Robust Hypothesis Testing with a Relative Entropy Tolerance

This paper considers the design of a minimax test for two hypotheses where the actual probability densities of the observations are located in neighborhoods obtained by placing a bound on the relative entropy between actual and nominal densities. The minimax problem admits a saddle point which is characterized. The robust test applies a nonlinear transformation which flattens the nominal likelihood ratio in the vicinity of one. Results are illustrated by considering the transmission of binary data in the presence of additive noise.Comment: 14 pages, 5 figures, submitted to the IEEE Transactions on Information Theory, July 2007, revised April 200

Robust Kalman Filtering: Asymptotic Analysis of the Least Favorable Model

We consider a robust filtering problem where the robust filter is designed according to the least favorable model belonging to a ball about the nominal model. In this approach, the ball radius specifies the modeling error tolerance and the least favorable model is computed by performing a Riccati-like backward recursion. We show that this recursion converges provided that the tolerance is sufficiently small

Robust State Space Filtering under Incremental Model Perturbations Subject to a Relative Entropy Tolerance

This paper considers robust filtering for a nominal Gaussian state-space model, when a relative entropy tolerance is applied to each time increment of a dynamical model. The problem is formulated as a dynamic minimax game where the maximizer adopts a myopic strategy. This game is shown to admit a saddle point whose structure is characterized by applying and extending results presented earlier in [1] for static least-squares estimation. The resulting minimax filter takes the form of a risk-sensitive filter with a time varying risk sensitivity parameter, which depends on the tolerance bound applied to the model dynamics and observations at the corresponding time index. The least-favorable model is constructed and used to evaluate the performance of alternative filters. Simulations comparing the proposed risk-sensitive filter to a standard Kalman filter show a significant performance advantage when applied to the least-favorable model, and only a small performance loss for the nominal model

A Contraction Analysis of the Convergence of Risk-Sensitive Filters

A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the N-fold composition of the Riccati map is strictly contractive with respect to the Riemannian metric of positive definite matrices, when N is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.Comment: 22 pages, 6 figure

Prediction of the Spectrum of a Digital Delta–Sigma Modulator Followed by a Polynomial Nonlinearity

This paper presents a mathematical analysis of the power spectral density of the output of a nonlinear block driven by a digital delta-sigma modulator. The nonlinearity is a memoryless third-order polynomial with real coefficients. The analysis yields expressions that predict the noise floor caused by the nonlinearity when the input is constant

Quantum Stochastic Processes: A Case Study

We present a detailed study of a simple quantum stochastic process, the quantum phase space Brownian motion, which we obtain as the Markovian limit of a simple model of open quantum system. We show that this physical description of the process allows us to specify and to construct the dilation of the quantum dynamical maps, including conditional quantum expectations. The quantum phase space Brownian motion possesses many properties similar to that of the classical Brownian motion, notably its increments are independent and identically distributed. Possible applications to dissipative phenomena in the quantum Hall effect are suggested.Comment: 35 pages, 1 figure

Kalman filtering and Riccati equations for descriptor systems

Projet META2The theory of Kalman filtering is extended to the case of systems with descriptor dynamics. Explicit expressions are obtained for this descriptor Kalman filter allowing for the possible singularity of the observation noise covariance. Asymptotic behavior of the filter in the time-invariant case is studied ; in particular, a method for constructing the solution of the algebraic descriptor Riccati equation is presented

Graph structure and recursive estimation of noisy linear relations

Projet META2This paper examines estimation problems specified by noisy linear relations describing either dynamical models or measurements. Each such problem has a graph structure, which can be exploited to derive recursive estimation algorithms only when the graph is acyclic, i.e. when it is obtained by combining disjoint trees. Aggregation techniques appropriate for reducing an arbitrary graph to an acyclic one are presented. The recursive maximum likelihood estimation procedures that we present are based on two elementary operations, called reduction and extraction, which are used to compress successive observations and discard unneeded variables. These elementary operations are used to derive filtering and smoothing formulas applicable to both linear and arbitrary trees, which are in turn applicable to estimation problems in settings ranging from 1-D descriptor systems to 2-D difference equations to multiscal models of random fields. These algorithms can beviewed as direct generalizations to a far richer setting of Kalman filtering and both two-filter and Rauch-Tung-Striebel smoothing for standard causal state space models