Appendix: the Multivariate Normal Distribution
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Comparison of Harmonic, Geometric and Arithmetic Means for Change Detection in SAR Time Series Guillaume Quin, Béatrice Pinel-Puysségur, Jean-Marie Nicolas
Comparison of Harmonic, Geometric and Arithmetic means for change detection in SAR time series Guillaume Quin, Béatrice Pinel-Puysségur, Jean-Marie Nicolas To cite this version: Guillaume Quin, Béatrice Pinel-Puysségur, Jean-Marie Nicolas. Comparison of Harmonic, Geometric and Arithmetic means for change detection in SAR time series. EUSAR. 9th European Conference on Synthetic Aperture Radar, 2012., Apr 2012, Germany. hal-00737524 HAL Id: hal-00737524 https://hal.archives-ouvertes.fr/hal-00737524 Submitted on 2 Oct 2012 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. EUSAR 2012 Comparison of Harmonic, Geometric and Arithmetic Means for Change Detection in SAR Time Series Guillaume Quin CEA, DAM, DIF, F-91297 Arpajon, France Béatrice Pinel-Puysségur CEA, DAM, DIF, F-91297 Arpajon, France Jean-Marie Nicolas Telecom ParisTech, CNRS LTCI, 75634 Paris Cedex 13, France Abstract The amplitude distribution in a SAR image can present a heavy tail. Indeed, very high–valued outliers can be observed. In this paper, we propose the usage of the Harmonic, Geometric and Arithmetic temporal means for amplitude statistical studies along time. In general, the arithmetic mean is used to compute the mean amplitude of time series. -
University of Cincinnati
UNIVERSITY OF CINCINNATI Date:___________________ I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ Gibbs Sampling and Expectation Maximization Methods for Estimation of Censored Values from Correlated Multivariate Distributions A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial ful…llment of the requirements for the degree of DOCTORATE OF PHILOSOPHY (Ph.D.) in the Department of Mathematical Sciences of the McMicken College of Arts and Sciences May 2008 by Tina D. Hunter B.S. Industrial and Systems Engineering The Ohio State University, Columbus, Ohio, 1984 M.S. Aerospace Engineering University of Cincinnati, Cincinnati, Ohio, 1989 M.S. Statistics University of Cincinnati, Cincinnati, Ohio, 2003 Committee Chair: Dr. Siva Sivaganesan Abstract Statisticians are often called upon to analyze censored data. Environmental and toxicological data is often left-censored due to reporting practices for mea- surements that are below a statistically de…ned detection limit. Although there is an abundance of literature on univariate methods for analyzing this type of data, a great need still exists for multivariate methods that take into account possible correlation amongst variables. Two methods are developed here for that purpose. One is a Markov Chain Monte Carlo method that uses a Gibbs sampler to es- timate censored data values as well as distributional and regression parameters. The second is an expectation maximization (EM) algorithm that solves for the distributional parameters that maximize the complete likelihood function in the presence of censored data. -
Moments of the Product and Ratio of Two Correlated Chi-Square Variables
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Stat Papers (2009) 50:581–592 DOI 10.1007/s00362-007-0105-0 REGULAR ARTICLE Moments of the product and ratio of two correlated chi-square variables Anwar H. Joarder Received: 2 June 2006 / Revised: 8 October 2007 / Published online: 20 November 2007 © The Author(s) 2007 Abstract The exact probability density function of a bivariate chi-square distribu- tion with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such applica- tion is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution. Keywords Bivariate chi-square distribution · Moments · Product of correlated chi-square variables · Ratio of correlated chi-square variables Mathematics Subject Classification (2000) 62E15 · 60E05 · 60E10 1 Introduction Fisher (1915) derived the distribution of mean-centered sum of squares and sum of products in order to study the distribution of correlation coefficient from a bivariate nor- mal sample. Let X1, X2,...,X N (N > 2) be two-dimensional independent random vectors where X j = (X1 j , X2 j ) , j = 1, 2,...,N is distributed as a bivariate normal distribution denoted by N2(θ, ) with θ = (θ1,θ2) and a 2 × 2 covariance matrix = (σik), i = 1, 2; k = 1, 2. The sample mean-centered sums of squares and sum of products are given by a = N (X − X¯ )2 = mS2, m = N − 1,(i = 1, 2) ii j=1 ij i i = N ( − ¯ )( − ¯ ) = and a12 j=1 X1 j X1 X2 j X2 mRS1 S2, respectively. -
Discrete Probability Distributions Uniform Distribution Bernoulli
Discrete Probability Distributions Uniform Distribution Experiment obeys: all outcomes equally probable Random variable: outcome Probability distribution: if k is the number of possible outcomes, 1 if x is a possible outcome p(x)= k ( 0 otherwise Example: tossing a fair die (k = 6) Bernoulli Distribution Experiment obeys: (1) a single trial with two possible outcomes (success and failure) (2) P trial is successful = p Random variable: number of successful trials (zero or one) Probability distribution: p(x)= px(1 − p)n−x Mean and variance: µ = p, σ2 = p(1 − p) Example: tossing a fair coin once Binomial Distribution Experiment obeys: (1) n repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: number of successful trials n x n−x Probability distribution: b(x; n,p)= x p (1 − p) Mean and variance: µ = np, σ2 = np(1 − p) Example: tossing a fair coin n times Approximations: (1) b(x; n,p) ≈ p(x; λ = pn) if p ≪ 1, x ≪ n (Poisson approximation) (2) b(x; n,p) ≈ n(x; µ = pn,σ = np(1 − p) ) if np ≫ 1, n(1 − p) ≫ 1 (Normal approximation) p Geometric Distribution Experiment obeys: (1) indeterminate number of repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: trial number of first successful trial Probability distribution: p(x)= p(1 − p)x−1 1 2 1−p Mean and variance: µ = p , σ = p2 Example: repeated attempts to start -
Chapter 5 Sections
Chapter 5 Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions Continuous univariate distributions: 5.6 Normal distributions 5.7 Gamma distributions Just skim 5.8 Beta distributions Multivariate distributions Just skim 5.9 Multinomial distributions 5.10 Bivariate normal distributions 1 / 43 Chapter 5 5.1 Introduction Families of distributions How: Parameter and Parameter space pf /pdf and cdf - new notation: f (xj parameters ) Mean, variance and the m.g.f. (t) Features, connections to other distributions, approximation Reasoning behind a distribution Why: Natural justification for certain experiments A model for the uncertainty in an experiment All models are wrong, but some are useful – George Box 2 / 43 Chapter 5 5.2 Bernoulli and Binomial distributions Bernoulli distributions Def: Bernoulli distributions – Bernoulli(p) A r.v. X has the Bernoulli distribution with parameter p if P(X = 1) = p and P(X = 0) = 1 − p. The pf of X is px (1 − p)1−x for x = 0; 1 f (xjp) = 0 otherwise Parameter space: p 2 [0; 1] In an experiment with only two possible outcomes, “success” and “failure”, let X = number successes. Then X ∼ Bernoulli(p) where p is the probability of success. E(X) = p, Var(X) = p(1 − p) and (t) = E(etX ) = pet + (1 − p) 8 < 0 for x < 0 The cdf is F(xjp) = 1 − p for 0 ≤ x < 1 : 1 for x ≥ 1 3 / 43 Chapter 5 5.2 Bernoulli and Binomial distributions Binomial distributions Def: Binomial distributions – Binomial(n; p) A r.v. -
Arcsine Laws for Random Walks Generated from Random Permutations with Applications to Genomics
Applied Probability Trust (4 February 2021) ARCSINE LAWS FOR RANDOM WALKS GENERATED FROM RANDOM PERMUTATIONS WITH APPLICATIONS TO GENOMICS XIAO FANG,1 The Chinese University of Hong Kong HAN LIANG GAN,2 Northwestern University SUSAN HOLMES,3 Stanford University HAIYAN HUANG,4 University of California, Berkeley EROL PEKOZ,¨ 5 Boston University ADRIAN ROLLIN,¨ 6 National University of Singapore WENPIN TANG,7 Columbia University 1 Email address: [email protected] 2 Email address: [email protected] 3 Email address: [email protected] 4 Email address: [email protected] 5 Email address: [email protected] 6 Email address: [email protected] 7 Email address: [email protected] 1 Postal address: Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong 2 Postal address: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208 2 Current address: University of Waikato, Private Bag 3105, Hamilton 3240, New Zealand 1 2 X. Fang et al. Abstract A classical result for the simple symmetric random walk with 2n steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows(q) permutation and show that they have the same asymptotic distributions as for the simple random walk. -
A Multivariate Student's T-Distribution
Open Journal of Statistics, 2016, 6, 443-450 Published Online June 2016 in SciRes. http://www.scirp.org/journal/ojs http://dx.doi.org/10.4236/ojs.2016.63040 A Multivariate Student’s t-Distribution Daniel T. Cassidy Department of Engineering Physics, McMaster University, Hamilton, ON, Canada Received 29 March 2016; accepted 14 June 2016; published 17 June 2016 Copyright © 2016 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract A multivariate Student’s t-distribution is derived by analogy to the derivation of a multivariate normal (Gaussian) probability density function. This multivariate Student’s t-distribution can have different shape parameters νi for the marginal probability density functions of the multi- variate distribution. Expressions for the probability density function, for the variances, and for the covariances of the multivariate t-distribution with arbitrary shape parameters for the marginals are given. Keywords Multivariate Student’s t, Variance, Covariance, Arbitrary Shape Parameters 1. Introduction An expression for a multivariate Student’s t-distribution is presented. This expression, which is different in form than the form that is commonly used, allows the shape parameter ν for each marginal probability density function (pdf) of the multivariate pdf to be different. The form that is typically used is [1] −+ν Γ+((ν n) 2) T ( n) 2 +Σ−1 n 2 (1.[xx] [ ]) (1) ΓΣ(νν2)(π ) This “typical” form attempts to generalize the univariate Student’s t-distribution and is valid when the n marginal distributions have the same shape parameter ν . -
Incorporating a Geometric Mean Formula Into The
Calculating the CPI Incorporating a geometric mean formula into the CPI Beginning in January 1999, a new geometric mean formula will replace the current Laspeyres formula in calculating most basic components of the Consumer Price Index; the new formula will better account for the economic substitution behavior of consumers 2 Kenneth V. Dalton, his article describes an important improve- bias” in the CPI. This upward bias was a techni- John S. Greenlees, ment in the calculation of the Consumer cal problem that tied the weight of a CPI sample and TPrice Index (CPI). The Bureau of Labor Sta- item to its expected price change. The flaw was Kenneth J. Stewart tistics plans to use a new geometric mean for- effectively eliminated by changes to the CPI mula for calculating most of the basic compo- sample rotation and substitution procedures and nents of the Consumer Price Index for all Urban to the functional form used to calculate changes Consumers (CPI-U) and the Consumer Price In- in the cost of shelter for homeowners. In 1997, a dex for Urban Wage Earners and Clerical Work- new approach to the measurement of consumer ers (CPI-W). This change will become effective prices for hospital services was introduced.3 Pric- with data for January 1999.1 ing procedures were revised, from pricing indi- The geometric mean formula will be used in vidual items (such as a unit of blood or a hospi- index categories that make up approximately 61 tal inpatient day) to pricing the combined sets of percent of total consumer spending represented goods and services provided on selected patient by the CPI-U. -
Wavelet Operators and Multiplicative Observation Models
Wavelet Operators and Multiplicative Observation Models - Application to Change-Enhanced Regularization of SAR Image Time Series Abdourrahmane Atto, Emmanuel Trouvé, Jean-Marie Nicolas, Thu Trang Le To cite this version: Abdourrahmane Atto, Emmanuel Trouvé, Jean-Marie Nicolas, Thu Trang Le. Wavelet Operators and Multiplicative Observation Models - Application to Change-Enhanced Regularization of SAR Image Time Series. 2016. hal-00950823v3 HAL Id: hal-00950823 https://hal.archives-ouvertes.fr/hal-00950823v3 Preprint submitted on 26 Jan 2016 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. 1 Wavelet Operators and Multiplicative Observation Models - Application to Change-Enhanced Regularization of SAR Image Time Series Abdourrahmane M. Atto1;∗, Emmanuel Trouve1, Jean-Marie Nicolas2, Thu-Trang Le^1 Abstract|This paper first provides statistical prop- I. Introduction - Motivation erties of wavelet operators when the observation model IGHLY resolved data such as Synthetic Aperture can be seen as the product of a deterministic piece- Radar (SAR) image time series issued from new wise regular function (signal) and a stationary random H field (noise). This multiplicative observation model is generation sensors show minute details. Indeed, the evo- analyzed in two standard frameworks by considering lution of SAR imaging systems is such that in less than 2 either (1) a direct wavelet transform of the model decades: or (2) a log-transform of the model prior to wavelet • high resolution sensors can achieve metric resolution, decomposition. -
A Guide on Probability Distributions
powered project A guide on probability distributions R-forge distributions Core Team University Year 2008-2009 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic ditribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 132 13 Misc 134 Conclusion 135 Bibliography 135 A Mathematical tools 138 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗). -
Free Infinite Divisibility and Free Multiplicative Mixtures of the Wigner Distribution
FREE INFINITE DIVISIBILITY AND FREE MULTIPLICATIVE MIXTURES OF THE WIGNER DISTRIBUTION Victor Pérez-Abreu and Noriyoshi Sakuma Comunicación del CIMAT No I-09-0715/ -10-2009 ( PE/CIMAT) Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution Victor P´erez-Abreu∗ Department of Probability and Statistics, CIMAT Apdo. Postal 402, Guanajuato Gto. 36000, Mexico [email protected] Noriyoshi Sakumay Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Yokohama 223-8522, Japan. [email protected] March 19, 2010 Abstract Let I∗ and I be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici-Pata bijection between I∗ and I: The class type W of symmetric distributions in I that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between sym- metric distributions in I and the free counterpart under Λ of the positive distributions in I∗ is established. It is shown that the class type W does not include all symmetric distributions in I and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I∗. Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented. AMS 2000 Subject Classification: 46L54, 15A52. Keywords: Free convolutions, type G law, free stable law, free compound distribution, Bercovici-Pata bijection. -
Location-Scale Distributions
Location–Scale Distributions Linear Estimation and Probability Plotting Using MATLAB Horst Rinne Copyright: Prof. em. Dr. Horst Rinne Department of Economics and Management Science Justus–Liebig–University, Giessen, Germany Contents Preface VII List of Figures IX List of Tables XII 1 The family of location–scale distributions 1 1.1 Properties of location–scale distributions . 1 1.2 Genuine location–scale distributions — A short listing . 5 1.3 Distributions transformable to location–scale type . 11 2 Order statistics 18 2.1 Distributional concepts . 18 2.2 Moments of order statistics . 21 2.2.1 Definitions and basic formulas . 21 2.2.2 Identities, recurrence relations and approximations . 26 2.3 Functions of order statistics . 32 3 Statistical graphics 36 3.1 Some historical remarks . 36 3.2 The role of graphical methods in statistics . 38 3.2.1 Graphical versus numerical techniques . 38 3.2.2 Manipulation with graphs and graphical perception . 39 3.2.3 Graphical displays in statistics . 41 3.3 Distribution assessment by graphs . 43 3.3.1 PP–plots and QQ–plots . 43 3.3.2 Probability paper and plotting positions . 47 3.3.3 Hazard plot . 54 3.3.4 TTT–plot . 56 4 Linear estimation — Theory and methods 59 4.1 Types of sampling data . 59 IV Contents 4.2 Estimators based on moments of order statistics . 63 4.2.1 GLS estimators . 64 4.2.1.1 GLS for a general location–scale distribution . 65 4.2.1.2 GLS for a symmetric location–scale distribution . 71 4.2.1.3 GLS and censored samples .