Robust Minimax Research Articles

Some Results on Guaranteeing Cost Strategies: Full-block Multipliers and Fictitious Games

Guaranteeing cost robust minimax controls are determined for a discrete-time linear system with uncertainty of linear fractional form over infinite horizon by means of the full-block multiplier technique. It is highlighted in what sense this technique can be considered as the extension of other approaches applying a single adjustable parameter in the derivation of the linear matrix inequalities to be solved. Moreover, it is shown how to construct a fictitious game with completely known dynamics and with a cost criterion parametrized by the full-block multipliers in such a way that the minimax strategy and the upper value for it turn out to be the robust minimax strategy and the guaranteed cost for the original uncertain system.

Robust multiclass kernel-based classifiers

In this research, a robust optimization approach applied to multiclass support vector machines (SVMs) is investigated. Two new kernel based-methods are developed to address data with input uncertainty where each data point is inside a sphere of uncertainty. The models are called robust SVM and robust feasibility approach model (Robust-FA) respectively. The two models are compared in terms of robustness and generalization error. The models are compared to robust Minimax Probability Machine (MPM) in terms of generalization behavior for several data sets. It is shown that the Robust-SVM performs better than robust MPM.

Robust optimal decisions with imprecise forecasts

A robust minimax approach for optimal investment decisions with imprecise return forecasts and risk estimations in financial portfolio management is considered. Single-period and multi-period mean–variance optimization models are extended to worst-case design with multiple rival risk estimations and return forecasts. In multi-period stochastic formulation of classical mean–variance portfolio optimization problem, an investor makes an investment decision based on expectations and/or scenarios up to some intermediate times prior to the horizon and, consequently, rebalances or restructures the portfolio. Multi-period portfolio optimization entails the construction of a scenario tree representing a discretized estimate of uncertainties and associated probabilities in future stages. It is well known that return forecasts and risk estimations are inherently inaccurate and there are different rival estimates, or scenario trees. Robust optimization models are presented and imprecise nature of moment forecasts to reduce the risk of making a decision based on the wrong scenario is addressed. The worst-case performance is guaranteed in view of all rival risk and return scenarios and will only improve when any scenario other than the worst-case is realized. The ex-ante performance of minimax models is tested using historical data and backtesting results are presented.

LQG optimal control of discrete stochastic systems under parametric and noise uncertainties

In this paper, the linear-quadratic-Gaussian (LQG) optimal control problem is considered and a robust minimax controller composed of the Kalman filter and the optimal regulator is synthesized to guarantee the asymptotic stability of the discrete time-delay systems under both parametric uncertainties and uncertain noise covariances. Designed procedures are finally elaborated with an illustrative example.

Robust minimax detection of a weak signal in noise with a bounded variance and density value at the center of symmetry

In practical communication environments, it is frequently observed that the underlying noise distribution is not Gaussian and may vary in a wide range from short-tailed to heavy-tailed forms. To describe partially known noise distribution densities, a distribution class characterized by the upper-bounds upon a noise variance and a density dispersion in the central part is used. The results on the minimax variance estimation in the Huber sense are applied to the problem of asymptotically minimax detection of a weak signal. The least favorable density minimizing Fisher information over this class is called the Weber-Hermite density and it has the Gaussian and Laplace densities as limiting cases. The subsequent minimax detector has the following form: i) with relatively small variances, it is the minimum L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm distance rule; ii) with relatively large variances, it is the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 </sub> -norm distance rule; iii) it is a compromise between these extremes with relatively moderate variances. It is shown that the proposed minimax detector is robust and close to Huber's for heavy-tailed distributions and more efficient than Huber's for short-tailed ones both in asymptotics and on finite samples

Robust mean-squared error estimation in the presence of model uncertainties

We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties, where the vector x is known to satisfy a weighted norm constraint. We first assume that the model is known exactly and seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible values of x. We show that for an arbitrary choice of weighting, the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved very efficiently. We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke, with a specific choice of shrinkage factor that explicitly takes the prior information into account. Next, we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst-case MSE across all possible values of x and all possible values of the model matrix. As we show, the robust minimax MSE estimator can also be formulated as a solution to an SDP. Finally, we demonstrate through several examples that the minimax MSE estimator can significantly increase the performance over the conventional least-squares estimator, and when the model matrix is subject to uncertainties, the robust minimax MSE estimator can lead to a considerable improvement in performance over the minimax MSE estimator.

Robust finite horizon minimax filtering for discrete-time stochastic uncertain systems

We study a finite-horizon robust minimax filtering problem for time-varying discrete-time stochastic uncertain systems. The uncertainty in the system is characterized by a set of probability measures under which the stochastic noises, driving the system, are defined. The optimal minimax filter has been found by applying techniques of risk-sensitive LQG control. The structure and properties of resulting filter are analyzed and compared to H ∞ and Kalman filters.

Wavelet designs for estimating nonparametric curves with heteroscedastic error

In this paper, we discuss the problem of constructing designs in order to maximize the accuracy of nonparametric curve estimation in the possible presence of heteroscedastic errors. Our approach is to exploit the flexibility of wavelet approximations to approximate the unknown response curve by its wavelet expansion thereby eliminating the mathematical difficulty associated with the unknown structure. It is expected that only finitely many parameters in the resulting wavelet response can be estimated by weighted least squares. The bias arising from this, compounds the natural variation of the estimates. Robust minimax designs and weights are then constructed to minimize mean-squared-error-based loss functions of the estimates. We find the periodic and symmetric properties of the Euclidean norm of the multiwavelet system useful in eliminating some of the mathematical difficulties involved. These properties lead us to restrict the search for robust minimax designs to a specific class of symmetric designs. We also construct minimum variance unbiased designs and weights which minimize the loss functions subject to a side condition of unbiasedness. We discuss an example from the nonparametric literature.

Aeroservoelastic Model Uncertainty Bound Estimation from Flight Data

Uncertainty modeling is a critical element in the estimation of robust stability margins for stability boundary prediction and robust flight control system development. There has been a serious deficiency to date in aeroservoelastic data analysis with attention to uncertainty modeling. Uncertainty can be estimated from flight data using both parametric and nonparametric identification techniques. The model validation problem addressed here is to identify aeroservoelastic models with associated uncertainty structures from a limited amount of controlled excitation inputs over an extensive flight envelope. The challenge is to update analytical models from flight data estimates while also deriving nonconservative uncertainty descriptions consistent with the flight data. Transfer function estimates are incorporated in a robust minimax estimation scheme to update models and get error bounds consistent with the data and model structure. Uncertainty estimates derived from the data in this manner provide an appropriate and relevant representation for model development and robust stability analysis. The method incorporates parametric and nonparamteric uncertainty into various uncertainty structures for quantitative measures of robust stability relating to parameter variations and unmodeled dynamics. This model-plus-uncertainty identification procedure is applied to aeroservoelastic flight data from the NASA Dryden Flight Research Center F-18 Systems Research Aircraft.

Robust H infinity suboptimal and guaranteed cost state feedbacks as solutions to linear-quadratic dynamic games under uncertainty

For linear-quadratic dynamic games in which norm-bounded time-varying uncertainty enters in all matrices of the system equation without assuming the matching conditions, the notions of robust minimax and maximin strategies are introduced. It is shown how to construct two auxiliary dynamic games with completely known equations in such a way that a minimax strategy for one of them and a maximin strategy for the other turn out to be the robust minimax and maximin strategies, respectively, for the original dynamic game. By assuming 'control' as the first player and 'disturbance' as the second player, we show that the robust minimax strategy in the appropriate dynamic game provides the uncertain system with a robust H infinity suboptimal control, which guarantees quadratic stability of the unforced uncertain system. The results presented in this paper generalize those obtained in previous works.

On the characterization of fisher information and stability of the least favorable lattice distributions

The paper is concerned with the stability properties of the least favorable distributions minimizing the Fisher information in a given class of distributions. The derivation of a least favorable distribution (the solution of a variational problem) is a necessary stage of the Huber minimax approach in robust estimation of a location parameter. Generally, the solutions of variational problems essentially depend on the regularity restrictions of a functional class. The stability of these optimal solutions to violations of the smoothness restrictions is studied under the lattice distribution classes. The discrete analogues of Fisher information are obtained in these cases. They have the form of the Hellinger metrics with the estimation of a real continuous location parameter and the form of the X2 metrics with the estimation of an integer discrete location parameter. The analytical expressions for the corresponding least favorable discrete distributions are derived in some classes of lattice distributions by means of generating functions and Bellman's recursive functional equations of dynamic programming. These classes include the class of nondegenerate distributions with a restriction on the value of the density in the center of symmetry, the class of finite distributions, and the class of contaminated distributions. The obtained least favorable lattice distributions preserve the structure of their prototypes in the continuous case. These results show the stability of robust minimax solutions under different types of transitions from the continuous distribution to the discrete one.

Robust speech recognition based on joint model and feature space optimization of hidden Markov models

The hidden Markov model (HMM) inversion algorithm, based on either the gradient search or the Baum-Welch reestimation of input speech features, is proposed and applied to the robust speech recognition tasks under general types of mismatch conditions. This algorithm stems from the gradient-based inversion algorithm of an artificial neural network (ANN) by viewing an HMM as a special type of ANN. Given input speech features s, the forward training of an HMM finds the model parameters lambda subject to an optimization criterion. On the other hand, the inversion of an HMM finds speech features, s, subject to an optimization criterion with given model parameters lambda. The gradient-based HMM inversion and the Baum-Welch HMM inversion algorithms can be successfully integrated with the model space optimization techniques, such as the robust MINIMAX technique, to compensate the mismatch in the joint model and feature space. The joint space mismatch compensation technique achieves better performance than the single space, i.e. either the model space or the feature space alone, mismatch compensation techniques. It is also demonstrated that approximately 10-dB signal-to-noise ratio (SNR) gain is obtained in the low SNR environments when the joint model and feature space mismatch compensation technique is used.

On dynamic programming for sequential decision problems under a general form of uncertainty

We study the applicability of the method of Dynamic Programming (DP) for the solution of a general class of sequential decision problems under uncertainty, that may more commonly be referred to as discrete-time control problems under uncertainty. The uncertainty is due to the fact that the evolution of the state of the controlled system is affected by disturbances that are only known to belong to random sets, whose distributions are given a-priori. This includes as special cases the well known stochastic control problem and the robust min-max problem.

Computational Techniques for Econometrics and Economic Analysis.

Part One: The Computer and Econometric Methods. A Nonparametric Simulation Estimator for Nonlinear Structural Models R. Bansal, A.R. Gallant, R. Hussey, G. Tauchen. On the Accuracy and Efficiency of GMM Estimators: a Monte Carlo Study A.J. Hughes Hallett, Yue Ma. A Bootstrap Estimator for Dynamic Optimization Models A.J. Reed, C. Hallahan. Computation of Optimum Control Functions by Lagrange Multipliers G.C. Chow. Part Two: The Computer and Economic Analysis. Computational Approaches to Learning with Control Theory D. Kendrick. Computability, Complexity and Economics A.L. Norman. Robust Min-Max Decisions with Rival Models B. Rustem. Part Three: Computational Techniques for Econometrics. Wavelets in Macroeconomics: an Introduction W.L. Goffe. MatClass: a Matrix Class for C++ C.R. Birchenhall. Parallel Implementations of Primal and Dual Algorithms for Matrix Balancing I. Chabini, I. Drissi-Kaitouni, M. Florian. Part Four: The Computer and Econometric Studies. Variational Inequalities for the Computation of Financial Equilibria in the Presence of Taxes and Price Controls N. Nagurney, J. Dong. Modeling Dynamic Resource Adjustment using Iterative Least Squares A. Somwaru, E. Ball, U. Vasavada. Intensity of Takeover Defenses: the Empirical Evidence A. Chakraborty, C.F. Baum. Index.

A Minimax Approach to Cost Variance Investigation with Imperfect Parameter Knowledge

ABSTRACTAlthough models of the multiperiod cost variance investigation decision problem have been extensively analyzed, their implementation is hindered by the manager's lack of knowledge of parameter values essential to determining the optimal investigation strategy. This paper develops a minimax approach to determine a reasonable investigation strategy, even without knowledge of certain parameters. Specifically, the Dittman‐Prakash Markovian variance investigation model is modified to allow specification of a suitable strategy when neither the probability of staying in the in‐control state nor the expected cost for the out‐of‐control state is known. An objective function based on controllable costs is derived and a solution technique for the resulting minimax problem is developed. Numerical results indicate that the inherently robust minimax approach captures most of the cost savings available from methods that place greater input demands on the manager.

Robust Fixed Size Confidence Procedures for a Restricted Parameter Space

Robust fixed size confidence procedures are derived for the location parameter $\theta$ of a sample of $N$ i.i.d. observations of a scalar random variable $Z$ with CDF $F(z - \theta)$. Here, $\theta$ is restricted to a closed interval $\Omega$ and the uncertainty in $F$ is modeled by an uncertainty class $\mathscr{F}$. These robust confidence procedures are, in turn, based on the solution of a related robust minimax decision problem that is characterized by a zero-one loss function, the parameter space $\Omega$ and the uncertainty class $\mathscr{F}$. Sufficient conditions for the existence of robust minimax and robust median-minimax estimators are delineated. Sufficient conditions on $\mathscr{F}$ are obtained such that (i) both types of rules are minimax within the class of nonrandomized odd monotone procedures and (ii) subject to additional conditions, both types of rules are globally minimax admissible Bayes procedures. The paper concludes with an examination of the asymptotic behavior of the robust median-minimax estimators and their extensions to the robust $\alpha$-minimax rules, which are based on the $\alpha$-trimmed mean.

The measurement of productive efficiency: A robust minimax approach

AbstractSome applied aspects of robustness are formulated and empirically tested so as to generalize the Farrell efficiency measure for the frontier production function. A minimax method is used to develop a Chebyshev efficiency measure along with a stochastic production frontier. The relative insensitivity of such a measure in respect of sample size variations and outliers are illustrated by an empirical application to educational production functions.

Robust Adaptive Approach to Identification of Regression Models

Abstract The paper deals with problems of estimating regression and autoregression models by the robust minimax approach. These problems amount to solving a variational problem of minimizing Fisher informa tion on some class of distributions. The optimal minimax algorithms of identification for various classes are obtained. Such optimal solu tions are the basis for heuristic adaptive procedures of robust estimation, which implies sample determination of the characteristics of a distribution class followed by the use of appropriate minimax algorithm. The Monte-Carlo technique is used to investigate the proposed adaptive procedures of estimation on samples of various sizes.