Information Theory and Predictability. Lecture 3: Stochastic Processes
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Entropy Rate of Stochastic Processes
Entropy Rate of Stochastic Processes Timo Mulder Jorn Peters [email protected] [email protected] February 8, 2015 The entropy rate of independent and identically distributed events can on average be encoded by H(X) bits per source symbol. However, in reality, series of events (or processes) are often randomly distributed and there can be arbitrary dependence between each event. Such processes with arbitrary dependence between variables are called stochastic processes. This report shows how to calculate the entropy rate for such processes, accompanied with some denitions, proofs and brief examples. 1 Introduction One may be interested in the uncertainty of an event or a series of events. Shannon entropy is a method to compute the uncertainty in an event. This can be used for single events or for several independent and identically distributed events. However, often events or series of events are not independent and identically distributed. In that case the events together form a stochastic process i.e. a process in which multiple outcomes are possible. These processes are omnipresent in all sorts of elds of study. As it turns out, Shannon entropy cannot directly be used to compute the uncertainty in a stochastic process. However, it is easy to extend such that it can be used. Also when a stochastic process satises certain properties, for example if the stochastic process is a Markov process, it is straightforward to compute the entropy of the process. The entropy rate of a stochastic process can be used in various ways. An example of this is given in Section 4.1. -
Entropy Rate Estimation for Markov Chains with Large State Space
Entropy Rate Estimation for Markov Chains with Large State Space Yanjun Han Department of Electrical Engineering Stanford University Stanford, CA 94305 [email protected] Jiantao Jiao Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720 [email protected] Chuan-Zheng Lee, Tsachy Weissman Yihong Wu Department of Electrical Engineering Department of Statistics and Data Science Stanford University Yale University Stanford, CA 94305 New Haven, CT 06511 {czlee, tsachy}@stanford.edu [email protected] Tiancheng Yu Department of Electronic Engineering Tsinghua University Haidian, Beijing 100084 [email protected] Abstract Entropy estimation is one of the prototypicalproblems in distribution property test- ing. To consistently estimate the Shannon entropy of a distribution on S elements with independent samples, the optimal sample complexity scales sublinearly with arXiv:1802.07889v3 [cs.LG] 24 Sep 2018 S S as Θ( log S ) as shown by Valiant and Valiant [41]. Extendingthe theory and algo- rithms for entropy estimation to dependent data, this paper considers the problem of estimating the entropy rate of a stationary reversible Markov chain with S states from a sample path of n observations. We show that Provided the Markov chain mixes not too slowly, i.e., the relaxation time is • S S2 at most O( 3 ), consistent estimation is achievable when n . ln S ≫ log S Provided the Markov chain has some slight dependency, i.e., the relaxation • 2 time is at least 1+Ω( ln S ), consistent estimation is impossible when n . √S S2 log S . Under both assumptions, the optimal estimation accuracy is shown to be S2 2 Θ( n log S ). -
A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation
fractal and fractional Article A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation Marylu L. Lagunes, Oscar Castillo * , Fevrier Valdez , Jose Soria and Patricia Melin Tijuana Institute of Technology, Calzada Tecnologico s/n, Fracc. Tomas Aquino, Tijuana 22379, Mexico; [email protected] (M.L.L.); [email protected] (F.V.); [email protected] (J.S.); [email protected] (P.M.) * Correspondence: [email protected] Abstract: Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion. Keywords: fractal search; fuzzy logic; parameter adaptation; CEC 2017 Citation: Lagunes, M.L.; Castillo, O.; Valdez, F.; Soria, J.; Melin, P. A New Approach for Dynamic Stochastic 1. Introduction Fractal Search with Fuzzy Logic for Metaheuristic algorithms are applied to optimization problems due to their charac- Parameter Adaptation. Fractal Fract. teristics that help in searching for the global optimum, while simple heuristics are mostly 2021, 5, 33. -
Introduction to Stochastic Processes - Lecture Notes (With 33 Illustrations)
Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Contents 1 Probability review 4 1.1 Random variables . 4 1.2 Countable sets . 5 1.3 Discrete random variables . 5 1.4 Expectation . 7 1.5 Events and probability . 8 1.6 Dependence and independence . 9 1.7 Conditional probability . 10 1.8 Examples . 12 2 Mathematica in 15 min 15 2.1 Basic Syntax . 15 2.2 Numerical Approximation . 16 2.3 Expression Manipulation . 16 2.4 Lists and Functions . 17 2.5 Linear Algebra . 19 2.6 Predefined Constants . 20 2.7 Calculus . 20 2.8 Solving Equations . 22 2.9 Graphics . 22 2.10 Probability Distributions and Simulation . 23 2.11 Help Commands . 24 2.12 Common Mistakes . 25 3 Stochastic Processes 26 3.1 The canonical probability space . 27 3.2 Constructing the Random Walk . 28 3.3 Simulation . 29 3.3.1 Random number generation . 29 3.3.2 Simulation of Random Variables . 30 3.4 Monte Carlo Integration . 33 4 The Simple Random Walk 35 4.1 Construction . 35 4.2 The maximum . 36 1 CONTENTS 5 Generating functions 40 5.1 Definition and first properties . 40 5.2 Convolution and moments . 42 5.3 Random sums and Wald’s identity . 44 6 Random walks - advanced methods 48 6.1 Stopping times . 48 6.2 Wald’s identity II . 50 6.3 The distribution of the first hitting time T1 .......................... 52 6.3.1 A recursive formula . 52 6.3.2 Generating-function approach . -
Entropy Power, Autoregressive Models, and Mutual Information
entropy Article Entropy Power, Autoregressive Models, and Mutual Information Jerry Gibson Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106, USA; [email protected] Received: 7 August 2018; Accepted: 17 September 2018; Published: 30 September 2018 Abstract: Autoregressive processes play a major role in speech processing (linear prediction), seismic signal processing, biological signal processing, and many other applications. We consider the quantity defined by Shannon in 1948, the entropy rate power, and show that the log ratio of entropy powers equals the difference in the differential entropy of the two processes. Furthermore, we use the log ratio of entropy powers to analyze the change in mutual information as the model order is increased for autoregressive processes. We examine when we can substitute the minimum mean squared prediction error for the entropy power in the log ratio of entropy powers, thus greatly simplifying the calculations to obtain the differential entropy and the change in mutual information and therefore increasing the utility of the approach. Applications to speech processing and coding are given and potential applications to seismic signal processing, EEG classification, and ECG classification are described. Keywords: autoregressive models; entropy rate power; mutual information 1. Introduction In time series analysis, the autoregressive (AR) model, also called the linear prediction model, has received considerable attention, with a host of results on fitting AR models, AR model prediction performance, and decision-making using time series analysis based on AR models [1–3]. The linear prediction or AR model also plays a major role in many digital signal processing applications, such as linear prediction modeling of speech signals [4–6], geophysical exploration [7,8], electrocardiogram (ECG) classification [9], and electroencephalogram (EEG) classification [10,11]. -
Source Coding: Part I of Fundamentals of Source and Video Coding
Foundations and Trends R in sample Vol. 1, No 1 (2011) 1–217 c 2011 Thomas Wiegand and Heiko Schwarz DOI: xxxxxx Source Coding: Part I of Fundamentals of Source and Video Coding Thomas Wiegand1 and Heiko Schwarz2 1 Berlin Institute of Technology and Fraunhofer Institute for Telecommunica- tions — Heinrich Hertz Institute, Germany, [email protected] 2 Fraunhofer Institute for Telecommunications — Heinrich Hertz Institute, Germany, [email protected] Abstract Digital media technologies have become an integral part of the way we create, communicate, and consume information. At the core of these technologies are source coding methods that are described in this text. Based on the fundamentals of information and rate distortion theory, the most relevant techniques used in source coding algorithms are de- scribed: entropy coding, quantization as well as predictive and trans- form coding. The emphasis is put onto algorithms that are also used in video coding, which will be described in the other text of this two-part monograph. To our families Contents 1 Introduction 1 1.1 The Communication Problem 3 1.2 Scope and Overview of the Text 4 1.3 The Source Coding Principle 5 2 Random Processes 7 2.1 Probability 8 2.2 Random Variables 9 2.2.1 Continuous Random Variables 10 2.2.2 Discrete Random Variables 11 2.2.3 Expectation 13 2.3 Random Processes 14 2.3.1 Markov Processes 16 2.3.2 Gaussian Processes 18 2.3.3 Gauss-Markov Processes 18 2.4 Summary of Random Processes 19 i ii Contents 3 Lossless Source Coding 20 3.1 Classification -
Relative Entropy Rate Between a Markov Chain and Its Corresponding Hidden Markov Chain
⃝c Statistical Research and Training Center دورهی ٨، ﺷﻤﺎرهی ١، ﺑﻬﺎر و ﺗﺎﺑﺴﺘﺎن ١٣٩٠، ﺻﺺ ٩٧–١٠٩ J. Statist. Res. Iran 8 (2011): 97–109 Relative Entropy Rate between a Markov Chain and Its Corresponding Hidden Markov Chain GH. Yariy and Z. Nikooraveshz;∗ y Iran University of Science and Technology z Birjand University of Technology Abstract. In this paper we study the relative entropy rate between a homo- geneous Markov chain and a hidden Markov chain defined by observing the output of a discrete stochastic channel whose input is the finite state space homogeneous stationary Markov chain. For this purpose, we obtain the rela- tive entropy between two finite subsequences of above mentioned chains with the help of the definition of relative entropy between two random variables then we define the relative entropy rate between these stochastic processes and study the convergence of it. Keywords. Relative entropy rate; stochastic channel; Markov chain; hid- den Markov chain. MSC 2010: 60J10, 94A17. 1 Introduction Suppose fXngn2N is a homogeneous stationary Markov chain with finite state space S = f0; 1; 2; :::; N − 1g and fYngn2N is a hidden Markov chain (HMC) which is observed through a discrete stochastic channel where the input of channel is the Markov chain. The output state space of channel is characterized by channel’s statistical properties. From now on we study ∗ Corresponding author 98 Relative Entropy Rate between a Markov Chain and Its ::: the channels state spaces which have been equal to the state spaces of input chains. Let P = fpabg be the one-step transition probability matrix of the Markov chain such that pab = P rfXn = bjXn−1 = ag for a; b 2 S and Q = fqabg be the noisy matrix of channel where qab = P rfYn = bjXn = ag for a; b 2 S. -
1 Stochastic Processes and Their Classification
1 1 STOCHASTIC PROCESSES AND THEIR CLASSIFICATION 1.1 DEFINITION AND EXAMPLES Definition 1. Stochastic process or random process is a collection of random variables ordered by an index set. ☛ Example 1. Random variables X0;X1;X2;::: form a stochastic process ordered by the discrete index set f0; 1; 2;::: g: Notation: fXn : n = 0; 1; 2;::: g: ☛ Example 2. Stochastic process fYt : t ¸ 0g: with continuous index set ft : t ¸ 0g: The indices n and t are often referred to as "time", so that Xn is a descrete-time process and Yt is a continuous-time process. Convention: the index set of a stochastic process is always infinite. The range (possible values) of the random variables in a stochastic process is called the state space of the process. We consider both discrete-state and continuous-state processes. Further examples: ☛ Example 3. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f0; 1; 2; 3; 4g representing which of four types of transactions a person submits to an on-line data- base service, and time n corresponds to the number of transactions submitted. ☛ Example 4. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f1; 2g re- presenting whether an electronic component is acceptable or defective, and time n corresponds to the number of components produced. ☛ Example 5. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2;::: g representing the number of accidents that have occurred at an intersection, and time t corresponds to weeks. ☛ Example 6. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2; : : : ; sg representing the number of copies of a software product in inventory, and time t corresponds to days. -
Monte Carlo Sampling Methods
[1] Monte Carlo Sampling Methods Jasmina L. Vujic Nuclear Engineering Department University of California, Berkeley Email: [email protected] phone: (510) 643-8085 fax: (510) 643-9685 UCBNE, J. Vujic [2] Monte Carlo Monte Carlo is a computational technique based on constructing a random process for a problem and carrying out a NUMERICAL EXPERIMENT by N-fold sampling from a random sequence of numbers with a PRESCRIBED probability distribution. x - random variable N 1 xˆ = ---- x N∑ i i = 1 Xˆ - the estimated or sample mean of x x - the expectation or true mean value of x If a problem can be given a PROBABILISTIC interpretation, then it can be modeled using RANDOM NUMBERS. UCBNE, J. Vujic [3] Monte Carlo • Monte Carlo techniques came from the complicated diffusion problems that were encountered in the early work on atomic energy. • 1772 Compte de Bufon - earliest documented use of random sampling to solve a mathematical problem. • 1786 Laplace suggested that π could be evaluated by random sampling. • Lord Kelvin used random sampling to aid in evaluating time integrals associated with the kinetic theory of gases. • Enrico Fermi was among the first to apply random sampling methods to study neutron moderation in Rome. • 1947 Fermi, John von Neuman, Stan Frankel, Nicholas Metropolis, Stan Ulam and others developed computer-oriented Monte Carlo methods at Los Alamos to trace neutrons through fissionable materials UCBNE, J. Vujic Monte Carlo [4] Monte Carlo methods can be used to solve: a) The problems that are stochastic (probabilistic) by nature: - particle transport, - telephone and other communication systems, - population studies based on the statistics of survival and reproduction. -
Entropy Rates & Markov Chains Information Theory Duke University, Fall 2020
ECE 587 / STA 563: Lecture 4 { Entropy Rates & Markov Chains Information Theory Duke University, Fall 2020 Author: Galen Reeves Last Modified: September 1, 2020 Outline of lecture: 4.1 Entropy Rate........................................1 4.2 Markov Chains.......................................3 4.3 Card Shuffling and MCMC................................4 4.4 Entropy of the English Language [Optional].......................6 4.1 Entropy Rate • A stochastic process fXig is an indexed sequence of random variables. They need not be in- n dependent nor identically distributed. For each n the distribution on X = (X1;X2; ··· ;Xn) is characterized by the joint pmf n pXn (x ) = p(x1; x2; ··· ; xn) • A stochastic process provides a useful probabilistic model. Examples include: ◦ the state of a deck of cards after each shuffle ◦ location of a molecule each millisecond. ◦ temperature each day ◦ sequence of letters in a book ◦ the closing price of the stock market each day. • With dependent sequences, hows does the entropy H(Xn) grow with n? (1) Definition 1: Average entropy per symbol H(X ;X ; ··· ;X ) H(X ) = lim 1 2 n n!1 n (2) Definition 2: Rate of information innovation 0 H (X ) = lim H(XnjX1;X2; ··· ;Xn−1) n!1 • If fXig are iid ∼ p(x), then both limits exists and are equal to H(X). nH(X) H(X ) = lim = H(X) n!1 n 0 H (X ) = lim H(XnjX1;X2; ··· ;Xn−1) = H(X) n!1 But what if the symbols are not iid? 2 ECE 587 / STA 563: Lecture 4 • A stochastic process is stationary if the joint distribution of subsets is invariant to shifts in the time index, i.e. -
Information Theory 1 Entropy 2 Mutual Information
CS769 Spring 2010 Advanced Natural Language Processing Information Theory Lecturer: Xiaojin Zhu [email protected] In this lecture we will learn about entropy, mutual information, KL-divergence, etc., which are useful concepts for information processing systems. 1 Entropy Entropy of a discrete distribution p(x) over the event space X is X H(p) = − p(x) log p(x). (1) x∈X When the log has base 2, entropy has unit bits. Properties: H(p) ≥ 0, with equality only if p is deterministic (use the fact 0 log 0 = 0). Entropy is the average number of 0/1 questions needed to describe an outcome from p(x) (the Twenty Questions game). Entropy is a concave function of p. 1 1 1 1 7 For example, let X = {x1, x2, x3, x4} and p(x1) = 2 , p(x2) = 4 , p(x3) = 8 , p(x4) = 8 . H(p) = 4 bits. This definition naturally extends to joint distributions. Assuming (x, y) ∼ p(x, y), X X H(p) = − p(x, y) log p(x, y). (2) x∈X y∈Y We sometimes write H(X) instead of H(p) with the understanding that p is the underlying distribution. The conditional entropy H(Y |X) is the amount of information needed to determine Y , if the other party knows X. X X X H(Y |X) = p(x)H(Y |X = x) = − p(x, y) log p(y|x). (3) x∈X x∈X y∈Y From above, we can derive the chain rule for entropy: H(X1:n) = H(X1) + H(X2|X1) + .. -
Arxiv:1711.03962V1 [Stat.ME] 10 Nov 2017 the Entropy Rate of an Individual’S Behavior
Estimating the Entropy Rate of Finite Markov Chains with Application to Behavior Studies Brian Vegetabile1;∗ and Jenny Molet2 and Tallie Z. Baram2;3;4 and Hal Stern1 1Department of Statistics, University of California, Irvine, CA, U.S.A. 2Department of Anatomy and Neurobiology, University of California, Irvine, CA, U.S.A. 3Department of Pediatrics, University of California, Irvine, CA, U.S.A. 4Department of Neurology, University of California, Irvine, CA, U.S.A. *email: [email protected] Summary: Predictability of behavior has emerged an an important characteristic in many fields including biology, medicine, and marketing. Behavior can be recorded as a sequence of actions performed by an individual over a given time period. This sequence of actions can often be modeled as a stationary time-homogeneous Markov chain and the predictability of the individual's behavior can be quantified by the entropy rate of the process. This paper provides a comprehensive investigation of three estimators of the entropy rate of finite Markov processes and a bootstrap procedure for providing standard errors. The first two methods directly estimate the entropy rate through estimates of the transition matrix and stationary distribution of the process; the methods differ in the technique used to estimate the stationary distribution. The third method is related to the sliding-window Lempel-Ziv (SWLZ) compression algorithm. The first two methods achieve consistent estimates of the true entropy rate for reasonably short observed sequences, but are limited by requiring a priori specification of the order of the process. The method based on the SWLZ algorithm does not require specifying the order of the process and is optimal in the limit of an infinite sequence, but is biased for short sequences.