DOCSLIB.ORG

A Rate Distortion Approach

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Rate Distortion Approach Opportunistic Scheduling with Limited Channel State Information: A Rate Distortion Approach Matthew Johnston, Eytan Modiano, Yury Polyanskiy Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA Email: fmrj, modiano, [email protected] Abstract—We consider an opportunistic communication The above minimization can be formulated as a rate system in which a transmitter selects one of multiple channels distortion optimization with an appropriately designed over which to schedule a transmission, based on partial distortion metric. The opportunistic communication frame- knowledge of the network state. We characterize a fun- damental limit on the rate that channel state information work, in contrast to traditional rate distortion, requires must be conveyed to the transmitter in order to meet a that the channel state information sequence be causally constraint on expected throughput. This problem is modeled encoded, as the receiver causally observes the channel as a causal rate distortion optimization of a Markov source. states. Consequently, restricting the rate distortion problem We introduce a novel distortion metric capturing the impact to causal encodings provides a tighter lower bound on the of imperfect channel state information on throughput. We compute a closed-form expression for the causal information required CSI that must be provided to the transmitter. rate distortion function for the case of two channels, as well Opportunistic scheduling is one of many network con- as an algorithmic upper bound on the causal rate distortion trol schemes that requires network state information (NSI) function. Finally, we characterize the gap between the causal in order to make control decisions. The performance of information rate distortion and the causal entropic rate- these schemes is directly affected by the availability and distortion functions. accuracy of this information. If the network state changes I. INTRODUCTION rapidly, there are more possibilities to take advantage of Consider a transmitter and a receiver connected by two an opportunistic performance gain, albeit at the cost of independent channels. The state of each channel is either additional overhead. For large networks, this overhead can ON or OFF, where transmissions over an ON channel become prohibitive. result in a unit throughput, and transmissions over an OFF This paper presents a novel rate distortion formulation channel fail. Channels evolve over time according to a to quantify the fundamental limit on the rate of overhead Markov process. At the beginning of each time slot, the required for opportunistic scheduling. We design a new receiver measures the channel states in the current slot, distortion metric for this setting that captures the impact and transmits (some) channel state information (CSI) to on network performance, and incorporate a causality con- the transmitter. Based on the CSI sent by the receiver, the straint to the rate distortion formulation to reflect practical transmitter chooses over which of the channels to transmit. constraints of a real-time communication system. We ana- In a system in which an ON channel and OFF channel lytically compute a closed-form expression for the causal are equally likely to occur, the transmitter can achieve an rate distortion lower bound for a two-channel system. 1 Additionally, we propose a practical encoding algorithm expected per-slot throughput of 2 without channel state in- 3 to achieve the required throughput with limited overhead. formation, and a per-slot throughput of 4 if the transmitter has full CSI before making scheduling decisions. How- Moreover, we show that for opportunistic scheduling, there ever, the transmitter does not need to maintain complete is a fundamental gap between the mutual information and knowledge of the channel state in order to achieve high entropy-rate-based rate distortion functions, and discuss throughput; it is sufficient to only maintain knowledge of scenarios under which this gap vanishes. Proofs have been which channel has the best state. Furthermore, the memory omitted for brevity. in the system can be used to further reduce the required II. PROBLEM FORMULATION CSI. We are interested in the minimum rate that CSI must be sent to the transmitter in order to guarantee a lower Consider a transmitter and a receiver, connected through bound on expected throughput. This quantity represents a M independent channels. Assume a time slotted system, fundamental limit on the overhead information required in where at time-slot t, each channel has a time-varying this setting. channel state Si(t) 2 fOFF; ONg, independent from all other channels. The notation Si(t) 2 f0; 1g is used This work was supported by NSF Grants CNS-0915988, CNS- interchangeably. 1217048, ARO MURI Grant W911NF-08-1-0238, ONR Grant N00014- 12-1-0064, and by the Center for Science of Information (CSoI), an NSF Let X(t) = Xt = fS1(t);S2(t);:::;SM (t)g represent Science and Technology Center, under grant agreement CCF-09-39370. the system state at time slot t. At each time slot, the p satisfying n n X X n n n n n 1 − p OFF ON 1 − q E[d(x1 ; z1 )] = p(x1 )q(z1 jx1 )d(x1 ; z1 ) ≤ D: n n x1 z1 q (3) where p(xn) is the PDF of the source, and the causality Fig. 1: Markov chain describing the channel state evolution of 1 each independent channel. constraint: i n i n n n i i q(z1jx1 ) = q(z1jy1 ) 8x1 ; y1 s.t. x1 = y1; (4) transmitter chooses a channel over which to transmit, with the goal of opportunistically transmitting over an ON Mathematically, the minimum rate that CSI must be trans- channel. Channel states evolve over time according to a mitted is given by Markov process described by the chain in Figure 1, with NG 1 n 1 1 Rc (D) = lim inf H(Z1 ) (5) n!1 transition probabilities p and q satisfying p ≤ 2 and q ≤ 2 , q2Qc(D) n corresponding to channels with “positive memory.” where 1 H(Zn) is the entropy rate of the encoded se- The transmitter does not observe the state of the system. n 1 quence in bits. Equation (5) is the causal rate distortion Instead, the receiver causally encodes the sequence of function, as defined by Neuhoff and Gilbert [1], and channel states Xn into the sequence Zn and sends the 1 1 is denoted using the superscript NG. This quantity is encoded sequence to the transmitter, where Xn is used to 1 an entropy rate distortion function, in contrast to the denote the vector of random variables [X(1);:::;X(n)]. information rate distortion function [2], [3], [4], which will The encoding Z(t) = Z 2 f1;:::;Mg represents the t be discussed in Section III. The decision to formulate this index of the channel over which to transmit. Since the problem as a minimization of entropy rate is based on the throughput-optimal transmission decision is to transmit intuition that the entropy rate should capture the average over the channel with the best state, it is sufficient for number of bits per channel use required to convey channel the transmitter to restrict its knowledge to the index of the state information. channel with the best state at each time. The expected throughput earned in slot t is E[thpt(t)] = B. Previous Work S (t) i = Z(t) Z(t) , since the transmitter uses channel , Among the earliest theoretical works to study commu- 1 and receives a throughput of if that channel is ON, nication overhead in networks is Gallager’s seminal paper and 0 otherwise. Clearly, a higher throughput is attainable [5], where fundamental lower bounds on the amount of with more accurate CSI, determined by the quality of overhead needed to keep track of source and destination Zn the encoding 1 . The average distortion between the addresses and message starting and stopping times are xn zn sequences 1 and 1 is defined in terms of the per-letter derived using rate-distortion theory. A discrete-time analog distortion, of Gallager’s model is considered in [6]. A similar frame- n 1 X work was considered in [7] and [8] for different forms of d(xn; zn) = d(x ; z ); (1) 1 1 n i i network state information. i=1 The traditional rate-distortion problem [9] has been where d(xi; zi) is the per-letter distortion between the ith extended to bounds for Markov Sources in [10], [11], source symbol and the ith encoded symbol at the transmit- [12]. Additionally, researchers have considered the causal ter. For the opportunistic communication framework, the source coding problem due to its application to real-time per-letter distortion is defined as processing. One of the first works in this field was [1], in which Neuhoff and Gilbert show that the best causal d(x ; z ) 1 − [thpt(t)] = 1 − S (t); (2) i i , E Z(t) encoding of a memoryless source is a memoryless coding, where SZ(t) is the state of the channel indexed by Z(t). or a time sharing between two memoryless codes. Neuhoff Thus, an upper bound on expected distortion translates and Gilbert focus on the minimization of entropy rate, as to a lower bound on expected throughput. Note that the in (5). The work in [13] studied the optimal finite-horizon traditional Hamming distortion metric is inappropriate in sequential quantization problem, and showed that the this setting, since the transmitter does not need to know optimal encoder for a kth-order Markov source depends the channel states of channels it will not transmit over. on the last k source symbols and the present state of the decoder’s memory (i.e. the history of decoded symbols). A. Problem Statement A causal (sequential) rate distortion theory was intro- duced in [3] and [14] for stationary sources.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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 ).
    [Show full text]
  • 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].
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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) + ..
    [Show full text]
  • 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.
    [Show full text]
  • Entropy Rate
    Lecture 6: Entropy Rate • Entropy rate H(X) • Random walk on graph Dr. Yao Xie, ECE587, Information Theory, Duke University Coin tossing versus poker • Toss a fair coin and see and sequence Head, Tail, Tail, Head ··· −nH(X) (x1; x2;:::; xn) ≈ 2 • Play card games with friend and see a sequence A | K r Q q J ♠ 10 | ··· (x1; x2;:::; xn) ≈? Dr. Yao Xie, ECE587, Information Theory, Duke University 1 How to model dependence: Markov chain • A stochastic process X1; X2; ··· – State fX1;:::; Xng, each state Xi 2 X – Next step only depends on the previous state p(xn+1jxn;:::; x1) = p(xn+1jxn): – Transition probability pi; j : the transition probability of i ! j P = j – p(xn+1) xn p(xn)p(xn+1 xn) – p(x1; x2; ··· ; xn) = p(x1)p(x2jx1) ··· p(xnjxn−1) Dr. Yao Xie, ECE587, Information Theory, Duke University 2 Hidden Markov model (HMM) • Used extensively in speech recognition, handwriting recognition, machine learning. • Markov process X1; X2;:::; Xn, unobservable • Observe a random process Y1; Y2;:::; Yn, such that Yi ∼ p(yijxi) • We can build a probability model Yn−1 Yn n n p(x ; y ) = p(x1) p(xi+1jxi) p(yijxi) i=1 i=1 Dr. Yao Xie, ECE587, Information Theory, Duke University 3 Time invariance Markov chain • A Markov chain is time invariant if the conditional probability p(xnjxn−1) does not depend on n p(Xn+1 = bjXn = a) = p(X2 = bjX1 = a); for all a; b 2 X • For this kind of Markov chain, define transition matrix 2 3 6 ··· 7 6P11 P1n7 = 6 ··· 7 P 46 57 Pn1 ··· Pnn Dr.
    [Show full text]
  • Redundancy Rates of Slepian-Wolf Coding∗
    Redundancy Rates of Slepian-Wolf Coding∗ Dror Baron, Mohammad Ali Khojastepour, and Richard G. Baraniuk Dept. of Electrical and Computer Engineering, Rice University, Houston, TX 77005 Abstract The results of Shannon theory are asymptotic and do not reflect the finite codeword lengths used in practice. Recent progress in design of distributed source codes, along with the need for distributed compression in sensor networks, has made it crucial to understand how quickly distributed communication systems can approach their ultimate limits. Slepian and Wolf considered the distributed encoding of length-n sequences x and y. If y is available at the decoder, then as n increases x can be encoded losslessly at rates arbitrarily close to the conditional entropy H(X Y ). However, for any finite n there is a positive probability that x and y are | not jointly typical, and so x cannot be decoded correctly. We examine a Bernoulli setup, where x is generated by passing y through a binary symmetric correlation channel. We prove that the finite n requires us to increase the rate above the conditional entropy by K()/√n, where is the probability of error. We also study the cost of universality in Slepian-Wolf coding, and propose a universal variable 1 rate scheme wherein the encoder for x receives PY = n i yi. For PY < 0.5, our redundancy rate is K0()/√n above the empirical conditional entropy. When 1/6 1/6 P PY 0.5 = O(n− ), K0()=Ω(n ), and another scheme with redundancy rate | −1/3 | O(n− ) should be used. Our results indicate that the penalties for finite n and unknown statistics can be large, especially for P 0.5.
    [Show full text]
  • Entropy Rate
    School of Information Science Entropy Rate 2009 2-2 Course - Information Theory - Tetsuo Asano and Tad matsumoto Email: {t-asano, matumoto}@jaist.ac.jp Japan Advanced Institute of Science and Technology Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan http://www.jaist.ac.jp School of Information Science Preliminary Experiment Information Destination Source Encoder Fax Tx+ Fax Rx Decoder Noise In the previous chapter, we learned that information can be evaluated by the uncertainty of the source output random variable. Æ Entropy Question: How does the entropy of the sequence grow with its length? Goes the growth rate have a limit, if the source outputs infinite length of information? School of Information Science Outline We have made a journey on the topics: 1. Revisit to Markov Chain 2. Entropy Rate - Stationary Process - Asymptotic Properties - Entropy Rate Chain Rule - Conditioning School of Information Science Revisit of Markov Chain (1) Definition 5.1.1: Markov Chain or Markov Process A discrete stochastic process X1, X2, …, Xk, … forms Markov Chain (or Markov Process), if for n=1, 2, …., Pr(X n+1 = xn+1 X n = xn , X n−1 = xn−1,L, X1 = x1 )= Pr(X n+1 = xn+1 X n = xn ) withx1, x2 ,L, xn , xn+1 ∈ X . Theorem 5.1.1: Joint Distribution: p()x1,L, xn−1, xn = p(x1 )p(x2 x1 )p(x3 x2 )L p(xn xn−1 ) Proof: Obvious from the definition. Definition 5.1.2: Time Invariance: The Markov process is time-invariant, if the conditional probability p(xn|xn-1) does not depend on n, i.e., .
    [Show full text]
  • Arxiv:1906.02570V2 [Cs.IT] 22 Apr 2020 on Orsodn Oagvnmliait Ore E [Kac12])
    ON TYPICAL ENCODINGS OF MULTIVARIATE ERGODIC SOURCES In memory of Fero Mat´uˇs Abstract. We show that the typical coordinate-wise encoding of multivari- ate ergodic source into prescribed alphabets has the entropy profile close to the convolution of the entropy profile of the source and the modular poly- matroid that is determined by the cardinalities of the output alphabets. We show that the proportion of the exceptional encodings that are not close to the convolution goes to zero doubly exponentially. The result holds for a class of multivariate sources that satisfy asymptotic equipartition property described via the mean fluctuation of the information functions. This class covers asymp- totically mean stationary processes with ergodic mean, ergodic processes, ir- reducible Markov chains with an arbitrary initial distribution. We also proved that typical encodings yield the asymptotic equipartition property for the out- put variables. These asymptotic results are based on an explicit lower bound of the proportion of encodings that transform a multivariate random variable into a variable with the entropy profile close to the suitable convolution. M. Kupsa 1 1. Introduction In Information theory and namely in Network coding theory, the encoding of multiple possibly correlated sources have been extensively studied in the last two decades (for the main framework, overview and references see [Yeu08] and [BMR+13]). The sources are usually assumed to be discrete and memory-less, so they are repre- sented by sequences of i.i.d. discrete random variables. The important character- istics of multivariate sources, which we focus on, is its entropy profile (an entropic point corresponding to a given multivariate source, see [Kac12]).
    [Show full text]