Minimax estimation of deterministic parameters in linear models with a random model matrix

We consider the problem of estimating an unknown deterministic parameter vector in a linear model with a random model matrix, with known second-order statistics. We first seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all parameter vectors whose (possibly weighted) norm is bounded above. We show that the minimax MSE estimator can be found by solving a semidefinite programming problem and develop necessary and sufficient optimality conditions on the minimax MSE estimator. Using these conditions, we derive closed-form expressions for the minimax MSE estimator in some special cases. We then demonstrate, through examples, that the minimax MSE estimator can improve the performance over both a Baysian approach and a least-squares method. We then consider the case in which the norm of the parameter vector is also bounded below. Since the minimax MSE approach cannot account for a nonzero lower bound, we consider, in this case, a minimax regret method in which we seek the estimator that minimizes the worst-case difference between the MSE attainable using a linear estimator that does not know the parameter vector, and the optimal MSE attained using a linear estimator that knows the parameter vector. For analytical tractability, we restrict our attention to the scalar case and develop a closed-form expression for the minimax regret estimator.     

Dual Gabor frames: theory and computational aspects

We consider a general method for constructing dual Gabor elements different from the canonical dual. Our approach is based on combining two Gabor frames such that the generated frame-type operator S/sub g,/spl gamma// is nonsingular. We provide necessary and sufficient conditions on the Gabor window functions g and /spl gamma/ such that S/sub g,/spl gamma// is nonsingular for rational oversampling, considering both the continuous-time and the discrete-time settings. In contrast to the frame operator, the operator S/sub g,/spl gamma// is, in general, not positive. Therefore, all results in Gabor analysis that are based on the positivity of the frame operator cannot be applied directly. The advantage of the proposed characterization is that the algebraic system for computing the Gabor dual elements preserves the high structure of usual Gabor frames, leading to computationally efficient algorithms. In particular, we consider examples in which both the condition number and the computational complexity in computing the proposed dual Gabor elements decrease in comparison to the canonical dual Gabor elements.     

Linear precoding via conic optimization for fixed MIMO receivers

In this paper, the problem of designing linear precoders for fixed multiple-input-multiple-output (MIMO) receivers is considered. Two different design criteria are considered. In the first, the transmitted power is minimized subject to signal-to-interference-plus-noise-ratio (SINR) constraints. In the second, the worst case SINR is maximized subject to a power constraint. It is shown that both problems can be solved using standard conic optimization packages. In addition, conditions are developed for the optimal precoder for both of these problems, and two simple fixed-point iterations are proposed to find the solutions that satisfy these conditions. The relation to the well-known uplink-downlink duality in the context of joint transmit beamforming and power control is also explored. The proposed precoder design is general, and as a special case, it solves the transmit rank-one beamforming problem. Simulation results in a multiuser system show that the resulting precoders can significantly outperform existing linear precoders.     

Minimax MSE-ratio estimation with signal covariance uncertainties

In continuation to an earlier work, we further consider the problem of robust estimation of a random vector (or signal), with an uncertain covariance matrix, that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix. While, in the earlier work, we developed and proposed a competitive minimax approach of minimizing the worst-case mean-squared error (MSE) difference regret criterion, here, we study, in the same spirit, the minimum worst-case MSE ratio regret criterion, namely, the worst-case ratio (rather than difference) between the MSE attainable using a linear estimator, ignorant of the exact signal covariance, and the minimum MSE (MMSE) attainable by optimum linear estimation with a known signal covariance. We present the optimal linear estimator, under this criterion, in two ways: The first is as a solution to a certain semidefinite programming (SDP) problem, and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure. We then show that the linear minimax ratio regret estimator can also be interpreted as the MMSE estimator that minimizes the MSE for a certain choice of signal covariance that depends on the uncertainty region. We demonstrate that in applications, the proposed minimax MSE ratio regret approach may outperform the well-known minimax MSE approach, the minimax MSE difference regret approach, and the "plug-in" approach, where in the latter, one uses the MMSE estimator with an estimated covariance matrix replacing the true unknown covariance.     

On the pursuit and misuse of useless information.

Decision makers often pursue noninstrumental information—information that appears relevant but, if simply available, would have no impact on choice. Once they pursue such information, people then use it to make their decision. Consequently, the pursuit of information that would have had no impact on choice leads people to make choices they would not otherwise have made. The pursuit of noninstrumental information is documented and its effects on ensuing decisions are explored in a variety of social, consumer, and strategic situations. The causes and implications of this pattern are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A Competitive Mean-Squared Error Approach to Beamforming

We treat the problem of beamforming for signal estimation where the goal is to estimate a signal amplitude from a set of array observations. Conventional beamforming methods typically aim at maximizing the signal-to-interference-plus-noise ratio (SINR). However, this does not guarantee a small mean-squared error (MSE), so that on average the resulting signal estimate can be far from the true signal. Here, we consider strategies that attempt to minimize the MSE between the estimated and unknown signal waveforms. The methods we suggest all maximize the SINR but at the same time are designed to have good MSE performance. Since the MSE depends on the signal power, which is unknown, we develop competitive beamforming approaches that minimize a robust MSE measure. Two design strategies are proposed: minimax MSE and minimax regret. We demonstrate through numerical examples that the suggested minimax beamformers can outperform several existing standard and robust methods, over a wide range of signal-to-noise ratio (SNR) values. Finally, we apply our techniques to subband beamforming and illustrate their advantage in estimating a wideband signal.     

Obituary: Amos Tversky (1937–1006).

Reports the death of Amos Tversky. Tversky received his bachelor of arts from Hebrew University in Jerusalem in 1961, majoring in philosophy and psychology. He received his doctor of philosophy degree from the University of Michigan in 1965. Tversky taught at Hebrew University (1966–1978) and at Stanford University. (1978–1996), where he was the inaugural Davis-Brack Professor of Behavioral Sciences and principal investigator at the Stanford Center on Conflict and negotiation. Since 1992 he held an appointment as senior visiting professor of economics and psychology and permanent fellow of the Sackler Institute of Advanced Studies at Tel Aviv University. Tversky's main research interests were in the study of similarity, judgment under uncertainty, and decision making. His last major contribution, support theory, focused on 2 observations that had been reported in the literature: The independently judged probabilities of an event and its complement add up to approximately one, whereas the judged probabilities of separate constituents of an inclusive event usually add up to much more than the judged probability of that event. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Quality of life in younger adults (17-49) after first stroke--a two year follow-up]

To study the effect of stroke on the quality of life in younger adults, 199 patients 17-49 years of age who had sustained a first stroke between 1.11.92 and 31.10.93 were followed up. They were interviewed by telephone at 3, 6, 12 and 24 months after the event. 2 died during the first year of follow-up, and 8 had recurrent strokes. After 2 years, 8 additional patients had died and 4 had sustained recurrent events. Gradual improvement was reported within all age groups and in all areas. During the 3-6 months period, a mean of 4% improvement occurred in functional capability, 15% in social and recreational activity and 8% in return-to-work. The 6-12 month period showed an increase of 3% in improvement in mean functional capability, 10% in social and recreational activity and 2% in return-to-work. 1 year after the stroke 27% remained with moderate to severe disability, but over 86% were functionally independent in their daily living activities. There were no significant changes during the second year of follow-up in these statistics. 67% of those employed prior to their stroke returned to work and approximately 70% reported a return to prestroke social and recreational activity. These results demonstrate that the relatively high recovery rate and functional improvement during a year of follow-up were not accompanied by similar rates of improvement in employment and in social integration. They indicate the need for increased emphasis on long-term psychosocial rehabilitation services within the community.     

Optimal detection of symmetric mixed quantum states

We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly non-Abelian generating group, and show that if the generator satisfies a weighted norm constraint, then the LSM is optimal. In particular, for pure-state GU ensembles, the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets, the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.     

A community-oriented programme for rehabilitation of persons with arthritis

A community-oriented programme for rehabilitation of persons with arthritis is described. It combines an 8-week ambulatory rehabilitation course (including patient education) at a rehabilitation institution, with long-term self-rehabilitation (at patients' homes), supervised by primary-care nurses. Sixty persons (mean age 54; 80% females) with a 13-year average duration of arthritis, were suffering pain, physical limitations, difficulties in social activities, tension and a tendency to depression. They had completed the course and a 12-month self-rehabilitation at home, and were assessed with the Arthritis Impact Measurement Scales (AIMS) questionnaire on admission to the course, at its completion and 1 year thereafter. Assessment upon completion of the course showed improvement in all areas studied. Measurements at follow-up showed that the majority carried out self-rehabilitation activities and preserved the improved functional status. The programme provided a much-needed service to arthritis sufferers living in the community, and promoted both the reorientation of institution-based rehabilitation professionals towards the community and self-rehabilitation and the cooperation between the institution and primary care.