Stochastic approximation algorithms for the local optimization of functions with nonunique stationary points

The aim of this paper is the provision of a framework for a practical stochastic unconstrained optimization theory. The results are based on certain concepts of stochastic approximation, although not restricted to those procedures, and aim at incorporating the great flexibility of currently available deterministic optimization ideas into the stochastic problem, whenever optimization must be done by Monte Carlo or sampling methods. Hills with nonunique stationary points are treated. A framework has been provided, with which convergence of stochastic versions of conjugate gradient, partan, etc., can be discussed and proved.     

Linear estimation of ARMA processes

The method of estimating ARMA parameters first discussed by Mayne and Firoozan is investigated. It is shown that if, at the first stage, the order of the fitted autoregression is allowed to depend on the number of time points, in a reasonable manner, then it will still be true that the final estimate of the parameter vector will converge, almost surely, to the true value. This is to be compared to the result in the original paper where the order is fixed and it is shown that, as the sample size increases, the estimate converges to a value which, if the order of the autoregression is high enough, will be arbitrarily near to the true value. Some comments are made on other extensions, on the law of the iterated logarithm, on the central limit theorem and on the choice of the order of the fitted autoregression. The innovation sequence need not be Gaussian and for the convergence result only a natural condition relating to prediction needs be imposed.     

Linear identification of ARMA processes

A new method for estimating the parameters of an ARMA process is presented. The method consists of three linear least-squares estimations. In the first an autoregressive model is fitted to the observation sequence, yielding an estimate of the values of the driving white noise sequence. Linear least squares is then used to fit an ARMA model to the observation and estimated white noise sequences. This model is used to filter the observation and estimated white noise sequences. Finally an ARMA model is fitted to the filtered sequences. It is shown that the resultant estimator is ‘p-consistent’ (the asymptotic bias tends to zero as the degree p of the autoregressive model tends to infinity) and is ‘p-efficient’ (the asymptotic efficiency approaches the theoretical maximum as p tends to infinity).     

A method for adaptive estimation of ARMA processes

When the noise process in adaptive identification of linear stochastic systems is correlated, and can be represented by a moving average model, extended least squares algorithms are commonly used, and converge under a strictly positive real (SPR) condition on the noise model. In this paper, we present an adaptive algorithm for the estimation of autoregressive moving average (ARMA) processes, and show that it is convergent without any SPR condition, and has a convergence rate of O({loglog t)/t}^1/2).     

Convergence analysis of ladder algorithms for AR and ARMA models

A class of exact fast algorithms originally introduced in the signal processing area is provided by the so-called recursive least squares ladder forms. The many nice numerical and structural properties of these algorithms have made them a very powerful alternative in a large variety of applications, yet the convergence properties of the algorithms have not received the necessary attention. This paper gives an asymptotic analysis of two ladder algorithms, designed for autoregressive (AR) and autoregressive moving average (ARMA) models. Convergence is studied based on the stability properties of an associated differential equation. It is shown that the convergence conditions obtained for the algorithms parallel those known for prediction error methods and for a particular type of pseudo-linear regression.     

Two stochastic approximation procedures for identifying linear systems

A Robbins-Monro [1] stochastic approximation procedure for identifying a finite memory time-discrete time-stationary linear system from noisy input-output measurements is developed. Two algorithms are presented to sequentially identify the linear system. The first one is derived, based on the minimization of the mean-square error between the unknown system and a model, and is shown to develop a bias which depends only on the variance of the input signal measurement error. Under the assumption that this variance is known a priori, a second algorithm is developed which, in the limit, identifies the unknown system exactly. The case when the covariance matrix of the random input sequence is not positive definite is also considered.     

Adaptive prediction for ARMA processes with Markov switching parameters

The problem of prediction for ARMA processes with switching parameters modelled as a finite-state Markov chain is considered. The Markov transition probability matrix is assumed to be unknown but constant and can take values only from a finite collection which contains the true transition matrix. A multiple-model prediction method is presented. The digital simulation shows a good performance of the proposed predictor.     

Improved deterministic approximation algorithms for Max TSP

We present an O(n³)-time approximation algorithm for the maximum traveling salesman problem whose approximation ratio is asymptotically , where n is the number of vertices in the input complete edge-weighted (undirected) graph. We also present an O(n³)-time approximation algorithm for the metric case of the problem whose approximation ratio is asymptotically . Both algorithms improve on the previous bests.     

Recursive identification for multidimensional ARMA processes with increasing variances

~~Recursive identification for multidimensional ARMA processes with increasing variances1. Anderson, T. W., The statistical Analysis of Time Series, New York: Wiley, 1971. 2. Brockwell, P. T., Davis, R. A., Time Series: Theory and Methods, New York: Springer, 1987. 3. Caines, P. E., Linear Stochastic Systems, New York: Wiley, 1988. 4. Hanann, E. J., Diestler, M., The Statistical Theory of Linear Systems, New York: Wiley, 1988. 5. Ljung, L., Soderstrom, T. S., T…     

Stochastic algorithms for robustness of control performances

In recent years, there has been a growing interest in developing statistical learning methods to provide approximate solutions to “difficult” control problems. In particular, randomized algorithms have become a very popular tool used for stability and performance analysis as well as for design of control systems. However, as randomized algorithms provide an efficient solution procedure to the “intractable” problems, stochastic methods bring closer to understanding the properties of the real systems. The topic of this paper is the use of stochastic methods in order to solve the problem of control robustness: the case of parametric stochastic uncertainty is considered. Necessary concepts regarding stochastic control theory and stochastic differential equations are introduced. Then a convergence analysis is provided by means of the Chernoff bounds, which guarantees robustness in mean and in probability. As an illustration, the robustness of control performances of example control systems is computed.     

基于ARMA的微惯性传感器随机误差建模方法

针对微惯性传感器随机误差建模效果不理想,影响微惯性组合导航系统性能的问题,提出了采用自回归滑动平均(ARMA)对微惯性传感器随机误差进行建模的方法。通过对随机误差模型应用于微惯性器件误差建模的深入分析,将Yule-Walker方程引入线性预测问题中,实现AR功率谱密度的估计,建立了基于随机过程有理功率谱密度的ARMA模型建立方法,并给出了ARMA建模准确性的LDA验证准则。通过微惯性传感器实测数据,对随机误差建模方法进行了有效性验证。该方法为微惯性器件的随机误差建模和分析提供了一种新的途径。     

Stochastic approximation methods for identification and control--A survey

Stochastic search techniques have been the essential part for most identification and self-organizing or learning control algorithms for stochastic systems. Stochastic approximation search algorithms have been very popular among the researchers in these areas because of their simplicity of implementation, convergence properties, as well as intuitive appeal to the investigator. This paper presents an exposition of the stochastic approximation algorithms and their application to various parameter identification and self-organizing control algorithms.     

Improvements on approximation algorithms for clustering probabilistic data

Knowledge and Information Systems - Uncertainty about data appears in many real-world applications and an important issue is how to manage, analyze and solve optimization problems over such data....     

Improved approximation algorithms for the Max Edge-Coloring problem

The Max Edge-Coloring problem asks for a proper edge-coloring of an edge-weighted graph minimizing the sum of the weights of the heaviest edges in the color classes. In this paper we present a PTAS for trees and a 1.74-approximation algorithm for bipartite graphs; we also adapt the last algorithm to one for general graphs of the same, asymptotically, approximation ratio.     

Heuristic evaluation techniques for bin packing approximation algorithms

Performance analysis of approximation algorithms purposes to give upper bounds on the ratio «approximate solution cost»/«optimal solution cost». We say that standard analysis brings to «a priori» evaluation since the bounds it provides refer to the overall worst-case performance: really those evaluations do not take into account useful information relevant to the performance that approximate solution might contain themselves. In this paper we show how the performances of the Any Fit Decreasing algorithms for Bin Packing can be evaluated more precisely than it is done in the standard «a priori» analysis, by means of demonstration techniques of «a posteriori» analysis. In fact, we show how approximate solutions of Bin Packing given by Any Fit Decreasing algorithms can be put in three classes of performances. Belongings to the first class imply really optimality of solutions (those events could remain unspected if one clings only to the 5/4 bound, as forecast by standard «a priori» evaluations). Also belonging to the second class of performances may involve remarkable revaluation of solutions in hand, although optimality cannot be recognized; an inequality is provided which permits to compute upper bounds on the evaluation ratios. The same inequality permits to recognize approximate solutions which could not be improved neither by First Fit nor Best Fit Decreasing algorithms. The third class refers to those performances for which the considered levels of inspection cannot offer a more precise evaluation that the «a priori» one.