Marked Poisson Process Research Articles

A flexible framework for spatial capture-recapture with unknown identities.

Camera traps or acoustic recorders are often used to sample wildlife populations. When animals can be individually identified, these data can be used with spatial capture-recapture (SCR) methods to assess populations. However, obtaining animal identities is often labor-intensive and not always possible for all detected animals. To address this problem, we formulate SCR, including acoustic SCR, as a marked Poisson process, comprising a single counting process for the detections of all animals and a mark distribution for what is observed (eg, animal identity, detector location). The counting process applies equally when it is animals appearing in front of camera traps and when vocalizations are captured by microphones, although the definition of a mark changes. When animals cannot be uniquely identified, the observed marks arise from a mixture of mark distributions defined by the animal activity centers and additional characteristics. Our method generalizes existing latent identity SCR models and provides an integrated framework that includes acoustic SCR. We apply our method to estimate density from a camera trap study of fisher (Pekania pennanti) and an acoustic survey of Cape Peninsula moss frog (Arthroleptella lightfooti). We also test it through simulation. We find latent identity SCR with additional marks such as sex or time of arrival to be a reliable method for estimating animal density.

Statistical characterization of erosion and sediment transport mechanics in shallow tidal environments – Part 1: Erosion dynamics

Abstract. Reliable descriptions of erosion events are foundational to effective frameworks relevant to the fate of tidal landscape evolution. Besides the rhythmic, predictable action of tidal currents, erosion in shallow tidal environments is strongly influenced by the stochastic wave-induced bottom shear stress (BSS), mainly responsible for sediment resuspension on tidal flats. However, the absence of sufficiently long, measured time series of BSS prevents a direct analysis of the combined tide- and wave-driven erosion dynamics and its proper representation in long-term morphodynamic models. Here we test the hypothesis of describing erosion dynamics in shallow tidal environments as a Poisson process by analysing, with the peak-over-threshold theory, the BSS time series computed using a fully coupled, bi-dimensional numerical model. We perform this analysis on the Venice Lagoon, Italy, taking advantage of several historical surveys done in the last 4 centuries, which allow us to investigate the effects of morphological modifications on spatial and temporal erosion patterns. Our analysis suggests that erosion events on intertidal flats can effectively be modelled as a marked Poisson process in different morphological configurations because the interarrival times, durations, and intensities of the over-threshold exceedances are always well described by exponentially distributed random variables. The resulting statistical characterization allows a straightforward computation of morphological indicators, such as the erosion work, and paves the way for a novel synthetic, yet reliable, approach for the long-term morphodynamic modelling of tidal environments.

Statistical characterization of erosion and sediment transport mechanics in shallow tidal environments – Part 2: Suspended sediment dynamics

Abstract. A proper understanding of sediment resuspension and transport processes is key to the morphodynamics of shallow tidal environments. However, a complete spatial and temporal coverage of suspended sediment concentration (SSC) to describe these processes is hardly available, preventing the effective representation of depositional dynamics in long-term modelling approaches. Through aiming to couple erosion and deposition dynamics in a unique synthetic theoretical framework, we here investigate SSC dynamics, following a similar approach to that adopted for erosion (D'Alpaos et al., 2024). The analysis with the peak-over-threshold theory of SSC time series computed using a fully coupled, bi-dimensional model allows us to identify interarrival times, intensities, and durations of over-threshold events and test the hypothesis of modelling SSC dynamics as a Poisson process. The effects of morphological modifications on spatial and temporal SSC patterns are investigated in the Venice Lagoon, for which several historical configurations in the last 4 centuries are available. Our results show that, similar to erosion events, SSC can be modelled as a marked Poisson process in the intertidal flats for all the analysed morphological lagoon configurations because exponentially distributed random variables describe over-threshold events well. Although erosion and resuspension are intimately intertwined, erosion alone does not suffice to describe also SSC because of the non-local dynamics due to advection and dispersion processes. The statistical characterization of SSC events completes the framework introduced for erosion mechanics, and together, they represent a promising tool to generate synthetic, yet realistic, time series of shear stress and SSC for the long-term modelling of tidal environments.

Ratemaking in a changing environment

AbstractIn pricing insurance contracts based on the individual policyholder’s aggregate losses for non-life insurers, the literature has mainly focused on using detailed information from policies and closed claims. However, the information on open claims can reflect shifts in the distribution of the expected claim payments better than closed claims. Such shifts may be needed to be reflected in the ratemaking process earlier rather than later, especially when insurers are experiencing environmental changes. In practice, actuaries use ad hoc techniques to adjust data to current levels to determine premiums. This paper presents an intuitive ratemaking model, employing a marked Poisson process framework, which ensures that the multivariate risk analysis is done more routinely using all reported claims and makes an adjustment for Incurred But Not Reported claims. Utilizing data from the Wisconsin Local Government Property Insurance Fund, we find that by determining rates based on current data, the proposed ratemaking model leads to better alignment of premiums and provides insurers with a more financially sound portfolio.

Predicting Time to and Average Quality of Future Offers for Kidney Transplant Candidates Declining a Current Deceased Donor Kidney Offer: A Retrospective Cohort Study.

At the time a kidney offer is made by an organ donation organization (ODO), transplant physicians must inform candidates on the pros and cons of accepting or declining the offer. Although physicians have a general idea of expected wait time to kidney transplantation by blood group in their ODO, there are no tools that provide quantitative estimates based on the allocation score used and donor/candidate characteristics. This limits the shared decision-making process at the time of kidney offer as (1) the consequences of declining an offer in terms of wait-time prolongation cannot be provided and (2) the quality of the current offer cannot be compared with that of offers that could be made to the specific candidate in the future. This is especially relevant to older transplant candidates as many ODOs use some form of utility matching in their allocation score. We aimed to develop a novel method to provide personalized estimates of wait time to next offer and quality of future offers for kidney transplant candidates if they refused a current deceased donor offer from an ODO. A retrospective cohort study. Administrative data from Transplant Quebec. All patients who were actively registered on the kidney transplant wait list at any point between March 29, 2012 and December 13, 2017. The time to next offer was defined as the number of days between the time of the current offer and the next offer if the current one were declined. The quality of the offers was measured with the 10-variable Kidney Donor Risk Index (KDRI) equation. Candidate-specific kidney offer arrival was modeled with a marked Poisson process. To derive the lambda parameter for the marked Poisson process for each candidate, the arrival of donors was examined in the 2 years prior to the time of the current offer. The Transplant Quebec allocation score was calculated for each ABO-compatible offer with the characteristics that the candidate presented at the time of the current offer. Offers where the candidate's score was lower than the scores of actual recipients of the second kidneys transplanted were filtered out from the candidate-specific kidney offer arrival. The KDRIs of offers that remained were averaged to provide an estimate of the quality of future offers, to be compared with that of the current offer. During the study period, there were 848 unique donors and 1696 transplant candidates actively registered. The models provide the following information: average time to next offer, time to which there is a 95% probability of receiving a next offer, average KDRI of future offers. The C-index of the model was 0.72. When compared with providing average group estimates of wait time and KDRI of future offers, the model reduced the root-mean-square error in the predicted time to next offer from 137 to 84 days and that of predicted KDRI of future offers from 0.64 to 0.55. The precision of the model's predictions was higher when observed times to next offer were 5 months or less. The models assume that patients declining an offer remain wait-listed until the next one. The model only updates wait time every year after the time of an offer and not in a continuous fashion. By providing personalized quantitative estimates of time to and quality of future offers, our new approach can inform the shared decision-making process between transplant candidates and physicians when a kidney offer from a deceased donor is made by an ODO.

Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks

In this paper, we study a reliability system subject to occasional random shocks hitting an underlying device in accordance with a general marked point process with position dependent marking. In addition, the system ages according to a linear path that eventually fails even without any external shocks that accelerate the total failure. The approach for obtaining the distribution of the failure time falls into the area of random walk analysis. The results obtained are in closed form. A special case of a marked Poisson process with exponentially distributed marks is discussed that supports our claim of analytical tractability. The example is further confirmed by simulation. We also provide a classification of the literature pertaining to various reliability systems with degradation and shocks.

Modeling wildfires via marked spatio-temporal Poisson processes

From a statistical viewpoint, characteristics such as ignition time, location and duration are relevant components for wildfire modeling. The observed ignition sites and starting times constitute a space-time point pattern, and a natural framework to model this type of data is via point processes. In this work, we propose a marked Poisson process to model fire patterns in space-time, considering durations as marks. The collected data correspond to fires observed in the Valencian Community, Spain, between 2010 and 2015. The methodology relies on writing the intensity function of such a process, jointly for starting times, locations and durations, as a weighted Dirichlet process mixture model. A particular choice of the kernel that determines such mixture was made, compatible with data features. We conducted posterior inference on some characteristics of interest for understanding wildfire behavior, showing high flexibility to emulate data patterns.

Asymptotics and renewal approximation in the online selection of increasing subsequence

We revisit the problem of maximising the expected length of increasing subsequence that can be selected from a marked Poisson process by an online strategy. Resorting to a natural size variable, we represent the problem in terms of a controlled piecewise deterministic Markov process with decreasing paths. We apply a comparison method to the optimality equation to obtain fairly complete asymptotic expansions for the moments of the maximal length, and, with the aid of a renewal approximation, give a novel proof to the central limit theorem for the length of selected subsequence under either the optimal strategy or a strategy sufficiently close to optimality.

Detecting Change between Urban Road Environments along a Route Based on Static Road Object Occurrences

For over a decade, urban road environment detection has been a target of intensive research. The topic is relevant for the design and implementation of advanced driver assistance systems. Typically, embedded systems are deployed in these for the operation. The environments can be categorized into road environment-types. Abrupt transitions between these pose a traffic safety risk. Road environment-type transitions along a route manifest themselves also in changes in the distribution of traffic signs and other road objects. Can the placement and the detection of traffic signs be modelled jointly with an easy-to-handle stochastic point process, e.g., an inhomogeneous marked Poisson process? Does this model lend itself for real-time application, e.g., via analysis of a log generated by a traffic sign detection and recognition system? How can the chosen change detector help in mitigating the traffic safety risk? A change detection method frequently used for Poisson processes is the cumulative sum (CUSUM) method. Herein, this method is tailored to the specific stochastic model and tested on realistic logs. The use of several change detectors is also considered. Results indicate that a traffic sign-based road environment-type change detection is feasible, though it is not suitable for an immediate intervention.

Jointly estimating spatial sampling effort and habitat suitability for multiple species from opportunistic presence‐only data

Abstract Building reliable species distribution models (SDMs) from presence‐only information requires a good understanding of the spatial variation in the sampling effort. However, in most cases, the sampling effort is unknown, leading to biases in SDMs. This study proposes a method to jointly estimate the parameters of sampling effort and species densities to avoid such biases. The method is particularly suited to the analysis of massive but highly heterogeneous presence‐only data. The proposed method is based on estimating the variation in sampling effort over units of a spatial mesh in parallel with the environmental density of multiple species using a marked Poisson process model. Based on simulations with realistic settings, we examined the performance and robustness of parameter estimations. We also analysed a large‐scale citizen science dataset with highly heterogeneous sampling (Pl@ntNet), including around 300,000 occurrences of 150 plant species. We found that sampling effort was correctly estimated when the true sampling effort was constant within the cells of a spatial mesh. Estimation bias arose when sampling effort and environmental drivers strongly covaried within cells. Otherwise, the inference was correct and robust to sampling variation within cells. Running the model on real occurrences of 150 plant species provided an estimated map of relative sampling effort for 15% of French territory. We also found that the density estimated for an exotic invasive plant was consistent with prior data. This is the first method jointly estimating species densities depending on environment, and sampling effort as an explicit spatial function, from occurrence data of multiple species. An asset of the method is that a few frequently observed species greatly contribute to correctly estimate sampling effort, thereby improving density estimation of all other species. This approach can thus provide reliable SDM for large opportunistic presence‐only datasets, with broad spatial variation in sampling effort but also many species, such as datasets from citizen science programmes.

Probabilistic structure of events controlling the after-storm recovery of coastal dunes

Coastal dunes protect beach communities and ecosystems from rising seas and storm flooding and influence the stability of barrier islands by preventing overwashes and limiting barrier migration. Therefore, the degree of dune recovery after a large storm provides a simple measure of the short-term resiliency (and potential long-term vulnerability) of barrier islands to external stresses. Dune recovery is modulated by low-intensity/high-frequency high-water events (HWEs), which remain poorly understood compared to the low-frequency extreme events eroding mature dunes and dominating the short-term socio-economic impacts on coastal communities. Here, we define HWEs and analyze their probabilistic structure using time series of still-water level and deep-water wave data from multiple locations around the world. We find that HWEs overtopping the beach can be modeled as a marked Poisson process with exponentially distributed sizes or marks and have a mean size that varies surprisingly little with location. This homogeneity of global HWEs is related to the distribution of the extreme values of a wave-runup parameter, [Formula: see text], defined in terms of deep-water significant wave height [Formula: see text] and peak wavelength [Formula: see text] Furthermore, the characteristic beach elevation at any given location seems to be tied to a constant HWE frequency of about one event per month, which suggests a stochastic dynamics behind beach stabilization. Our results open the door to the development of stochastic models of beach, dune, and barrier dynamics, as well as a better understanding of wave-driven nuisance flooding.

A spatio-temporal point process model for particle growth

AbstractA spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson–Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

A Bayesian multiple changepoint model for marked poisson processes with applications to deep earthquakes

A multiple changepoint model for marked Poisson process is formulated as a continuous time hidden Markov model, which is an extension of Chib’s multiple changepoint models (J Econ 86:221–241, 1998). The inference on the locations of changepoints and other model parameters is based on a two-block Gibbs sampling scheme. We suggest a continuous time version of forward-filtering backward-sampling algorithm for simulating the full trajectories of the latent Markov chain without utilizing the uniformization method. To retrieve the optimal posterior path of the latent Markov chain, i.e. the maximum a posteriori estimation of changepoint locations, a continuous-time version of Viterbi algorithm (CT-Viterbi) is proposed. The set of changepoint locations is obtainable either from the CT-Viterbi algorithm or the posterior samples of the latent Markov chain. The number of changepoints is determined according to a modified BIC criterion tailored particularly for the multiple changepoint problems of a marked Poisson process. We then perform a simulation study to demonstrate the methods. The methods are applied to investigate the temporal variabilities of seismicity rates and the magnitude-frequency distributions of medium size deep earthquakes in New Zealand.

Ergodic Poisson splittings

In this paper, we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random measure invariant under the action of the transformation.

On Queues with Priority Determined by Feedback

Customers arrive to a two-priority queueing system according to a marked Poisson process. Both waiting rooms have infinite capacity. Customers are served one at a time according to FIFO discipline on priority basis: those in waiting line 1([Formula: see text]) are given priority over the ones in line 2([Formula: see text]). The service time is class-dependent phase type. After completion of service, high priority ([Formula: see text]) customers may feed back for service according to a Bernoulli process. Feed back customers are sent to the low priority ([Formula: see text]) queue. When at a service completion epoch of a [Formula: see text] customer, if there is none left behind in [Formula: see text] line, then the server goes to serve [Formula: see text] class. For the two-priority queueing system, we assume that [Formula: see text] customers are not allowed an additional feed back. Both preemptive and non-preemptive service disciplines are analysed. Waiting time distribution of both type of customers are derived. As a special case, the situation where there is no external entry to the [Formula: see text] line is discussed.

Stochastic modelling of the hydrologic operation of rainwater harvesting systems

Rainwater harvesting (RWH) systems are an effective low impact development practice that provides both water supply and runoff reduction benefits. A stochastic modelling approach is proposed in this paper to quantify the water supply reliability and stormwater capture efficiency of RWH systems. The input rainfall series is represented as a marked Poisson process and two typical water use patterns are analytically described. The stochastic mass balance equation is solved analytically, and based on this, explicit expressions relating system performance to system characteristics are derived. The performances of a wide variety of RWH systems located in five representative climatic regions of the United States are examined using the newly derived analytical equations. Close agreements between analytical and continuous simulation results are shown for all the compared cases. In addition, an analytical equation is obtained expressing the required storage size as a function of the desired water supply reliability, average water use rate, as well as rainfall and catchment characteristics. The equations developed herein constitute a convenient and effective tool for sizing RWH systems and evaluating their performances.

Latent Marked Poisson Process with Applications to Object Segmentation

In difficult object segmentation tasks, utilizing image information alone is not sufficient; incorporation of object shape prior models is necessary to obtain competitive segmentation performance. Most formulations that incorporate both shape and image information are in the form of energy functional optimization problems. This paper introduces a Bayesian latent marked Poisson process for segmenting multiple objects in an image. The model takes both shape and image feature/appearance into account—it generates object locations from a spatial Poisson process, then generates shape parameters from a shape prior model as the latent marks. Inferentially, this partitions the image: pixels inside objects are assumed to be generated from an object observation/appearance model and pixels outside objects come from a background model. The Poisson process provides (non-homogeneous) spatial priors for object locations and the marks allow the incorporation of shape priors. We develop a hybrid Gibbs sampler that addresses the variation in model order and nonconjugacy that arise in this setting and we present experimental results on synthetic images and two diverse domains in real images: cell segmentation in biological images and pedestrian and car detection in traffic images.

Time and magnitude monitoring based on the renewal reward process

Time between event (TBE) charts are SPC tools for monitoring the occurrence of unwanted events, such as the appearance of a defective item or a failure of a piece of equipment. In some cases, a magnitude, indicating the severity of the event, is also measured. Time and magnitude charts, which are based on the assumption that the stochastic process underlying the occurrence of events is the marked Poisson process, are the preferred option. However, these charts are not suitable to deal with damage events caused by repeatedly occurring shocks or stress conditions. To bridge this gap, we introduce a new control chart based on the assumption of a renewal process with rewards, where the reward represents magnitude, and a magnitude-over-threshold condition represents the occurrence of an event. In particular, we consider two cases for magnitude: (i) magnitude is cumulative over time and (ii) magnitude is non-cumulative or independent over time. We use known results in renewal theory to provide expressions of the probability distributions needed to compute the control limits and perform a simulation analysis of the control chart performance.

Inferring the actual urban road environment from traffic sign data using a minimum description length approach

Abstract In our paper, we focus on a group of traffic signs and use traffic sign logs to statistically infer the type of urban environment in which a car is being driven. The traffic signs are either perceived and logged by a human data entry assistant, or preferably automatically detected and logged by an on-board traffic sign recognition and logging system. An entry in the log-file records the traffic sign type and the along-the-route location of the sign. Furthermore, in case of collecting training data it records also the actual road environment category entered by the data entry assistant. The logs are seen as realizations of an inhomogeneous marked Poisson process, and the minimum description length (MDL) principle is applied to infer the actual environment. The aim of this approach is to encode the current data in the shortest possible way ‒ assuming stochastic models derived from data collected earlier ‒ and thereby accept the corresponding model and environment as actual. To evaluate the quality of classification, the inferred environment categories are compared to the ground truth data.

Locating roadworks sites via detecting change in lateral positions of traffic signs measured relative to the ego-car

Abstract Roadworks can be hazardous for both road workers and road users. Even with state-of-the-art safety measures in place, serious accidents do happen there, particularly when drivers do not heed roadwork signs and speed limits. Crashes at roadworks that involve killed or seriously injured (KSI) casualties account for about 2% of all KSI crashes in developed countries, even though the roadworks are normally well-signaled and are also marked in quick-reaction road/traffic maps. These media provide several means for the drivers ‒ and for the on-board advanced driving assistance systems (ADAS) helping them ‒ to duly detect roadworks. In the paper, an approach based on statistical inference is presented for detecting roadwork zones. The approach takes into account the engineering regulations and practice concerning setting up temporary road configurations near and along roadworks. Such configurations often involve narrower traffic lanes and traffic signs installed closer to traffic. The approach detects change in ‒ among other type of collected data ‒ the lateral positions of the traffic signs measured relative to the ego-car along the road. In a practical implementation, the traffic sign detection and recognition, the lateral distance measurement and the data recording are carried out by some traffic sign recognition (TSR) system. The traffic sign data is seen as a realization of a marked Poisson process and the minimum description length (MDL) principle – set to work in the form of Page-Hinkley change detectors – is applied for detecting roadworks.