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A Variational Formula for Nonzero-Sum Stochastic Differential Games of FBSDEs and Applications

MATHEMATICAL PROBLEMS IN ENGINEERING 2014年第1期2014卷 1-9页

作者： Tang, Maoning Huzhou Univ Dept Math Sci Huzhou City 313000 Zhejiang Peoples R China

A nonzero-sum stochastic differential game problem is investigated for fully coupled forward-backward stochastic differential equations (FBSDEs in short) where the control domain is not necessarily convex. A variational formula for the cost functional in a given spike perturbation direction of control processes is derived by the Hamiltonian and associated adjoint systems. As an application, a global stochastic maximum principle of Pontryagin's type for open-loop Nash equilibrium points is established. Finally, an example of a linear quadratic nonzero-sum game problem is presented to illustrate that the theories may have interesting practical applications and the corresponding Nash equilibrium point is characterized by the optimality system. Here the optimality system is a fully coupled FBSDE with double dimensions (DFBSDEs in short) which consists of the state equation, the adjoint equation, and the optimality conditions.

关键词： NUMERICAL solutions to stochastic differential equations MATHEMATICAL formulas CALCULUS of variations forward-backward algorithm NASH equilibrium

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Bayes and Empirical Bayes Inference in Changepoint Problems

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2009年第3期38卷 419-430页

作者： Lian, Heng Nanyang Technol Univ Div Math Sci SPMS Singapore Singapore Nanyang Technol Univ Div Math Sci SPMS Singapore 637616 Singapore

We generalize the approach of Liu and Lawrence (1999) for multiple changepoint problems where the number of changepoints is unknown. The approach is based on dynamic programming recursion for efficient calculation of the marginal distribution of the data with the hidden parameters integrated out. For the estimation of the hyperparameters, we propose to use Monte Carlo EM when training data are available. The samples from the posterior obtained by our algorithm are independent, getting rid of the convergence issue associated with the MCMC approach. We illustrate our approach on limited simulations and some real data set.

关键词： Empirical Bayes forward-backward algorithm Hierarchical Bayesian model Monte Carlo EM

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Convergence analyses and comparisons of Markov chain Monte Carlo algorithms in digital communications

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2002年第2期50卷 255-270页

作者： Chen, R Liu, JS Wang, XD Univ Illinois Dept Informat & Decis Sci Chicago IL USA Harvard Univ Dept Stat Cambridge MA 02138 USA Columbia Univ Dept Elect Engn New York NY 10027 USA

Recently, Markov chain Monte Carlo (MCMC) methods have been applied to the design of blind Bayesian receivers in a number of digital communications applications. The salient features of these MCMC receivers include the following: a) They are optimal in the sense of achieving minimum symbol error rate;b) they do not require the knowledge of the channel states, nor do they explicitly estimate the channel by employing training signals or decision-feedback;and c) they are well suited for iterative (turbo) processing in coded systems. In this paper, we investigate the convergence behaviors of several MCMC algorithms (both existing and new ones) in,digital communication applications. The geometric convergence property of these algorithms is established by considering only the chains or the marginal chains corresponding to the transmitted digital symbols, which take values from a finite discrete set. We then focus on three specific applications, namely, the MCMC decoders in AWGN channels, ISI channels, and CDMA channels. The convergence rates for these algorithms are computed for small simulated datasets. Different convergence behaviors are observed. It is seen that differential encoding, parameter constraining, collapsing, and grouping are efficient ways of accelerating the convergence of the MCMC algorithms, especially in the presence of channel phase ambiguity.

关键词： Bayesian receiver duality eigenvalue forward-backward algorithm Gibbs sampler Metropolis algorithm transition function

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MAP symbol decoding in channels with error bursts

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IEEE TRANSACTIONS ON INFORMATION THEORY 2001年第5期47卷 1832-1838页

作者： Turin, W AT&T Labs Res Shannon Lab Florham Pk NJ 07932 USA

In this paper, we analyze the forward-backward algorithm for the maximum a posteriori (MAP) decoding of information transmitted over channels with memory and propose a suboptimal forward-only algorithm, We assume that both the information source and channel are described using hidden Markov models (HMMs), The algorithm lends itself to parallel implementation and pipelining. We apply the algorithm to MAP decoding of symbols which were trellis-code modulated and transmitted over channels with error bursts.

关键词： forward-backward algorithm hidden Markov models maximum aposteriori (MAP) decoding

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Exact Bayesian curve fitting and signal segmentation

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2005年第6期53卷 2160-2166页

作者： Fearnhead, P Univ Lancaster Dept Math & Stat Lancaster LA1 4YF England

We consider regression models where the underlying functional relationship between the response and the explanatory variable is modeled as independent linear regressions on disjoint segments. We present an algorithm for perfect simulation from the posterior distribution of such a model, even allowing for an unknown number of segments and an unknown model order for the linear regressions within each segment. The algorithm is simple, can scale well to large data sets, and avoids the problem of diagnosing convergence that is present with Monte Carlo Markov Chain (MCMC) approaches to this problem. We demonstrate our algorithm on standard denoising problems, on a piecewise constant AR model, and on a speech segmentation problem.

关键词： changepoints denoising forward-backward algorithm linear regression model uncertainty perfect simulation

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Conical averagedness and convergence analysis of fixed point algorithms

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第2期82卷 351-373页

作者： Bartz, Sedi Dao, Minh N. Phan, Hung M. Univ Massachusetts Dept Math Sci Kennedy Coll Sci Lowell MA 01854 USA Federat Univ Australia Sch Engn Informat Technol & Phys Sci Ballarat Vic 3353 Australia

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.

关键词： Adaptive Douglas-Rachford algorithm Cocoercivity Conically averaged operator forward-backward algorithm Proximal point algorithm Strong monotonicity Weak monotonicity

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Sequence Detection algorithms for PHY-Layer Sensing in Dynamic Spectrum Access Networks

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IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING 2011年第1期5卷 97-109页

作者： Sun, Zhanwei Bradford, Glenn J. Laneman, J. Nicholas Univ Notre Dame Dept Elect Engn Notre Dame IN 46556 USA

Spectrum sensing is a critical function for enabling dynamic spectrum access (DSA) in wireless networks that utilize cognitive radio (CR). In DSA networks, unlicensed secondary users can gain access to a licensed spectrum band as long as they do not cause harmful interfere to primary users. Spectrum sensing is subject to errors in the form of false alarms and missed detections. False alarms cause spectrum under-use by secondary users, and missed detections cause interference to primary users. Although existing research has demonstrated the utility of a Markov chain for modeling the spectrum access pattern of primary users over time, little effort has been directed toward spectrum sensing based upon such models. In this paper, we develop general sequence detection algorithms for Markov sources in noise for spectrum sensing in DSA networks. We assign different Bayesian cost factors for missed detections and false alarms, and we show that a suitably modified forward-backward sequence detection algorithm is optimal in minimizing the detection risk. Two advanced sequence detection algorithms, the complete forward algorithm and the complete forward partial backward algorithm are introduced and their performances are compared as well. Along the way, we observe new fundamental limitations on sensing performance that we term the risk floor and the window length limitation of energy detection and coherent detection that arise from mismatch of their observation window with the PU's spectrum access pattern.

关键词： Cognitive radio (CR) dynamic spectrum access (DSA) forward-backward algorithm hidden Markov model (HMM) spectrum sensing Viterbi algorithm

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Gradient computation in linear-chain conditional random fields using the entropy message passing algorithm

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PATTERN RECOGNITION LETTERS 2012年第13期33卷 1776-1784页

作者： Ilic, Velimir M. Mancev, Dejan I. Todorovic, Branimir T. Stankovic, Miomir S. Serbian Acad Arts & Sci Math Inst Belgrade 11000 Serbia Univ Nis Fac Sci & Math Nish 18000 Serbia Univ Nis Fac Occupat Safety Nish 18000 Serbia

The paper proposes a numerically stable recursive algorithm for the exact computation of the linear-chain conditional random field gradient. It operates as a forward algorithm over the log-domain expectation semiring and has the purpose of enhancing memory efficiency when applied to long observation sequences. Unlike the traditional algorithm based on the forward-backward recursions, the memory complexity of our algorithm does not depend on the sequence length. The experiments on real data show that it can be useful for the problems which deal with long sequences. (c) 2012 Elsevier B.V. All rights reserved.

关键词： Conditional random fields Expectation semiring forward-backward algorithm Gradient computation Graphical models Message passing

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Diagnosis and Prognosis of Degradation Process via Hidden Semi-Markov Model

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IEEE-ASME TRANSACTIONS ON MECHATRONICS 2018年第3期23卷 1456-1466页

作者： Liu, Tongshun Zhu, Kunpeng Zeng, Liangcai Univ Sci & Technol China Dept Automat Hefei 230026 Anhui Peoples R China Chinese Acad Sci Hefei Inst Phys Sci Inst Adv Mfg Technol Changzhou 213164 Peoples R China Wuhan Univ Sci & Technol Sch Machinery & Automat Wuhan 430081 Hubei Peoples R China

The intelligent estimation of degradation state and the prediction of remaining useful life (RUL) are important for the maintenance of industrial equipment. In this study, the degradation process of equipment is modeled as an improved hidden semi-Markov model (HSMM), in which the dependence of durations of adjacent degradation states is described and modeled in the HSMM. To avoid underflow problem in computing the forward and backward variables, a modified forward-backward algorithm is proposed in the HSMM. Based on the improved algorithm, online estimation of degradation state and the distribution of RUL can be obtained. Case studies on tool wearing diagnosis and prognosis have verified the effectiveness of this model.

关键词： Degradation process forward-backward algorithm health monitoring Hidden semi-Markov remaining useful life (RUL)

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A hidden semi-Markov model with missing data and multiple observation sequences for mobility tracking

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SIGNAL PROCESSING 2003年第2期83卷 235-250页

作者： Yu, SZ Kobayashi, H Princeton Univ Dept Elect Engn Princeton NJ 08544 USA

A hidden Markov model (HMM) encompasses a large class of stochastic process models and has been successfully applied to a number of scientific and engineering problems, including speech and other pattern recognition problems, and DNA sequence comparison. A hidden semi-Markov model (HSMM) is an extension of HMM, designed to remove the constant or geometric distributions of the state durations assumed in HMM. A larger class of practical problems can be appropriately modeled in the setting of HSMM. A major restriction is found, however, in both conventional HMM and HSMM, i.e., it is generally assumed that there exists at least one observation associated with every state that the hidden Markov chain takes on. We will remove this assumption and consider the following situations: (i) observation data may be missing for some intervals;and (ii) there are multiple observation streams that are not necessarily synchronous to each other and may have different "emission distributions" for the same state. We propose a new and computationally efficient forward-backward algorithm for HSMM with missing observations and multiple observation sequences. The required computational amount for the forward and backward variables is reduced to O(D), where D is the maximum allowed duration in a state. Finally, we will apply the extended HSMM to estimate the mobility model parameters for the Internet service provisioning in wireless networks. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： hidden Markov model (HMM) HMM with explicit duration hidden semi-Markov Model (HSMM) missing data multiple observations Ferguson algorithm forward-backward algorithm expectation-maximization (EM) algorithm maximum likelihood (ML) estimation maximum a posteriori probability (MAP) estimation mobility modeling wireless Internet service

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