under the point process paradigm, the times of the relocations are also considered to be random rather than fixed. 3. We show the point process models we propose are a generalization of the weighted distribution telemetry models. By generalizing the weighted model, we can access several numerical techniques for evaluating point process likelihoods that make use of common statistical software. Thus, the analysis methods can be readily implemented by animal ecologists. 4. In addition to ease of computation, the point process models can be aggregated over time by marginalizing over the temporal component of the model. This allows a full range of models to be constructed for RSF analysis at the individual movement level up to the study area level. 5. To demonstrate the analysis of telemetry data with the point process approach, we analysed a data set of telemetry locations from northern fur seals (Callorhinus ursinus) in the Pribilof Islands, Alaska. Both a space-time and an aggregated space-only model were fitted. At the individual level, the space-time analysis showed little selection relative to the habitat covariates. However, at the study area level, the space-only model showed strong selection relative to the covariates.

Transient Access to the Protein Interior: Simulation versus NMR

Many proteins rely on rare structural fluctuations for their function, whereby solvent and other small molecules gain transient access to internal cavities. In magnetic relaxation dispersion (MRD) experiments, water molecules buried in such cavities are used as intrinsic probes of the intermittent protein motions that govern their exchange with external solvent. While this has allowed a detailed characterization of exchange kinetics for several proteins, little is known about the exchange mechanism. Here, we use a millisecond all-atom MD trajectory produced by Shaw et al. (Science2010, 330, 341) to characterize water exchange from the four internal hydration sites in the protein bovine pancreatic trypsin inhibitor. Using a recently developed stochastic point process approach, we compute the survival correlation function probed by MRD experiments as well as other quantities designed to validate the exchange-mediated orientational randomization (EMOR) model used to interpret the MRD data. The EMOR model is found to be quantitatively accurate, and the simulation reproduces the experimental mean survival times for all four sites with activation energy discrepancies in the range 0-3 kBT. On the other hand, the simulated hydration sites are somewhat too flexible, and the water flip barrier is underestimated by up to 6 kBT. The simulation reveals that water molecules gain access to the internal sites by a transient aqueduct mechanism, migrating as single-file water chains through transient (<5 ns) tunnels or pores. The present study illustrates the power of state-of-the-art molecular dynamics simulations in validating and extending experimental results.

Analysis of Protein Dynamics Simulations by a Stochastic Point Process Approach.

MD simulations can now explore the complex dynamics of proteins and their associated solvent in atomic detail on a millisecond time scale. Among the phenomena that thereby become amenable to detailed study are intermittent conformational transitions where the protein accesses transient high-energy states that often play key roles in biology. Here, we present a coherent theoretical framework, based on the stochastic theory of stationary point processes, that allows the essential dynamical characteristics of such processes to be efficiently extracted from the MD trajectory without assuming Poisson statistics. Since the complete information content of a point process is contained in the sequence of residence or interevent times, the experimentally relevant survival correlation function can be computed several orders of magnitude more efficiently than with the conventional approach, involving averaging over initial times. We also present a detailed analysis of the statistical and binning errors, of particular importance when MD results are compared with experiment. As an illustration of the general theoretical framework, we use a 1 ms MD trajectory of the protein BPTI to analyze the exchange kinetics of an internal water molecule and the dynamics of the rare conformational fluctuations that govern the rate of this exchange process.

Quantification of annual wildfire risk; A spatio-temporal point process approach.

Policy responses for local and global firemanagement depend heavily on the proper understanding of the fire extent as well as its spatio-temporal variation across any given study area. Annual fire risk maps are important tools for such policy responses, supporting strategic decisions such as location-allocation of equipment and human resources. Here, we define risk of fire in the narrow sense as the probability of its occurrence without addressing the loss component. In this paper, we study the spatio-temporal point patterns of wildfires and model them by a log Gaussian Cox processes. Themean of predictive distribution of randomintensity function is used in the narrow sense, as the annual fire risk map for next year.

Stochastic Gene Expression in Cells: A Point Process Approach

This paper investigates the stochastic fluctuations of the number of copies of a given protein in a cell. This problem has already been addressed in the past, and closed-form expressions of the mean and variance have been obtained for a simplified stochastic model of the gene expression. These results have been obtained under the assumption that the durations of all the protein production steps are exponentially distributed. In such a case, a Markovian approach (via Fokker--Planck equations) is used to derive analytic formulas of the mean and the variance of the number of proteins at equilibrium. This assumption is, however, not totally satisfactory from a modeling point of view since the distribution of the duration of some steps is more likely to be Gaussian, if not almost deterministic. In such a setting, Markovian methods can no longer be used. A finer characterization of the fluctuations of the number of proteins is therefore of primary interest to understand the general economy of the cell. In this paper, we propose a new approach, based on marked Poisson point processes, which allows us to remove the exponential assumption. This is applied in the framework of the classical three-stage models of the literature: transcription, translation, and degradation. The interest of the method is shown by recovering the classical results under the assumptions that all the durations are exponentially distributed but also by deriving new analytic formulas when some of the distributions are no longer exponential. Our results show, in particular, that the exponential assumption may, surprisingly, underestimate significantly the variance of the number of proteins when some steps are, in fact, not exponentially distributed. This counterintuitive result stresses the importance of the statistical assumptions in the protein production process. Finally, our approach can also be used to consider more detailed models of the gene expression.

Mapping local density of young Eucalyptus plantations by individual tree detection in high spatial resolution satellite images

Local tree density may vary in young Eucalyptus plantations under the effects of environmental conditions or inadequate management, and these variations need to be mapped over large areas as they have a significant impact on the final biomass harvested. High spatial resolution optical satellite images have the potential to provide crucial information on tree density at an affordable cost for forest management. Here, we test the capacity of this promising technique to map the local density of young and small Eucalyptus trees in a large plantation in Brazil. We use three Worldview panchromatic images acquired at a 50cm resolution on different dates corresponding to trees aged 6, 9 and 13months and define an overall accuracy index to evaluate the quality of the detection results. The best agreement between the local densities obtained by visual detection and by marked point process modeling was found at 9months, with only small omission and commission errors and a stable 4% underestimation of the number of trees across the density gradient. We validated the capability of the MPP approach to detect trees aged 9months by making a comparison with local densities recorded on 112 plots of ∼590m2 and ranging between 1360 and 1700 trees per hectare. We obtained a good correlation (r2=0.88) with a root mean square error of 31trees/ha. We generalized detection by computing a consistent map over the whole plantation. Our results showed that local tree density was not uniformly distributed even in a well-controlled intensively-managed Eucalyptus plantation and therefore needed to be monitored and mapped. Use of the marked point process approach is then discussed with respect to stand characteristics (canopy closure), acquisition dates and recommendations for algorithm parameterization.

Spatial modeling for risk assessment of extreme values from environmental time series: a Bayesian nonparametric approach

We propose an approach to modeling and risk assessment for extremes of environmental processes evolving over time and recorded at a number of spatial locations. We follow an extension of the point process approach to analysis of extremes under which the times of exceedances over a given threshold are assumed to arise from a non‐homogeneous Poisson process. To achieve flexible shapes and temporal heterogeneity for the intensity of extremes at any particular spatial location, we utilize a logit‐normal mixture model for the corresponding Poisson process density. A spatial Dirichlet process prior for the mixing distributions completes the nonparametric spatio‐temporal model formulation. We discuss methods for posterior simulation, using Markov chain Monte Carlo techniques, and develop inference for spatial interpolation of risk assessment quantities for high‐level exceedances of the environmental process. The methodology is tested with a synthetic data example and is further illustrated with analysis of rainfall exceedances recorded over a period of 50 years from a region in South Africa. Copyright © 2012 John Wiley &amp; Sons, Ltd.

The point process approach for fractionally differentiated random walks under heavy traffic

We prove some heavy-traffic limit theorems for some nonstationary linear processes which encompass the fractionally differentiated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a non-Gaussian stable distribution. The results are based on an extension of the point process methodology to linear processes with nonsummable coefficients and make use of a new maximal type inequality.

Modelling maintenance and repair costs using stochastic point processes for life cycle costing of repairable systems

The objective of this article is to present a methodology based on reliability and maintainability (R & M) parameters for effective implementation of life cycle costing in design and procurement of repairable systems. For this purpose, a number of life cycle cost models developed over the years have been reviewed, the important life cycle stages for repairable systems are identified and a generalised model for life cycle cost analysis is first proposed. The mathematical equations have been formulated for the life cycle stages, such as acquisition, installation and commissioning, operation, maintenance and repair and disposal. The focus is mainly on modelling the maintenance and repair costs, which are the major elements of repairable system life cycle cost. To model maintenance and repair costs, the stochastic point process approach is employed. The lifetime of repairable system is modelled using a two parameter Weibull distribution. The expected number of failures are estimated based on the assumption that the number of replacements of the components in an interval (0, t) follow renewal process (RP) in the first case and minimal repair process in the second case. Based on the expected number of failures, the lifetime maintenance and repair costs are estimated for the RP and the minimal repair process. A methodology to decide whether a renewal approach or minimal repair approach should be planned for a particular component is also presented. The proposed technique is then illustrated through its application to a typical repairable system, namely an industrial pump and the results obtained are presented along with a review for future work. The proposed model is believed to be a simple way for system designers to estimate and compare the life cycle cost of their different design alternatives at system design stage using system R & M parameters.