POISSON SUPERPOSITION PROCESSES 1. Introduction
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Introduction to Lévy Processes
Introduction to Lévy Processes Huang Lorick [email protected] Document type These are lecture notes. Typos, errors, and imprecisions are expected. Comments are welcome! This version is available at http://perso.math.univ-toulouse.fr/lhuang/enseignements/ Year of publication 2021 Terms of use This work is licensed under a Creative Commons Attribution 4.0 International license: https://creativecommons.org/licenses/by/4.0/ Contents Contents 1 1 Introduction and Examples 2 1.1 Infinitely divisible distributions . 2 1.2 Examples of infinitely divisible distributions . 2 1.3 The Lévy Khintchine formula . 4 1.4 Digression on Relativity . 6 2 Lévy processes 8 2.1 Definition of a Lévy process . 8 2.2 Examples of Lévy processes . 9 2.3 Exploring the Jumps of a Lévy Process . 11 3 Proof of the Levy Khintchine formula 19 3.1 The Lévy-Itô Decomposition . 19 3.2 Consequences of the Lévy-Itô Decomposition . 21 3.3 Exercises . 23 3.4 Discussion . 23 4 Lévy processes as Markov Processes 24 4.1 Properties of the Semi-group . 24 4.2 The Generator . 26 4.3 Recurrence and Transience . 28 4.4 Fractional Derivatives . 29 5 Elements of Stochastic Calculus with Jumps 31 5.1 Example of Use in Applications . 31 5.2 Stochastic Integration . 32 5.3 Construction of the Stochastic Integral . 33 5.4 Quadratic Variation and Itô Formula with jumps . 34 5.5 Stochastic Differential Equation . 35 Bibliography 38 1 Chapter 1 Introduction and Examples In this introductive chapter, we start by defining the notion of infinitely divisible distributions. We then give examples of such distributions and end this chapter by stating the celebrated Lévy-Khintchine formula. -
Poisson Processes Stochastic Processes
Poisson Processes Stochastic Processes UC3M Feb. 2012 Exponential random variables A random variable T has exponential distribution with rate λ > 0 if its probability density function can been written as −λt f (t) = λe 1(0;+1)(t) We summarize the above by T ∼ exp(λ): The cumulative distribution function of a exponential random variable is −λt F (t) = P(T ≤ t) = 1 − e 1(0;+1)(t) And the tail, expectation and variance are P(T > t) = e−λt ; E[T ] = λ−1; and Var(T ) = E[T ] = λ−2 The exponential random variable has the lack of memory property P(T > t + sjT > t) = P(T > s) Exponencial races In what follows, T1;:::; Tn are independent r.v., with Ti ∼ exp(λi ). P1: min(T1;:::; Tn) ∼ exp(λ1 + ··· + λn) . P2 λ1 P(T1 < T2) = λ1 + λ2 P3: λi P(Ti = min(T1;:::; Tn)) = λ1 + ··· + λn P4: If λi = λ and Sn = T1 + ··· + Tn ∼ Γ(n; λ). That is, Sn has probability density function (λs)n−1 f (s) = λe−λs 1 (s) Sn (n − 1)! (0;+1) The Poisson Process as a renewal process Let T1; T2;::: be a sequence of i.i.d. nonnegative r.v. (interarrival times). Define the arrival times Sn = T1 + ··· + Tn if n ≥ 1 and S0 = 0: The process N(t) = maxfn : Sn ≤ tg; is called Renewal Process. If the common distribution of the times is the exponential distribution with rate λ then process is called Poisson Process of with rate λ. Lemma. N(t) ∼ Poisson(λt) and N(t + s) − N(s); t ≥ 0; is a Poisson process independent of N(s); t ≥ 0 The Poisson Process as a L´evy Process A stochastic process fX (t); t ≥ 0g is a L´evyProcess if it verifies the following properties: 1. -
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. -
Introduction to Lévy Processes
Introduction to L´evyprocesses Graduate lecture 22 January 2004 Matthias Winkel Departmental lecturer (Institute of Actuaries and Aon lecturer in Statistics) 1. Random walks and continuous-time limits 2. Examples 3. Classification and construction of L´evy processes 4. Examples 5. Poisson point processes and simulation 1 1. Random walks and continuous-time limits 4 Definition 1 Let Yk, k ≥ 1, be i.i.d. Then n X 0 Sn = Yk, n ∈ N, k=1 is called a random walk. -4 0 8 16 Random walks have stationary and independent increments Yk = Sk − Sk−1, k ≥ 1. Stationarity means the Yk have identical distribution. Definition 2 A right-continuous process Xt, t ∈ R+, with stationary independent increments is called L´evy process. 2 Page 1 What are Sn, n ≥ 0, and Xt, t ≥ 0? Stochastic processes; mathematical objects, well-defined, with many nice properties that can be studied. If you don’t like this, think of a model for a stock price evolving with time. There are also many other applications. If you worry about negative values, think of log’s of prices. What does Definition 2 mean? Increments , = 1 , are independent and Xtk − Xtk−1 k , . , n , = 1 for all 0 = . Xtk − Xtk−1 ∼ Xtk−tk−1 k , . , n t0 < . < tn Right-continuity refers to the sample paths (realisations). 3 Can we obtain L´evyprocesses from random walks? What happens e.g. if we let the time unit tend to zero, i.e. take a more and more remote look at our random walk? If we focus at a fixed time, 1 say, and speed up the process so as to make n steps per time unit, we know what happens, the answer is given by the Central Limit Theorem: 2 Theorem 1 (Lindeberg-L´evy) If σ = V ar(Y1) < ∞, then Sn − (Sn) √E → Z ∼ N(0, σ2) in distribution, as n → ∞. -
Applied Mathematics 1
Applied Mathematics 1 MATH 0180 Intermediate Calculus 1 Applied Mathematics MATH 0520 Linear Algebra 2 1 APMA 0350 Applied Ordinary Differential Equations 2 & APMA 0360 and Applied Partial Differential Equations Chair I 3 4 Bjorn Sandstede Select one course on programming from the following: 1 APMA 0090 Introduction to Mathematical Modeling Associate Chair APMA 0160 Introduction to Computing Sciences Kavita Ramanan CSCI 0040 Introduction to Scientific Computing and The Division of Applied Mathematics at Brown University is one of Problem Solving the most prominent departments at Brown, and is also one of the CSCI 0111 Computing Foundations: Data oldest and strongest of its type in the country. The Division of Applied CSCI 0150 Introduction to Object-Oriented Mathematics is a world renowned center of research activity in a Programming and Computer Science wide spectrum of traditional and modern mathematics. It explores the connections between mathematics and its applications at both CSCI 0170 Computer Science: An Integrated the research and educational levels. The principal areas of research Introduction activities are ordinary, functional, and partial differential equations: Five additional courses, of which four should be chosen from 5 stochastic control theory; applied probability, statistics and stochastic the 1000-level courses taught by the Division of Applied systems theory; neuroscience and computational molecular biology; Mathematics. APMA 1910 cannot be used as an elective. numerical analysis and scientific computation; and the mechanics of Total Credits 10 solids, materials science and fluids. The effort in virtually all research 1 ranges from applied and algorithmic problems to the study of fundamental Substitution of alternate courses for the specific requirements is mathematical questions. -
Alireza Sheikh-Zadeh, Ph.D
Alireza Sheikh-Zadeh, Ph.D. Assistant Professor of Practice in Data Science Rawls College of Business, Texas Tech University Area of Information Systems and Quantitative Sciences (ISQS) 703 Flint Ave, Lubbock, TX 79409 (806) 834-8569, [email protected] EDUCATION Ph.D. in Industrial Engineering, 2017 Industrial Engineering Department at the University of Arkansas, Fayetteville, AR Dissertation: “Developing New Inventory Segmentation Methods for Large-Scale Multi-Echelon In- ventory Systems” (Advisor: Manuel D. Rossetti) M.Sc. in Industrial Engineering (with concentration in Systems Management), 2008 Industrial Engineering & Systems Management Department at Tehran Polytechnic, Tehran, Iran Thesis: “System Dynamics Modeling for Study and Design of Science and Technology Parks for The Development of Deprived Regions” (Advisor: Reza Ramazani) AREA of INTEREST Data Science & Machine Learning Supply Chain Analytics Applied Operations Research System Dynamics Simulation and Stochastic Modeling Logistic Contracting PUBLICATIONS Journal Papers (including under review papers) • Sheikhzadeh, A., & Farhangi, H., & Rossetti, M. D. (under review), ”Inventory Grouping and Sensitivity Analysis in Large-Scale Spare Part Systems”, submitted work. • Sheikhzadeh, A., Rossetti, M. D., & Scott, M. (2nd review), ”Clustering, Aggregation, and Size-Reduction for Large-Scale Multi-Echelon Spare-Part Replenishment Systems,” Omega: The International Journal of Management Science. • Sheikhzadeh, A., & Rossetti, M. D. (4th review), ”Classification Methods for Problem Size Reduction in Spare Part Provisioning,” International Journal of Production Economics. • Al-Rifai, M. H., Rossetti, M. D., & Sheikhzadeh, A. (2016), ”A Heuristic Optimization Algorithm for Two- Echelon (r; Q) Inventory Systems with Non-Identical Retailers,” International Journal of Inventory Re- search. • Karimi-Nasab, M., Bahalke, U., Feili, H. R., Sheikhzadeh, A., & Dolatkhahi, K. -
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. -
Chapter 1 Poisson Processes
Chapter 1 Poisson Processes 1.1 The Basic Poisson Process The Poisson Process is basically a counting processs. A Poisson Process on the interval [0 , ∞) counts the number of times some primitive event has occurred during the time interval [0 , t]. The following assumptions are made about the ‘Process’ N(t). (i). The distribution of N(t + h) − N(t) is the same for each h> 0, i.e. is independent of t. ′ (ii). The random variables N(tj ) − N(tj) are mutually independent if the ′ intervals [tj , tj] are nonoverlapping. (iii). N(0) = 0, N(t) is integer valued, right continuous and nondecreasing in t, with Probability 1. (iv). P [N(t + h) − N(t) ≥ 2]= P [N(h) ≥ 2]= o(h) as h → 0. Theorem 1.1. Under the above assumptions, the process N(·) has the fol- lowing additional properties. (1). With probability 1, N(t) is a step function that increases in steps of size 1. (2). There exists a number λ ≥ 0 such that the distribution of N(t+s)−N(s) has a Poisson distribution with parameter λ t. 1 2 CHAPTER 1. POISSON PROCESSES (3). The gaps τ1, τ2, ··· between successive jumps are independent identically distributed random variables with the exponential distribution exp[−λ x] for x ≥ 0 P {τj ≥ x} = (1.1) (1 for x ≤ 0 Proof. Let us divide the interval [0 , T ] into n equal parts and compute the (k+1)T kT expected number of intervals with N( n ) − N( n ) ≥ 2. This expected value is equal to T 1 nP N( ) ≥ 2 = n.o( )= o(1) as n →∞ n n there by proving property (1). -
Notes on Stochastic Processes
Notes on stochastic processes Paul Keeler March 20, 2018 This work is licensed under a “CC BY-SA 3.0” license. Abstract A stochastic process is a type of mathematical object studied in mathemat- ics, particularly in probability theory, which can be used to represent some type of random evolution or change of a system. There are many types of stochastic processes with applications in various fields outside of mathematics, including the physical sciences, social sciences, finance and economics as well as engineer- ing and technology. This survey aims to give an accessible but detailed account of various stochastic processes by covering their history, various mathematical definitions, and key properties as well detailing various terminology and appli- cations of the process. An emphasis is placed on non-mathematical descriptions of key concepts, with recommendations for further reading. 1 Introduction In probability and related fields, a stochastic or random process, which is also called a random function, is a mathematical object usually defined as a collection of random variables. Historically, the random variables were indexed by some set of increasing numbers, usually viewed as time, giving the interpretation of a stochastic process representing numerical values of some random system evolv- ing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule [120, page 7][51, page 46 and 47][66, page 1]. Stochastic processes are widely used as math- ematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including physical sciences such as biology [67, 34], chemistry [156], ecology [16][104], neuroscience [102], and physics [63] as well as technology and engineering fields such as image and signal processing [53], computer science [15], information theory [43, page 71], and telecommunications [97][11][12]. -
Compound Poisson Processes, Latent Shrinkage Priors and Bayesian
Bayesian Analysis (2015) 10, Number 2, pp. 247–274 Compound Poisson Processes, Latent Shrinkage Priors and Bayesian Nonconvex Penalization Zhihua Zhang∗ and Jin Li† Abstract. In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional L´evy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete compound Poisson subordinators. We exemplify four specific subordinators: Gamma, Poisson, negative binomial and squared Bessel subordi- nators. The Laplace exponents of the subordinators are Bernstein functions, so they can be used as sparsity-inducing nonconvex penalty functions. We exploit these subordinators in regression problems, yielding a hierarchical model with multiple regularization parameters. We devise ECME (Expectation/Conditional Maximization Either) algorithms to simultaneously estimate regression coefficients and regularization parameters. The empirical evaluation of simulated data shows that our approach is feasible and effective in high-dimensional data analysis. Keywords: nonconvex penalization, subordinators, latent shrinkage parameters, Bernstein functions, ECME algorithms. 1 Introduction Variable selection methods based on penalty theory have received great attention in high-dimensional data analysis. A principled approach is due to the lasso of Tibshirani (1996), which uses the ℓ1-norm penalty. Tibshirani (1996) also pointed out that the lasso estimate can be viewed as the mode of the posterior distribution. Indeed, the ℓ1 penalty can be transformed into the Laplace prior. Moreover, this prior can be expressed as a Gaussian scale mixture. This has thus led to Bayesian developments of the lasso and its variants (Figueiredo, 2003; Park and Casella, 2008; Hans, 2009; Kyung et al., 2010; Griffin and Brown, 2010; Li and Lin, 2010). -
Bayesian Nonparametric Modeling and Its Applications
UNIVERSITY OF TECHNOLOGY,SYDNEY DOCTORAL THESIS Bayesian Nonparametric Modeling and Its Applications By Minqi LI A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the School of Computing and Communications Faculty of Engineering and Information Technology October 2016 Declaration of Authorship I certify that the work in this thesis has not previously been submitted for a degree nor has it been submitted as part of requirements for a degree except as fully acknowledged within the text. I also certify that the thesis has been written by me. Any help that I have received in my research work and the preparation of the thesis itself has been acknowledged. In addition, I certify that all information sources and literature used are indicated in the thesis. Signed: Date: i ii Abstract Bayesian nonparametric methods (or nonparametric Bayesian methods) take the benefit of un- limited parameters and unbounded dimensions to reduce the constraints on the parameter as- sumption and avoid over-fitting implicitly. They have proven to be extremely useful due to their flexibility and applicability to a wide range of problems. In this thesis, we study the Baye- sain nonparametric theory with Levy´ process and completely random measures (CRM). Several Bayesian nonparametric techniques are presented for computer vision and pattern recognition problems. In particular, our research and contributions focus on the following problems. Firstly, we propose a novel example-based face hallucination method, based on a nonparametric Bayesian model with the assumption that all human faces have similar local pixel structures. We use distance dependent Chinese restaurant process (ddCRP) to cluster the low-resolution (LR) face image patches and give a matrix-normal prior for learning the mapping dictionaries from LR to the corresponding high-resolution (HR) patches. -
Math 550-700 Applied Probability and Statistics, Summer 2019
Math 550-700 Applied Probability and Statistics, Summer 2019 MTWTh, 9:00am to 11:45am, synchronous online (via Zoom) In Person, June 10th through June 13th, in Ross 3275 Student Hours: MTWTh 1:00pm to 5:00pm in Ross 2250D Instructor: Neil Hatfield Email: [email protected] Office Number: Ross 2250D Course Website: Canvas https://unco.instructure.com All Course Materials, homework assignments, and tests will be posted to the course website. Catalog Description: Methods related to descriptive and inferential statistics and the concept of probability are investigated in depth. Course Description: This course addresses the statistical processes of formulating questions, collecting and analyzing data, and interpreting results. Methods related to descriptive and inferential statistics and the concept of probability are investigated in depth. This course will act as a first course in the concepts and methods of statistics, with emphasis on data visualizations, the distribution conception, descriptive statistics, and statistical inference. Inference topics include parametric and non-parametric point and interval estimation, hypothesis testing, and measures of effect size. Prerequisites and Placement: Graduates only. Textbook and Reading Materials: • Required Textbook: None. • Required Readings: There will be several required readings; PDFs of these readings will be made available to you at no cost through the course website. These readings fall under Educational Fair Use guidelines. • Recommended Readings: A list of additional readings, sometimes including PDFS, will be posted to the course website. These readings will be optional (strongly recommended) and are meant to help enrich your experience in the course. These readings fall under Educational Fair Use guidelines. If you have a suggestion for the list, please share the details with the instructor.