Binary Markov Research Articles

Inference for a binary lattice Markov process

Problems of inference for a binary Markov process defined on a two-dimensional lattice are discussed. A test of independence is derived via the Rao score statistic. Generalized least-squares and maximum likelihood estimators are derived and their asymptotic properties are established. Results on asymptotic inference based on several independent realizations of the process are also discussed.

The influence of initial conditions on maximum likelihood estimation of the parameters of a binary hidden Markov model

The Baum-Welch (EM) algorithm is a familiar tool for calculation of the maximum likelihood estimate of the parameters in hidden Markov chain models. For the particular case of a binary Markov chain corrupted by binary channel noise a detailed study is carried out of the influence that the initial conditions impose on the results produced by the algorithm.

A transient discrete-time queueing analysis of the ATM multiplexer

In this paper, we present a transient discrete-time queueing analysis of the ATM multiplexer whose arrival process consists of the superposition of the traffic generated by independent binary Markov sources. The functional equation describing the ATM multiplexer has been transformed into a mathematically tractable form. This allows derivation of the transient probability generating functions of the queue length and the number of active sources in the system. Then, application of the final value theorem results in the corresponding steady-state probability generating functions, as well as packet delay. We also present closed form expressions for the transient and steady-state moments of the queue length. The pure transform approach used in the present analysis is an extension of the well-known classical method used in the transient analysis of single server queues with uncorrelated arrivals. As a result, the analysis is relatively easy to follow and gives an alternative solution of the ATM multiplexer that does not involve matrix operations. The matrix solutions usually assume that the probability generating matrix of the system has distinct eigenvalues, where the solution presented here does not have such restrictions. The paper presents significant new simple results on the transient analysis of the ATM multiplexer.

Delay jitter first-order and second-order statistical functions of general traffic on high-speed multimedia networks

We develop analytical approximations for the first order and second-order statistics of the delay jitter experienced by a stationary traffic stream multiplexed at a major communication node; these approximations are then used to gain insight into the behavior of jitter at a single node under diverse system and traffic conditions. Our construction applies when the node can be modeled as a finite quasi-birth-death (QBD) process and the interarrival time probability density functions (pdfs) for the tagged stream are obtainable. We compare the jitter performance of three different tagged streams: periodic, Poisson, and on/off binary Markov process. Because jitter behavior is most deleterious for the former, we study the individual effects of various system and traffic parameters on the jitter statistics of a periodic stream multiplexed in a Markov-modulated Poisson process (MMPP)/M/1/K system. To demonstrate the flexibility of this analytical technique, we also consider a JPEG video stream multiplexed with JPEG, MPEG, and data sources on an OC-12 line.

Goodness-of-fit tests for stationary distributions of Markov chains based on Rao's divergence

Goodness-of-fit tests based on Rao's divergence between observed and theoretical frequencies for stationary distributions of Markov chains are considered. The asymptotic distribution of these tests under uniform hypothesis is obtained. A power comparison of the proposed tests is carried out in the case of binary Markov data.

Analysis and design of trellis codes optimized for a binary symmetric Markov source with MAP detection

We consider the problem of transmitting a binary symmetric Markov source (BSMS), over the additive white Gaussian noise (AWGN) channel. The coding technique considered is trellis-coded modulation (TCM), where we utilize decoders which implement the maximum-likelihood (ML) and maximum a posteriori (MAP) criteria. Employing 8-PSK Ungerboeck codes on a BSMS with state transition probability 0.1, we first show that the MAP decoder realizes a 0.8-2.1-dB coding gain over the ML decoder. Motivated by these gains, we consider the design of trellis codes optimized for the BSMS/AWGN/MAP system. An approximate union bound is established for this system. Using this bound, we found codes which exhibit additional 0.4-1.1-dB gains over Ungerboeck codes. Finally, we compare the proposed TCM system with a tandem coding system. At normalized signal-to-noise ratio (SNR) of 10.8 dB and below, the proposed system significantly outperforms the tandem system.

Performance of a Delta ATM switch

In this study, we investigate the performance of an ATM switch with a Delta switching fabric loaded with Poisson as well as Binary Markov input sources using Monte Carlo simulation. The performance measures used are switch latency, buffer occupancy and cell loss probabilities. Various improvements to the switch design, such as increasing the buffer size and implementing a modified Delta switching fabric were studied. A system consisting of two switches linked by a single line was also simulated. In general, the performance of the ATM switch is reasonable given its simple and low-cost architecture which were the design goals. This is especially true for the case of Poisson sources. Using the modified Delta switching fabric and increasing buffer sizes could improve performance significantly at the expense of cost and complexity.

Eddy current tomography using a binary Markov model : Nikolova, M.; MohammadDjafari, A. Signal Processing, Vol. 49, No. 2, pp. 119–132 (1996)

Eddy current tomography using a binary Markov model : Nikolova, M.; MohammadDjafari, A. Signal Processing, Vol. 49, No. 2, pp. 119–132 (1996)

Quantization of memoryless and Gauss-Markov sources over binary Markov channels

Joint source-channel coding for stationary memoryless and Gauss-Markov sources and binary Markov channels is considered. The channel is an additive-noise channel where the noise process is an Mth-order Markov chain. Two joint source-channel coding schemes are considered. The first is a channel-optimized vector quantizer-optimized for both source and channel. The second scheme consists of a scalar quantizer and a maximum a posteriori detector. In this scheme, it is assumed that the scalar quantizer output has residual redundancy that can be exploited by the maximum a posteriori detector to combat the correlated channel noise. These two schemes are then compared against two schemes which use channel interleaving. Numerical results show that the proposed schemes outperform the interleaving schemes. For very noisy channels with high noise correlation, gains of 4-5 dB in signal-to-noise ratio are possible.

An Exact McNemar Test for Paired Binary Markov Chains

A straightforward extension of the McNemar test for paired binary data yields an exact test for the equality of the limiting marginal distributions for bivariate binary Markov chains. The exact distribution of the test statistics under the null hypothesis of equal marginals depends on the classical cell occupancy statistics for the Bose-Einstein model. Exact p-values are computed for the one-sided test, and the mean and variance of the test statistic are found. The power of the Markov-McNemar test is found to be close to the power of the classical McNemar test for independent paired observations when the independence assumption holds. The method is applied to the comparison of ribosomal DNA sequences.

Eddy current tomography using a binary Markov model

The non-destructive evaluation (NDE) problem we treat is the testing of a globally homogeneous conductive medium for anomalies such as cracks and notches. The medium is illuminated with a monochromatic electric field; the anomalies induce eddy currents and they modify the total field which can be measured. The tomographic approach, aimed to draw up an image of the medium, is recent in this area. It corresponds to an extremely difficult ill-posed inverse problem and its resolution needs the use of pertinent prior information. The considered anomalies can be represented using images whose pixels can only take the values 0 and 1. Our main contribution lies in the regularization of a large-support ill-posed observation operator using a locally constant binary image Markov random field. The resulting high-dimensional combinatorial optimization problem is tedious: neither exact resolution nor simulated annealing are feasible. Instead, we establish an equivalent continuous-valued optimization problem. A nearly optimal solution is then calculated using a graduated non-convexity algorithm adapted for this purpose. The proposed inversion technique surpasses the particular NDE problem and can be applied whenever a binary image is observed using a linear system and corrupted by Gaussian noise.

A binary Markov process model for random testing

A binary Markov process model is proposed for the random testing of software. This model is suggested for replacing the standard binomial distribution model, which is based on the easily-violated assumption of test runs being statistically independent of each other. In addition to a general result on the probability of having any specific number of software failures during testing, practical implications of the new model are also discussed. In particular, we demonstrate that, in general, the effect of a possible correlation between test runs cannot be ignored in estimating software reliability.

Association in time of a finite semi-Markov process

We derive three equivalent sufficient conditions for association in time of a finite state semi-Markov process in terms of transition probabilities and crude hazard rates. This result generalizes the earlier results of Esary and Proschan (1970), for a binary Markov process and Hjort et al. (1985), for a multistate Markov process.

Detection of binary Markov sources over channels with additive Markov noise

We consider maximum a posteriori (MAP) detection of a binary asymmetric Markov source transmitted over a binary Markov channel. The MAP detector observes a long (but finite) sequence of channel outputs and determines the most probable source sequence. In some cases, the MAP detector can be implemented by simple rules such as the "believe what you see" rule or the "guess zero (or one) regardless of what you see" rule. We provide necessary and sufficient conditions under which this is true. When these conditions are satisfied, the exact bit error probability of the sequence MAP detector can be determined. We examine in detail two special cases of the above source: (i) binary independent and identically distributed (i.i.d.) source and (ii) binary symmetric Markov source. In case (i), our simulations show that the performance of the MAP detector improves as the channel noise becomes more correlated. Furthermore, a comparison of the proposed system with a (substantially more complex) traditional tandem source-channel coding scheme portrays superior performance for the proposed scheme at relatively high channel bit error rates. In case (ii), analytical as well as simulation results show the existence of a "mismatch" between the source and the channel (the performance degrades as the channel noise becomes more correlated). This mismatch is reduced by the use of a simple rate-one convolutional encoder.

NOTE: Blind Restoration of Degraded Binary Markov Random Field Images

The problem of restoring binary images degraded by region-dependent flip-flap noises is considered. The real image is described by a Markov random field (MRF). The iterative conditional modes (ICM) algorithm is adopted. It is shown that under certain conditions the ICM algorithm is insensitive to either the MRF model or the noise parameters. Using this property, we propose a blind restoration algorithm that has the following two basic properties: (a) it does not require full knowledge about the model and noise parameters; (b) it can be fully implemented in parallel. To show the effectiveness of the algorithm, simulation results are presented on the applications of the algorithm to degraded hand-drawn and synthetic images.

Optimal Smoothing of Binary Markov Sequences by Genetic Algorithm and its Application to Image Restoration

Optimal Smoothing of Binary Markov Sequences by Genetic Algorithm and its Application to Image Restoration

Identification of individual channel kinetics from recordings containing many identical channels

Given a discrete-time signal consisting of N identical, independent, binary Markov chains observed in white noise, we consider the problem of estimating the non-zero state level, the number of chains and the elementary transition probability matrix. We derive formulae for the central moments, first- and second-order auto-correlation functions and the power spectrum of a first-order, discrete-time Markov chain. We show that the mean, variance, third central moment and power spectrum provide sufficient information for the estimation of the parameters of the signal in question. We demonstrate the estimation procedure with numerical examples for both simulated and real biological data, and describe a method for estimating the non-unity eigenvalue of the transition matrix as well as the noise variance from the power spectrum of the noisy signal.

On the asymptotic analysis of statistical multiplexers with hyper-bursty arrivals

We consider multiplexers in discrete time fed by the superposition of “Ternary Markov Sources”. Such sources are the natural extension of the Binary Markov Sources (BMS) recently used to model bursty arrivals in a high speed environment. Unlike BMS, we allow sources to have arbitrary (large) variance in the duration of their OFF (silence) or ON (burst) periods. This paper focuses mainly on the impact of large variability either in the ON or OFF period on the performance. Following some asymptotic analysis, simple results on the tail behavior of the number of cells queued in the multiplexer are given. Our results indicate that ignoring the variability in the ON period may grossly underestimate the cell buildup in the multiplexer queue for all levels of the utilization. Furthermore, the impact of large variability of the OFF period depends very much on the utilization of the system. For a lightly-loaded multiplexer (utilization below a given threshold), the impact of large variability of the OFF period is minimal. However, for a heavy-loaded multiplexer (utilization above the threshold) the impact of the large variability in the OFF period is similar to that of the ON period.

On the performance of bursty and modulated sources subject to leaky bucket rate-based access control schemes

Provides an analysis of a rate-based access control scheme in high speed environments based on a buffered leaky bucket algorithm. The analysis is carried out in discrete time which is representative of an ATM environment. For the cell arrivals to the leaky bucket the authors consider a general discrete Markovian arrival process which models bursty and modulated sources. The key of the analysis is the introduction of the deficit junction that allows the reduction of the original problem to a more standard discrete time queueing system with the same arrival process. As an important special case, the detailed analysis of the binary Markov source throttled by such rate-based access control schemes is presented. Along with explicit recursions for computation of state probabilities and simple characterisation of the asymptotic behavior of the queue build up, some guidelines for the parameter selection of these schemes are provided. The results indicate that for sources with relatively large active periods, for an acceptable grade-of-service at the input queue, the token generation rate should be chosen to be close to the peak rate of the source, and increasing the bucket size of the leaky bucket does not improve substantially the performance at the input queue.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal detection of discrete Markov sources over discrete memoryless channels-applications to combined source-channel coding

The authors consider the problem of detecting a discrete Markov source which is transmitted across a discrete memoryless channel. Two maximum a posteriori (MAP) formulations are considered: (i) a sequence MAP detection in which the objective is to determine the most probable transmitted sequence given the observed sequence and (ii) an instantaneous MAP detection which is to determine the most probable transmitted symbol at time n given all the observations prior to and including time n. The solution to the first problem results in a "Viterbi-like" implementation of the MAP detector (with Large delay) while the latter problem results in a recursive implementation (with no delay). For the special case of the binary symmetric Markov source and binary symmetric channel, simulation results are presented and an analysis of these two systems yields explicit critical channel bit error rates above which the MAP detectors become useful. Applications of the MAP detection problem in a combined source-channel coding system are considered. Here, it is assumed that the source is highly correlated and that the source encoder (a vector quantizer (VQ)) fails to remove all of the source redundancy. The remaining redundancy at the output of the source encoder is referred to as the "residual" redundancy. It is shown, through simulation, that the residual redundancy can be used by the MAP detectors to combat channel errors. For small block sizes, the proposed system beats Farvardin and Vaishampayan's channel-optimized VQ by wide margins. Finally, it is shown that the instantaneous MAP detector can be combined with the VQ decoder to form an approximate minimum mean-squared error decoder.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>