Solutions of the Multivariate Inverse Frobenius–Perron Problem
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Deterministic Brownian Motion: the Effects of Perturbing a Dynamical
1 Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System Michael C. Mackey∗and Marta Tyran-Kami´nska † October 26, 2018 Abstract Here we review and extend central limit theorems for highly chaotic but deterministic semi- dynamical discrete time systems. We then apply these results show how Brownian motion-like results are recovered, and how an Ornstein-Uhlenbeck process results within a totally determin- istic framework. These results illustrate that the contamination of experimental data by “noise” may, under certain circumstances, be alternately interpreted as the signature of an underlying chaotic process. Contents 1 Introduction 2 2 Semi-dynamical systems 5 2.1 Density evolution operators . ......... 5 2.2 Probabilistic and ergodic properties of density evolution ................ 12 2.3 Brownian motion from deterministic perturbations . ............... 16 2.3.1 Centrallimittheorems. ..... 16 2.3.2 FCLTfornoninvertiblemaps . ..... 20 2.4 Weak convergence criteria . ........ 31 3 Analysis 33 3.1 Weak convergence of v(tn) and vn. ............................ 35 3.2 Thelinearcaseinonedimension . ........ 36 3.2.1 Behaviour of the velocity variable . ......... 36 3.2.2 Behaviour of the position variable . ........ 38 4 Identifying the Limiting Velocity Distribution 41 4.1 Dyadicmap....................................... 42 arXiv:cond-mat/0408330v1 [cond-mat.stat-mech] 13 Aug 2004 4.2 Graphical illustration of the velocity density evolution with dyadic map perturbations 45 4.3 r-dyadicmap ..................................... 45 ∗e-mail: [email protected], Departments of Physiology, Physics & Mathematics and Centre for Nonlinear Dynamics, McGill University, 3655 Promenade Sir William Osler, Montreal, QC, CANADA, H3G 1Y6 †Corresponding author, email: [email protected], Institute of Mathematics, Silesian University, ul. -
A Tail Quantile Approximation Formula for the Student T and the Symmetric Generalized Hyperbolic Distribution
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Schlüter, Stephan; Fischer, Matthias J. Working Paper A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution IWQW Discussion Papers, No. 05/2009 Provided in Cooperation with: Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics Suggested Citation: Schlüter, Stephan; Fischer, Matthias J. (2009) : A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution, IWQW Discussion Papers, No. 05/2009, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung (IWQW), Nürnberg This Version is available at: http://hdl.handle.net/10419/29554 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu IWQW Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung Diskussionspapier Discussion Papers No. -
A Pseudo Random Numbers Generator Based on Chaotic Iterations
A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking Christophe Guyeux, Qianxue Wang, Jacques Bahi To cite this version: Christophe Guyeux, Qianxue Wang, Jacques Bahi. A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking. WISM 2010, Int. Conf. on Web Information Systems and Mining, 2010, China. pp.202–211. hal-00563317 HAL Id: hal-00563317 https://hal.archives-ouvertes.fr/hal-00563317 Submitted on 4 Feb 2011 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. A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking Christophe Guyeux, Qianxue Wang, and Jacques M. Bahi University of Franche-Comte, Computer Science Laboratory LIFC, 25030 Besanc¸on Cedex, France {christophe.guyeux,qianxue.wang, jacques.bahi}@univ-fcomte.fr Abstract. In this paper, a new chaotic pseudo-random number generator (PRNG) is proposed. It combines the well-known ISAAC and XORshift generators with chaotic iterations. This PRNG possesses important properties of topological chaos and can successfully pass NIST and TestU01 batteries of tests. This makes our generator suitable for information security applications like cryptography. As an illustrative example, an application in the field of watermarking is presented. -
Algorithms for Operations on Probability Distributions in a Computer Algebra System
W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 2001 Algorithms for operations on probability distributions in a computer algebra system Diane Lynn Evans College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Evans, Diane Lynn, "Algorithms for operations on probability distributions in a computer algebra system" (2001). Dissertations, Theses, and Masters Projects. Paper 1539623382. https://dx.doi.org/doi:10.21220/s2-bath-8582 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproductionFurther reproduction prohibited without prohibited permission. without permission. ALGORITHMS FOR OPERATIONS ON PROBABILITY DISTRIBUTIONS IN A COMPUTER ALGEBRA SYSTEM A Dissertation Presented to The Faculty of the Department of Applied Science The College of William & Mary in Virginia In Partial Fulfillment Of the Requirements for the Degree of Doctor of Philosophy by Diane Lynn Evans July 2001 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3026405 Copyright 2001 by Evans, Diane Lynn All rights reserved. ___ ® UMI UMI Microform 3026405 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. -
Generating Random Samples from User-Defined Distributions
The Stata Journal (2011) 11, Number 2, pp. 299–304 Generating random samples from user-defined distributions Katar´ına Luk´acsy Central European University Budapest, Hungary lukacsy [email protected] Abstract. Generating random samples in Stata is very straightforward if the distribution drawn from is uniform or normal. With any other distribution, an inverse method can be used; but even in this case, the user is limited to the built- in functions. For any other distribution functions, their inverse must be derived analytically or numerical methods must be used if analytical derivation of the inverse function is tedious or impossible. In this article, I introduce a command that generates a random sample from any user-specified distribution function using numeric methods that make this command very generic. Keywords: st0229, rsample, random sample, user-defined distribution function, inverse method, Monte Carlo exercise 1 Introduction In statistics, a probability distribution identifies the probability of a random variable. If the random variable is discrete, it identifies the probability of each value of this vari- able. If the random variable is continuous, it defines the probability that this variable’s value falls within a particular interval. The probability distribution describes the range of possible values that a random variable can attain, further referred to as the sup- port interval, and the probability that the value of the random variable is within any (measurable) subset of that range. Random sampling refers to taking a number of independent observations from a probability distribution. Typically, the parameters of the probability distribution (of- ten referred to as true parameters) are unknown, and the aim is to retrieve them using various estimation methods on the random sample generated from this probability dis- tribution. -
Random Variables Generation
Random Variables Generation Revised version of the slides based on the book Discrete-Event Simulation: a first course L.L. Leemis & S.K. Park Section(s) 6.1, 6.2, 7.1, 7.2 c 2006 Pearson Ed., Inc. 0-13-142917-5 Discrete-Event Simulation Random Variables Generation 1/80 Introduction Monte Carlo Simulators differ from Trace Driven simulators because of the use of Random Number Generators to represent the variability that affects the behaviors of real systems. Uniformly distributed random variables are the most elementary representations that we can use in Monte Carlo simulation, but are not enough to capture the complexity of real systems. We must thus devise methods for generating instances (variates) of arbitrary random variables Properly using uniform random numbers, it is possible to obtain this result. In the sequel we will first recall some basic properties of Discrete and Continuous random variables and then we will discuss several methods to obtain their variates Discrete-Event Simulation Random Variables Generation 2/80 Basic Probability Concepts Empirical Probability, derives from performing an experiment many times n and counting the number of occurrences na of an event A The relative frequency of occurrence of event is na/n The frequency theory of probability asserts thatA the relative frequency converges as n → ∞ n Pr( )= lim a A n→∞ n Axiomatic Probability is a formal, set-theoretic approach Mathematically construct the sample space and calculate the number of events A The two are complementary! Discrete-Event Simulation Random -
Transformations)
TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof. -
On Half-Cauchy Distribution and Process
International Journal of Statistika and Mathematika, ISSN: 2277- 2790 E-ISSN: 2249-8605, Volume 3, Issue 2, 2012 pp 77-81 On Half-Cauchy Distribution and Process Elsamma Jacob1 , K Jayakumar2 1Malabar Christian College, Calicut, Kerala, INDIA. 2Department of Statistics, University of Calicut, Kerala, INDIA. Corresponding addresses: [email protected], [email protected] Research Article Abstract: A new form of half- Cauchy distribution using sin ξ cos ξ Marshall-Olkin transformation is introduced. The properties of () = − ξ, () = ξ, ≥ 0 the new distribution such as density, cumulative distribution ξ ξ function, quantiles, measure of skewness and distribution of the extremes are obtained. Time series models with half-Cauchy Remark 1.1. For the HC distribution the moments do distribution as stationary marginal distributions are not not exist. developed so far. We develop first order autoregressive process Remark 1.2. The HC distribution is infinitely divisible with the new distribution as stationary marginal distribution and (Bondesson (1987)) and self decomposable (Diedhiou the properties of the process are studied. Application of the (1998)). distribution in various fields is also discussed. Keywords: Autoregressive Process, Geometric Extreme Stable, Relationship with other distributions: Half-Cauchy Distribution, Skewness, Stationarity, Quantiles. 1. Let Y be a folded t variable with pdf given by 1. Introduction: ( ) > 0, (1.5) The half Cauchy (HC) distribution is derived from 2Γ( ) () = 1 + , ν ∈ , the standard Cauchy distribution by folding the ν ()√νπ curve on the origin so that only positive values can be observed. A continuous random variable X is said When ν = 1 , (1.5) reduces to 2 1 > 0 to have the half Cauchy distribution if its survival () = , function is given by 1 + (x)=1 − tan , > 0 (1.1) Thus, HC distribution coincides with the folded t distribution with = 1 degree of freedom. -
On the Topological Conjugacy Problem for Interval Maps
On the topological conjugacy problem for interval maps Roberto Venegeroles1, ∗ 1Centro de Matem´atica, Computa¸c~aoe Cogni¸c~ao,UFABC, 09210-170, Santo Andr´e,SP, Brazil (Dated: November 6, 2018) We propose an inverse approach for dealing with interval maps based on the manner whereby their branches are related (folding property), instead of addressing the map equations as a whole. As a main result, we provide a symmetry-breaking framework for determining topological conjugacy of interval maps, a well-known open problem in ergodic theory. Implications thereof for the spectrum and eigenfunctions of the Perron-Frobenius operator are also discussed. PACS numbers: 05.45.Ac, 02.30.Tb, 02.30.Zz Introduction - From the ergodic point of view, a mea- there exists a homeomorphism ! such that sure describing a chaotic map T is rightly one that, after S ! = ! T: (2) iterate a randomly chosen initial point, the iterates will ◦ ◦ be distributed according to this measure almost surely. Of course, the problem refers to the task of finding !, if This means that such a measure µ, called invariant, is it exists, when T and S are given beforehand. −1 preserved under application of T , i.e., µ[T (A)] = µ(A) Fundamental aspects of chaotic dynamics are pre- for any measurable subset A of the phase space [1, 2]. served under conjugacy. A map is chaotic if is topo- Physical measures are absolutely continuous with respect logical transitive and has sensitive dependence on initial to the Lebesgue measure, i.e., dµ(x) = ρ(x)dx, being the conditions. Both properties are preserved under conju- invariant density ρ(x) an eigenfunction of the Perron- gacy, then T is chaotic if and only if S is also. -
Arxiv:2108.08461V1 [Math.DS] 19 Aug 2021 References 48
THE BOOTSTRAP FOR DYNAMICAL SYSTEMS KASUN FERNANDO AND NAN ZOU Abstract. Despite their deterministic nature, dynamical systems often exhibit seemingly random behaviour. Consequently, a dynamical system is usually represented by a probabilistic model of which the unknown parameters must be estimated using statistical methods. When measuring the uncertainty of such parameter estimation, the bootstrap stands out as a simple but powerful technique. In this paper, we develop the bootstrap for dynamical systems and establish not only its consistency but also its second-order efficiency via a novel continuous Edgeworth expansion for dynamical systems. This is the first time such continuous Edgeworth expansions have been studied. Moreover, we verify the theoretical results about the bootstrap using computer simulations. Contents 1. Introduction 1 2. Notation and Preliminaries5 I Continuous Edgeworth Expansions 7 3. Spectral Assumptions8 4. First-order Continuous Edgeworth Expansions 19 5. Examples 24 II The Bootstrap 33 6. The Bootstrap Algorithm 33 7. Asymptotic Accuracy of the Bootstrap 35 8. Application of the Bootstrap to our Examples 39 9. Computer Simulation of the Bootstrap 42 arXiv:2108.08461v1 [math.DS] 19 Aug 2021 References 48 1. Introduction After its establishment in late 19th century through the efforts of Poincar´e[64] and Lyapunov [49], the theory of dynamical systems was applied to study the qualitative behaviour of dynamical processes in the real world. For example, the theory of dynamical systems has proved useful in, 2010 Mathematics Subject Classification. 62F40, 37A50, 62M05. Key words and phrases. dynamical systems, the bootstrap, continuous Edgeworth expansions, expanding maps, Markov chains. 1 2 KASUN FERNANDO AND NAN ZOU among many others, astronomy [12], chemical engineering [3], biology [51], ecology [74], demography [81], economics [75], language processing [65], neural activity [71], and machine learning [9], [16]. -
Handbook on Probability Distributions
R powered R-forge project Handbook on probability distributions R-forge distributions Core Team University Year 2009-2010 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 distribution 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 133 13 Misc 135 Conclusion 137 Bibliography 137 A Mathematical tools 141 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∗). -
5 Introduction to the Theory of Order Statistics and Rank Statistics • This
5 Introduction to the Theory of Order Statistics and Rank Statistics • This section will contain a summary of important definitions and theorems that will be useful for understanding the theory of order and rank statistics. In particular, results will be presented for linear rank statistics. • Many nonparametric tests are based on test statistics that are linear rank statistics. { For one sample: The Wilcoxon-Signed Rank Test is based on a linear rank statistic. { For two samples: The Mann-Whitney-Wilcoxon Test, the Median Test, the Ansari- Bradley Test, and the Siegel-Tukey Test are based on linear rank statistics. • Most of the information in this section can be found in Randles and Wolfe (1979). 5.1 Order Statistics • Let X1;X2;:::;Xn be a random sample of continuous random variables having cdf F (x) and pdf f(x). th • Let X(i) be the i smallest random variable (i = 1; 2; : : : ; n). • X(1);X(2);:::;X(n) are referred to as the order statistics for X1;X2;:::;Xn. By defini- tion, X(1) < X(2) < ··· < X(n). Theorem 5.1: Let X(1) < X(2) < ··· < X(n) be the order statistics for a random sample from a distribution with cdf F (x) and pdf f(x). The joint density for the order statistics is n Y g(x(1); x(2); : : : ; x(n)) = n! f(x(i)) for − 1 < x(1) < x(2) < ··· < x(n) < 1 (16) i=1 = 0 otherwise th Theorem 5.2: The marginal density for the j order statistic X(j) (j = 1; 2; : : : ; n) is n! g (t) = [F (t)]j−1 [1 − F (t)]n−j f(t) − 1 < t < 1: j (j − 1)!(n − j)! • For random variable X with cdf F (x), the inverse distribution F −1(·) is defined as F −1(y) = inffx : F (x) ≥ yg 0 < y < 1: • If F (x) is strictly increasing between 0 and 1, then there is only one x such that F (x) = y.