Rates of Convergence of Ordinal Comparison for Dependent Discrete Event Dynamic Systems

Recent research has demonstrated that ordinal comparison, i.e., comparing relative orders of performance measures, converges much faster than the performance measures themselves do. Sometimes, the rate of convergence can be exponential. However, the actual rate is affected by the dependence among systems under consideration. In this paper, we investigate convergence rates of ordinal comparison for dependent discrete event dynamic systems. Although counterexamples show that positive dependence is not necessarily helpful for ordinal comparison, there does exist some dependence that increases the convergence rate of ordinal comparison. It is shown that positive quadrant dependence increases the convergence rate of ordinal comparison, while negative quadrant dependence decreases the rate. The results of this paper also show that the rate is maximized by using the scheme of common random numbers, a widely-used technique for variance reduction.     

Ordinal optimisation and simulation

Simulation plays a vital role in designing and analysing stochastic systems, particularly, in comparing alternative system designs with a view to optimise system performance. Using simulation to analyse complex systems, however, can be both prohibitively expensive and time consuming. Efficiency is a key concern for the application of simulation to optimisation problems. Ordinal optimisation has emerged as an effective approach to significantly improve efficiency of simulation and optimisation. Ordinal optimisation for simulation problems achieves an exponential convergence rate. There are already several success stories of ordinal optimisation. This paper introduces the idea of ordinal optimisation, and reports some recent advances in this research. It also gives details of an extension of ordinal optimisation to a class of resource application problems.     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.     

Empirical likelihood estimators for the error distribution in nonparametric regression models

The aim of this paper is to show that existing estimators for the error distribution in non-parametric regression models can be improved when additional information about the distribution is included by the empirical likelihood method. The weak convergence of the resulting new estimator to a Gaussian process is shown and the performance is investigated by comparison of asymptotic mean squared errors and by means of a simulation study.      

A heuristic method to rectify intransitive judgments in pairwise comparison matrices

This paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices. Statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a Monte-Carlo simulation, which confirms that relatively high percentage of comparison matrices, satisfying Saaty’s CR criterion are ordinally inconsistent. It is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process, in contrast with cardinal inconsistency. A heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices. The proposed algorithm generates near-optimal solutions and outperforms other tested approaches with respect to computation time.     

基于非均匀变异的进化算法对高维多峰函数的收敛性分析

对基于非均匀变异算子的进化算法的实验和机理分析已经证明了该算法模型的良好特性,最近基于非均匀变异算子的进化算法模型求解一维多峰函数问题的收敛性已经得到证明.基于马尔科夫过程理论,对基于非均匀变异算子的一般性进化算法模型和一般性高维多峰函数的收敛性给出证明,并基于典型算例与同类典型算法进行性能比较,数值试验表明算法模型具有很好的性能表现和应用前景.     

How does a stochastic optimization/approximation algorithm adapt to a randomly evolving optimum/root with jump Markov sample paths

Stochastic optimization/approximation algorithms are widely used to recursively estimate the optimum of a suitable function or its root under noisy observations when this optimum or root is a constant or evolves randomly according to slowly time-varying continuous sample paths. In comparison, this paper analyzes the asymptotic properties of stochastic optimization/approximation algorithms for recursively estimating the optimum or root when it evolves rapidly with nonsmooth (jump-changing) sample paths. The resulting problem falls into the category of regime-switching stochastic approximation algorithms with two-time scales. Motivated by emerging applications in wireless communications, and system identification, we analyze asymptotic behavior of such algorithms. Our analysis assumes that the noisy observations contain a (nonsmooth) jump process modeled by a discrete-time Markov chain whose transition frequency varies much faster than the adaptation rate of the stochastic optimization algorithm. Using stochastic averaging, we prove convergence of the algorithm. Rate of convergence of the algorithm is obtained via bounds on the estimation errors and diffusion approximations. Remarks on improving the convergence rates through iterate averaging, and limit mean dynamics represented by differential inclusions are also presented. The research of G. Yin was supported in part by the National Science Foundation under DMS-0603287, in part by the National Security Agency under MSPF-068-029, and in part by the National Natural Science Foundation of China under #60574069. The research of C. Ion was supported in part by the Wayne State University Rumble Fellowship. The research of V. Krishnamurthy was supported in part by NSERC (Canada).     

Bayesian subset selection for threshold autoregressive moving-average models

Optimal subset selection among a general family of threshold autoregressive moving-average (TARMA) models is considered. The usual complexity of model/order selection is increased by capturing the uncertainty of unknown threshold levels and an unknown delay lag. The Monte Carlo method of Bayesian model averaging provides a possible way to overcome such model uncertainty. Incorporating with the idea of Bayesian model averaging, a modified stochastic search variable selection method is adapted to consider subset selection in TARMA models, by adding latent indicator variables for all potential model lags as part of the proposed Markov chain Monte Carlo sampling scheme. Metropolis–Hastings methods are employed to deal with the well-known difficulty of including moving-average terms in the model and a novel proposal mechanism is designed for this purpose. Bayesian comparison of two hyper-parameter settings is carried out via a simulation study. The results demonstrate that the modified method has favourable performance under reasonable sample size and appropriate settings of the necessary hyper-parameters. Finally, the application to four real datasets illustrates that the proposed method can provide promising and parsimonious models from more than 16 million possible subsets.     

理想收敛的若干研究及推广

在Banach空间X中利用序列的I-收敛与I*-收敛给出理想I具可加性质(AP)的等价刻画,并进一步研究弱I-收敛、弱I*-收敛、一致弱I*-收敛之间,以及弱I-收敛与收敛之间的关系,最后基于I-λ-统计收敛给出其推广:I-A-统计收敛,并以次微分映射为工具定义一族有限可加测度,用于等价刻画I-A-统计收敛,这亦是有限可加测度的一个应用体现.     

粒子群优化的收敛分析及在广义预测控制中的应用

在进行粒子群优化的收敛性理论分析的基础上,推出了保证粒子群优化算法收敛性的参数设置区域,合理选择粒子群算法的关键参数,将粒子群优化与广义预测控制有机融合,用粒子群算法来解决广义预测控制的优化问题,提出基于粒子群优化的广义预测控制算法,通过工业过程对象的仿真并和传统的广义预测控制算法进行了对比分析,表明了该算法的有效性,特别是算法具有良好的输出跟踪精度和较强的鲁棒性.