Deterministic Sampling and Range Counting in Geometric Data Streams

We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.Comment: 12 pages, 1 figur

Linear estimation in Krein spaces. Part II. Applications

We have shown that several interesting problems in H∞-filtering, quadratic game theory, and risk sensitive control and estimation follow as special cases of the Krein-space linear estimation theory developed in Part I. We show that all these problems can be cast into the problem of calculating the stationary point of certain second-order forms, and that by considering the appropriate state space models and error Gramians, we can use the Krein-space estimation theory to calculate the stationary points and study their properties. The approach discussed here allows for interesting generalizations, such as finite memory adaptive filtering with varying sliding patterns

Adaptive estimation and equalisation of the high frequency communications channel

SIGLEAvailable from British Library Document Supply Centre- DSC:D94945 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Estimation-based synthesis of H∞-optimal adaptive FIR filtersfor filtered-LMS problems

This paper presents a systematic synthesis procedure for H∞-optimal adaptive FIR filters in the context of an active noise cancellation (ANC) problem. An estimation interpretation of the adaptive control problem is introduced first. Based on this interpretation, an H∞ estimation problem is formulated, and its finite horizon prediction (filtering) solution is discussed. The solution minimizes the maximum energy gain from the disturbances to the predicted (filtered) estimation error and serves as the adaptation criterion for the weight vector in the adaptive FIR filter. We refer to this adaptation scheme as estimation-based adaptive filtering (EBAF). We show that the steady-state gain vector in the EBAF algorithm approaches that of the classical (normalized) filtered-X LMS algorithm. The error terms, however, are shown to be different. Thus, these classical algorithms can be considered to be approximations of our algorithm. We examine the performance of the proposed EBAF algorithm (both experimentally and in simulation) in an active noise cancellation problem of a one-dimensional (1-D) acoustic duct for both narrowband and broadband cases. Comparisons to the results from a conventional filtered-LMS (FxLMS) algorithm show faster convergence without compromising steady-state performance and/or robustness of the algorithm to feedback contamination of the reference signal

Linear MMSE-Optimal Turbo Equalization Using Context Trees

Formulations of the turbo equalization approach to iterative equalization and decoding vary greatly when channel knowledge is either partially or completely unknown. Maximum aposteriori probability (MAP) and minimum mean square error (MMSE) approaches leverage channel knowledge to make explicit use of soft information (priors over the transmitted data bits) in a manner that is distinctly nonlinear, appearing either in a trellis formulation (MAP) or inside an inverted matrix (MMSE). To date, nearly all adaptive turbo equalization methods either estimate the channel or use a direct adaptation equalizer in which estimates of the transmitted data are formed from an expressly linear function of the received data and soft information, with this latter formulation being most common. We study a class of direct adaptation turbo equalizers that are both adaptive and nonlinear functions of the soft information from the decoder. We introduce piecewise linear models based on context trees that can adaptively approximate the nonlinear dependence of the equalizer on the soft information such that it can choose both the partition regions as well as the locally linear equalizer coefficients in each region independently, with computational complexity that remains of the order of a traditional direct adaptive linear equalizer. This approach is guaranteed to asymptotically achieve the performance of the best piecewise linear equalizer and we quantify the MSE performance of the resulting algorithm and the convergence of its MSE to that of the linear minimum MSE estimator as the depth of the context tree and the data length increase.Comment: Submitted to the IEEE Transactions on Signal Processin

Single mode excitation in the shallow water acoustic channel using feedback control

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution June 1996The shallow water acoustic channel supports far-field propagation in a discrete set of modes. Ocean experiments have confirmed the modal nature of acoustic propagation, but no experiment has successfully excited only one of the suite of mid-frequency propagating modes propagating in a coastal environment. The ability to excite a single mode would be a powerful tool for investigating shallow water ocean processes. A feedback control algorithm incorporating elements of adaptive estimation, underwater acoustics, array processing and control theory to generate a high-fidelity single mode is presented. This approach also yields a cohesive framework for evaluating the feasibility of generating a single mode with given array geometries, noise characteristics and source power limitations. Simulations and laboratory waveguide experiments indicate the proposed algorithm holds promise for ocean experiments.Josko Catipovic funded my research for summer of 1992 on the Office of Naval Research Grant Number N00014-92-J-1661 and from June 1993 through August 1995 on Defense Advanced Research Projects Agency Grant Number MDA972-92-J- 1041. The Office of Naval Research Grant N00014-95-1-0362 to MIT supported the computer facilities used to do much of this work

Forecasting Time Series with VARMA Recursions on Graphs

Graph-based techniques emerged as a choice to deal with the dimensionality issues in modeling multivariate time series. However, there is yet no complete understanding of how the underlying structure could be exploited to ease this task. This work provides contributions in this direction by considering the forecasting of a process evolving over a graph. We make use of the (approximate) time-vertex stationarity assumption, i.e., timevarying graph signals whose first and second order statistical moments are invariant over time and correlated to a known graph topology. The latter is combined with VAR and VARMA models to tackle the dimensionality issues present in predicting the temporal evolution of multivariate time series. We find out that by projecting the data to the graph spectral domain: (i) the multivariate model estimation reduces to that of fitting a number of uncorrelated univariate ARMA models and (ii) an optimal low-rank data representation can be exploited so as to further reduce the estimation costs. In the case that the multivariate process can be observed at a subset of nodes, the proposed models extend naturally to Kalman filtering on graphs allowing for optimal tracking. Numerical experiments with both synthetic and real data validate the proposed approach and highlight its benefits over state-of-the-art alternatives.Comment: submitted to the IEEE Transactions on Signal Processin

Indefinite metric spaces in estimation, control and adaptive filtering

The goal of this thesis is two-fold: first to present a unified mathematical framework (based upon optimization in indefinite metric spaces) for a wide range of problems in estimation and control, and second, to motivate and introduce the problem of robust estimation and control, and to study its implications to the area of adaptive signal processing. Robust estimation (and control) is concerned with the design of estimators (and controllers that have acceptable performance in the face of model uncertainties and lack of statistical information, and can be considered an outgrowth and extension of (the now classical) LQG theory, developed in the 1950's and 1960's which assumed perfect models and complete statistical knowledge. It has particular significance in adaptive signal processing where one needs to cope with time-variations of system parameters and to compensate for lack of a priori knowledge of the statistics of the input data and disturbances. One method of addressing the above problem is the so-called H∞ approach, which was introduced by G. Zames in 1980 and that has been recently solved by various authors. Despite the "fundamental differences" between the philosophies of the H∞ and LQG approaches to control and estimation, there are striking "formal similarities" between the controllers and estimators obtained from these two methodologies. In an attempt to explain these similarities, we shall describe a new approach to H∞ estimation (and control), different from the existing (e.g., interpolation-theoretic-based, game-theoretic-based, etc) approaches, that is based upon setting up estimation (and control problems) not in the usual Hilbert space of random variables, but in an indefinite (so-called Krein) space

Adaptive equalisation for fading digital communication channels

This thesis considers the design of new adaptive equalisers for fading digital communication channels. The role of equalisation is discussed in the context of the functions of a digital radio communication system and both conventional and more recent novel equaliser designs are described. The application of recurrent neural networks to the problem of equalisation is developed from a theoretical study of a single node structure to the design of multinode structures. These neural networks are shown to cancel intersymbol interference in a manner mimicking conventional techniques and simulations demonstrate their sensitivity to symbol estimation errors. In addition the error mechanisms of conventional maximum likelihood equalisers operating on rapidly time-varying channels are investigated and highlight the problems of channel estimation using delayed and often incorrect symbol estimates. The relative sensitivity of Bayesian equalisation techniques to errors in the channel estimate is studied and demonstrates that the structure's equalisation capability is also susceptible to such errors. Applications of multiple channel estimator methods are developed, leading to reduced complexity structures which trade performance for a smaller computational load. These novel structures are shown to provide an improvement over the conventional techniques, especially for rapidly time-varying channels, by reducing the time delay in the channel estimation process. Finally, the use of confidence measures of the equaliser's symbol estimates in order to improve channel estimation is studied and isolates the critical areas in the development of the technique — the production of reliable confidence measures by the equalisers and the statistics of symbol estimation error bursts