Computing S Estimators for Regression and Multivariate Location/Dispersion

Abstract

An improved resampling algorithm for S estimators reduces the number of times the objective function is evaluated and increases the speed of convergence. With this algorithm, S estimates can be computed in less time than least median squares (LMS) for regression and minimum volume ellipsoid (MVE) for location/scatter estimates with the same accuracy. Here accuracy refers to the randomness due to the algorithm. S estimators are also more statistically efficient than the LMS and MVE estimators, that is, they have less variability due to the randomness of the data.     

Robust Estimation for Generalized Additive Models

This article studies M-type estimators for fitting robust generalized additive models in the presence of anomalous data. A new theoretical construct is developed to connect the costly M-type estimation with least-squares type calculations. Its asymptotic properties are studied and used to motivate a computational algorithm. The main idea is to decompose the overall M-type estimation problem into a sequence of well-studied conventional additive model fittings. The resulting algorithm is fast and stable, can be paired with different nonparametric smoothers, and can also be applied to cases with multiple covariates. As another contribution of this article, automatic methods for smoothing parameter selection are proposed. These methods are designed to be resistant to outliers. The empirical performance of the proposed methodology is illustrated via both simulation experiments and real data analysis. Supplementary materials are available online.     

Diagnostic Plots for Robust Multivariate Methods

Robust techniques for multivariate statistical methods—such as principal component analysis, canonical correlation analysis, and factor analysis—have been recently constructed. In contrast to the classical approach, these robust techniques are able to resist the effect of outliers. However, there does not yet exist a graphical tool to identify in a comprehensive way the data points that do not obey the model assumptions. Our goal is to construct such graphics based on empirical influence functions. These graphics not only detect the influential points but also classify the observations according to their robust distances. In this way the observations are divided into four different classes which are regular points, nonoutlying influential points, influential outliers, and noninfluential outliers. We thus gain additional insight in the data by detecting different types of deviating observations. Some real data examples will be given to show how these plots can be used in practice.     

Budget balancing in a two-dimensional macroeconomic model

A two-person nonlinear dynamic game is presented to model the government's strategy to decrease the budget deficit, where Player 1 is the government using fiscal control and Player 2 represents the private sector. In our macroeconomic model the growth rate of the labour force is not known, but its lower and upper bounds are given a priori. This means that the system is uncertain, which makes the determination of an optimal solution (in a Nash, Stackelberg, etc. sense) impossible. Therefore, only a guaranteeing cost control is determined for Player 1. It is shown that the balancing by a guaranteeing cost control is possible even in the most unfavourable situation, when the governmental debt is higher and the volume of fixed capital stock is lower than the equilibrium value.     

LMI criteria for robust chaos synchronization of a class of chaotic systems

Based on the Lyapunov stability theory and LMI technique, a new sufficient criterion, formulated in the LMI form, is established in this paper for chaos robust synchronization by linear-state-feedback approach for a class of uncertain chaotic systems with different parameters perturbation and different external disturbances on both master system and slave system. The new sufficient criterion can guarantee that the slave system will robustly synchronize to the master system at an exponential convergence rate. Meanwhile, we also provide a criterion to find out proper feedback gain matrix K$K$ that is still a pending problem in literature [H. Zhang, X.K. Ma, Synchronization of uncertain chaotic systems with parameters perturbation via active control, Chaos, Solitons and Fractals 21 (2004) 39–47]. Finally, the effectiveness of the two criteria proposed herein is verified and illustrated by the chaotic Murali–Lakshmanan–Chua system and Lorenz systems, respectively.     

Design and implementation of a robust and nonlinear flight control system for an unmanned helicopter

In this work, we focus on the design and implementation of a robust flight control system for an unmanned helicopter. A comprehensive nonlinear model for an unmanned helicopter system, which is built by our research team at the National University of Singapore, is first presented. A three-layer control architecture is then adopted to construct an automatic flight control system for the aircraft, which includes (1) an inner-loop controller designed using the H_∞ control technique to internally stabilize the aircraft and at the same time yield good robustness properties with respect to external disturbances, (2) a nonlinear outer-loop controller to effectively control the helicopter position and yaw angle in the overall flight envelope, and lastly, (3) a flight-scheduling layer for coordinating flight missions. Design specifications for military rotorcraft set for the US army aviation are utilized throughout the whole process to guarantee a top level performance. The result of actual flight tests shows our design is very successful. The unmanned helicopter system is capable of achieving the desired performance in accordance with the military standard under examination.     

Robust reliable stabilization of stochastic switched nonlinear systems under asynchronous switching

This paper is concerned with the problem of robust reliable control for a class of uncertain stochastic switched nonlinear systems under asynchronous switching, where the switching instants of the controller experience delays with respect to those of the system. A design scheme for the reliable controller is proposed to guarantee almost surely exponential stability for stochastic switched systems with actuator failures, and the dwell time approach is utilized for the stability analysis. Then the approach is extended to take into account stochastic switched system with Lipschitz nonlinearities and structured uncertainties. Finally, a numerical example is employed to verify the proposed method.     

Filtering-based robust fault detection of fuzzy jump systems

This paper studies the robust fault detection filter (RFDF) design problems for uncertain nonlinear Markov jump systems with state delays and parameter uncertainties. By means of Takagi-Sugeno fuzzy models, the dynamics of filtering error generator and the fuzzy RFDF system are constructed. With the aid of the selected weighting matrix function, the design objective is to find an optimal RFDF which results in a minimal difference between the reference model (ideal solution) and the RFDF (real solution) to be designed. A sufficient condition is firstly established on the stochastic stability by using stochastic Lyapunov-Krasovskii functional approach. Then in terms of linear matrix inequalities techniques, sufficient conditions on the existence of fuzzy RFDF are presented and proved. Finally, the design problem is formulated as an optimization algorithm. Simulation results illustrate that the proposed RFDF can detect the faults shortly after the occurrences.     

A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty

The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the columnwise affine parameter uncertainty.     

Robust estimation for nonparametric generalized regression

This paper focuses on nonparametric regression estimation for the parameters of a discrete or continuous distribution, such as the Poisson or Gamma distributions, when anomalous data are present. The proposal is a natural extension of robust methods developed in the setting of parametric generalized linear models. Robust estimators bounding either large values of the deviance or of the Pearson residuals are introduced and their asymptotic behaviour is derived. Through a Monte Carlo study, for the Poisson and Gamma distributions, the finite properties of the proposed procedures are investigated and their performance is compared with that of the classical ones. A resistant cross-validation method to choose the smoothing parameter is also considered.