Spatial Point Processes and Their Applications
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A J-Function for Marked Point Patterns
AISM (2006) 58: 235–259 DOI 10.1007/s10463-005-0015-7 M.N.M. van Lieshout A J -function for marked point patterns Received: 5 July 2004 / Revised: 7 February 2005 / Published online: 30 May 2006 © The Institute of Statistical Mathematics, Tokyo 2006 Abstract We propose a new summary statistic for marked point patterns. The underlying principle is to compare the distance from a marked point to the near- est other marked point in the pattern to the same distance seen from an arbitrary point in space. Information about the range of interaction can be inferred, and the statistic is well-behaved under random mark allocation. We develop a range of Hanisch style kernel estimators to tackle the problems of exploding tail variance earlier associated with J -function plug-in estimators, and carry out an exploratory analysis of a forestry data set. Keywords Empty space function · J -function · Marked point pattern · Mark correlation function · Nearest neighbour distance distribution function · Product density · Random labelling · Reduced second moment measure · Spatial interaction · Spatial statistics 1 Introduction Marked point patterns are spatial point configurations with a mark attached to each point (Stoyan and Stoyan, 1994). The points could represent the locations in (Euclidean) space of objects, while the marks capture additional information. The latter could be a type label, in which case we also speak of a multivariate point pattern (Cox and Lewis, 1972), a continuous measurement or shape descriptor, or a combination of these. The statistical analysis of such a pattern in general begins by plotting a few summary statistics. Which statistic is used depends on taste and the type of mark. -
Generalized Factorial Cumulants Applied to Coulomb-Blockade Systems Signal
Generalized factorial cumulants applied to Coulomb-blockade systems signal time Von der Fakultät für Physik der Universität Duisburg-Essen genehmigte Dissertation zur Erlangung des Grades Dr. rer. nat. von Philipp Stegmann aus Bottrop Tag der Disputation: 05.07.2017 Referent: Prof. Dr. Jürgen König Korreferent: Prof. Dr. Christian Flindt Korreferent: Prof. Dr. Thomas Guhr Summary Tunneling of electrons through a Coulomb-blockade system is a stochastic (i.e., random) process. The number of the transferred electrons per time interval is determined by a prob- ability distribution. The form of this distribution can be characterized by quantities called cumulants. Recently developed electrometers allow for the observation of each electron transported through a Coulomb-blockade system in real time. Therefore, the probability distribution can be directly measured. In this thesis, we introduce generalized factorial cumulants as a new tool to analyze the information contained in the probability distribution. For any kind of Coulomb-blockade system, these cumulants can be used as follows: First, correlations between the tunneling electrons are proven by a certain sign of the cumulants. In the limit of short time intervals, additional criteria indicate correlations, respectively. The cumulants allow for the detection of correlations which cannot be noticed by commonly used quantities such as the current noise. We comment in detail on the necessary ingredients for the presence of correlations in the short-time limit and thereby explain recent experimental observations. Second, we introduce a mathematical procedure called inverse counting statistics. The procedure reconstructs, solely from a few experimentally measured cumulants, character- istic features of an otherwise unknown Coulomb-blockade system, e.g., a lower bound for the system dimension and the full spectrum of relaxation rates. -
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. -
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. -
Completely Random Measures and Related Models
CRMs Sinead Williamson Background Completely random measures and related L´evyprocesses Completely models random measures Applications Normalized Sinead Williamson random measures Neutral-to-the- right processes Computational and Biological Learning Laboratory Exchangeable University of Cambridge matrices January 20, 2011 Outline CRMs Sinead Williamson 1 Background Background L´evyprocesses Completely 2 L´evyprocesses random measures Applications 3 Completely random measures Normalized random measures Neutral-to-the- right processes 4 Applications Exchangeable matrices Normalized random measures Neutral-to-the-right processes Exchangeable matrices A little measure theory CRMs Sinead Williamson Set: e.g. Integers, real numbers, people called James. Background May be finite, countably infinite, or uncountably infinite. L´evyprocesses Completely Algebra: Class T of subsets of a set T s.t. random measures 1 T 2 T . 2 If A 2 T , then Ac 2 T . Applications K Normalized 3 If A1;:::; AK 2 T , then [ Ak = A1 [ A2 [ ::: AK 2 T random k=1 measures (closed under finite unions). Neutral-to-the- right K processes 4 If A1;:::; AK 2 T , then \k=1Ak = A1 \ A2 \ ::: AK 2 T Exchangeable matrices (closed under finite intersections). σ-Algebra: Algebra that is closed under countably infinite unions and intersections. A little measure theory CRMs Sinead Williamson Background L´evyprocesses Measurable space: Combination (T ; T ) of a set and a Completely σ-algebra on that set. random measures Measure: Function µ between a σ-field and the positive Applications reals (+ 1) s.t. Normalized random measures 1 µ(;) = 0. Neutral-to-the- right 2 For all countable collections of disjoint sets processes P Exchangeable A1; A2; · · · 2 T , µ([k Ak ) = µ(Ak ). -
POISSON PROCESSES 1.1. the Rutherford-Chadwick-Ellis
POISSON PROCESSES 1. THE LAW OF SMALL NUMBERS 1.1. The Rutherford-Chadwick-Ellis Experiment. About 90 years ago Ernest Rutherford and his collaborators at the Cavendish Laboratory in Cambridge conducted a series of pathbreaking experiments on radioactive decay. In one of these, a radioactive substance was observed in N = 2608 time intervals of 7.5 seconds each, and the number of decay particles reaching a counter during each period was recorded. The table below shows the number Nk of these time periods in which exactly k decays were observed for k = 0,1,2,...,9. Also shown is N pk where k pk = (3.87) exp 3.87 =k! {− g The parameter value 3.87 was chosen because it is the mean number of decays/period for Rutherford’s data. k Nk N pk k Nk N pk 0 57 54.4 6 273 253.8 1 203 210.5 7 139 140.3 2 383 407.4 8 45 67.9 3 525 525.5 9 27 29.2 4 532 508.4 10 16 17.1 5 408 393.5 ≥ This is typical of what happens in many situations where counts of occurences of some sort are recorded: the Poisson distribution often provides an accurate – sometimes remarkably ac- curate – fit. Why? 1.2. Poisson Approximation to the Binomial Distribution. The ubiquity of the Poisson distri- bution in nature stems in large part from its connection to the Binomial and Hypergeometric distributions. The Binomial-(N ,p) distribution is the distribution of the number of successes in N independent Bernoulli trials, each with success probability p. -
The Q-Factorial Moments of Discrete Q-Distributions and a Characterization of the Euler Distribution
3 The q-Factorial Moments of Discrete q-Distributions and a Characterization of the Euler Distribution Ch. A. Charalambides and N. Papadatos Department of Mathematics, University of Athens, Athens, Greece ABSTRACT The classical discrete distributions binomial, geometric and negative binomial are defined on the stochastic model of a sequence of indepen- dent and identical Bernoulli trials. The Poisson distribution may be defined as an approximation of the binomial (or negative binomial) distribution. The cor- responding q-distributions are defined on the more general stochastic model of a sequence of Bernoulli trials with probability of success at any trial depending on the number of trials. In this paper targeting to the problem of calculating the moments of q-distributions, we introduce and study q-factorial moments, the calculation of which is as ease as the calculation of the factorial moments of the classical distributions. The usual factorial moments are connected with the q-factorial moments through the q-Stirling numbers of the first kind. Several ex- amples, illustrating the method, are presented. Further, the Euler distribution is characterized through its q-factorial moments. Keywords and phrases: q-distributions, q-moments, q-Stirling numbers 3.1 INTRODUCTION Consider a sequence of independent Bernoulli trials with probability of success at the ith trial pi, i =1, 2,.... The study of the distribution of the number Xn of successes up to the nth trial, as well as the closely related to it distribution of the number Yk of trials until the occurrence of the kth success, have attracted i−1 i−1 special attention. -
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. -
Functions of Random Variables
Names for Eg(X ) Generating Functions Topic 8 The Expected Value Functions of Random Variables 1 / 19 Names for Eg(X ) Generating Functions Outline Names for Eg(X ) Means Moments Factorial Moments Variance and Standard Deviation Generating Functions 2 / 19 Names for Eg(X ) Generating Functions Means If g(x) = x, then µX = EX is called variously the distributional mean, and the first moment. • Sn, the number of successes in n Bernoulli trials with success parameter p, has mean np. • The mean of a geometric random variable with parameter p is (1 − p)=p . • The mean of an exponential random variable with parameter β is1 /β. • A standard normal random variable has mean0. Exercise. Find the mean of a Pareto random variable. Z 1 Z 1 βαβ Z 1 αββ 1 αβ xf (x) dx = x dx = βαβ x−β dx = x1−β = ; β > 1 x β+1 −∞ α x α 1 − β α β − 1 3 / 19 Names for Eg(X ) Generating Functions Moments In analogy to a similar concept in physics, EX m is called the m-th moment. The second moment in physics is associated to the moment of inertia. • If X is a Bernoulli random variable, then X = X m. Thus, EX m = EX = p. • For a uniform random variable on [0; 1], the m-th moment is R 1 m 0 x dx = 1=(m + 1). • The third moment for Z, a standard normal random, is0. The fourth moment, 1 Z 1 z2 1 z2 1 4 4 3 EZ = p z exp − dz = −p z exp − 2π −∞ 2 2π 2 −∞ 1 Z 1 z2 +p 3z2 exp − dz = 3EZ 2 = 3 2π −∞ 2 3 z2 u(z) = z v(t) = − exp − 2 0 2 0 z2 u (z) = 3z v (t) = z exp − 2 4 / 19 Names for Eg(X ) Generating Functions Factorial Moments If g(x) = (x)k , where( x)k = x(x − 1) ··· (x − k + 1), then E(X )k is called the k-th factorial moment. -
Some Theory for the Analysis of Random Fields Diplomarbeit
Philipp Pluch Some Theory for the Analysis of Random Fields With Applications to Geostatistics Diplomarbeit zur Erlangung des akademischen Grades Diplom-Ingenieur Studium der Technischen Mathematik Universit¨at Klagenfurt Fakult¨at fur¨ Wirtschaftswissenschaften und Informatik Begutachter: O.Univ.-Prof.Dr. Jurgen¨ Pilz Institut fur¨ Mathematik September 2004 To my parents, Verena and all my friends Ehrenw¨ortliche Erkl¨arung Ich erkl¨are ehrenw¨ortlich, dass ich die vorliegende Schrift verfasst und die mit ihr unmittelbar verbundenen Arbeiten selbst durchgefuhrt¨ habe. Die in der Schrift verwendete Literatur sowie das Ausmaß der mir im gesamten Arbeitsvorgang gew¨ahrten Unterstutzung¨ sind ausnahmslos angegeben. Die Schrift ist noch keiner anderen Prufungsb¨ eh¨orde vorgelegt worden. St. Urban, 29 September 2004 Preface I remember when I first was at our univeristy - I walked inside this large corridor called ’Aula’ and had no idea what I should do, didn’t know what I should study, I had interest in Psychology or Media Studies, and now I’m sitting in my office at the university, five years later, writing my final lines for my master degree theses in mathematics. A long and also hard but so beautiful way was gone, I remember at the beginning, the first mathematic courses in discrete mathematics, how difficult that was for me, the abstract thinking, my first exams and now I have finished them all, I mastered them. I have to thank so many people and I will do so now. First I have to thank my parents, who always believed in me, who gave me financial support and who had to fight with my mood when I was working hard. -
Completely Random Measures
Pacific Journal of Mathematics COMPLETELY RANDOM MEASURES JOHN FRANK CHARLES KINGMAN Vol. 21, No. 1 November 1967 PACIFIC JOURNAL OF MATHEMATICS Vol. 21, No. 1, 1967 COMPLETELY RANDOM MEASURES J. F. C. KlNGMAN The theory of stochastic processes is concerned with random functions defined on some parameter set. This paper is con- cerned with the case, which occurs naturally in some practical situations, in which the parameter set is a ^-algebra of subsets of some space, and the random functions are all measures on this space. Among all such random measures are distinguished some which are called completely random, which have the property that the values they take on disjoint subsets are independent. A representation theorem is proved for all completely random measures satisfying a weak finiteness condi- tion, and as a consequence it is shown that all such measures are necessarily purely atomic. 1. Stochastic processes X(t) whose realisation are nondecreasing functions of a real parameter t occur in a number of applications of probability theory. For instance, the number of events of a point process in the interval (0, t), the 'load function' of a queue input [2], the amount of water entering a reservoir in time t, and the local time process in a Markov chain ([8], [7] §14), are all processes of this type. In many applications the function X(t) enters as a convenient way of representing a measure on the real line, the Stieltjes measure Φ defined as the unique Borel measure for which (1) Φ(a, b] = X(b + ) - X(a + ), (-co <α< b< oo).