A game theoretic approach to robust filtering

A game theoretic approach to the filtering or smoothing problem is presented. A family of stationary information carrying processes and generalized models for the noise channel and the filter is considered. Sufficient conditions for the existence of saddle-point type solutions are stated. In addition, the solution for a special case of noise channel, a family of information carrying processes, and a nonlinear filter are found

Outlier resistant filtering and smoothing

AbstractWe consider a stationary Gaussian information process transmitted through an additive noise channel. We assume that the noise and information processes are mutually independent, and we model the noise process as nominally Gaussian with additive outliers. For the above system model, we first develop a theory for outlier resistant filtering and smoothing operations. We then design specific such nonlinear operations, and we study their performance. The performance criteria are the asymptotic mean squared error at the Gaussian nominal model, the breakdown point, and the influence function. We find that the proposed operations combine excellent performance at the nominal model with strong resistance to outliers

A Unified Multi-Functional Dynamic Spectrum Access Framework: Tutorial, Theory and Multi-GHz Wideband Testbed

Dynamic spectrum access is a must-have ingredient for future sensors that are ideally cognitive. The goal of this paper is a tutorial treatment of wideband cognitive radio and radar—a convergence of (1) algorithms survey, (2) hardware platforms survey, (3) challenges for multi-function (radar/communications) multi-GHz front end, (4) compressed sensing for multi-GHz waveforms—revolutionary A/D, (5) machine learning for cognitive radio/radar, (6) quickest detection, and (7) overlay/underlay cognitive radio waveforms. One focus of this paper is to address the multi-GHz front end, which is the challenge for the next-generation cognitive sensors. The unifying theme of this paper is to spell out the convergence for cognitive radio, radar, and anti-jamming. Moore’s law drives the system functions into digital parts. From a system viewpoint, this paper gives the first comprehensive treatment for the functions and the challenges of this multi-function (wideband) system. This paper brings together the inter-disciplinary knowledge

Some New Performance Criteria in Robust Statistics-Small Sample Robustness

Tech ReportIn this paper the analysis of estimates operating on dependent data is considered. Special dependent data structures are considered and the analysis is made for three different choices of contamination and performance distance measures. For contamination and performance measures both being Levy (Hampel model), an analysis that is particularly oriented toward fast convergence of the estimate to a value that is stable (robust) inside the contaminated family is undertaken. The minimum sample size to satisfy certain performance is investigated and it is found that the problem reduces to finding continuous, absolutely bounded estimates with logarithms of their moment generating function slowly increasing with the absolute value of the argument for all data distributions considered.Air Force Office of Scientific Researc

p-Efficient Estimators

Tech ReportThis paper is concerned with estimates of an unknown vector parameter S based on observations X. A generalized error autocorrelation matrix with components the error moments of order 2p, is defined. A lower bound for this matrix is found and the p-efficient P(X/S) statistics realizing it are determined. The cases of i) X and S real, ii) S scalar real, and iii) X real and S complex are examined.National Science Foundatio

Some Distance Measures and Their Use in Feature Selection

Tech ReportThe Bhattacharyya, I-divergence, Vasershtein, variational and Levy distances are evaluated, compared and used for the reduction of n data to one feature. This reduction is obtained through a restricted linear transformation and the original data are assumed to be originating from two different jointly Gaussian classes. It is found that the Bhattacharyya, I-divergence and Vasershtein distances give the same "optimal" linear transformation that applied on the original n data result in one feature with maximum possible distance between classes. The distortion measures considered in the Vasershtein distance are |x-y| and (x-y)². For the same distance measures and classes with equal covariances the Levy distance results in the same "optimal" linear transformation.Air Force Office of Scientific Researc

Moments and Error Expressions in Polynomial Minimum Mean Square Estimation

Tech ReportThe mathematical complexity of the minimum mean square estimators made inevitable the consideration of suboptimal solutions, such as the linear minimum mean square estimators. The compromise between performance and complexity can be in general less serious if the estimator that will substitute the optimum one is polynomial. If the minimum mean square estimator happens to be equal to a polynomial one, the polynomial substitution does not involve any compromise with respect to performance. Balakrishnan [1] found a necessary and sufficient condition satisfied by the joint characteristic functions of observations and variable to be estimated, so that the m.m.s. estimiate is a polynomial. The equivalent relationships in this case were found in the present paper. A matrix expression of the error difference from two different m.m.s. polynomial estimators was also found. This form involves much fewer calculations than required for finding separately the two errors

Robustness in Parameter Estimation

Tech ReportDue to vagueness in the definition of robustness, there has been no natural transition between robust and nonparametric parameter estimators. In this work, qualitative ideas first expressed by Hampel [1] are extended in an effort to present a theory that unifies the nonparametric and robust concepts. Robustness is definied in a precise mathematical way that transists to nonparametricness naturally. As a result, some general constructive characteristics of robust estimators are studied