Markov Sources Research Articles

Optimal Design of Sequential Real-Time Communication Systems

Optimal design of sequential real-time communication of a Markov source over a noisy channel is investigated. In such a system, the delay between the source output and its reconstruction at the receiver should equal a fixed prespecified amount. An optimal communication strategy must minimize the total expected symbol-by-symbol distortion between the source output and its reconstruction. Design techniques or performance bounds for such real-time communication systems are unknown. In this paper a systematic methodology, based on the concepts of information structures and information states, to search for an optimal real-time communication strategy is presented. This methodology trades off complexity in communication length (linear in contrast to doubly exponential) with complexity in alphabet sizes (doubly exponential in contrast to exponential). As the communication length is usually order of magnitudes bigger than the alphabet sizes, the proposed methodology simplifies the search for an optimal communication strategy. In spite of this simplification, the resultant optimality equations cannot be solved efficiently using existing algorithmic techniques. The main idea is to formulate a zero-delay communication problem as a dynamic team with nonclassical information structure. Then, an appropriate choice of information states converts the dynamic team problem into a centralized stochastic control problem in function space. Thereafter, Markov decision theory is used to derive nested optimality equations for choosing an optimal design. For infinite horizon problems, these optimality equations give rise to a fixed point functional equation. Communication systems with fixed finite delay constraint, a higher-order Markov source, and channels with memory are treated in the same manner after an appropriate expansion of the state space. Thus, this paper presents a comprehensive methodology to study different variations of real-time communication.

Some Research on Shannon–McMillan Theorem for mth-Order Nonhomogeneous Markov Information Source

In this article, a class of strong limit theorems for relative entropy density of arbitrary stochastic sequence, expressed by inequalities, are obtained by comparing arbitrary dependent distribution with and the mth-order Markov distribution on probability space. As corollaries, some Shannon–McMillan theorems of mth-order nonhomogeneous Markov information source are obtained. Some results of nonhomogeneous Markov information source obtained are generalized.

Distributed Source Coding Using Raptor Codes for Hidden Markov Sources

In this correspondence, the problem of distributed source coding (DSC) of binary sources with side information at the decoder is addressed. A scheme is proposed based on raptor codes which are a new class of rateless codes. The decoding scheme is adapted to this problem by implementing a message passing strategy between the constituent decoders of raptor codes at each decoding iteration. The case in which the sources are modeled as independent and identically distributed (i.i.d) processes as well as the more general case in which the sources are modeled as hidden Markov processes (HMPs) are considered. The proposed approach achieves better performance than those achieved by the solutions based on turbo codes, and by the solutions based on regular low density parity check (LDPC) codes when i.i.d. sources are considered. On the other hand, when modeling sources as HMPs, an additional module to exploit the underlying Markovian nature is necessary to achieve good performance.

On Analytic Properties of Entropy Rate

Entropy rate is a real valued functional on the space of discrete random sources for which it exists. However, it lacks existence proofs and/or closed formulas even for classes of random sources which have intuitive parameterizations. A good way to overcome this problem is to examine its analytic properties relative to some reasonable topology. A canonical choice of a topology is that of the norm of total variation as it immediately arises with the idea of a discrete random source as a probability measure on sequence space. It is shown that both upper and lower entropy rate, hence entropy rate itself if it exists, are Lipschitzian relative to this topology, which, by well known facts, is close to differentiability. An application of this theorem leads to a simple and elementary proof of the existence of entropy rate of random sources with finite evolution dimension. This class of sources encompasses arbitrary hidden Markov sources and quantum random walks.

Characterization of Ergodic Hidden Markov Sources

An algebraic criterion for the ergodicity of discrete random sources is presented. For finite-dimensional sources, which contain hidden Markov sources as a subclass, the criterion can be effectively computed. This result is obtained on the background of a novel, elementary theory of discrete random sources, which is based on linear spaces spanned by word functions, and linear operators on these spaces. An outline of basic elements of this theory is provided.

Blind Separation of Nonstationary Markovian Sources Using an Equivariant Newton–Raphson Algorithm

This letter presents a new maximum likelihood method for blindly separating linear instantaneous source mixtures, where source signals are assumed to be mutually independent, Markovian and possibly nonstationary. The proposed approach first extends previous works, by Hosseini to possibly nonstationary sources using two approaches based on blocking and kernel smoothing, respectively. Moreover, to reduce time consumption, we propose an equivariant modified Newton-Raphson algorithm to solve the estimating equations, and we introduce polynomial estimators for the conditional score functions used in our method. Experimental results, both for artificial and real (speech) signals, prove the better performance of our method as compared to various classical blind separation algorithms.

Markov Incremental Constructions

A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be “locally” random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology. We generalize Mulmuley’s theory of Θ-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson–Shor-type bounds for this new model; (iii) applying the results to classical geometric problems. We hope that this work will pioneer a new approach to randomized analysis in computational geometry.

A performance analysis of discrete-time tandem queues with Markovian sources

We present an exact performance analysis of tandem networks with an arbitrary number of multiplexers. Each multiplexer is fed by the output of its upstream neighbor as well as the traffic generated by a number of independent binary Markovian sources. We model the tandem network as a discrete-time queueing system. After determining the unknown boundary functions, the probability generating function (PGF) of the joint distribution of queue length and number of O n sources for each multiplexer is derived as a function of the transform of the busy period of its upstream neighbor. From this PGF, we determine closed form expressions for the mean and variance of queue length, as well as the mean packet delay. The solution has also been extended to tandem networks in which each multiplexer is fed by multiple types of traffic. Then, numerical results are presented and it has been shown that the multiplexing smooths the traffic. The mean queue lengths are very easy to calculate and the computation complexity does not increase with the number of sources in the network.

Multiuser Detection Using Hidden Markov Model

Many existing multiuser detection algorithms assume that the user sequences are independent and identically distributed (i.i.d.). These algorithms, however, may not be efficient when the user sequences sent to a multiuser system are time correlated due to signal processing procedures such as channel coding. In this paper, we assume that the user sequences are time correlated and can be modeled as first-order, finite-state Markov chains. The proposed algorithm applies the decision feedback framework in which a linear filter based on the maximum target likelihood (MTL) criterion is derived to remove the interferences. A hidden Markov model (HMM) estimator is applied to the output of the MTL filter to estimate the user data, noise variance, and state transition probabilities. The estimated user data in turn are applied to update the parameters of the MTL filter. By exploiting the transmission of training symbols, the proposed algorithm requires neither knowledge of the user codes nor the timing information. Simulation results show the performance improvement of the proposed algorithm by exploiting the time-correlated redundancy of the Markov sources.

Resource-Scalable Joint Source-Channel MAP and MMSE Estimation of Multiple Descriptions

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A joint source-channel multiple description (JSC-MD) framework for signal estimation and communication in resource-constrained lossy networks is presented. To keep the encoder complexity at a minimum, a signal is coded by a multiple description quantizer (MDQ) with neither entropy nor channel coding. The code diversity of MDQ and the path diversity of the network are exploited by decoders to combat transmission errors. A key design objective is resource scalability: powerful nodes in the network can perform JSC-MD estimation under the criteria of maximum <emphasis emphasistype="boldital">a posteriori</emphasis> probability (MAP) or minimum mean-square error (MMSE), while primitive nodes resort to simpler MD decoding, all working with the same MDQ code. The application of JSC-MD to distributed estimation of hidden Markov models in a sensor network is demonstrated. The proposed JSC-MD MAP estimator is an algorithm of the longest path in a weighted directed acyclic graph, while the JSC-MD MMSE decoder is an extension of the well-known forward-backward algorithm to multiple descriptions. Both algorithms simultaneously exploit the source memory, the redundancy of the fixed-rate MDQ and the inter-description correlations. They outperform the existing hard-decision MDQ decoders by large margins (up to 8 dB). For Gaussian Markov sources, the complexity of JSC-MD distributed MAP sequence estimation can be made as low as that of typical single description Viterbi-type algorithms. </para>

Universal Simulation With Fidelity Criteria

We consider the problem of universal simulation of a memoryless source (with some partial extensions to Markov sources), based on a training sequence emitted from the source. The objective is to maximize the conditional entropy of the simulated sequence given the training sequence, subject to a certain distance constraint between the probability distribution of the output sequence and the probability distribution of the input, training sequence. We derive, for several distance criteria, single-letter expressions for the maximum attainable conditional entropy as well as corresponding universal simulation schemes that asymptotically attain these maxima.

Universal Estimation of Erasure Entropy

Erasure entropy rate differs from Shannon's entropy rate in that the conditioning occurs with respect to both the past and the future, as opposed to only the past (or the future). In this paper, consistent universal algorithms for estimating erasure entropy rate are proposed based on the basic and extended context-tree weighting (CTW) algorithms. Simulation results for those algorithms applied to Markov sources, tree sources, and English texts are compared to those obtained by fixed-order plug-in estimators with different orders.

Universal Delay-Limited Simulation

Universal, delay-limited simulation of an unknown information source of a certain parametric family (e.g., the family of memoryless sources or Markov sources of a given order), given a training sequence from that source and a stream of independent and uniformly distributed bits, is considered. The goal of universal simulation is that the probability law of the generated sequence be identical to that of the training sequence, with minimum mutual information between the random processes generating both sequences. In the delay-limited setting, the simulation algorithm generates a random sequence sequentially, by delivering one symbol for each training symbol that is made available after a given initial delay, whereas the random bits are assumed to be available on demand. In this paper, the optimal universal delay-limited simulation scheme is characterized for broad parametric families, and the mutual information achieved by the proposed scheme is analyzed. The results are extended to a setting of variable delay.

On the Construction of (Explicit) Khodak's Code and Its Analysis

Variable-to-variable (VV) codes are very attractive yet not well understood data compression schemes. In 1972, Khodak claimed to provide upper and lower bounds for the achievable redundancy rate, however, he did not offer explicit construction of such codes. In this paper, we first present a constructive and transparent proof of Khodak's result showing that for memoryless sources there exists a code with the average redundancy bounded by <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5/3</sup> , where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> is the average delay (e.g., the average length of a dictionary entry). We also describe an algorithm that constructs a VV length code with a small redundancy rate for large <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> . Then, we discuss several generalizations. We prove that the worst case redundancy does not exceed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4/3</sup> . Furthermore, we provide similar upper bound for Markov sources (of order 1). Finally, we consider bounds that are valid for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">almost</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">all</i> memoryless and Markov sources for which the set of exceptional source parameters has zero measure. In particular, for all memoryless sources outside this exceptional class, we prove there exists a VV code with the average redundancy rate bounded by <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1-m/3+</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">epsiv</sup> and the worst case redundancy rate bounded by <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1-m/3+</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">epsiv</sup> , where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> is the cardinality of the alphabet. We complete our analysis with a lower bound showing that for all VV codes the average and the worst case redundancy rates are at least <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2m-1-</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">epsiv</sup> for almost all memoryless sources in the sense that the set of exceptional source parameters has zero measure. We prove these results using techniques of Diophantine approximations.

On the Operational Rate-Distortion Performance of Uniform Scalar Quantization-Based Wyner–Ziv Coding of Laplace–Markov Sources

Wyner-Ziv (WZ) coding has recently been proposed as a low encoding complexity alternative to traditional DPCM coding for compression of sources with memory, in particular, in applications like multimedia compression. The viability of this alternative approach clearly depends on the compression performance of WZ coding compared to that of DPCM coding. In an attempt to understand the performance gap between WZ coding and DPCM coding, this paper studies the operational rate-distortion performance of WZ coding, using uniform scalar quantization followed by perfect Slepian-Wolf coding, for compression of a Laplace-Markov (LM) source. It is shown that at low rates or for weakly correlated LM sources, WZ coding is indeed a competitive alternative to DPCM coding. However, at high rates the performance gap becomes non-negligible for strongly correlated LM sources. In order to reduce the gap at high rates, a hybrid approach that combines DPCM coding and WZ coding is further investigated. It is shown that the hybrid approach is indeed competitive to DPCM coding at all rates even for strongly correlated LM sources.

On Unique Decodability

In this paper we propose a revisitation of the topic of unique decodability and of some fundamental theorems of lossless coding. It is widely believed that, for any discrete source X, every "uniquely decodable" block code satisfies E[l(X_1 X_2 ... X_n)]>= H(X_1,X_2,...,X_n), where X_1, X_2,...,X_n are the first n symbols of the source, E[l(X_1 X_2 ... X_n)] is the expected length of the code for those symbols and H(X_1,X_2,...,X_n) is their joint entropy. We show that, for certain sources with memory, the above inequality only holds when a limiting definition of "uniquely decodable code" is considered. In particular, the above inequality is usually assumed to hold for any "practical code" due to a debatable application of McMillan's theorem to sources with memory. We thus propose a clarification of the topic, also providing an extended version of McMillan's theorem to be used for Markovian sources.

Group Testing for Binary Markov Sources: Data-Driven Group Queries for Cooperative Sensor Networks

Group testing has been used in many applications to efficiently identify rare events in a large population. In this paper, the concept of group testing is generalized to applications with correlated source models to derive scheduling policies for sensors' adopting cooperative transmissions. The tenet of our work is that in a wireless sensor network it is advantageous to allocate the same channel dimensions to all sensor sources that have the same response to a sequence of queries or tests. That is, nodes that have the same data attributes should transmit as a cooperative super-source. Specifically, we consider the case where sensors' data are modeled spatially as a one-dimensional Markov chain. Two strategies are considered: the recursive algorithm and the tree-based algorithm. The recursive scheme allows us to illustrate the performance of group testing for finite populations while the tree-based algorithm is used to derive the achievable scaling performances of the class of group testing strategies as the number of sensors increases. We show that the total number of queries required to gather all sensors' data scales in the order of the joint entropy. A further generalization of this concept provides the basis of deriving efficient data-gathering algorithms for correlated sources.

A minimum description length approach to hidden Markov models with Poisson and Gaussian emissions. Application to order identification

We address the issue of order identification for hidden Markov models with Poisson and Gaussian emissions. We prove information-theoretic BIC-like mixture inequalities in the spirit of Finesso [1991. Consistent estimation of the order for Markov and hidden Markov chains. Ph.D. Thesis, University of Maryland]; Liu and Narayan [1994. Order estimation and sequential universal data compression of a hidden Markov source by the method of mixtures. Canad. J. Statist. 30(4), 573–589]; Gassiat and Boucheron [2003. Optimal error exponents in hidden Markov models order estimation. IEEE Trans. Inform. Theory 49(4), 964–980]. These inequalities lead to consistent penalized estimators that need no prior bound on the order. A simulation study and an application to postural analysis in humans are provided.

On the complexity of joint source-channel decoding of Markov sequences over memoryless channels

We investigate the complexity of joint source- channel maximum a posteriori (MAP) decoding of a Markov sequence which is first encoded by a source code, then encoded by a convolutional code, and sent through a noisy memoryless channel. As established previously the MAP decoding can be performed by a Viterbi-like algorithm on a trellis whose states are triples of the states of the Markov source, source coder and convolutional coder. The large size of the product space (in the order of K2N, where K is the number of source symbols and N is the number of states of the convolutional coder) appears to prohibit such a scheme. We show that for finite impulse response convolutional codes, the state space size of joint source-channel decoding can be reduced to O(K2+N log N), hence the decoding time becomes O(TK2 +TN log N), where T is the length in bits of the decoded bitstream. We further prove that an additional complexity reduction can be achieved when K > N, if the logarithm of the source transition probabilities satisfy the so- called Monge property. This decrease becomes more significant as the tree structure of the source code is more unbalanced. The reduction factor ranges between O(K/N) (for a fixed-length source code) and O(K / log N) (for Golomb-Rice code).

A Universal Online Caching Algorithm Based on Pattern Matching

We present a universal algorithm for the classical online problem of caching or demand paging. We consider the caching problem when the page request sequence is drawn from an unknown probability distribution and the goal is to devise an efficient algorithm whose performance is close to the optimal online algorithm which has full knowledge of the underlying distribution. Most previous works have devised such algorithms for specific classes of distributions with the assumption that the algorithm has full knowledge of the source. In this paper, we present a universal and simple algorithm based on pattern matching for mixing sources (includes Markov sources). The expected performance of our algorithm is within 4+o(1) times the optimal online algorithm (which has full knowledge of the input model and can use unbounded resources).