Negative Binomial Process Count and Mixture Modeling
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 ). -
STOCHASTIC COMPARISONS and AGING PROPERTIES of an EXTENDED GAMMA PROCESS Zeina Al Masry, Sophie Mercier, Ghislain Verdier
STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS Zeina Al Masry, Sophie Mercier, Ghislain Verdier To cite this version: Zeina Al Masry, Sophie Mercier, Ghislain Verdier. STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS. Journal of Applied Probability, Cambridge University press, 2021, 58 (1), pp.140-163. 10.1017/jpr.2020.74. hal-02894591 HAL Id: hal-02894591 https://hal.archives-ouvertes.fr/hal-02894591 Submitted on 9 Jul 2020 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. Applied Probability Trust (4 July 2020) STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS ZEINA AL MASRY,∗ FEMTO-ST, Univ. Bourgogne Franche-Comt´e,CNRS, ENSMM SOPHIE MERCIER & GHISLAIN VERDIER,∗∗ Universite de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, Pau, France Abstract Extended gamma processes have been seen to be a flexible extension of standard gamma processes in the recent reliability literature, for cumulative deterioration modeling purpose. The probabilistic properties of the standard gamma process have been well explored since the 1970's, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and aging properties for the corresponding level-crossing times are nowadays well understood. -
Benford's Law As a Logarithmic Transformation
Benford’s Law as a Logarithmic Transformation J. M. Pimbley Maxwell Consulting, LLC Abstract We review the history and description of Benford’s Law - the observation that many naturally occurring numerical data collections exhibit a logarithmic distribution of digits. By creating numerous hypothetical examples, we identify several cases that satisfy Benford’s Law and several that do not. Building on prior work, we create and demonstrate a “Benford Test” to determine from the functional form of a probability density function whether the resulting distribution of digits will find close agreement with Benford’s Law. We also discover that one may generalize the first-digit distribution algorithm to include non-logarithmic transformations. This generalization shows that Benford’s Law rests on the tendency of the transformed and shifted data to exhibit a uniform distribution. Introduction What the world calls “Benford’s Law” is a marvelous mélange of mathematics, data, philosophy, empiricism, theory, mysticism, and fraud. Even with all these qualifiers, one easily describes this law and then asks the simple questions that have challenged investigators for more than a century. Take a large collection of positive numbers which may be integers or real numbers or both. Focus only on the first (non-zero) digit of each number. Count how many numbers of the large collection have each of the nine possibilities (1-9) as the first digit. For typical J. M. Pimbley, “Benford’s Law as a Logarithmic Transformation,” Maxwell Consulting Archives, 2014. number collections – which we’ll generally call “datasets” – the first digit is not equally distributed among the values 1-9. -
Hyperwage Theory
Hyperwage Theory About the cover The cover is a glowing red-hot bark of a tree embedded with the greatest equations of nature that tremendously impacted the thought of mankind and the course of civilization. These are the Maxwell’s electromagnetic field equations, the Navier-Stokes Theorem, Euler’s identity, Newton’s second law of motion, Newton’s law of gravitation, the ideal gas equation of state, Stefan’s law, the Second Law of Thermodynamics, Einstein’s mass-energy equivalence, Einstein’s gravity tensor equation, Lorenz time dilation, Planck’s Law, Heisenberg’s Uncertainty Principle, Schrodinger’s equation, Shannon’s Theorem, and Feynman diagrams in quantum electrodynamics. Concept and design by Thads Bentulan Hyperwage Theory The Misadventures of the Street Strategist Volume 10 Thads Bentulan Street Strategist First Edition 2008 Street Strategist Publications Limited Hyperwage Theory The Misadventures of the Street Strategist Volume 10 by Thads Bentulan ISBN 978-988-17536-9-4 This book is a compilation of articles that first appeared in BusinessWorld, unless otherwise noted. While the author endeavored to credit sources whenever possible, he welcomes corrections for any unacknowledged material. No part of this book may be reproduced, stored, or utilized in any form or by any means, electronic or mechanical, including photocopying, or recording, or by any information storage and retrieval system, including internet websites, without prior written permission. Copyright 2005-2008 by Thads Bentulan [email protected] 08/24/09 -
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 → ∞. -
The 7Th Workshop on MARKOV PROCESSES and RELATED TOPICS
The 7th Workshop on MARKOV PROCESSES AND RELATED TOPICS July 19 - 23, 2010 No.6 Lecture room on 3th floor, Jingshi Building (京京京师师师大大大厦厦厦) Beijing Normal University Organizers: Mu-Fa Chen, Zeng-Hu Li, Feng-Yu Wang Supported by National Natural Science Foundation of China (No. 10721091) Probability Group, Stochastic Research Center School of Mathematical Sciences, Beijing Normal University No.19, Xinjiekouwai St., Beijing 100875, China Phone and Fax: 86-10-58809447 E-mail: [email protected] Website: http://math.bnu.edu.cn/probab/Workshop2010 Schedule 1 July 19 July 20 July 21 July 22 July 23 Chairman Mu-Fa Chen Fu-Zhou Gong Shui Feng Jie Xiong Dayue Chen Opening Shuenn-Jyi Sheu Anyue Chen Leonid Mytnik Zengjing Chen 8:30{8:40 8:30{9:00 8:30{9:00 8:30{9:00 8:30{9:00 M. Fukushima Feng-Yu Wang Jiashan Tang Quansheng Liu Dong Han 8:40{9:30 9:00{9:30 9:00{9:30 9:00{9:30 9:00{9:30 Speaker Tea Break Zhen-Qing Chen Renming Song Xia Chen Hao Wang Zongxia Liang 10:00{10:30 10:00{10:30 10:00{10:30 10:00{10:305 10:00{10:30 Panki Kim Christian Leonard Yutao Ma Xiaowen Zhou Fubao Xi 10:30{11:00 10:30{11:00 10:30{11:00 10:30{11:00 10:30{11:00 Jinghai Shao Xu Zhang Liang-Hui Xia Hui He Liqun Niu 11:00{11:30 11:00{11:20 11:00{11:20 11:00{11:20 11:00{11:20 Lunch Chairman Feng-Yu Wang Shizan Fang Zengjing Chen Zeng-Hu Li Chii-Ruey Hwang Tusheng Zhang Dayue Chen Shizan Fang 14:30{15:00 14:30{15:00 14:30{15:00 14:30{15:00 Ivan Gentil Xiang-Dong Li Alok Goswami Yimin Xiao 15:00{15:30 15:00{15:30 15:00{15:30 15:00{15:30 Speaker Tea Break Fu-Zhou Gong Jinwen -
1988: Logarithmic Series Distribution and Its Use In
LOGARITHMIC SERIES DISTRIBUTION AND ITS USE IN ANALYZING DISCRETE DATA Jeffrey R, Wilson, Arizona State University Tempe, Arizona 85287 1. Introduction is assumed to be 7. for any unit and any trial. In a previous paper, Wilson and Koehler Furthermore, for ~ach unit the m trials are (1988) used the generalized-Dirichlet identical and independent so upon summing across multinomial model to account for extra trials X. = (XI~. X 2 ...... X~.)', the vector of variation. The model allows for a second order counts ~r the j th ~nit has ~3multinomial of pairwise correlation among units, a type of distribution with probability vector ~ = (~1' assumption found reasonable in some biological 72 ..... 71) ' and sample size m. Howe~er, data. In that paper, the two-way crossed responses given by the J units at a particular generalized Dirichlet Multinomial model was used trial may be correlated, producing a set of J to analyze repeated measure on the categorical correlated multinomial random vectors, X I, X 2, preferences of insurance customers. The number • o.~ X • of respondents was assumed to be fixed and Ta~is (1962) developed a model for this known. situation, which he called the generalized- In this paper a generalization of the model multinomial distribution in which a single is made allowing the number of respondents m, to parameter P, is used to reflect the common be random. Thus both the number of units m, and dependency between any two of the dependent the underlying probability vector are allowed to multinomial random vectors. The distribution of vary. The model presented here uses the m logarithmic series distribution to account for the category total Xi3. -
Notes on Scale-Invariance and Base-Invariance for Benford's
NOTES ON SCALE-INVARIANCE AND BASE-INVARIANCE FOR BENFORD’S LAW MICHAŁ RYSZARD WÓJCIK Abstract. It is known that if X is uniformly distributed modulo 1 and Y is an arbitrary random variable independent of X then Y + X is also uniformly distributed modulo 1. We prove a converse for any continuous random variable Y (or a reasonable approximation to a continuous random variable) so that if X and Y +X are equally distributed modulo 1 and Y is independent of X then X is uniformly distributed modulo 1 (or approximates the uniform distribution equally reasonably). This translates into a characterization of Benford’s law through a generalization of scale-invariance: from multiplication by a constant to multiplication by an independent random variable. We also show a base-invariance characterization: if a positive continuous random variable has the same significand distribution for two bases then it is Benford for both bases. The set of bases for which a random variable is Benford is characterized through characteristic functions. 1. Introduction Before the early 1970s, handheld electronic calculators were not yet in widespread use and scientists routinely used in their calculations books with tables containing the decimal logarithms of numbers between 1 and 10 spaced evenly with small increments like 0.01 or 0.001. For example, the first page would be filled with numbers 1.01, 1.02, 1.03, . , 1.99 in the left column and their decimal logarithms in the right column, while the second page with 2.00, 2.01, 2.02, 2.03, ..., 2.99, and so on till the ninth page with 9.00, 9.01, 9.02, 9.03, . -
Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality
Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality Nikolai N. Leonenko School of Mathematics Cardiff University The second conference on Ambit Fields and Related Topics Aarhus August 15 Abstract Superpositions of stationary processes of Ornstein-Uhlenbeck (supOU) type have been introduced by Barndorff-Nielsen. We consider the constructions producing processes with long-range dependence and infinitely divisible marginal distributions. We consider additive functionals of supOU processes that satisfy the property referred to as intermittency. We investigate the properties of multifractal products of supOU processes. We present the general conditions for the Lq convergence of cumulative processes and investigate their q-th order moments and R´enyifunctions. These functions are nonlinear, hence displaying the multifractality of the processes. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios. Acknowledgments Joint work with Denis Denisov, University of Manchester, UK Danijel Grahovac, University of Osijek, Croatia Alla Sikorskii, Michigan State University, USA Murad Taqqu, Boston University, USA OU type process I OU Type process is the unique strong solution of the SDE: dX (t) = −λX (t)dt + dZ(λt) where λ > 0, fZ(t)gt≥0 is a (non-decreasing, for this talk) L´eviprocess, and an initial condition X0 is taken to be independent of Z(t). Note, in general Zt doesn't have to be a non-decreasing L´evyprocess. I For properties of OU type processes and their generalizations see Mandrekar & Rudiger (2007), Barndorff-Nielsen (2001), Barndorff-Nielsen & Stelzer (2011). I The a.s. -
Package 'Distributional'
Package ‘distributional’ February 2, 2021 Title Vectorised Probability Distributions Version 0.2.2 Description Vectorised distribution objects with tools for manipulating, visualising, and using probability distributions. Designed to allow model prediction outputs to return distributions rather than their parameters, allowing users to directly interact with predictive distributions in a data-oriented workflow. In addition to providing generic replacements for p/d/q/r functions, other useful statistics can be computed including means, variances, intervals, and highest density regions. License GPL-3 Imports vctrs (>= 0.3.0), rlang (>= 0.4.5), generics, ellipsis, stats, numDeriv, ggplot2, scales, farver, digest, utils, lifecycle Suggests testthat (>= 2.1.0), covr, mvtnorm, actuar, ggdist RdMacros lifecycle URL https://pkg.mitchelloharawild.com/distributional/, https: //github.com/mitchelloharawild/distributional BugReports https://github.com/mitchelloharawild/distributional/issues Encoding UTF-8 Language en-GB LazyData true Roxygen list(markdown = TRUE, roclets=c('rd', 'collate', 'namespace')) RoxygenNote 7.1.1 1 2 R topics documented: R topics documented: autoplot.distribution . .3 cdf..............................................4 density.distribution . .4 dist_bernoulli . .5 dist_beta . .6 dist_binomial . .7 dist_burr . .8 dist_cauchy . .9 dist_chisq . 10 dist_degenerate . 11 dist_exponential . 12 dist_f . 13 dist_gamma . 14 dist_geometric . 16 dist_gumbel . 17 dist_hypergeometric . 18 dist_inflated . 20 dist_inverse_exponential . 20 dist_inverse_gamma -
Nonparametric Bayesian Latent Factor Models for Network Reconstruction
NONPARAMETRIC BAYESIAN LATENT FACTOR MODELS FORNETWORKRECONSTRUCTION Dem Fachbereich Elektrotechnik und Informationstechnik der TECHNISCHE UNIVERSITÄT DARMSTADT zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) vorgelegte Dissertation von SIKUN YANG, M.PHIL. Referent: Prof. Dr. techn. Heinz Köppl Korreferent: Prof. Dr. Kristian Kersting Darmstadt 2019 The work of Sikun Yang was supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement 668858 at the Technische Universsität Darmstadt, Germany. Yang, Sikun : Nonparametric Bayesian Latent Factor Models for Network Reconstruction, Darmstadt, Technische Universität Darmstadt Jahr der Veröffentlichung der Dissertation auf TUprints: 2019 URN: urn:nbn:de:tuda-tuprints-96957 Tag der mündlichen Prüfung: 11.12.2019 Veröffentlicht unter CC BY-SA 4.0 International https://creativecommons.org/licenses/by-sa/4.0/ ABSTRACT This thesis is concerned with the statistical learning of probabilistic models for graph-structured data. It addresses both the theoretical aspects of network modelling–like the learning of appropriate representations for networks–and the practical difficulties in developing the algorithms to perform inference for the proposed models. The first part of the thesis addresses the problem of discrete-time dynamic network modeling. The objective is to learn the common structure and the underlying interaction dynamics among the entities involved in the observed temporal network. Two probabilistic modeling frameworks are developed. First, a Bayesian nonparametric framework is proposed to capture the static latent community structure and the evolving node-community memberships over time. More specifi- cally, the hierarchical gamma process is utilized to capture the underlying intra-community and inter-community interactions. The appropriate number of latent communities can be automatically estimated via the inherent shrinkage mechanism of the hierarchical gamma process prior. -
Option Pricing in Bilateral Gamma Stock Models
OPTION PRICING IN BILATERAL GAMMA STOCK MODELS UWE KUCHLER¨ AND STEFAN TAPPE Abstract. In the framework of bilateral Gamma stock models we seek for adequate option pricing measures, which have an economic interpretation and allow numerical calculations of option prices. Our investigations encompass Es- scher transforms, minimal entropy martingale measures, p-optimal martingale measures, bilateral Esscher transforms and the minimal martingale measure. We illustrate our theory by a numerical example. Key Words: Bilateral Gamma stock model, bilateral Esscher transform, minimal martingale measure, option pricing. 91G20, 60G51 1. Introduction An issue in continuous time finance is to find realistic and analytically tractable models for price evolutions of risky financial assets. In this text, we consider expo- nential L´evymodels Xt St = S0e (1.1) rt Bt = e consisting of two financial assets (S; B), one dividend paying stock S with dividend rate q ≥ 0, where X denotes a L´evyprocess and one risk free asset B, the bank account with fixed interest rate r ≥ 0. Note that the classical Black-Scholes model σ2 is a special case by choosing Xt = σWt + (µ − 2 )t, where W is a Wiener process, µ 2 R denotes the drift and σ > 0 the volatility. Although the Black-Scholes model is a standard model in financial mathematics, it is well-known that it only provides a poor fit to observed market data, because typically the empirical densities possess heavier tails and much higher located modes than fitted normal distributions. Several authors therefore suggested more sophisticated L´evyprocesses with a jump component. We mention the Variance Gamma processes [30, 31], hyperbolic processes [8], normal inverse Gaussian processes [1], generalized hyperbolic pro- cesses [9, 10, 34], CGMY processes [3] and Meixner processes [39].