An iterative numerical algorithm for modeling a class of Wiener nonlinear systems

This letter presents an iterative estimation algorithm for modeling a class of output nonlinear systems. The basic idea is to derive an estimation model and to solve an optimization problem using the gradient search. The proposed iterative numerical algorithm can estimate the parameters of a class of Wiener nonlinear systems from input–output measurement data. The proposed algorithm has faster convergence rates compared with the stochastic gradient algorithm. The numerical simulation results indicate that the proposed algorithm works well.     

Optimal Control Approach to Nonlinear Diffusion Equations Driven by Wiener Noise

The stochastic nonlinear infinite-dimensional equations of gradient type and with additive Wiener noise can be reduced to an optimal convex control problem via Brezis–Ekeland duality device. This approach is illustrated here on a few classes of nonlinear stochastic parabolic equations which are relevant as diffusion models under stochastic Gaussian perturbations, and image restoring technique.     

On the Convergence of the P-Algorithm for One-Dimensional Global Optimization of Smooth Functions

The Wiener process is a widely used statistical model for stochastic global optimization. One of the first optimization algorithms based on a statistical model, the so-called P-algorithm, was based on the Wiener process. Despite many advantages, this process does not give a realistic model for many optimization problems, particularly from the point of view of local behavior. In the present paper, a version of the P-algorithm is constructed based on a stochastic process with smooth sampling functions. It is shown that, in such a case, the algorithm has a better convergence rate than in the case of the Wiener process. A similar convergence rate is proved for a combination of the Wiener model-based P-algorithm with quadratic fit-based local search.     

Gradient based iterative parameter identification for Wiener nonlinear systems

This paper focuses on the identification problem of Wiener nonlinear output error systems. The application of the key-term decomposition technique provides a special form of the Wiener model with polynomials, where all the model parameters to be estimated are separated. To solve the identification problem of Wiener nonlinear output error systems with the unmeasurable variables in the information vector, an auxiliary model-based gradient iterative algorithm is presented by replacing the unmeasurable variables with their corresponding iterative estimates. The performances of the proposed algorithm are analyzed and compared by using numerical examples.     

A variable step-size control algorithm for the weak approximation of stochastic differential equations

In this paper, we propose two local error estimates based on drift and diffusion terms of the stochastic differential equations in order to determine the optimal step-size for the next stage in an adaptive variable step-size algorithm. These local error estimates are based on the weak approximation solution of stochastic differential equations with one-dimensional and multi-dimensional Wiener processes. Numerical experiments are presented to illustrate the effectiveness of this approach in the weak approximation of several standard test problems including SDEs with small noise and scalar and multi-dimensional Wiener processes.     

梯度法简述

梯度法是一类求解优化问题的一阶方法。梯度法形式简单、计算开销小,在大规模问题的求解中得到了广泛应用。系统地介绍了光滑无约束问题梯度法的迭代格式、理论框架。梯度法中最重要的参数是步长,步长的选取直接决定了梯度法的收敛性质与收敛速度。从线搜索框架、近似技巧、随机技巧和交替重复步长四方面介绍了梯度步长的构造思想及相应梯度法的收敛性结果,还对非光滑及约束问题的梯度法、梯度法加速技巧和随机梯度法等扩展方向做了简要介绍。     

A sparsity preserving stochastic gradient methods for sparse regression

We propose a new stochastic first-order algorithm for solving sparse regression problems. In each iteration, our algorithm utilizes a stochastic oracle of the subgradient of the objective function. Our algorithm is based on a stochastic version of the estimate sequence technique introduced by Nesterov (Introductory lectures on convex optimization: a basic course, Kluwer, Amsterdam, 2003). The convergence rate of our algorithm depends continuously on the noise level of the gradient. In particular, in the limiting case of noiseless gradient, the convergence rate of our algorithm is the same as that of optimal deterministic gradient algorithms. We also establish some large deviation properties of our algorithm. Unlike existing stochastic gradient methods with optimal convergence rates, our algorithm has the advantage of readily enforcing sparsity at all iterations, which is a critical property for applications of sparse regressions.     

A combined direction stochastic approximation algorithm

The stochastic approximation problem is to find some root or minimum of a nonlinear function in the presence of noisy measurements. The classical algorithm for stochastic approximation problem is the Robbins-Monro (RM) algorithm, which uses the noisy negative gradient direction as the iterative direction. In order to accelerate the classical RM algorithm, this paper gives a new combined direction stochastic approximation algorithm which employs a weighted combination of the current noisy negative gradient and some former noisy negative gradient as iterative direction. Both the almost sure convergence and the asymptotic rate of convergence of the new algorithm are established. Numerical experiments show that the new algorithm outperforms the classical RM algorithm.     

基于信息压缩矩阵特征值映射的时变系统UD分解辨识算法

在系统辨识领域遗忘因子UD分解算法(一种通过对系统数据矩阵进行UD分解的在线辨识算法)具有对时变系统阶次和参数同步估计的优异性能,但传统的遗忘策略不能从根本上解决信息压缩矩阵数据过饱和问题,为了拓展现有UD分解算法在时变系统的适用范围,同时针对数据空间分布不均匀性,提出一种基于信息压缩矩阵特征值映射的UD分解辨识算法....     

球面上$\ell_1$正则优化的随机临近梯度方法

本文研究球面上的$\ell_1$正则优化问题,其目标函数由一般光滑函数项和非光滑$\ell_1$正则项构成,且假设光滑函数的随机梯度可由随机一阶oracle估计.这类优化问题被广泛应用在机器学习,图像、信号处理和统计等领域.根据流形临近梯度法和随机梯度估计技术,提出一种球面随机临近梯度算法.基于非光滑函数的全局隐函数定理,分析了子问题解关于参数的Lipschtiz连续性,进而证明了算法的全局收敛性.在基于随机数据集和实际数据集的球面$\ell_1$正则二次规划问题、有限和SPCA问题和球面$\ell_1$正则逻辑回归问题上数值实验结果显示所提出的算法与流形临近梯度法、黎曼随机临近梯度法相比CPU时间上具有一定的优越性.     

Recursive computational formulas of the least squares criterion functions for scalar system identification

The paper discusses recursive computation problems of the criterion functions of several least squares type parameter estimation methods for linear regression models, including the well-known recursive least squares (RLS) algorithm, the weighted RLS algorithm, the forgetting factor RLS algorithm and the finite-data-window RLS algorithm without or with a forgetting factor. The recursive computation formulas of the criterion functions are derived by using the recursive parameter estimation equations. The proposed recursive computation formulas can be extended to the estimation algorithms of the pseudo-linear regression models for equation error systems and output error systems. Finally, the simulation example is provided.     

A new hybrid stochastic approximation algorithm

In this paper, we consider optimizing the performance of a stochastic system that is too complex for theoretical analysis to be possible, but can be evaluated by using simulation or direct experimentation. To optimize the expected performance of such system as a function of several input parameters, we propose a hybrid stochastic approximation algorithm for finding the root of the gradient of the response function. At each iteration of the hybrid algorithm, alternatively, either an average of two independent noisy negative gradient directions or a scaled noisy negative gradient direction is selected. The almost sure convergence of the hybrid algorithm is established. Numerical comparisons of the hybrid algorithm with two other existing algorithms in a simple queueing system and five nonlinear unconstrained stochastic optimization problems show the advantage of the hybrid algorithm.     

A weak stochastic integral in Banach space with application to a linear stochastic differential equation

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved.A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.Research supported by National Science Foundation under Grant No. ECS-8005960.     

Various topologies in the Wiener space and Lévy's stochastic area

Summary In this paper, we observe how Lévy's stochastic area looks when we see it through various topologies in the Wiener space. Our theorem implies that it is quite natural from the viewpoint of topology to define a distinct skeleton of Lévy's stochastic areaS(w) for each distinct topology in the Wiener space, or equivalently, for each distinct abstract Wiener space on which the Wiener measure andS(w) are realized. Thus we cannot determine its intrinsic skeleton in the theory of abstract Wiener spaces.     

Multi-innovation stochastic gradient algorithms for dual-rate sampled systems with preload nonlinearity

Since the stochastic gradient algorithm has a slower convergence rate, this letter presents a multi-innovation stochastic gradient algorithm for a class of dual-rate sampled systems with preload nonlinearity. The basic idea is to transform the dual-rate system model into an identification model which can use dual-rate data by using the polynomial transformation technique. A simulation example is provided to verify the effectiveness of the proposed method.     

Modeling of degradation data via wiener stochastic process based on acceleration factor constant principle

This paper proposes a systematic method of modeling accelerated degradation data based on the acceleration factor constant principle. Wiener stochastic process is considered because it is the most extensively used for degradation modeling. For the Wiener stochastic processes with three different time functions, the parameter relationships, which should be satisfied under any two different stress levels, are deduced according to the acceleration factor constant principle. The deduced parameter relationships indicate the stress-related parameters, which are applied to establish accurate accelerated degradation models. In addition, the deduced parameter relationships provide a guidance to test the consistency of the degradation mechanisms under different stress levels. A hypothesis method based on Analysis of Variance is adopted to identify the accelerated stress levels with different degradation mechanism. The degradation data under these stress levels should not be used to assess the product's reliability. The methods of validating accelerated degradation models and reliability assessments are also proposed. The simulation results prove the feasibility and effectiveness of the proposed methods. From the numerical example, it is concluded that the accelerated degradation model established based on the acceleration factor constant principle is more credible and accurate.     

实值可料过程关于模糊集值Wiener过程的伊藤积分

通过利用水平集和承集将实值可料过程关于模糊集值W iener过程的伊藤积分与实值可料过程关于可积有界紧凸集值W iener过程的伊藤积分联系起来,给出相应的定义和性质,并初步探讨了该伊藤积分在随机微分方程方面的应用。     

Stochastic optimization on Bayesian nets

In this paper we are concerned with stochastic optimization problems in the case when the joint probability distribution, associated with random parameters, can be described by means of a Bayesian net. In such a case we suggest that the structured nature of the probability distribution can be exploited for designing efficient gradient estimation algorithm. Such gradient estimates can be used within the general framework of stochastic gradient (quasi-gradient) solution procedures in order to solve complex non-linear stochastic optimization problems. We describe a gradient estimation algorithm and present a case study related to the reliability of semiconductor manufacturing together with numerical experiments.     

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Itô integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.     

The residual based interactive stochastic gradient algorithms for controlled moving average models

This paper derives a residual based interactive stochastic gradient (ISG) parameter estimation algorithm for controlled moving average (CMA) models and studied the performance of the residual based ISG algorithm under weaker conditions on statistical properties of the noise. Compared with the residual based extended stochastic gradient algorithm for identifying CMA models, the proposed ISG algorithm can give highly accurate parameter estimates by the simulation example.