DOCSLIB.ORG

Gaussian Process Modulated Cox Processes Under Linear Inequality Constraints

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gaussian Process Modulated Cox Processes Under Linear Inequality Constraints Gaussian Process Modulated Cox Processes under Linear Inequality Constraints Andr´esF. L´opez-Lopera ST John Nicolas Durrande Mines Saint-Etienne´ ∗ PROWLER.io PROWLER.io Abstract Poisson processes are the foundation for modelling point patterns (Kingman, 1992). Their extension Gaussian process (GP) modulated Cox pro- to stochastic intensity functions, known as doubly cesses are widely used to model point pat- stochastic Poisson processes or Cox processes (Cox, terns. Existing approaches require a mapping 1955), enables non-parametric inference on the in- (link function) between the unconstrained tensity function and allows expressing uncertainties GP and the positive intensity function. This (Møller and Waagepetersen, 2004). Moreover, pre- commonly yields solutions that do not have a vious studies have shown that other classes of point closed form or that are restricted to specific processes may also be seen as Cox processes. For ex- covariance functions. We introduce a novel ample, Yannaros (1988) proved that Gamma renewal finite approximation of GP-modulated Cox processes are Cox processes under non-increasing con- processes where positiveness conditions can ditions. A similar analysis was made later for Weibull be imposed directly on the GP, with no re- processes (Yannaros, 1994). strictions on the covariance function. Our ap- Gaussian processes (GPs) form a flexible prior over proach can also ensure other types of inequal- functions, and are widely used to model the intensity ity constraints (e.g. monotonicity, convexity), process Λ( ) (Møller et al., 2001; Adams et al., 2009; resulting in more versatile models that can Teh and Rao,· 2011; Gunter et al., 2014; Lasko, 2014; be used for other classes of point processes Lloyd et al., 2015; Fernandez et al., 2016; Donner and (e.g. renewal processes). We demonstrate Opper, 2018). However, to ensure positive intensities, on both synthetic and real-world data that this commonly requires link functions between the our framework accurately infers the intensity intensity process and the GP g( ). Typical examples functions. Where monotonicity is a feature of mappings are Λ(x) = exp(g(x))· (Møller et al., 2001; of the process, our ability to include this in Diggle et al., 2013; Flaxman et al., 2015) or Λ(x) = the inference improves results. g(x)2 (Lloyd et al., 2015; Kozachenko et al., 2016). The exponential transformation has the drawback that there is no closed-form expression for some of the 1 INTRODUCTION integrals required to compute the likelihood. Although the square inverse link function allows closed-form Point processes are used in a variety of real-world expressions for certain kernels, it leads to models problems for modelling temporal or spatiotemporal exhibiting \nodal lines" with zero intensity due to point patterns in fields such as astronomy, geogra- the non-monotonicity of the transformation (see John phy, and ecology (Baddeley et al., 2015; Møller and and Hensman, 2018, for a discussion). Furthermore, Waagepetersen, 2004). In reliability analysis, they are current approaches to Cox process inference cannot used as renewal processes to model the lifetime of items be used in applications such as renewal processes that or failure (hazard) rates (Cha and Finkelstein, 2018). require both positivity and monotonicity constraints. Here, we introduce a novel approximation of GP- ∗Part of this work was completed during an internship of modulated Cox processes that does not rely on a A. F. L´opez-Lopera at PROWLER.io. mapping to obtain the intensity. In our approach we impose the constraints (e.g. non-negativeness or Proceedings of the 22nd International Conference on Ar- monotonicity) directly on Λ( ) by sampling from a tificial Intelligence and Statistics (AISTATS) 2019, Naha, · Okinawa, Japan. PMLR: Volume 89. Copyright 2019 by truncated Gaussian vector. This has the advantage the author(s). that the likelihood can be computed in closed form. Gaussian Process Modulated Cox Processes under Inequality Constraints Moreover, our approach can ensure any type of linear 3 APPROXIMATION OF GP inequality constraint everywhere, which allows mod- MODULATED COX PROCESSES elling of a broader range of point processes. This paper is organised as follows. In Section 2, In this work, we approximate the intensity Λ( ) of the Cox process by a finite-dimensional GP Λ· ( ) we briefly describe inhomogeneous Poisson processes m · and some of their extensions. In Sections 3 and 4, subject to some inequality constraints (e.g. bounded- we introduce a finite representation of GP-modulated ness, monotonicity, convexity). Since positiveness con- straints are imposed directly on Λ ( ), a link function Cox processes and the corresponding Cox process m · inference under inequality constraints. In Section is no longer necessary. This has two main advantages. 5, we apply our framework to 1D and 2D inference First, the likelihood (1) can be computed analytically. examples under different inequality conditions. We Second, as our approach ensures any linear inequality also test its performance in reliability applications constraint, it can be used for modelling a broader with hazard rates exhibiting monotonic behaviours. range of point processes. Finally, in Section 6, we summarise our results and outline potential future work. 3.1 Finite Approximation of 1D GPs Let Λ( ) be a zero-mean GP on R with arbitrary 2 POISSON POINT PROCESSES covariance· function k. Consider x , with compact space = [0; 1], and a set of knots2 St ; ; t . S 1 ··· m 2 S A Poisson process X is a random countable subset of Here we consider equispaced knots tj = (j 1)∆m Rd where points occur independently (Baddeley with ∆ = 1=(m 1). We define Λ ( ) as the− finite- S ⊆ m m et al., 2006). Let N N be a random variable (r.v.) dimensional approximation− of Λ( ) consisting· of its 2 · denoting the number of points in X. Let X1; ;Xn piecewise-linear interpolation at knots t ; ; t , i.e., ··· 1 ··· m be a set of n independent and identically distributed m (i.i.d.) r.v.'s on . The likelihood of (N = n; X = X S 1 Λm(x) = φj(x)ξj; (3) x1; ;Xn = xn) under an inhomogeneous Poisson j=1 process··· with non-negative intensity λ( ) is given by (Møller and Waagepetersen, 2004) · where ξj := Λ(tj) for j = 1; ; m, and φ1; ; φm are hat basis functions given by··· ··· n exp( µ) Y ( x tj x tj f(N;X ; ;X )(n; x1; ; xn) = − λ(xi); (1) 1 − if − 1; 1 n ∆m ∆m ··· ··· n! φj(x) := − ≤ (4) i=1 0 otherwise: where Z Similarly to spline-based approaches (e.g., Sleeper and µ = λ(s) ds (2) Harrington, 1990), we assume that Λ( ) is piecewise · S defined by (first-order) polynomials. The striking is the intensity measure or overall intensity. property of this basis is that satisfying the inequality When is the real line, the distance (inter-arrival constraints (e.g. boundedness, monotonicity, convex- time) betweenS consecutive points of a Poisson process ity) at the knots implies that the constraints are follows an exponential distribution. Renewal processes satisfied everywhere in the input space (Maatouk and are a generalisation of Poisson processes where inter- Bay, 2017). Although it is tempting to generalise arrival times are i.i.d. but not necessarily exponentially the above construction to smoother basis functions, distributed. An example is the Weibull process where it makes this property difficult to enforce. inter-arrival times are distributed following λ(x) = We aim at computing the distribution of Λ ( ) under αβxβ 1 (Cha and Finkelstein, 2018). m − the condition that it belongs to a convex· set of Cox processes (Cox, 1955) are a natural extension functions defined by some inequality constraints (e.g. E of inhomogeneous Poisson processes where λ( ) is positivity). This piecewise-linear representation has · sampled from a non-negative stochastic process Λ( ). the benefit that satisfying Λm( ) is equivalent · 2 E Previous studies have shown that many classes· of to satisfying only a finite number of inequality con- point processes can be seen as Cox processes under straints. More precisely, certain conditions (Møller and Waagepetersen, 2004; Λ ( ) ξ ; (5) Yannaros, 1988, 1994). For example, Weibull renewal m · 2 E , 2 C m processes are Cox processes for β (0; 1] (Yannaros, where ξ = [ξ ; ; ξ ]>, and is a convex set on R . 1 ··· m C 1994). This motivates the construction2 of GPs with For non-negativeness conditions , is given by non-negative and monotonic constraints, so that they E+ C can be used as intensities Λ( ) of Cox processes. := c Rm; j = 1; ; m : c 0 ; (6) · C+ f 2 8 ··· j ≥ g Andr´esF. L´opez-Lopera, ST John, Nicolas Durrande 2 2 2 1 1 1 ↓ + + m 0 ∈ E 0 ∈ E 0 Λ m m -1 Λ -1 Λ -1 -2 -2 -2 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 # (a) unconstrained GP (b) GP with C+ constraints (c) GP with C+ constraints Figure 1: Samples from the prior Λm( ) under (a) no constraints, (b) non-negativeness constraints, (c) both non-negativeness and non-increasing constraints.· The grey region shows the 95% confidence interval. and for non-increasing conditions , is given by Since (8) depends on r.v.'s ξ1; ; ξm, it can be E# C approximated using samples of ξ···. To estimate the Rm := c ; j = 2; ; m : cj 1 cj : (7) covariance parameters θ of the vector ξ, we can use C# f 2 8 ··· − ≥ g stochastic global optimisation (Jones et al., 1998). Constraints can be composed, e.g. the convex set of non-negativeness and non-increasing conditions is 3.3 Extension to Higher Dimensions given by +# = + . C C \C# Assuming that ξ is zero-mean Gaussian-distributed The approximation in (3) can be extended to grids in d dimensions by tensorisation.
Load more

Recommended publications
  • Cox Process Representation and Inference for Stochastic Reaction-Diffusion Processes
    Edinburgh Research Explorer Cox process representation and inference for stochastic reaction-diffusion processes Citation for published version: Schnoerr, D, Grima, R & Sanguinetti, G 2016, 'Cox process representation and inference for stochastic reaction-diffusion processes', Nature Communications, vol. 7, 11729. https://doi.org/10.1038/ncomms11729 Digital Object Identifier (DOI): 10.1038/ncomms11729 Link: Link to publication record in Edinburgh Research Explorer Document Version: Publisher's PDF, also known as Version of record Published In: Nature Communications General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 02. Oct. 2021 ARTICLE Received 16 Dec 2015 | Accepted 26 Apr 2016 | Published 25 May 2016 DOI: 10.1038/ncomms11729 OPEN Cox process representation and inference for stochastic reaction–diffusion processes David Schnoerr1,2,3, Ramon Grima1,3 & Guido Sanguinetti2,3 Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction–diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data.
    [Show full text]
  • Poisson Representations of Branching Markov and Measure-Valued
    The Annals of Probability 2011, Vol. 39, No. 3, 939–984 DOI: 10.1214/10-AOP574 c Institute of Mathematical Statistics, 2011 POISSON REPRESENTATIONS OF BRANCHING MARKOV AND MEASURE-VALUED BRANCHING PROCESSES By Thomas G. Kurtz1 and Eliane R. Rodrigues2 University of Wisconsin, Madison and UNAM Representations of branching Markov processes and their measure- valued limits in terms of countable systems of particles are con- structed for models with spatially varying birth and death rates. Each particle has a location and a “level,” but unlike earlier con- structions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level r, or for the limiting models, hits infinity. For branching Markov pro- cesses, at each time t, conditioned on the state of the process, the levels are independent and uniformly distributed on [0,r]. For the limiting measure-valued process, at each time t, the joint distribu- tion of locations and levels is conditionally Poisson distributed with mean measure K(t) × Λ, where Λ denotes Lebesgue measure, and K is the desired measure-valued process. The representation simplifies or gives alternative proofs for a vari- ety of calculations and results including conditioning on extinction or nonextinction, Harris’s convergence theorem for supercritical branch- ing processes, and diffusion approximations for processes in random environments. 1. Introduction. Measure-valued processes arise naturally as infinite sys- tem limits of empirical measures of finite particle systems. A number of ap- proaches have been developed which preserve distinct particles in the limit and which give a representation of the measure-valued process as a transfor- mation of the limiting infinite particle system.
    [Show full text]
  • Local Conditioning in Dawson–Watanabe Superprocesses
    The Annals of Probability 2013, Vol. 41, No. 1, 385–443 DOI: 10.1214/11-AOP702 c Institute of Mathematical Statistics, 2013 LOCAL CONDITIONING IN DAWSON–WATANABE SUPERPROCESSES By Olav Kallenberg Auburn University Consider a locally finite Dawson–Watanabe superprocess ξ =(ξt) in Rd with d ≥ 2. Our main results include some recursive formulas for the moment measures of ξ, with connections to the uniform Brown- ian tree, a Brownian snake representation of Palm measures, continu- ity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of ξt by a stationary clusterη ˜ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of ξt for a fixed t> 0, given d that ξt charges the ε-neighborhoods of some points x1,...,xn ∈ R . In the limit as ε → 0, the restrictions to those sets are conditionally in- dependent and given by the pseudo-random measures ξ˜ orη ˜, whereas the contribution to the exterior is given by the Palm distribution of ξt at x1,...,xn. Our proofs are based on the Cox cluster representa- tions of the historical process and involve some delicate estimates of moment densities. 1. Introduction. This paper may be regarded as a continuation of [19], where we considered some local properties of a Dawson–Watanabe super- process (henceforth referred to as a DW-process) at a fixed time t> 0. Recall that a DW-process ξ = (ξt) is a vaguely continuous, measure-valued diffu- d ξtf µvt sion process in R with Laplace functionals Eµe− = e− for suitable functions f 0, where v = (vt) is the unique solution to the evolution equa- 1 ≥ 2 tion v˙ = 2 ∆v v with initial condition v0 = f.
    [Show full text]
  • Cox Process Functional Learning Gérard Biau, Benoît Cadre, Quentin Paris
    Cox process functional learning Gérard Biau, Benoît Cadre, Quentin Paris To cite this version: Gérard Biau, Benoît Cadre, Quentin Paris. Cox process functional learning. Statistical Inference for Stochastic Processes, Springer Verlag, 2015, 18 (3), pp.257-277. 10.1007/s11203-015-9115-z. hal- 00820838 HAL Id: hal-00820838 https://hal.archives-ouvertes.fr/hal-00820838 Submitted on 6 May 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Cox Process Learning G´erard Biau Universit´ePierre et Marie Curie1 & Ecole Normale Sup´erieure2, France [email protected] Benoˆıt Cadre IRMAR, ENS Cachan Bretagne, CNRS, UEB, France3 [email protected] Quentin Paris IRMAR, ENS Cachan Bretagne, CNRS, UEB, France [email protected] Abstract This article addresses the problem of supervised classification of Cox process trajectories, whose random intensity is driven by some exoge- nous random covariable. The classification task is achieved through a regularized convex empirical risk minimization procedure, and a nonasymptotic oracle inequality is derived. We show that the algo- rithm provides a Bayes-risk consistent classifier. Furthermore, it is proved that the classifier converges at a rate which adapts to the un- known regularity of the intensity process.
    [Show full text]
  • Spatio-Temporal Cluster Detection and Local Moran Statistics of Point Processes
    Old Dominion University ODU Digital Commons Mathematics & Statistics Theses & Dissertations Mathematics & Statistics Spring 2019 Spatio-Temporal Cluster Detection and Local Moran Statistics of Point Processes Jennifer L. Matthews Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_etds Part of the Applied Statistics Commons, and the Biostatistics Commons Recommended Citation Matthews, Jennifer L.. "Spatio-Temporal Cluster Detection and Local Moran Statistics of Point Processes" (2019). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/3mps-rk62 https://digitalcommons.odu.edu/mathstat_etds/46 This Dissertation is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. ABSTRACT Approved for public release; distribution is unlimited SPATIO-TEMPORAL CLUSTER DETECTION AND LOCAL MORAN STATISTICS OF POINT PROCESSES Jennifer L. Matthews Commander, United States Navy Old Dominion University, 2019 Director: Dr. Norou Diawara Moran's index is a statistic that measures spatial dependence, quantifying the degree of dispersion or clustering of point processes and events in some location/area. Recognizing that a single Moran's index may not give a sufficient summary of the spatial autocorrelation measure, a local
    [Show full text]
  • Log Gaussian Cox Processes
    Log Gaussian Cox processes by Jesper Møller, Anne Randi Syversveen, and Rasmus Plenge Waagepetersen. Log Gaussian Cox processes JESPER MØLLER Aalborg University ANNE RANDI SYVERSVEEN The Norwegian University of Science and Technology RASMUS PLENGE WAAGEPETERSEN University of Aarhus ABSTRACT. Planar Cox processes directed by a log Gaussian intensity process are investigated in the univariate and multivariate cases. The appealing properties of such models are demonstrated theoretically as well as through data examples and simulations. In particular, the first, second and third-order properties are studied and utilized in the statistical analysis of clustered point patterns. Also empirical Bayesian inference for the underlying intensity surface is considered. Key words: empirical Bayesian inference; ergodicity; Markov chain Monte Carlo; Metropolis-adjusted Langevin algorithm; multivariate Cox processes; Neyman-Scott processes; pair correlation function; parameter estimation; spatial point processes; third- order properties. AMS 1991 subject classification: Primary 60G55, 62M30. Secondary 60D05. 1 Introduction Cox processes provide useful and frequently applied models for aggregated spatial point patterns where the aggregation is due to a stochastic environmental heterogeneity, see e.g. Diggle (1983), Cressie (1993), Stoyan et al. (1995), and the references therein. A Cox process is ’doubly stochastic’ as it arises as an inhomogeneous Poisson process with a random intensity measure. The random intensity measure is often specified by a random intensity function or as we prefer to call it an intensity process or surface. There may indeed be other sources of aggregation in a spatial point pattern than spatial heterogeneity. Cluster processes is a well-known class of models where clusters are generated by an unseen point process, cf.
    [Show full text]
  • Cox Process Representation and Inference for Stochastic Reaction&Ndash
    ARTICLE Received 16 Dec 2015 | Accepted 26 Apr 2016 | Published 25 May 2016 DOI: 10.1038/ncomms11729 OPEN Cox process representation and inference for stochastic reaction–diffusion processes David Schnoerr1,2,3, Ramon Grima1,3 & Guido Sanguinetti2,3 Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction–diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data. Here we use ideas from statistical physics and machine learning to provide a solution to the inverse problem of learning a stochastic reaction–diffusion process from data. Our solution relies on a non-trivial connection between stochastic reaction–diffusion processes and spatio-temporal Cox processes, a well-studied class of models from computational statistics. This connection leads to an efficient and flexible algorithm for parameter inference and model selection. Our approach shows excellent accuracy on numeric and real data examples from systems biology and epidemiology. Our work provides both insights into spatio-temporal stochastic systems, and a practical solution to a long-standing problem in computational modelling. 1 School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, UK. 2 School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, UK. 3 SynthSys, University of Edinburgh, Edinburgh EH9 3JD, UK. Correspondence and requests for materials should be addressed to G.S. (email: [email protected]). NATURE COMMUNICATIONS | 7:11729 | DOI: 10.1038/ncomms11729 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11729 any complex behaviours in several disciplines originate process in terms of (stochastic) partial differential equations from a common mechanism: the dynamics of locally (SPDEs).
    [Show full text]
  • Exact Bayesian Inference in Spatio-Temporal Cox Processes Driven by Multivariate Gaussian Processes
    Exact Bayesian inference in spatio-temporal Cox processes driven by multivariate Gaussian processes Fl´avioB. Gon¸calvesa, Dani Gamermanb a Departamento de Estat´ıstica,Universidade Federal de Minas Gerais, Brazil b Departamento de M´etodos Estat´ısticos,Universidade Federal do Rio de Janeiro, Brazil Abstract In this paper we present a novel inference methodology to perform Bayesian in- ference for spatio-temporal Cox processes where the intensity function depends on a multivariate Gaussian process. Dynamic Gaussian processes are introduced to allow for evolution of the intensity function over discrete time. The novelty of the method lies on the fact that no discretisation error is involved despite the non-tractability of the likelihood function and infinite dimensionality of the problem. The method is based on a Markov chain Monte Carlo algorithm that samples from the joint pos- terior distribution of the parameters and latent variables of the model. A particular choice of the dominating measure to obtain the likelihood function is shown to be crucial to devise a valid MCMC. The models are defined in a general and flexible way but they are amenable to direct sampling from the relevant distributions, due to careful characterisation of its components. The models also allow for the inclu- sion of regression covariates and/or temporal components to explain the variability of the intensity function. These components may be subject to relevant interaction with space and/or time. Simulated examples illustrate the methodology, followed by concluding remarks. Key Words: Cox process, spatio-temporal, dynamic Gaussian process, exact inference, MCMC. 1Address: Av. Ant^onioCarlos, 6627 - DEST/ICEx/UFMG - Belo Horizonte, Minas Gerais, 31270-901, Brazil.
    [Show full text]
  • Spatial Point Process Modeling of Vehicles in Large and Small Cities
    Spatial Point Process Modeling of Vehicles in Large and Small Cities Qimei Cui, Ning Wang Martin Haenggi School of Information and Communication Engineering Department of Electrical Engineering Beijng University of Posts and Telecommunications University of Notre Dame Beijing, 100876, China Notre Dame, IN 46556, USA Email: {cuiqimei, wangning}@bupt.edu.cn Email: [email protected] Abstract—The uncertainty in the locations of vehicles on streets nodes are modeled as linear Poisson processes. Reference [3] induced by vehicles passing and queueing make the spatial observes that the vehicles that enter the highway through one modeling of vehicles a difficult task. To analyze the performance of the traffic entry points form a Poisson process and studies of vehicle-to-vehicle (V2V) communication for vehicular ad hoc networks (VANETs), accurate spatial modeling is of great im- the impact of vehicle mobility on the connectivity of VANETs. portance. In this paper, we concentrate on spatial point process Connectivity is also analyzed in [4] and [5], where the same modeling for random vehicle locations in large and small cities, model for vehicles entering the highway is used and, in performing empirical experiments with real location data of addition, it is assumed that the speed of the vehicles is constant mobile taxi trajectories recorded by the global positioning system or normally distributed. In [6] and [7], the vehicle locations (GPS) in Beijing city of China and Porto city of Portugal. We find that the empirical probability mass functions (PMFs) of the are assumed to form a Poisson point process in one dimension, number of taxis in test sets in different regions of Beijing or in and the authors perform an analytical evaluation of broadcast Porto all follow a negative binomial (NB) distribution.
    [Show full text]
  • Notes on the Poisson Point Process
    Notes on the Poisson point process Paul Keeler January 5, 2016 This work is licensed under a “CC BY-SA 3.0” license. Abstract The Poisson point process is a type of random object known as a point pro- cess that has been the focus of much study and application. This survey aims to give an accessible but detailed account of the Poisson point process by covering its history, mathematical definitions in a number of settings, and key properties as well detailing various terminology and applications of the process, with rec- ommendations for further reading. 1 Introduction In probability, statistics and related fields, a Poisson point process or a Poisson process or a Poisson point field is a type of random object known as a point pro- cess or point field that consists of randomly positioned points located on some underlying mathematical space [68]. The process has convenient mathematical properties [39], which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numer- ous disciplines such as astronomy [5], biology [53], ecology [70], geology [16], physics [61], image processing [12], and telecommunications [7][28]. The Poisson point process is often defined on the real line playing an impor- tant role in the field of queueing theory [40] where it is used to model certain random events happening in time such as the arrival of customers at a store or phone calls at an exchange. In the plane, the point process, also known as a spa- tial Poisson process [9], may represent scattered objects such as users in a wireless network [2, 6, 7, 27], particles colliding into a detector, or trees in a forest [68].
    [Show full text]
  • Autoregressive Point Processes As Latent State-Space Models: a Moment-Closure Approach to Fluctuations and Autocorrelations
    LETTER Communicated by Jonathan Pillow Autoregressive Point Processes as Latent State-Space Models: A Moment-Closure Approach to Fluctuations and Autocorrelations Michael Rule [email protected] Guido Sanguinetti [email protected] Institute for Adaptive and Neural Computation, School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, U.K. Modeling and interpreting spike train data is a task of central impor- tance in computational neuroscience, with significant translational im- plications. Two popular classes of data-driven models for this task are autoregressive point-process generalized linear models (PPGLM) and la- tent state-space models (SSM) with point-process observations. In this letter, we derive a mathematical connection between these two classes of models. By introducing an auxiliary history process, we represent exactly a PPGLM in terms of a latent, infinite-dimensional dynamical system, which can then be mapped onto an SSM by basis function projections and moment closure. This representation provides a new perspective on widely used methods for modeling spike data and also suggests novel algorithmic approaches to fitting such models. We illustrate our results on a phasic bursting neuron model, showing that our proposed approach provides an accurate and efficient way to capture neural dynamics. 1 Introduction Connecting single-neuron spiking to the collective dynamics that emerge in neural populations remains a central challenge in systems neuroscience. As well as representing a major barrier in our understanding of fundamental neural function, this challenge has recently acquired new salience due to the rapid improvements in technologies that can measure neural population ac- tivity in vitro and in vivo at unprecedented temporal and spatial resolution (Jun et al., 2017; Maccione et al., 2014).
    [Show full text]
  • Large-Scale Cox Process Inference Using Variational Fourier Features
    Large-Scale Cox Process Inference using Variational Fourier Features ST John 1 James Hensman 1 Abstract spatiotemporal domain, which cannot be computed in gen- eral. There are three potential remedies to this issue. First, Gaussian process modulated Poisson processes the classic approach is to discretize the input domain (see, provide a flexible framework for modeling spa- e.g., Taylor et al., 2015), assuming that the rate is constant tiotemporal point patterns. So far this was re- over each grid cell. However, this discretized structure stricted to one dimension, binning to a pre- scales poorly with the number of grid cells. Second, we determined grid, or small data sets of up to a can use a thinning strategy to construct an exact MCMC few thousand data points. Here we introduce Cox sampler (Adams et al., 2009; Gunter et al., 2014), which process inference based on Fourier features. This demands the inverse link function to be bounded, e.g., sig- sparse representation induces global rather than moidal. However, this is prohibitively expensive for more local constraints on the function space and is com- than a few thousand points. Finally, we can use a square- putationally efficient. This allows us to formulate root link function λ(·) = f(·)2, as proposed by Lloyd et al. a grid-free approximation that scales well with the (2015). Combined with variational inference for the latent number of data points and the size of the domain. function, this can make the integral tractable. Although their We demonstrate that this allows MCMC approxi- approach scales linearly with the number of events, it scales mations to the non-Gaussian posterior.
    [Show full text]