This paper derives the optimal test channel distribution and the complete characterization of the classical Gorbunov and Pinsker (1973), Gorbunov and Pinsker (1974) nonanticipatory epsilon entropy of multivariate Gaussian Markov sources with square-error fidelity, which remained an open problem since 1974. The paper also formulates a state dependent nonanticipatory epsilon entropy, in which past reproductions are available to the decoder and not to the encoder, the test channel is specified with respect to an auxiliary (state) random process, and the reproduction process is a causal function of past reproduction and the auxiliary random process. This variation is analogous to the Wyner and Ziv (1976) and Wyner (1978) rate distortion function (RDF), of memoryless sources. It is shown that the operational rate of zero-delay codes, with past reproductions available to the decoder but not to the encoder is bounded below by the state dependent nonanticipatory epsilon entropy rate. For the case of multivariate Gaussian Markov sources with square-error fidelity, the optimal test channel distribution and the complete characterization of the state dependent of nonanticipatory epsilon entropy are derived, and also shown that that the two nonanticipatory epsilon entropies coincide. The derivations are new; they are based on structural properties of the stochastic realizations of the reproduction process that induce the optimal test channel distributions. They are derived using, achievable lower bounds on information theoretic measures, properties of mean-square estimation theory, Hadamard’s inequality, and canonical correlation coefficients of a tuple of multivariate jointly Gaussian random processes. Applications of the nonanticipatory epsilon entropy and its state dependent variation are discussed to the areas of control of unstable Gaussian systems over limited memory channels, design of causal estimators for Gaussian Markov sources with a fidelity criterion, computation of the rate loss of causal and zero-delay codes of Gaussian Markov sources with respect to non-causal codes.

The complete characterization of the Gorbunov and Pinsker [1], [2] nonanticipatory epsilon entropy of multivariate Gauss-Markov sources with square-error fidelity is derived, which remained an open problem since 1974. Specifically, it is shown that the optimal matrices of the stochastic realization of the optimal test channel or reproduction distribution, admit spectral representations with respect to the same unitary matrices, and that the optimal reproduction process is generated, subject to pre-processing and post-processing by memoryless parallel additive Gaussian noise channels. The derivations and analyses are new and bring out several properties of such optimization problems over the space of conditional distributions and their realizations.

The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) X1: Ω → Rp1and X2: Ω → Rp2, with respect to square-error distortion at the two decoders is reexamined using (1) Hotelling's geometric approach of Gaussian RVs-the canonical variable form, and (2) van Putten's and van Schuppen's parametrization of joint distributions PX1,X2,Wby Gaussian RVs W : Ω → Rnwhich make (X1,X2) conditionally independent, and the weak stochastic realization of (X1,X2). Item (2) is used to parametrize the lossy rate region of the Gray and Wyner source coding problem for joint decoding with mean-square error distortions E{||Xi- Xi||Rpi2} ≤ Δi∈ [0,∞],i = 1,2, by the covariance matrix of RV W. From this then follows Wyner's common information CW(X1,X2) (information definition) is achieved by W with identity covariance matrix, while a formula for Wyner's lossy common information (operational definition) is derived, given by CWL(X1,X2) =CW (X1,X2) = 1/2 Σj=1nIn (1+dj/1-dj), for the distortion region 0 ≤ Δ1≤ n(1-d1), 0 ≤ Δ2≤ n(1 - d1), and where 1 > d1≥ d2≥ ... ≥ dn> 0 in (0,1) are the canonical correlation coefficients computed from the canonical variable form of the tuple (X1,X2). The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.

For online traffic control at traffic control centers, there is a need for predictions of the traffic flow during a short horizon, for example, 30 min ahead. For this effort, predictions are needed of the traffic inflow into the network at motorways on the network boundary and at on-ramps. This paper presents an adaptive prediction algorithm for the inflows into the network in regular traffic situations based on stochastic control theory. The prediction algorithm is based on an adaptive prediction algorithm of T. Bohlin. The algorithm is designed and tested on traffic flow data of the ring road of Amsterdam. The results show that the algorithm provides robust predictions of traffic demand with relatively small errors for the next 30 min in a large-scale real-time environment.

In this paper we study the identifiability of linear switched systems (LSSs) in discrete-time.The question of identifiability is central to system identification, as it sets the boundaries of applicability of any system identification method; no system identification algorithm can properly estimate the parameters of a system which is not identifiable. We present necessary and sufficient conditions that guarantee structural identifiability for parametrized LSSs. We also introduce the class of semi-algebraic parametrizations, for which these conditions can be checked effectively.

Control of traffic in a network of roads may involve a large number of individual control loops, with various scopes and time scales, some of them working locally, such as traffic signals at a crossing, some of them coordinating a number of local control loops within its scope. In order to organize such a set of control loops, this paper proposes a hierarchical network model which is primarily based on a modularity property of networks. One of the prime applications of this hierarchical model is to derive an implementation architecture for the development of operational control systems, including the necessary hardware and software infrastructure, in order to achieve highly flexible control systems and thereby support the research in road traffic network control. The primary goal of this paper is to list the research issues of this approach.

Some open-loop and closed-loop control algorithms are discussed for an example of a discrete-event system, namely, the routing of arriving tasks from different arrival streams among several possible service stations. It is shown that it is possible to design open-loop policies that give good performance in a way which is very robust with respect to large changes in the arrival rates. This is possible even though it is assumed that there is no online coordination between the routing algorithms for the different arrival streams. Some further improvements of the performance are possible when a simple feedback policy, namely, overflow routing, is implemented. This also gives reasonable robustness of performance with respect to changes in the service rates.>

The evaluation of error-probability bounds for binary detection problems involving continuons-time stochastic processes as signals is considered. These bounds are of interest because, in even the simplest detection problems, the computation of the exact probabilities of error is usually mathematically intractable. The method used consists of applying some results from martingale theory to detection and estimation problems. Only discontinuous observations that contain the rate process associated with a counting process are considered. The problem addressed is to evaluate Chernoff bounds on error probabilities for the likelihood-ratio test. The solution procedure consists of a measure transformation technique that makes it possible to obtain an expression for the Chernoff bound in terms of an expectation of a multiplicative functional of the conditional mean signal (rate process) estimates. If the processes involved are Markov, it is then possible to represent the above expression as a solution to a partial differential equation that is derived from the backward equation of Kolmogorov. The above procedure is repeated when the optimal estimates are replaced by suboptimal estimates. Examples are given to illustrate the technique.