Minimax Estimation Research Articles

Resilient distribution network planning under the severe windstorms using a risk-based approach

This paper proposes a risk-based approach for planning resilient radial distribution networks to severe windstorms. The proposed approach is based on the concept of interval analysis, Relative Distance Measure (RDM) interval arithmetic, mixed integer linear programming, and risk analysis. It enables obtaining a number of different resilient network plans taking into account dependence among the following uncertain inputs: maximum wind speed, storm duration and its annual frequency of occurrence, fragility of network components, repair duration, forecasted load and generation from renewable generators. In order to improve the computational efficiency of the proposed approach a hybrid simulated annealing and mixed integer linear programming algorithm is introduced. The best resilient plan is selected by employing the Minimal risk (Minimax) criterion for measuring and managing risk. In this way, the proposed approach provides a decision-maker with a means of determining the resilient network plan that minimizes the risk of significant costs due to severe windstorms.

Minimax Linear Estimation with the Probability Criterion under Unimodal Noise and Bounded Parameters

We consider a linear regression model with a vector of bounded parameters and a centered noise vector that has an uncertain unimodal distribution but known covariance matrix. We pose the minimax estimation problem for a linear combination of unknown parameters with the use of the probability criterion. The minimax estimate is determined as a result of minimizing a probability bound over the region of possible values of the variance and squared bias for all possible linear estimates. We establish that the resulting robust solution is less conservative in comparison with wider classes of distributions.

On estimation of $$L_{r}$$-norms in Gaussian white noise models

We provide a complete picture of asymptotically minimax estimation of $$L_r$$ -norms (for any $$r\ge 1$$ ) of the mean in Gaussian white noise model over Nikolskii–Besov spaces. In this regard, we complement the work of Lepski et al. (Probab Theory Relat Fields 113(2):221–253, 1999), who considered the cases of $$r=1$$ (with poly-logarithmic gap between upper and lower bounds) and r even (with asymptotically sharp upper and lower bounds) over Hölder spaces. We additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non-even r in terms of an investigator’s ability to produce asymptotically adaptive minimax estimators without paying a penalty.

Effect of Co-channel Interference on Two-Way Untrustworthy Relay Selection

This study investigates the impact of co-channel interference on two-way untrustworthy relay selection over Rayleigh fading environment. The investigation considers that two source terminals make information exchange over J number of full-duplex based amplify-and-forward bi-directional untrustworthy relay in the presence of co-channel interference. In addition, limited number of friendly jammers generate artificial interference/noise signals to mitigate information leakage. So as to minimize the system overhead and also enhancing secrecy performance, the relay selection is made by means of max–min criteria, which is based on maximizing the minimum secrecy rate over the relay terminals. The system model fundamental performance limits are measured by means of secrecy outage probability, secrecy achievable sum rate, intercept probability, and secrecy throughput analysis metrics. According to Monte-Carlo based simulations and numerical results, the max–min based selection strategy achieves cooperative diversity order, which is equal to total number of possible paths, in high signal-to-noise ratio regimes. Results also reveal that the co-channel interference dominates the process of the information exchange and degrades the achievable cooperative diversity order from J to 0 and leads system coding gain losses on free-interference cases in high signal-to-noise ratio regimes.

Minimax estimation of large precision matrices with bandable Cholesky factor

The last decade has witnessed significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the variables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this paper we focus on two commonly used types of parameter spaces and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found. Two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed. For the operator norm our optimal procedure is based on a novel local cropping estimator, targeting on all principle submatrices of the precision matrix, while for the Frobenius norm our optimal procedure relies on a delicate regression-based thresholding rule. Lepski’s method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model. Numerical studies are carried out to confirm our theoretical findings.

Robust Minimax Variance Estimation of Location under Bounded Distribution Interquantile Ranges

The performance of the minimax variance (in the Huber sense) robust M-estimates of a location parameter designed in the class of symmetric distributions with bounded interquantile ranges is studied both on large and small samples. Although the general structure of these minimax M-estimates has been described by Huber (1981) long ago, they were not studied as main attention was focused on the estimates designed for the various neighborhoods of a Gaussian. In this work, the proposed estimates are compared to the sample mean, sample median, and the classical robust Huber and Hampel M-estimates under the Gaussian, contaminated Gaussian, Laplace, and Cauchy distributions. The performance of an estimate is measured by its efficiency, bias, and mean squared error. The following conclusions are made: (i) under the Gaussian, Laplace, Cauchy, and moderately contaminated Gaussian distributions, the proposed minimax M-estimates outperform the robust Huber and Hampel M-estimates with respect to asymptotic efficiency; (ii) under heavily contaminated Gaussian distributions, the Huber and Hampel M-estimates are slightly better; (iii) on small samples, these classical robust estimates also slightly outperform the proposed minimax estimates in mean squared error.

Optimal, minimax and admissible two-stage design for phase II oncology clinical trials

BackgroundThe article aims to compare the efficiency of minimax, optimal and admissible criteria in Simon’s and Fleming’s two-stage design.MethodsThree parameter settings (p1-p0 = 0.25–0.05, 0.30–0.10, 0.50–0.30) are designed to compare the maximum sample size, the critical values and the expected sample size for minimax, optimal and admissible designs. Type I & II error constraints (α, β) vary across (0.10, 0.10), (0.05, 0.20) and (0.05, 0.10), respectively.ResultsIn both Simon’s and Fleming’s two-stage designs, the maximum sample size of admissible design is smaller than optimal design but larger than minimax design. Meanwhile, the expected samples size of admissible design is smaller than minimax design but larger than optimal design. Mostly, the maximum sample size and expected sample size in Fleming’s designs are considerably smaller than that of Simon’s designs.ConclusionsWhenever (p0, p1) is pre-specified, it is better to explore in the range of probability q, based on relative importance between maximum sample size and expected sample size, and determine which design to choose. When q is unknown, optimal design may be more favorable for drugs with limited efficacy. Contrarily, minimax design is recommended if treatment demonstrates impressive efficacy.

Graph-based multi-user scheduling for indoor cooperative visible light transmission.

Recent progress in the indoor visible light communication (VLC) has shown promising signs of alleviating an increasing strain on the radio frequency spectrum and enhancing transmission capacity. Nevertheless, the indoor VLC usually suffers from inter-channel interference (ICI) due to the dense light-emitting diode (LED) deployment. The ICI is considered as a key factor affecting signal to interference and noise ratio (SINR) and spectral efficiency. To address this challenge, an efficient multi-user scheduling framework that employs interference coordination and cooperative transmission is investigated based on the graph theory. To effectively mitigate ICI and maximize benefit of the cooperative transmission, the cell-centric (CC) and user-centric (UC) clustering are introduced for cooperative transmission. For the CC clustering, the multi-user scheduling problem under the proportional fairness criterion is formulated to maximize spectral efficiency while ensuring user fairness. Such a problem is solved by linear programming and greedy algorithms after transforming it into a maximum weighted independent set problem with the aid of a modified interference graph. For the UC clustering, the multi-user scheduling problem under the max-min criterion is formulated and solved by a proposed heuristic approach based on the bipartite graph theory. Numerical results show that the proposed graph-based scheduling is capable of providing up to 7.7 dB gain in SINR over the non-cooperative transmission. The bipartite graph scheduling offers high spectral efficiency and service fairness index. In the worst case with an occlusion probability of 1, a small SINR penalty of up to 1 dB is observed with the greedy algorithm. It is shown that the graph-based scheduling is robust to receiver rotation and occlusion in terms of spectral efficiency, SINR, and user fairness.

Optimal Design of a Composite Reinforced by Unidirectional Fibers

On the basis of a minimax criterion and stress intensity factor minimization, the optimal preload is determined for the case of elastic fibers fit into a doubly periodic system of holes in an isotropic elastic binder. A selection criterion is proposed for an interference fit that prevents the fracture of a composite reinforced with unidirectional fibers and ensures optimal stress distribution.

Methods of searching for systems of complex nonlinear discrete signals optimal by the minimax criterion

Among the main directions of improving information security indicators, noise immunity and secrecy of information and communication systems (ICS), we can single out areas related to the use of channels with high frequency redundancy, significant spatial, structural, energy and temporal secrecy. To ensure frequency redundancy at the physical level, discrete signals are widely used, in which the manipulated parameters (amplitude, phase, frequency) change at strictly fixed time intervals. The law of variation of these parameters is set by discrete sequences that completely determine the properties of discrete signals and are often identified with them. In the synthesis of signal systems (for use in protected ICS), they strive to provide certain properties of signals, first of all: a given structural secrecy in determining the laws of signal formation; improved ensemble properties (existence for almost any period value, a significant amount of the signal system); necessary (to ensure the desired value of the noise immunity of the reception) correlation properties. Based on the criteria of maximum likelihood, minimax criterion, the fundamental boundary of the Cramer-Rao assessment, the requirements to the choice of signal systems for a wide range of multi-user ICS applications are formulated. In particular, large ensembles of nonlinear complex signals are proposed as signals — physical data carriers in ICS. It is shown that such signals have improved (in comparison with the widely used classes of linear signals) ensemble, correlation, structural and other properties. The aforementioned allows to improve such performance indicators of the ICS as noise immunity, electromagnetic compatibility, secrecy and information security, which is very important for some general and critical ICS applications. The possibility of using the proposed signal systems in solving classical problems of optimal reception is shown: detection and discrimination of signals, estimation of signal parameters. In this case (due to the good correlation properties of the proposed signal systems), the necessary (for the corresponding tasks) indicators of noise immunity of signal reception and the accuracy of the estimation of signal parameters are provided.

Double-channel resistance-to-voltage converter for cable teraohmmeters

The paper considers a teraohmmeter resistance converter to monitor cable insulation with an additional input amplifier that emits a low-frequency interference signal. Adaptive algorithms for a double-channel converter circuit to compensate for low-frequency interference are proposed. There are considered algorithms using minimax criteria and linear approximation method for estimation of interference influence. It is shown classification of algorithms according to industrial frequency interference filtering method and signal observation interval. There were investigated two ways of interference application: step signal from a DC voltage source up to 300 V and fading harmonic signal from an AC voltage source and amplifier up to 300 V. A doublechannel circuit of the resistance-to-voltage converter is found to provide a 2-fold increase in the signal-tonoise ratio in comparison with a single-channel circuit. It is shown that the maximum deviation of readings for the single-channel circuit exceeds 20 % (up to 32 %) in short-term exposure to interference with amplitude of up to 300 V. At the same time, the maximum deviation for the double-channel circuit can attain 17 %, but it does not exceed 20 %. According to GOST 3345–76, the insulation resistance measuring error in therange of 10 GΩ to 100 TΩ should not exceed 20 %.The advantage of the proposed double-channel converter is the possibility to develop new algorithms to eliminate the dependence of readings on interference effects.

A Risk Management Procedure for Island Partitioning of Automated Radial Distribution Networks With Distributed Generators

This paper presents an approach to deal with the uncertainty of generation and load in the optimal island partitioning of the automated radial distribution networks. The proposed approach is based on the concept of interval analysis, Relative Distance Measure (RDM) interval arithmetic, mixed integer linear programming with interval coefficients, and risk management tools. It defines a number of different island partitioning scenarios taking into account dependence (correlation) between loads and DGs. The best island partitioning scenario is obtained by employing the Minimal risk (Minimax) criterion for risk measurement. In this way, the presented approach provides a tool for a decision-maker to determine the island partitioning scenario that minimizes the risk of unsuccessful island operations in the uncertain environment.

A map of ecologically rational heuristics for uncertain strategic worlds.

The ecological rationality of heuristics has been extensively investigated in the domain of individual decision making. In strategic decision making, however, the focus has been on repeated games, and there is a lack of research on 1-shot games, where opponents and the game itself can vary from one interaction to another. Mapping the performance of simple versus more complex decision policies (or strategies) from the experimental game theory literature is an important first step in this direction. We investigate how 10 policies fare conditional on strategic properties of the games and 2 classes of uncertainty. The strategic properties are the complexity (number of actions) and the degree of harmony (competitiveness) of the games. The first class of uncertainty is environmental (or payoff) uncertainty, arising from missing payoff values. The second class is strategic uncertainty about the type of opponent a player is facing. Policies' performance was measured by 3 criteria: a mean criterion averaging over the whole set of opponent policies, a maxmin criterion capturing the worst-case scenario and another criterion measuring robustness to different distributions of opponent policies. Heuristics performed well and were more robust than complex policies such as pure-strategy Nash equilibria, while simultaneously requiring significantly less information and fewer computational resources. Our ranking of the decision policies' performance was closely aligned to their prevalence in experimental studies of games. In particular, the Level-1 policy, which completely ignores an opponent's payoffs and uses equal weighting to determine the expected payoffs of different actions, exhibited a robust beauty. (PsycINFO Database Record (c) 2020 APA, all rights reserved).

ABER performance investigation of LDPC-coded multi-hop parallel underwater wireless optical communication system.

The average bit error rate (ABER) performance of a low-density parity-check (LDPC)-coded multi-hop parallel underwater wireless optical communication (UWOC) system is investigated with the combined effects of absorption, scattering, the misalignment characterized by the beam spread function, and the ocean turbulence-induced fading modeled by log-normal distribution. With the max-min criterion as the best path selection scheme, the cumulative distribution function for identically and independently distributed and non-identically and independently distributed links are derived, respectively. Then, the analytical ABER expressions of binary phase shift keying and multiple phase shift keying subcarrier intensity modulation schemes are deduced with the help of the Gauss-Laguerre quadrature rule, and they are also confirmed by Monte Carlo simulation. In addition, LDPC codes are applied in the simulation to improve the system performance. The results show that the combined degrading effects are mainly limited by the link length, especially under the coastal ocean condition. And the multi-hop parallel transmission demonstrates good ABER performance and can expand the communication range in ocean. Furthermore, LDPC codes can significantly improve the ABER performance of the UWOC system, and the coding gain is strongly affected by channel conditions and the corresponding parameters of LDPC codes. This work is beneficial for the UWOC system design.

Controllability and actuator placement optimization for active damping of a thin rotating ring with piezo-patch transducers

A study on optimal actuator placement for controlling flexural vibration of a thin rotating ring is reported. Piezoelectric patch actuators and sensors may be applied with feedback control to provide active damping of structural vibration involving circumferential travelling waves. To determine the optimum patch positions, a model-based cost function is defined involving a time-weighted controllability Gramian, with balanced model realization for combined treatment of sensors and actuators in collocated pairs. Analytical and numerical results indicate that, without rotation, at least two actuator/sensor pairs are required to achieve controllability of any given set of natural vibration modes. Also, the optimal angular separation varies with rotational speed due to the combined influence of initial damping and Coriolis forces on the vibratory dynamics. To obtain a practical solution, a finite range of speeds is considered within a mini-max optimization criterion for the placement problem. Experiments have been conducted on a thin steel ring where a synthetic proof-mass-damper control law was used to suppress vibration involving the six lowest frequency vibration modes (with natural frequencies of 161, 442 and 846 Hz without rotation). The results show that, although a single actuator-sensor pair can achieve improved damping at high rotational speeds, the optimized configuration of two pairs provides effective damping over the full range of operating speeds.

Minimax Nonparametric Estimation on Maxisets

We study nonparametric estimation of a signal in Gaussian white noise on maxisets. We point out minimax estimators in the class of all linear estimators and strong asymptotically minimax estimators in the class of all estimators. We show that balls in Sobolev spaces are maxisets for the Pinsker estimators.

Анализ инвестиционных рисков на фондовом рынке на основе критерия минимакса

Анализ инвестиционных рисков на фондовом рынке на основе критерия минимакса

ОБЧИСЛЕННЯ РЕЗУЛЬТУЮЧОГО РАНЖУВАННЯ АЛЬТЕРНАТИВ НА ОСНОВІ ВИКОРИСТАННЯ НЕПОВНИХ ЕКСПЕРТНИХ РАНЖУВАНЬ

Incomplete information is a characteristic feature of organizational systems. Incomplete data accompanies the decision-maker in all components of corporate security, namely the management of the organization, staff activities, company assets, implemented business processes, information and other resources, financial resources, used technologies, the company's reputation, etc. Nevertheless, a reasonable decision should be made. In particular, a common practical task is to rank alternatives of different nature. This is done by experts of high competence within the areas of responsibility. Naturally, there is a situation of decision-making with incomplete data, on the basis of which it is necessary to find a complete resulting ranking of alternatives, which best approximates the information obtained from experts, ie is in some sense closest to the given incomplete expert rankings. In order to compare different ways to achieve the resulting ranking of alternatives, the formalization of the problem in the classes of single-criteria and multicriteria models for the metrics of Cook, Heming, Euclid and Litvak is considered. To solve the problems that arise in a situation of incomplete information, a number of heuristics that are empirical methodological rules that help to find solutions and contribute to the definition of mathematically incorrect problems are introduced. The notion of the modified Litvak median and the Litvak compromise median, which is used using the minimax criterion, is introduced. The algorithms developed by the authors for determining the medians of expert rankings of alternatives, namely the genetic algorithm and the heuristic algorithm are described. To illustrate the results the schemes of the genetic algorithm are given. The main results of the application of the described algorithms, which illustrate the efficiency of their application to ranking problems, that are characterized by incomplete information are given.

ИГРОВАЯ МОДЕЛЬ ИНФОРМАЦИОННОГО КОНФЛИКТА СИСТЕМЫ СВЯЗИ С СИСТЕМОЙ ДЕСТАБИЛИЗИРУЮЩИХ ВОЗДЕЙСТВИЙ

The article proposes a model of information conflict between the application specific communications system and the system of destabilizing effects based on the game theory formalism. The model takes into account strategies of destabilizing effects and of communications system protection, as well as their change during the information conflict. It shows that based on the game with direct or mixed strategies, it is possible to substantiate the most probably variants of destabilizing effects, as well as the rational structure of communications system in the conflict. The authors propose decision-making formalized by the opposing parties, when choosing strategies at each stage of the information conflict. The paper describes the rules for choosing the strategy of protection and corresponding structure of the communications system in the form of a minimax criterion. These rules allow searching for a rational structure of the communications system both in the current and in the preventive decision-making for countering the system of destabilizing effects.

АДАПТИВНОЕ МИНИМАКСНОЕ ОЦЕНИВАНИЕ ВИДЕОИНФОРМАЦИИ В ЗАДАЧАХ АЭРОКОСМИЧЕСКОГО МОНИТОРИНГА

This article is dedicated to the problem of increasing the performance of aerospace monitoring and automatic video-data processing under prior uncertainty. The development of adaptive minimax estimate for the image model parameters is proposed, and the computational algorithm for adaptive estimation of video-data is described.