Shot Noise Cox Processes Research Articles

Optimal insurance contracts for a shot-noise Cox claim process and persistent insured's actions

We consider a continuous-time model in which an insurer proposes an insurance contract to a potential insured. Motivated by climate change and catastrophic events, we assume that the number of claims process is a shot-noise Cox process. The insurer selects the premium to be paid by the potential insured and the amount to be paid for each claim. In addition, the insurer can request some actions from the potential insured to reduce the number of claims. The insurer wants to maximize his expected total utility, while the potential insured signs the contract if his expected total utility for signing the contract is greater than or equal to his expected total utility when he does not sign the contract. We obtain an analytical solution for the optimal premium, the optimal amount to be paid for each claim, and the optimal actions of the insured. This leads to interesting managerial insights.

Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.

Condition-based maintenance for a system subject to multiple degradation processes with stochastic arrival intensity

In this work, a system subject to different deterioration processes is analysed. The arrival of the degradation processes to the system is modelled using a shot-noise Cox process. The degradation processes grow according to an homogeneous gamma process. The system fails when a degradation process exceeds a failure threshold. The combined process of initiation and growth of the degradation processes is modelled and the system reliability is obtained. Heterogeneities are also integrated in the model assuming that the inverse of the scale parameter follows a uniform distribution. A maintenance strategy is implemented in this system and the state of the system is checked in inspection times. If the system is working at inspection time, a preventive replacement is performed if the deterioration level of a degradation process exceeds a certain threshold. A corrective replacement is performed if the system is down at inspection time. Under this maintenance strategy, the expected cost rate is obtained. Sensitivity analysis on the main parameters of the gamma process is performed.

Modelling columnarity of pyramidal cells in the human cerebral cortex

SummaryFor modelling the location of pyramidal cells in the human cerebral cortex, we suggest a hierarchical point process in that exhibits anisotropy in the form of cylinders extending along the z‐axis. The model consists first of a generalised shot noise Cox process for the xy‐coordinates, providing cylindrical clusters, and next of a Markov random field model for the z‐coordinates conditioned on the xy‐coordinates, providing either repulsion, aggregation or both within specified areas of interaction. Several cases of these hierarchical point processes are fitted to two pyramidal cell data sets, and of these a final model allowing for both repulsion and attraction between the points seem adequate. We discuss how the final model relates to the so‐called minicolumn hypothesis in neuroscience.

A Bivariate Compound Dynamic Contagion Process for Cyber Insurance

As corporates and governments become more digital, they become vulnerable to various forms of cyber attack. Cyber insurance products have been used as risk management tools, yet their pricing does not reflect actual risk, including that of multiple, catastrophic and contagious losses. For the modelling of aggregate losses from cyber events, in this paper we introduce a bivariate compound dynamic contagion process, where the bivariate dynamic contagion process is a point process that includes both externally excited joint jumps, which are distributed according to a shot noise Cox process and two separate self-excited jumps, which are distributed according to the branching structure of a Hawkes process with an exponential fertility rate, respectively. We analyse the theoretical distributional properties for these processes systematically, based on the piecewise deterministic Markov process developed by Davis (1984) and the univariate dynamic contagion process theory developed by Dassios and Zhao (2011). The analytic expression of the Laplace transform of the compound process and its moments are presented, which have the potential to be applicable to a variety of problems in credit, insurance, market and other operational risks. As an application of this process, we provide insurance premium calculations based on its moments. Numerical examples show that this compound process can be used for the modelling of aggregate losses from cyber events. We also provide the simulation algorithm for statistical analysis, further business applications and research.

Macroscopic analysis of shot-noise Cox random balls

In this paper, we consider a cluster model of weighted Euclidean random balls generated by a shot-noise Cox process. It is an example of cluster point process. We perform a scaling on the model by shrinking the radii of the balls and compensate this effect by increasing the (mean) number of balls in each cluster, or/and increasing the (mean) number of clusters. We consider two different scenarios, say a local and a global scenario. Heuristically, in the first scenario, we focus on the mean number of large balls in a cluster while in the second one, we focus on the global mean number of large balls in the model. According to the different scenarios, the cluster structure can persist at the limit or disappear.

Using a spatial point process framework to characterize lung computed tomography scans

Pulmonary emphysema is a destructive disease of the lungs that is currently diagnosed via visual assessment of lung Computed Tomography (CT) scans by a radiologist. Visual assessment can have poor inter-rater reliability, is time consuming, and requires access to trained assessors. Quantitative methods that reliably summarize the biologically relevant characteristics of an image are needed to improve the way lung diseases are characterized. The goal of this work was to show how spatial point process models can be used to create a set of radiologically derived quantitative lung biomarkers of emphysema. We formalized a general framework for applying spatial point processes to lung CT scans, and developed a Shot Noise Cox Process to quantify how radiologically based emphysematous tissue clusters into larger structures. Bayesian estimation of model parameters was done using spatial Birth–Death MCMC (BD-MCMC). In simulations, we showed the BD-MCMC estimation algorithm is able to accurately recover model parameters. In an application to real lung CT scans from the COPDGene cohort, we showed variability in the clustering characteristics of emphysematous tissue across disease subtypes that were based on visual assessments of the CT scans.

Precise deviations for Cox processes with a shot noise intensity

We consider a Cox process with Poisson shot noise intensity which has been widely applied in insurance, finance, queue theory, statistic, and many other fields. Cox process is flexible because its intensity not only depends on the time but also can be considered as a stochastic process and so it can be considered as a two-step randomization procedure. Due to the structure of such models, a number of useful and general results can easily be established. In this paper, we study the fluctuations and precise deviations for shot noise Cox process using the recent mod- convergence method.

Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a stepwise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity parameters. In the second and third steps, the minimum contrast estimation based on K-functions of the projected processes is used to estimate the interaction parameters. We study the asymptotic properties of the resulting estimators and formulate a set of conditions sufficient for establishing consistency and asymptotic normality of the estimators under the increasing domain asymptotics.

Two-step estimation procedures for inhomogeneous shot-noise Cox processes

In the present paper, we discuss and compare several two-step estimation procedures for inhomogeneous shot-noise Cox processes. The intensity function is parametrized by the inhomogeneity parameters while the pair-correlation function is parametrized by the interaction parameters. The suggested procedures are based on a combination of Poisson likelihood estimation of the inhomogeneity parameters in the first step and an adaptation of a method from the homogeneous case for estimation of the interaction parameters in the second step. The adapted methods, based on minimum contrast estimation, composite likelihood and Palm likelihood, are compared both theoretically and by means of a simulation study. The general conclusion from the simulation study is that the three estimation methods have similar performance. Two-step estimation with Palm likelihood has not been considered before and is motivated by the superior performance of the Palm likelihood in the stationary case for estimation of certain parameters of interest. Asymptotic normality of the two-step estimator with Palm likelihood is proved.

Matérn thinned Cox processes

A class of spatial point process models that combine short range repulsion with medium range clustering is introduced. The model is motivated by patterns of centres of non-overlapping spherical cells in biological tissue which tend to have a clustering behaviour. Such a combination of clustering and hard core behaviour can be achieved by applying a dependent Matérn thinning to a Cox process. An exact formula for the intensity of a Matérn thinned shot noise Cox process is derived from the Palm distribution. For the more general class of Matérn thinned Cox processes, formulae for the intensity and second-order characteristics are established using the conditional Poisson assumption. These formulae include more complicated integrals for which approximations are suggested to simplify calculations. An example from pathology illustrates the applicability of the models.

A risk model with renewal shot-noise Cox process

In this paper we generalise the risk models beyond the ordinary framework of affine processes or Markov processes and study a risk process where the claim arrivals are driven by a Cox process with renewal shot-noise intensity. The upper bounds of the finite-horizon and infinite-horizon ruin probabilities are investigated and an efficient and exact Monte Carlo simulation algorithm for this new process is developed. A more efficient estimation method for the infinite-horizon ruin probability based on importance sampling via a suitable change of probability measure is also provided; illustrative numerical examples are also provided.

Modeling forest fires in Mazandaran Province, Iran

We used a spatio-temporal shot-noise Cox process to study the distribution of forest fires reported between 2006 and 2010 in the Mazandaran Province’s forests. The fitted model shows that daily temperature, altitude, and slope-exposure impacted fire occurrence. Forest fire occurred in the region had an aggregated behavior, which increased in radius below 1-km away from fired areas; a periodic pattern of fire occurrence in the region was verified. The risk of forest fire is significantly higher for areas with southern exposure and slope between 30° and 50°, northern exposure and slope between 0° and 50°, and eastern exposure and slope between 0° and 30°. The risk of fire was also significantly higher at altitudes between 1350 and 3000 m asl. Human causes were the main ignition source for forest fires in the region. The fire occurrence rate stayed above average during the drought period from September 2008 to September 2009. Our findings could lead to the development of fire-response and fire-suppression strategies appropriate to specific regions.

Bayesian Inference on a Cox Process Associated with a Dirichlet Process

In ecology and epidemiology, spatio-temporal distributions of events can be described by Cox processes. Situations for which there exists a hidden process which contributes to random effects on the intensity of the observed Cox process are considered. The observed process is a generalized shot noise Cox process and the hidden process is a Poisson process associated with a Dirichlet process. The distributional properties of quadrat counts are presented and bayesian inference is proposed for estimating and predicting parameters of interest in the model. Illustrations are given from weed spatial count data and disease mortality data.

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

ABSTRACTWe consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.

A bivariate shot noise self-exciting process for insurance

In this paper, we study a bivariate shot noise self-exciting process. This process includes both externally excited joint jumps, which are distributed according to a shot noise Cox process, and two separate self-excited jumps, which are distributed according to the branching structure of a Hawkes process with an exponential fertility rate, respectively. A constant rate of exponential decay is included in this process as it can play a role as the time value of money in economics, finance and insurance applications. We analyse this process systematically for its theoretical distributional properties, based on the piecewise deterministic Markov process theory developed by Davis (1984), and the martingale methodology used by Dassios and Jang (2003). The analytic expressions of the Laplace transforms of this process and the moments are presented, which have the potential to be applicable to a variety of problems in economics, finance and insurance. In this paper, as an application of this process, we provide insurance premium calculations based on its moments. Numerical examples show that this point process can be used for the modelling of discounted aggregate losses from catastrophic events.

Statistics for Inhomogeneous Space-Time Shot-Noise Cox Processes

In the paper we introduce a flexible inhomogeneous space-time shot-noise Cox process model and derive a two-step estimation procedure for it. In the first step the inhomogeneity is estimated by means of a Poisson score estimating equation and in the second step we use minimum contrast estimation based on second order properties to obtain estimates of the clustering parameters. The suggested model is not separable but it has a special interaction structure which enables to use the spatial and temporal projections of the process for parameter estimation. Efficiency of the introduced method is investigated by means of a simulation study and it is compared to a previously used method.

Downscaling species occupancy from coarse spatial scales

The measurement and prediction of species' populations at different spatial scales is crucial to spatial ecology as well as conservation biology. An efficient yet challenging goal to achieve such population estimates consists of recording empirical species' presence and absence at a specific regional scale and then trying to predict occupancies at finer scales. So far the majority of the methods have been based on particular species' distributional features deemed to be crucial for downscaling occupancy. However, only a minority of them have dealt explicitly with specific spatial features. Here we employ a wide class of spatial point processes, the shot noise Cox processes (SNCP), to model species occupancies at different spatial scales and show that species' spatial aggregation is crucial for predicting population estimates at fine scales starting from coarser ones. These models are formulated in continuous space and locate points regardless of the arbitrary resolution that one employs to study the spatial pattern. We compare the performances of nine models, calibrated at regional scales and demonstrate that a very simple class of SNCP, the Thomas process, is able to outperform other published models in predicting occupancies down to areas four orders of magnitude smaller than the ones employed for the parameterization. We conclude by explaining the ability of the approach to infer spatially explicit information from spatially implicit measures, the potential of the framework to combine niche and spatial models, and the possibility of reversing the method to allow upscaling.

Importance of habitat heterogeneity and biotic processes in the spatial distribution of a riparian herb (Carex remota L.): a point process approach

This study attempts to understand the dependence on abiotic factors and on the biotic process of the population development. We used three spatial point process models (Poisson, Area-Interaction and shot-noise Cox processes) in both homogenous and inhomogeneous versions to model the distribution of three Carex remota cohorts in wet zones of a temperate forest in the north of Spain. The cohorts studied were adults and seedlings born in two consecutive years. With the use of these models we are able to simulate separately and jointly the effect on plant distribution of a homogeneous or heterogeneous habitat, and the absence or presence of some biotic processes, as seed dispersal and/or density-dependent interactions. The result of the bivariate function analysis does not reveal sufficient evidences, but suggests a weak positive relation between adults and seedlings that survived a dry period in the first summer. Models from the three cohorts show a decreasing degree of clustering from seedlings to adults. Besides, the results show that the importance of the main factors that explain the population structure changes along the development of Carex stages. Compared to seedlings, the adults pattern shows an increasing dependence on abiotic factors.

Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Shot-Noise Cox Process

In this paper, we introduce tail dependene measures for collateral losses from catastrophic events. To calculate these measures, we use bivariate compound process where a Cox process with shot noise intensity is used to count collateral losses. A homogeneous Poisson process is also examined as its counterpart for the case where the catastrophic loss frequency rate is deterministic. Joint Laplace transform of the distribution of the aggregate collateral losses is derived and joint Fast Fourier transform is used to obtain the joint distributions of aggregate collateral losses. For numerical illustrations, a member of Farlie-Gumbel-Morgenstern copula with exponential margins is used. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are also provided.