Random spherical uncertainty in estimation and robustness

A theorem is formulated that gives an exact probability distribution for a linear function of a random vector uniformly distributed over a ball in n-dimensional space. This mathematical result is illustrated via applications to a number of important problems of estimation and robustness under spherical uncertainty. These include parameter estimation, characterization of attainability sets of dynamical systems, and robust stability of affine polynomial families.     

Performance bounds for estimation under uncertainty

For the class of linear Gauss-Markov systems with binary parameters uncertainty, the minimum variance estimate of the state and associated covariance of error expressions were derived in a closed form in [1, 2]. In this paper expressions for the conditional and unconditional covariances of error matrices are presented for the M-ary case. Useful upper and lower bounds for the unconditional covariance of error are also presented which are valid, on the average, for any measurement sequence. A geophysical seismic data filtering example applying these results is also given.     

Common fallacies in estimation and control under uncertainty

Examples from the literature are used to illustrate certain common fallacies in the development of estimators and controllers under uncertainty. It is shown that the common notions of process, model, objective, and method must carefully be distinguished from one another and from additional concepts of attitude and best strategy. Failure to do so can readily lead to false claims and sub-optimal, or even useless, schemes. Concrete guidelines are provided to avoid the fallacies and to better focus the research effort.     

Managing uncertainty in project portfolio cost estimation

Although typically a software development organisation is involved in more than one project simultaneously, the available tools in the area of software cost estimation deal mostly with single software projects. In order to calculate the possible cost of the entire project portfolio, one must combine the single project estimates taking into account the uncertainty involved. In this paper, statistical simulation techniques are used to calculate confidence intervals for the effort needed for a project portfolio. The overall approach is illustrated through the adaptation of the analogy-based method for software cost estimation to cover multiple projects.     

Global optimal attitude estimation using uncertainty ellipsoids

A deterministic attitude estimation problem for a rigid body in a potential field, with bounded attitude and angular velocity measurement errors is considered. An attitude estimation algorithm that globally minimizes the attitude estimation error is obtained. Assuming that the initial attitude, the initial angular velocity and measurement noise lie within given ellipsoidal bounds, an uncertainty ellipsoid that bounds the attitude and the angular velocity of the rigid body is obtained. The center of the uncertainty ellipsoid provides point estimates, and the size of the uncertainty ellipsoid measures the accuracy of the estimates. The point estimates and the uncertainty ellipsoids are propagated using a Lie group variational integrator and its linearization, respectively. The attitude and angular velocity estimates are optimal in the sense that the sizes of the uncertainty ellipsoids are minimized.     

On resampling and uncertainty estimation in Linear System Identification

Linear System Identification yields a nominal model parameter, which minimizes a specific criterion based on the single input-output data set. Here we investigate the utility of various methods for estimating the probability distribution of this nominal parameter using only the data from this single experiment. The results are compared to the actual parameter distribution generated by many Monte Carlo runs of the data-collection experiment. The methods considered are collectively known as resampling schemes, which include Subsampling, the Jackknife, and the Bootstrap. The broad aim is to generate an empirical parameter distribution function via the construction of a large number of new data records from the original single set of data, based on an assumption that this data is representative of all possible data, and then to run the parameter estimator on each of these new records to develop the distribution function. The performance of these schemes is evaluated on a difficult, almost unidentifiable system, and compared to the standard results based on asymptotic normality. In addition to the exploration of this example as a means to evaluate the strengths and weaknesses of these resampling schemes, some new theoretical results are proven and demonstrated for Subsampling schemes.     

Parameter estimation and uncertainty analysis for a watershed model

Where numerical models are employed as an aid to environmental management, the uncertainty associated with predictions made by such models must be assessed. A number of different methods are available to make such an assessment. This paper explores the use of three such methods, and compares their performance when used in conjunction with a lumped parameter model for surface water flow (HSPF) in a large watershed.Linear (or first-order) uncertainty analysis has the advantage that it can be implemented with virtually no computational burden. While the results of such an analysis can be extremely useful for assessing parameter uncertainty in a relative sense, and ascertaining the degree of correlation between model parameters, its use in analyzing predictive uncertainty is often limited. Markov Chain Monte Carlo (MCMC) methods are far more robust, and can produce reliable estimates of parameter and predictive uncertainty. As well as this, they can provide the modeler with valuable qualitative information on the shape of parameter and predictive probability distributions; these shapes can be quite complex, especially where local objective function optima lie within those parts of parameter space that are considered probable after calibration has been undertaken. Nonlinear calibration-constrained optimization can also provide good estimates of parameter and predictive uncertainty, even in situations where the objective function surface is complex. Furthermore, they can achieve these estimates using far fewer model runs than MCMC methods. However, they do not provide the same amount of qualitative information on the probability structure of parameter space as do MCMC methods, a situation that can be partially rectified by combining their use with an efficient gradient-based search method that is specifically designed to locate different local optima.All methods of parameter and predictive uncertainty analysis discussed herein are implemented using freely-available software. Hence similar studies, or extensions of the present study, can be easily undertaken in other modeling contexts by other modelers.     

Watershed model parameter estimation and uncertainty in data-limited environments

Parameter uncertainty and sensitivity for a watershed-scale simulation model in Portugal were explored to identify the most critical model parameters in terms of model calibration and prediction. The research is intended to help provide guidance regarding allocation of limited data collection and model parameterization resources for modelers working in any data and resource limited environment. The watershed-scale hydrology and water quality simulation model, Hydrologic Simulation Program – FORTRAN (HSPF), was used to predict the hydrology of Lis River basin in Portugal. The model was calibrated for a 5-year period 1985–1989 and validated for a 4-year period 2003–2006. Agreement between simulated and observed streamflow data was satisfactory considering the performance measures such as Nash–Sutcliffe efficiency (E), deviation runoff (Dv) and coefficient of determination (R²). The Generalized Likelihood Uncertainty Estimation (GLUE) method was used to establish uncertainty bounds for the simulated flow using the Nash–Sutcliffe coefficient as a performance likelihood measure. Sensitivity analysis results indicate that runoff estimations are most sensitive to parameters related to climate conditions, soil and land use. These results state that even though climate conditions are generally most significant in water balance modeling, attention should also focus on land use characteristics as well. Specifically with respect to HSPF, the two most sensitive parameters, INFILT and LZSN, are both directly dependent on soil and land use characteristics.     

Speech enhancement using MMSE estimation under phase uncertainty

Most of the speech enhancement algorithms process the amplitudes of speech, but the phase of noisy speech is left unprocessed as it may cause undesired artifacts. Recently, short time Fourier transform based single channel speech enhancement algorithms are developed by considering uncertain prior knowledge of phase. The uncertain knowledge of the phase is obtained from the phase reconstruction algorithms. The goal of this paper is to develop joint minimum mean square error estimate of complex speech coefficients given uncertainty phase (CUP) information by considering Nagakami probability density function (PDF) and gamma PDF as speech spectral amplitude priors and generalized gamma PDF for noise prior. Estimators like amplitudes given uncertainty phase, which uses uncertain phase only for amplitude estimation and not for phase improvement are developed. Experimental results shows that incorporating uncertain phase information improves quality and intelligibility of speech. Also novel phase-blind estimators are developed using Nagakami PDF/gamma as speech priors and generalized gamma as noise prior. Finally comparison of all estimators using uncertain prior phase information is discussed and how initial phase information affects the enhancement process is analyzed with novel estimators. For comparison of all the derived estimators, the speech signals uttered by male and female speakers are taken from TIMIT database. The proposed CUP estimators outperforms the existing algorithms in terms of objective performance measure segmental signal to noise ratio, phase signal to noise ratio, perceptual evaluation of speech quality, short time objective intelligibility.     

Methods for minimax estimation under elementwise covariance uncertainty

We consider the minimax estimation problem in the linear regression model under elementwise constraints imposed on the covariance matrix of the random parameters vector. Minimax estimates are designed using several approaches to the numerical solution of the dual problem, namely, the semidefinite programming method, the conditional gradient method and its modification with the Lagrange multipliers and regularization. The efficiency of the suggested methods is illustrated by the example of path restoration for a maneuvering target with a statistically uncertain acceleration.     

Trajectory reconstruction with uncertainty estimation using mosaic registration

This paper addresses the problem of estimating the 3D trajectory and associated uncertainty of an underwater autonomous vehicle from a set of images of the seabed taken by an onboard camera. The presented algorithms resort to the use of video mosaics and build upon previous work on image registration and visual pose estimation. The pose estimation is accomplished in two steps. Firstly, a video mosaic is created automatically, covering a region of interest of the seabed. Then, after associating a 3D referential for the mosaic, the estimation of the camera position from a new view of the scene becomes possible.

The main contribution of this paper lies on the assessment of the performance of the 3D pose algorithms. In order to do this, an image sequence with available ground-truth is used for precise error measuring. A first-order error propagation analysis is presented, relating the uncertainty in the location of the match points with the uncertainty in the pose parameters. The importance of predicting the estimate uncertainty is emphasized by the fact that it can be used for comparing algorithms and for the on-line monitoring of the vehicle trajectory reconstruction quality.

Several iterative and non-iterative pose estimation methods are discussed, differing both on the criteria being minimized and on the required information about the camera intrinsic parameters. This information ranges from the full knowledge of the parameters, to the case where they are estimated using self-calibration from an image sequence under pure rotation. The implemented pose algorithms are compared for the accuracy and estimate covariance.     

Ellipsoidal parameter or state estimation under model uncertainty

Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.     

Robust control of robot manipulators based on uncertainty and disturbance estimation

In this work, uncertainty and disturbance estimation (UDE) based robust trajectory tracking controller for rigid link manipulators was proposed. The UDE was employed to estimate the composite uncertainty that comprises the effects of system nonlinearities, external disturbances, and parametric uncertainties. A feedback linearization based controller was designed for trajectory tracking, and the same was augmented by the UDE‐estimated uncertainties to achieve robustness. The resulting controller however required measurement of joint velocities apart from the joint positions. To address the issue, an observer that employed the UDE‐estimated uncertainties for robustness was proposed, giving rise to the UDE‐based controller–observer structure. Closed‐loop stability of the overall system was established. The notable feature of the proposed design was that it neither required accurate plant model nor any information about the uncertainty. Also, the design needed only joint position measurements for its implementation. To demonstrate the effectiveness, simulation results of the proposed approach as applied to the trajectory tracking control of two‐link robotic manipulator and comparison of its performance with some of the well‐known existing controllers were presented. Lastly, hardware implementation of the proposed design for trajectory control of Quanser's single‐link flexible joint module was carried out, and it was shown that the proposed strategy offered a viable approach for designing implementable robust controllers for robots. Copyright © 2011 John Wiley & Sons, Ltd.     

Integrating perceptual level of detail with head-pose estimation and its uncertainty

Immersive virtual environments with life-like interaction capabilities can provide a high fidelity view of the virtual world and seamless interaction methods to the user. High demanding requirements, however, raise many challenges in the development of sensing technologies and display systems. The focus of this study is on improving the performance of human–computer interaction by rendering optimizations guided by head pose estimates and their uncertainties. This work is part of a larger study currently being under investigation at NASA Ames, called “Virtual GloveboX” (VGX). VGX is a virtual simulator that aims to provide advanced training and simulation capabilities for astronauts to perform precise biological experiments in a glovebox aboard the International Space Station (ISS). Our objective is to enhance the virtual experience by incorporating information about the user’s viewing direction into the rendering process. In our system, viewing direction is approximated by estimating head orientation using markers placed on a pair of polarized eye-glasses. Using eye-glasses does not pose any constraints in our operational environment since they are an integral part of a stereo display used in VGX. During rendering, perceptual level of detail methods are coupled with head-pose estimation to improve the visual experience. A key contribution of our work is incorporating head pose estimation uncertainties into the level of detail computations to account for head pose estimation errors. Subject tests designed to quantify user satisfaction under different modes of operation indicate that incorporating uncertainty information during rendering improves the visual experience of the user.     

容错调度算法中反向调度与正向调度性能分析

分析了软件容错模型中的BCE容错调度算法,针对该算法中的反向调度和正向调度两个过程,给出了RMB、DMB、EDFB 3种反向调度算法和RMF、EDFF 2种正向调度算法,指出了反向调度和正向调度相互协调的特性。将各种算法在BCE算法中进行模拟,结果表明EDFF正向调度算法能够与3种反向调度算法更好地协调,从而获得比RMF正向调度算法更高的调度性能。模拟结果表明,3种反向调度算法在BCE算法中的性能相近。得出RMB(或DMB)反向调度算法与EDFF正向调度算法的组合较适用于软件容错模型的结论。     

On the use of second-order derivatives and metamodel-based Monte-Carlo for uncertainty estimation in aerodynamics

This article addresses the delicate issue of estimating physical uncertainties in aerodynamics. Usually, flow simulations are performed in a fully deterministic approach, although in real life operational uncertainty arises due to unpredictable factors that alter the flow conditions. In this article, we present and compare two methods to account for uncertainty in aerodynamic simulation. Firstly, automatic differentiation tools are used to estimate first- and second-order derivatives of aerodynamic coefficients with respect to uncertain variables, yielding an estimate of expectation and variance values (Method of Moments). Secondly, metamodelling techniques (radial basis functions, kriging) are employed in conjunction with Monte-Carlo simulations to derive statistical information. These methods are demonstrated for 3D Eulerian flows around the wing of a business aircraft at different regimes subject to uncertain Mach number and angle of attack.     

Bounds on estimation errors of discrete-time filters under modeling uncertainty

The performance of Kalman-type linear discrete-time filters in the presence of modeling errors is considered, and bounds are obtained for the performance index, the mean-squared error of estimates for suboptimal filters. The computation of these bounds requires information on only the model matrices and the range of errors for these matrices. Consequently, a designer can easily evaluate the performance of a suboptimal filter when only the range of errors in the elements of the model matrices is available.     

Model-based prediction error uncertainty estimation for k-nn method

The k-nearest neighbour estimation method is one of the main tools used in multi-source forest inventories. It is a powerful non-parametric method for which estimates are easy to compute and relatively accurate. One downside of this method is that it lacks an uncertainty measure for predicted values and for areas of an arbitrary size. We present a method to estimate the prediction uncertainty based on the variogram model which derives the necessary formula for the k-nn method. A data application is illustrated for multi-source forest inventory data, and the results are compared at pixel level to the conventional RMSE method. We find that the variogram model-based method which is analytic, is competitive with the RMSE method.