Adrian Segall

Distributed network protocols

A unified approach to the formal description and validation of several distributed protocols is presented. After introducing two basic protocols, a series of known and new protocols for connectivity test, shortest path, and path updating are described and validated. All protocols are extended to networks with changing topology.

Bit allocation and encoding for vector sources

This paper considers the problem of efficient transmission of vector sources over a digital noiseless channel. It treats the problem of optimal allocation of the total number of available bits to the components of a memoryless stationary vector source with independent components. This allocation is applied to various encoding schemes, such as minimum mean-square error, sample-by-sample quantization, or entropy quantization. We also give the optimally decorrelating scheme for a source whose components are dependent and treat the problems of selecting the optimum characteristic of the encoding scheme such that the overall mean-squared error is minimized. Several examples of encoding schemes, including the ideal encoder that achieves the rated istortion bound, and of sources related to a practical problem are discussed.

Recursive estimation from discrete-time point processes

The paper presents models for discrete-time point processes (DTPP) and schemes for recursive estimation of signals randomly influencing their rates. Although the models are similar to the better known models of signals in additive Gaussian noise, DTPP differ from these in that it is possible for DTPPs to find recursive representations for the nonlinear filters. If the signal can be modeled as a finite state Markov process, then these representations reduce to explicit recursive finite-dimensional filters. The derivation of the estimation schemes, as well as the filters themselves, present a surprising similarity to the Kalman filters for signals in Gaussian noise. We present finally an application of the estimation schemes derived in the paper to the estimation of the state of a random time-division multiple access (ALOHA-type) computer network.

Nonlinear filtering with counting observations

We apply some recent results in martingale theory and the innovations method to obtain the evolution of the conditional mean and conditional density of a process that modulates the rate of a counting process.

Stochastic processes in estimation theory

We describe the role of various stochastic processes, especially martingales and related concepts, in estimation theory. It is shown, in the simplest context, that in nonlinear estimation theory martingales play the same fundamental role as uncorrelation and white noise do in linear estimation.

Orthogonal functionals of independent-increment processes

In analogy with the Wiener-Ito theory of multiple integrals and orthogonal polynominals, a set of functionals of general square-integrable martingales is presented which, in the case of independent-increments processes, is orthogonal and complete in the sense that every L^{2} -functional of the independent-increment process can be represented as an infinite sum of these elementary functionals. The functionals are iterated integrals of the basic martingales, similar to the multiple iterated integrals of Ito and can be also thought of as being the analogs of the powers 1,x,x^{2}, \cdots of the usual calculus. The analogy is made even clearer by observing that expanding the Doleans-Dade formula for the exponential of the process in a Taylor-like series leads again to the above elementary functionals. A recursive formula for these functionals in terms of the basic martingale and of lower order functionals is given, and several connections with the theory of reproducing kernel Hilbert spaces associated with independent-increment processes are obtained.

A further note on innovations, martingales and nonlinear estimation

We present a rigorous approach to nonlinear estimation of signals in additive white Gaussian noise using the innovations method. The general argument is deliberately chosen to follow the same lines as in the linear case given in a previous note. Although a much heavier use of martingales is necessary here, the parallelism is quite striking.

A martingale approach to modeling, estimation, and detection of jump processes (Ph.D. Thesis abstr.)

Abstract : The study contains a systematic approach to problems of modeling, nonlinear estimation and detection of signals in jump-type observations, namely processes whose paths are discontinuous. It is shown that modern martingale theory provides a powerful tool for attacking these problems in a unified and rigorous manner. A general model for describing signals in jump observations is presented. It is shown that a martingale model includes all the previously proposed ones and also covers the difficult case of past-dependent signals that arises in feedback communication and control problems. (Modified author abstract)