Structured Space-Sphere Point Processes and K-Functions

Methodology and Computing in Applied Probability

Published On 2021/6

This paper concerns space-sphere point processes, that is, point processes on the product space of (the d-dimensional Euclidean space) and (the k-dimensional sphere). We consider specific classes of models for space-sphere point processes, which are adaptations of existing models for either spherical or spatial point processes. For model checking or fitting, we present the space-sphere K-function which is a natural extension of the inhomogeneous K-function for point processes on to the case of space-sphere point processes. Under the assumption that the intensity and pair correlation function both have a certain separable structure, the space-sphere K-function is shown to be proportional to the product of the inhomogeneous spatial and spherical K-functions. For the presented space-sphere point process models, we discuss cases where such a separable structure can be obtained. The …

Journal

Methodology and Computing in Applied Probability

Published On

2021/6

Volume

23

Page

569-591

Authors

Jesper Møller

Jesper Møller

Aalborg Universitet

Position

Professor in Statistics

H-Index(all)

46

H-Index(since 2020)

23

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Statistics

Probability Theory

University Profile Page

Francisco Cuevas-Pacheco

Francisco Cuevas-Pacheco

Aalborg Universitet

Position

H-Index(all)

8

H-Index(since 2020)

8

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Geoestatistics

Statistics

Point processes

Random fields

University Profile Page

Other Articles from authors

Francisco Cuevas-Pacheco

Francisco Cuevas-Pacheco

Aalborg Universitet

arXiv preprint arXiv:2402.12548

Composite likelihood inference for space-time point processes

The dynamics of a rain forest is extremely complex involving births, deaths and growth of trees with complex interactions between trees, animals, climate, and environment. We consider the patterns of recruits (new trees) and dead trees between rain forest censuses. For a current census we specify regression models for the conditional intensity of recruits and the conditional probabilities of death given the current trees and spatial covariates. We estimate regression parameters using conditional composite likelihood functions that only involve the conditional first order properties of the data. When constructing assumption lean estimators of covariance matrices of parameter estimates we only need mild assumptions of decaying conditional correlations in space while assumptions regarding correlations over time are avoided by exploiting conditional centering of composite likelihood score functions. Time series of point patterns from rain forest censuses are quite short while each point pattern covers a fairly big spatial region. To obtain asymptotic results we therefore use a central limit theorem for the fixed timespan - increasing spatial domain asymptotic setting. This also allows us to handle the challenge of using stochastic covariates constructed from past point patterns. Conveniently, it suffices to impose weak dependence assumptions on the innovations of the space-time process. We investigate the proposed methodology by simulation studies and applications to rain forest data.

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.09525

Coupling results and Markovian structures for number representations of continuous random variables

A general setting for nested subdivisions of a bounded real set into intervals defining the digits of a random variable with a probability density function is considered. Under the weak condition that is almost everywhere lower semi-continuous, a coupling between and a non-negative integer-valued random variable is established so that have an interpretation as the ``sufficient digits'', since the distribution of conditioned on does not depend on . Adding a condition about a Markovian structure of the lengths of the intervals in the nested subdivisions, becomes a Markov chain of a certain order . If then are IID with a known distribution. When and the Markov chain is uniformly geometric ergodic, a coupling is established between and a random time so that the chain after time is stationary and follows a simple known distribution. The results are related to several examples of number representations generated by a dynamical system, including base- expansions, generalized L\"uroth series, -expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.08387

The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let be the base- expansion of a continuous random variable on the unit interval where is the positive solution to for an integer (i.e., is a generalization of the golden mean for which ). We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

Jesper Møller

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

Francisco Cuevas-Pacheco

Francisco Cuevas-Pacheco

Aalborg Universitet

The Annals of Statistics

Pairwise interaction function estimation of stationary Gibbs point processes using basis expansion

This Supplementary Material contains an application to real data sets, proofs of Theorem 1, auxiliary lemmas and additional results including a multivariate extension of Theorem 1.

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2312.09652

The asymptotic distribution of the remainder in a certain base- expansion

Let be the base- expansion of a continuous random variable on the unit interval where is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

2023/12/15

Article Details
Jesper Møller

Jesper Møller

Aalborg Universitet

Proceedings of the London Mathematical Society

Realizability and tameness of fusion systems

A saturated fusion system over a finite p$p$‐group S$S$ is a category whose objects are the subgroups of S$S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over S$S$ is realized by a finite group G$G$ if S$S$ is a Sylow p$p$‐subgroup of G$G$ and morphisms in the category are those induced by conjugation in G$G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the …

Jesper Møller

Jesper Møller

Aalborg Universitet

ACM Transactions on Spatial Algorithms and Systems

Stochastic Routing with Arrival Windows

Arriving at a destination within a specific time window is important in many transportation settings. For example, trucks may be penalized for early or late arrivals at compact terminals, and early and late arrivals at general practitioners, dentists, and so on, are also discouraged, in part due to COVID. We propose foundations for routing with arrival-window constraints. In a setting where the travel time of a road segment is modeled by a probability distribution, we define two problems where the aim is to find a route from a source to a destination that optimizes or yields a high probability of arriving within a time window while departing as late as possible. In this setting, a core challenge is to enable comparison between paths that may potentially be part of a result path with the goal of determining whether a path is uninteresting and can be disregarded given the existence of another path. We show that existing solutions …

2023/11/21

Article Details
Jesper Møller

Jesper Møller

Aalborg Universitet

Spatial Statistics

Fitting the grain orientation distribution of a polycrystalline material conditioned on a Laguerre tessellation

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel …

Jesper Møller

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

Singular distribution functions for random variables with stationary digits

Let F be the cumulative distribution function (CDF) of the base-q expansion , where is an integer and is a stationary stochastic process with state space . In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

Francisco Cuevas-Pacheco

Francisco Cuevas-Pacheco

Aalborg Universitet

A convolution type model for the intensity of spatial point processes applied to eye-movement data

Estimating the first-order intensity function in point pattern analysis is an important problem, and it has been approached so far from different perspectives: parametrically, semiparametrically or nonparametrically. Our approach is close to a semiparametric one. Motivated by eye-movement data, we introduce a convolution type model where the log-intensity is modeled as the convolution of a function β (⋅), to be estimated, and a single spatial covariate (the image an individual is looking at for eye-movement data). Based on a Fourier series expansion, we show that the proposed model can be viewed as a log-linear model with an infinite number of coefficients, which correspond to the spectral decomposition of β (⋅). After truncation, we estimate these coefficients through a penalized Poisson likelihood. We illustrate the efficiency of the proposed methodology on simulated data and on eye-movement data.

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2212.08402

Cox processes driven by transformed Gaussian processes on linear networks

There is a lack of point process models on linear networks. For an arbitrary linear network, we use isotropic covariance functions with respect to the geodesic metric or the resistance metric to construct new models for isotropic Gaussian processes and hence new models for various Cox processes with isotropic pair correlation functions. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.

2022/12/16

Article Details
Jesper Møller

Jesper Møller

Aalborg Universitet

Stat

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.

Jesper Møller

Jesper Møller

Aalborg Universitet

International Statistical Review

Should we condition on the number of points when modelling spatial point patterns?

We discuss the practice of directly or indirectly assuming a model for the number of points when modelling spatial point patterns even though it is rarely possible to validate such a model in practice because most point pattern data consist of only one pattern. We therefore explore the possibility to condition on the number of points instead when fitting and validating spatial point process models. In a simulation study with different popular spatial point process models, we consider model validation using global envelope tests based on functional summary statistics. We find that conditioning on the number of points will for some functional summary statistics lead to more narrow envelopes and thus stronger tests and that it can also be useful for correcting for some conservativeness in the tests when testing composite hypothesis. However, for other functional summary statistics, it makes little or no difference to condition …

Jesper Møller

Jesper Møller

Aalborg Universitet

Journal of Applied Probability

Characterization of random variables with stationary digits

Let be an integer, a stochastic process with state space , and F the cumulative distribution function (CDF) of . We show that stationarity of is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

Jesper Møller

Jesper Møller

Aalborg Universitet

Spatial Statistics

Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points, we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points, we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step, parameter estimation based on maximum pseudolikelihood methods is tractable. For model selection, we consider maximized log pseudolikelihood functions for models of the radii conditioned on the points. Model checking is performed using global envelopes and corresponding tests in both steps and moreover by comparing observed and simulated tessellation characteristics in the second step …

Francisco Cuevas-Pacheco

Francisco Cuevas-Pacheco

Aalborg Universitet

Package ‘SpatialPack’

Description Tools to assess the association between two spatial processes. Currently, several methodologies are implemented: A modified t-test to perform hypothesis testing about the independence between the processes, a suitable nonparametric correlation coefficient, the codispersion coefficient, and an F test for assessing the multiple correlation between one spatial process and several others. Functions for image processing and computing the spatial association between images are also provided. Functions contained in the package are intended to accompany Vallejos, R., Osorio, F., Bevilacqua, M.(2020). Spatial Relationships Between Two Georeferenced Variables: With Applications in R. Springer, Cham< doi: 10.1007/978-3-030-56681-4>.

Jesper Møller

Jesper Møller

Aalborg Universitet

Translational psychiatry

Layer III pyramidal cells in the prefrontal cortex reveal morphological changes in subjects with depression, schizophrenia, and suicide

Brodmann Area 46 (BA46) has long been regarded as a hotspot of disease pathology in individuals with schizophrenia (SCH) and major depressive disorder (MDD). Pyramidal neurons in layer III of the Brodmann Area 46 (BA46) project to other cortical regions and play a fundamental role in corticocortical and thalamocortical circuits. The AutoCUTS-LM pipeline was used to study the 3-dimensional structural morphology and spatial organization of pyramidal cells. Using quantitative light microscopy, we used stereology to calculate the entire volume of layer III in BA46 and the total number and density of pyramidal cells. Volume tensors estimated by the planar rotator quantified the volume, shape, and nucleus displacement of pyramidal cells. All of these assessments were carried out in four groups of subjects: controls (C, n = 10), SCH (n = 10), MDD (n = 8), and suicide subjects with a history of depression (SU …

Jesper Møller

Jesper Møller

Aalborg Universitet

Graphs and Combinatorics

Equivariant Euler characteristics of symplectic buildings

We compute the equivariant Euler characteristics of the buildings for the symplectic groups over finite fields.

Jesper Møller

Jesper Møller

Aalborg Universitet

Journal of Computational and Graphical Statistics

MCMC computations for Bayesian mixture models using repulsive point processes

Repulsive mixture models have recently gained popularity for Bayesian cluster detection. Compared to more traditional mixture models, repulsive mixture models produce a smaller number of well-separated clusters. The most commonly used methods for posterior inference either require to fix a priori the number of components or are based on reversible jump MCMC computation. We present a general framework for mixture models, when the prior of the “cluster centers” is a finite repulsive point process depending on a hyperparameter, specified by a density which may depend on an intractable normalizing constant. By investigating the posterior characterization of this class of mixture models, we derive a MCMC algorithm which avoids the well-known difficulties associated to reversible jump MCMC computation. In particular, we use an ancillary variable method, which eliminates the problem of having intractable …

Other articles from Methodology and Computing in Applied Probability journal

Krzysztof Dębicki

Krzysztof Dębicki

Uniwersytet Wroclawski

Methodology and Computing in Applied Probability

On Berman Functions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}$$\end{document} with a standard fractional Brownian motion (fBm) with Hurst parameter and define for x non-negative the Berman function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}}{ \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{aligned …

Gracia Yunruo Dong

Gracia Yunruo Dong

University of Waterloo

Methodology and Computing in Applied Probability

Randomized quasi-Monte Carlo methods on triangles: extensible lattices and sequences

Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of …

Christiane Lemieux

Christiane Lemieux

University of Waterloo

Methodology and Computing in Applied Probability

Randomized quasi-Monte Carlo methods on triangles: extensible lattices and sequences

Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of …

Farzad Eskandari

Farzad Eskandari

Allameh Tabataba'i University

Methodology and Computing in Applied Probability

The Multivariate Generalized Linear Hawkes Process in High Dimensions with Applications in Neuroscience

The Hawkes process models have been recently become a popular tool for modeling and analysis of neural spike trains. In this article, motivated by neuronal spike trains study, we propose a novel multivariate generalized linear Hawkes process model, where covariates are included in the intensity function. We consider the problem of simultaneous variable selection and estimation for the multivariate generalized linear Hawkes process in the high-dimensional regime. Estimation of the intensity function of the high-dimensional point process is considered within a nonparametric framework, applying B-splines and the SCAD penalty for matters of sparsity. We apply the Doob-Kolmogorov inequality and the martingale central limit theory to establish the consistency and asymptotic normality of the resulting estimators. Finally, we illustrate the performance of our proposal through simulation and demonstrate its utility by …

Marius Hofert

Marius Hofert

University of Waterloo

Methodology and Computing in Applied Probability

Randomized quasi-Monte Carlo methods on triangles: extensible lattices and sequences

Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of …

Zbigniew Michna

Zbigniew Michna

Uniwersytet Ekonomiczny we Wroclawiu

Methodology and Computing in Applied Probability

On Berman Functions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}$$\end{document} with a standard fractional Brownian motion (fBm) with Hurst parameter and define for x non-negative the Berman function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}}{ \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{aligned …

Ira Herbst

Ira Herbst

University of Virginia

Methodology and Computing in Applied Probability

How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

Nour Eddine BERRAHOU

Nour Eddine BERRAHOU

Université Cadi Ayyad

Methodology and Computing in Applied Probability

The Bahadur Representation for Empirical and Smooth Quantile Estimators Under Association

In this paper, the Bahadur representation of the empirical and Bernstein polynomials estimators of the quantile function based on associated sequences are investigated. The rate of approximation depends on the rate of decay in covariances, and it converges to the optimal rate observed under independence when the covariances quickly approach zero. As an application, we establish a Berry-Esseen bound with the rate assuming polynomial decay of covariances. All these results are established under fairly general conditions on the underlying distributions. Additionally, we perform Monte Carlo simulations to evaluate the finite sample performance of the suggested estimators, utilizing an associated and non-mixing model.

Jesper Møller

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

Maxim Finkelstein

Maxim Finkelstein

University of the Free State

Methodology and Computing in Applied Probability

On survival of coherent systems subject to random shocks

We consider coherent systems subject to random shocks that can damage a random number of components of a system. Based on the distribution of the number of failed components, we discuss three models, namely, (i) a shock can damage any number of components (including zero) with the same probability, (ii) each shock damages, at least, one component, and (iii) a shock can damage, at most, one component. Shocks arrival times are modeled using three important counting processes, namely, the Poisson generalized gamma process, the Poisson phase-type process and the renewal process with matrix Mittag-Leffler distributed inter-arrival times. For the defined shock models, we discuss relevant reliability properties of coherent systems. An optimal replacement policy for repairable systems is considered as an application of the proposed modeling.

Nil Kamal Hazra

Nil Kamal Hazra

Indian Institute of Technology Jodhpur

Methodology and Computing in Applied Probability

On survival of coherent systems subject to random shocks

We consider coherent systems subject to random shocks that can damage a random number of components of a system. Based on the distribution of the number of failed components, we discuss three models, namely, (i) a shock can damage any number of components (including zero) with the same probability, (ii) each shock damages, at least, one component, and (iii) a shock can damage, at most, one component. Shocks arrival times are modeled using three important counting processes, namely, the Poisson generalized gamma process, the Poisson phase-type process and the renewal process with matrix Mittag-Leffler distributed inter-arrival times. For the defined shock models, we discuss relevant reliability properties of coherent systems. An optimal replacement policy for repairable systems is considered as an application of the proposed modeling.

Savas Dayanik

Savas Dayanik

Bilkent Üniversitesi

Methodology and Computing in Applied Probability

Model misspecification in discrete time Bayesian online change detection

We revisit the classical formulation of the discrete time Bayesian online change detection problem in which the common distribution of an observed sequence of random variables changes at an unknown point in time. The objective is to detect the change with a stopping time of the observations and minimize a given Bayes risk. When the change time has a zero-modified geometric prior distribution, the first crossing time of the odds-ratio process over a threshold is known to be an optimal solution. In the current paper, we consider a modeler who misspecifies some of the elements of this formulation. Because of this misspecification, the modeler computes a wrong stopping threshold and follows an incorrect odds-ratio process in implementation. To find her actual Bayes risk, which is different from the value function evaluated with the wrong choices, one needs to compute the expected costs accumulated by the true …

Sonya Javadi

Sonya Javadi

Dogus Üniversitesi

Methodology and Computing in Applied Probability

On computing the multivariate poisson probability distribution

Within the theory of non-negative integer valued multivariate infinitely divisible distributions, the multivariate Poisson distribution plays a key role. As in the univariate case, any non-negative integer valued infinitely divisible multivariate distribution can be approximated by a multivariate distribution belonging to the compound Poisson family. The multivariate Poisson distribution is an important member of this family. In recent years, the multivariate Poisson distributions also has gained practical importance, since they serve as models to describe counting data having a positive covariance structure. However, due to the computational complexity of computing the multivariate Poisson probability mass function (pmf) and its corresponding cumulative distribution function (cdf), their use within these counting models is limited. Since most of the theoretical properties of the multivariate Poisson probability distribution seem …

Pablo Azcue

Pablo Azcue

Universidad Torcuato di Tella

Methodology and Computing in Applied Probability

Optimal Strategies in a Production Inventory Control Model

We consider a production-inventory control model with finite capacity and two different production rates, assuming that the cumulative process of customer demand is given by a compound Poisson process. It is possible at any time to switch over from the different production rates but it is mandatory to switch-off when the inventory process reaches the storage maximum capacity. We consider holding, production, shortage penalty and switching costs. This model was introduced by Doshi, Van Der Duyn Schouten and Talman in 1978. In their paper they found a formula for the long-run average expected cost per unit time as a function of two critical levels, in this paper we consider expected discounted cumulative costs instead. We seek to minimize this discounted cost over all admissible switching strategies. We show that the optimal cost functions for the different production rates satisfy the corresponding Hamilton …

Sidney Resnick

Sidney Resnick

Cornell University

Methodology and Computing in Applied Probability

Poisson edge growth and preferential attachment networks

When modeling a directed social network, one choice is to use the traditional preferential attachment model, which generates power-law tail distributions. In traditional directed preferential attachment, every new edge is added sequentially into the network. However, real datasets often have only coarse timestamps, which means several new edges are created at the same timestamp. Previous analyses on the evolution of social networks reveal that after reaching a stable phase, the growth of edge counts in a network follows a non-homogeneous Poisson process with a constant rate across the day but varying rates from day to day. Taking such empirical observations into account, we propose a modified preferential attachment model with Poisson edge growth, and study its asymptotic behavior. This new model is then fitted to real datasets using an extreme value estimation approach.

Xuejun Wang

Xuejun Wang

Anhui University

Methodology and Computing in Applied Probability

Strong Convergence for Weighted Sums of Widely Orthant Dependent Random Variables and Applications

In this paper, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of widely orthant dependent random variables are presented. Some applications to simple linear errors-in-variables model, nonparametric regression model, and quasi-renewal counting process are provided. Simulation studies are also carried out to confirm the theoretical results.

Spiros Dafnis

Spiros Dafnis

Agricultural University of Athens

Methodology and Computing in Applied Probability

Distributions Related to Weak Runs With a Minimum and a Maximum Number of Successes: A Unified Approach

In the present paper we extend the definition of r-weak runs in sequences of binary trials so that such a run contains both a minimum and a maximum number of successes and we study the distribution of the statistic enumerating such (r-weak) runs. We are also investigating the distribution of the total number of successes in all the (r-weak) runs that are enumerated. The new introduced distributions may be of great applicability in scientific fields such as Agriculture. Our study requires an appropriate generalization of the Markov chain imbedding technique. More specifically, we introduce and study the family of Markov chain imbeddable variables of returnable-polynomial type, which generalizes the families of Markov chain imbeddable variables of binomial type, returnable type and polynomial type. The new family allows a unified approach for the study of the distributions of the statistics of current interest and other …

Gabriel Martos Venturini

Gabriel Martos Venturini

Universidad Torcuato di Tella

Methodology and Computing in Applied Probability

Level Sets Semimetrics for Probability Measures with applications in hypothesis testing

In this paper we introduce a novel family of level sets semimetrics for density functions and address subtleties entailed in the estimation and computation of such semimetrics. Given data drawn from f and q, two unknown density functions, we consider different level set semimetrics so to test the null hypothesis . The performance of such testing procedure is showcased in a Monte Carlo simulation study. Using the methods developed in the paper, we assess differences in gene expression profiles between two groups of patients with different respiratory recovery patterns in a clinical study; and find significant differences between the 15 top–ranked genes density profiles corresponding to the two groups.

Sujit Kumar Samanta

Sujit Kumar Samanta

National Institute of Technology, Raipur

Methodology and Computing in Applied Probability

Detailed Analytical and Computational Studies of D-BMAP/D-BMSP/1 Queueing System

This paper studies a discrete-time single server batch arrival and batch service queueing model with unlimited waiting space. The discrete-time batch Markovian arrival process and discrete-time batch Markovian service process, respectively, manage the arrival and service processes. We adopt the UL-type RG-factorization approach based on censoring methodology for variable size batch service queue to calculate the stationary probability vectors of the transition probability matrix with general structure Markov chain at outside observer’s epoch. We reblock the transition probability matrix to its desired M/G/1 structure to find the stationary probability vectors at outside observer’s epoch for fixed size batch service queue using the matrix analytic method. We also develop relationships to determine probability vector expressions for other important time epochs such as pre-arrival, intermediate, post-departure, and …

N Balakrishnan

N Balakrishnan

McMaster University

Methodology and Computing in Applied Probability

Joint Reliability of Two Consecutive-(1, l) or (2, k)-out-of-(2, n): F Type Systems and Its Application in Smart Street Light Deployment

To investigate some complex practical systems, such as a smart street light system composed of symmetrically deployed lighting and sensing equipment on both sides of a road segment, two consecutive-(1,l) or (2,k)-out-of-(2,n): F type systems that share components are quite useful and are therefore studied in this paper. The joint reliability of the two systems is presented by using the finite Markov chain imbedding approach (FMCIA), which also presents a new computational method for joint signatures of such systems. Compared to the direct method, the new method is not only computationally more efficient, but also presents a unified mathematical form. Finally, some numerical examples are presented to show the computational process, and some further applications and extensions of the models and the methods developed here are mentioned.