Factorial Hidden Markov Models

Hidden Markov models (HMMs) have proven to be one of the most widely used tools for learning probabilistic models of time series data. In an HMM, information about the past is conveyed through a single discrete variable—the hidden state. We discuss a generalization of HMMs in which this state is factored into multiple state variables and is therefore represented in a distributed manner. We describe an exact algorithm for inferring the posterior probabilities of the hidden state variables given the observations, and relate it to the forward–backward algorithm for HMMs and to algorithms for more general graphical models. Due to the combinatorial nature of the hidden state representation, this exact algorithm is intractable. As in other intractable systems, approximate inference can be carried out using Gibbs sampling or variational methods. Within the variational framework, we present a structured approximation in which the the state variables are decoupled, yielding a tractable algorithm for learning the parameters of the model. Empirical comparisons suggest that these approximations are efficient and provide accurate alternatives to the exact methods. Finally, we use the structured approximation to model Bach's chorales and show that factorial HMMs can capture statistical structure in this data set which an unconstrained HMM cannot.     

Reduced Stochastic Models for Complex Molecular Systems

We present a new numerical method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The approach is based on the representation of the effective dynamics of the full system by a Markov jump process between metastable states, and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of hidden Markov models (HMMs) with output behavior given by SDEs. A first complete algorithm including an explicit Euler–Maruyama-based likelihood estimator has already been presented in Horenko et al. (MMS, 2006a). Herein, we present a semi-implicit exponential estimator that, in contrast to the Euler–Maruyama-based estimator, also allows for reliable parameter optimization for time series where the time steps between single observations are large. The performance of the resulting method is demonstrated for some generic examples, in detail compared to the Euler–Maruyama-based estimator, and finally applied to time series originating from a 100 ns B-DNA molecular dynamics simulation.Dedicated to Peter Deuflhard on the occassion of his sixtieth birthday.Supported by the SfB 450 and DFG research center “Mathematics for key technologies” (FZT 86) in Berlin.     

Sequential Activity Profiling: Latent Dirichlet Allocation of Markov Chains

To provide a parsimonious generative representation of the sequential activity of a number of individuals within a population there is a necessary tradeoff between the definition of individual specific and global representations. A linear-time algorithm is proposed that defines a distributed predictive model for finite state symbolic sequences which represent the traces of the activity of a number of individuals within a group. The algorithm is based on a straightforward generalization of latent Dirichlet allocation to time-invariant Markov chains of arbitrary order. The modelling assumption made is that the possibly heterogeneous behavior of individuals may be represented by a relatively small number of simple and common behavioral traits which may interleave randomly according to an individual-specific distribution. The results of an empirical study on three different application domains indicate that this modelling approach provides an efficient low-complexity and intuitively interpretable representation scheme which is reflected by improved prediction performance over comparable models.Mark Girolami is a Reader in the Department of Computing Science at the University of Glasgow. In 2000 he was the TEKES visiting professor at the Laboratory of Computing and Information Science in Helsinki University of Technology. In 1998 and 1999 Dr. Girolami was a research fellow at the Laboratory for Advanced Brain Signal Processing in the Brain Science Institute, RIKEN, Wako-Shi, Japan. He has been a visiting researcher at the Computational Neurobiology Laboratory (CNL) of the Salk Institute. This year (2005) he will take up an MRC funded Discipline Hopping Award in the Department of Biochemistry. Mark Girolami holds a degree in Mechanical Engineering from the University of Glasgow (1985), and a Ph.D. in Computing Science from the University of Paisley (1998). Dr. Girolami was a development engineer with IBM from 1985 until 1995 when he left to pursue an academic career.Ata Kabán received the B.Sc. degree with honours (1999) in computer science from the University “Babes-Bolyai” of Cluj-Napoca, Romania, and the Ph.D. degree in computer science (2001) from the University of Paisley, UK. She is a lecturer in the School of Computer Science of the University of Birmingham. Her current interests concern probabilistic modelling, machine learning and their applications to automated data analysis. She has been a visiting researcher at Helsinki University of Technology (June–December 2000 and in the summer of 2003). Prior to her career in Computer Science, she received the B.A. degree in musical composition (1994) and the M.A. (1995) and Ph.D. (1999) degrees in musicology from the Music Academy “Gh. Dima” of Cluj-Napoca, Romania.     

Multivariate time series anomaly detection: A framework of Hidden Markov Models

In this study, we develop an approach to multivariate time series anomaly detection focused on the transformation of multivariate time series to univariate time series. Several transformation techniques involving Fuzzy C-Means (FCM) clustering and fuzzy integral are studied. In the sequel, a Hidden Markov Model (HMM), one of the commonly encountered statistical methods, is engaged here to detect anomalies in multivariate time series. We construct HMM-based anomaly detectors and in this context compare several transformation methods. A suite of experimental studies along with some comparative analysis is reported.     

Stochastic stability analysis of integral non-homogeneous Markov jump systems

This paper investigates the problem of stability analysis for time-delay integral Markov jump systems with time-varying transition rates. Some free-weight matrices are addressed and sufficient conditions are established under which the system is stochastically stable. The bound of delay is larger than those in other results obtained, which guarantees that the proposed conditions are tighter. Numerical examples show the effectiveness of the method proposed.     

Dynamic clustering of energy markets: An extended hidden Markov approach

This paper studies the synchronization of energy markets using an extended hidden Markov model that captures between- and within-heterogeneity in time series by defining clusters and hidden states, respectively. The model is applied to U.S. data in the period from 1999 to 2012. While oil and natural gas returns are well portrayed by two volatility states, electricity markets need three additional states: two transitory and one to capture a period of abnormally high volatility. Although some states are common to both clusters, results favor the segmentation of energy markets as they are not in the same state at the same time.     

Comparison of MCMC Methods for Estimating Stochastic Volatility Models

This article investigates performances of MCMC methods to estimate stochastic volatility models on simulated and real data. There are two efficient MCMC methods to generate latent volatilities from their full conditional distribution. One is the mixture sampler and the other is the multi-move sampler. There is another efficient method for latent volatilities and all parameters called the integration sampler, which is based on the mixture sampler. This article proposes an alternative method based on the multi-move sampler and finds evidence that it is the best method among them.JEL classification C22     

Simulation-based sequential analysis of Markov switching stochastic volatility models

We propose a simulation-based algorithm for inference in stochastic volatility models with possible regime switching in which the regime state is governed by a first-order Markov process. Using auxiliary particle filters we developed a strategy to sequentially learn about states and parameters of the model. The methodology is tested against a synthetic time series and validated with a real financial time series: the IBOVESPA stock index (São Paulo Stock Exchange).     

Structural Hidden Markov Models Using a Relation of Equivalence: Application to Automotive Designs

Standard hidden Markov models (HMM's) have been studied extensively in the last two decades. It is well known that these models assume state conditional independence of the observations. Therefore, they are inadequate for classification of complex and highly structured patterns. Nowadays, the need for new statistical models that are capable to cope with structural time series data is increasing. We propose in this paper a novel paradigm that we named “structural hidden Markov model” (SHMM). It extends traditional HMM's by partitioning the set of observation sequences into classes of equivalences. These observation sequences are related in the sense they all contribute to produce a particular local structure. We describe four basic problems that are assigned to a structural hidden Markov model: (1) probability evaluation, (2) statistical decoding, (3) local structure decoding, and (4) parameter estimation. We have applied SHMM in order to mine customers' preferences for automotive designs. The results reported in this application show that SHMM's outperform the traditional hidden Markov model with a 9% of increase in accuracy. Note In other words, it is possible to decrease the resolution level of a complex pattern.     

Using Hidden Markov Models for the accurate linguistic analysis of process model activity labels

Many process model analysis techniques rely on the accurate analysis of the natural language contents captured in the models’ activity labels. Since these labels are typically short and diverse in terms of their grammatical style, standard natural language processing tools are not suitable to analyze them. While a dedicated technique for the analysis of process model activity labels was proposed in the past, it suffers from considerable limitations. First of all, its performance varies greatly among data sets with different characteristics and it cannot handle uncommon grammatical styles. What is more, adapting the technique requires in-depth domain knowledge. We use this paper to propose a machine learning-based technique for activity label analysis that overcomes the issues associated with this rule-based state of the art. Our technique conceptualizes activity label analysis as a tagging task based on a Hidden Markov Model. By doing so, the analysis of activity labels no longer requires the manual specification of rules. An evaluation using a collection of 15,000 activity labels demonstrates that our machine learning-based technique outperforms the state of the art in all aspects.     

基于参数优化隐马尔可夫模型的网络取证证据融合方法

为了克服隐马尔可夫模型(hidden Markov model,HMM)在训练时波氏算法(Baum-Welch,B-W)易陷入局部最优解的不足,采用自适应遗传算法对其进行参数优化,设计了染色体编码方法和遗传操作方式。利用Viterbi算法选择最有可能的元证据序列,用疑似证据替换元证据回溯得到证据链。实验结果表明,自适应遗传算法优化的HMM具有更好的状态,采用Viterbi算法得到的证据链能够较精确地重现网络入侵的犯罪现场。     

AR 型非线性时间序列模型的稳定性分析

在工程中,振幅依赖指数自回归模型、门限自回归模型和多项式自回归模型等一类具有ＡＲ型的非线性时间序列模型具有广泛的应用,为此给出了ＡＲ型非线性时间序列模型的稳定性条件及极限环存在条件,并对一些特殊模型进行了讨论。     

复杂系统数据挖掘的多尺度混合算法

任何复杂系统都要受到某些基本规律的约束,包括宏观、中观与微观的多层次规律的约束.怎样从一个系统的这些偶然现象(观测数据)中找出它的必然规律,是知识发现(KDD)与数据挖掘(DM)的首要任务,也是研究目标.建立了一个基于演化计算与自然分形相结合的多尺度的动态预测系统.它以微分方程描述系统的宏观行为,以自然分形刻画系统的微观行为.同时,以股票市场数据(君安证券股票数据)和科学观测数据(武汉汛期雨量数据)为例,进行了分析与预测模拟.数值实验表明,该系统的描述(拟合)性能优越,即使是对起伏波动很大的时间序列,也能拟合得很好,预测效果也较好.     

On Detectability and Observability of Discrete‐Time Stochastic Markov Jump Systems with State‐Dependent Noise

This paper mainly studies the notions of detectability and observability for discrete‐time stochastic Markov jump systems with state‐dependent noise. Two concepts, called “W‐detectability” and “W‐observability,” for such systems are introduced, and are shown to coincide with the other concepts on detectability and observability reported recently in literature. Moreover, some criteria and interesting properties for both W‐detectability and W‐observability are obtained.     

Tests of Long Memory: A Bootstrap Approach

Many time series in diverse fields have been found to exhibit long memory. This paper analyzes the behaviour of some of the most used tests of long memory: the R/S analysis, the modified R/S, the Geweke and Porter-Hudak (GPH) test and the detrended fluctuation analysis (DFA). Some of these tests exhibit size distortions in small samples. It is well known that the bootstrap procedure may correct this fact. Here I examine the size and power of those tests for finite samples and different distributions, such as the normal, uniform, and lognormal. In the short-memory processes such as AR, MA and ARCH and long memory ones such as ARFIMA, p-values are calculated using the post-blackening moving-block bootstrap. The Monte Carlo study suggests that the bootstrap critical values perform better. The results are applied to financial return time series.     

Predicting Chaotic Time Series Using Neural and Neurofuzzy Models: A Comparative Study

The prediction accuracy and generalization ability of neural/neurofuzzy models for chaotic time series prediction highly depends on employed network model as well as learning algorithm. In this study, several neural and neurofuzzy models with different learning algorithms are examined for prediction of several benchmark chaotic systems and time series. The prediction performance of locally linear neurofuzzy models with recently developed Locally Linear Model Tree (LoLiMoT) learning algorithm is compared with that of Radial Basis Function (RBF) neural network with Orthogonal Least Squares (OLS) learning algorithm, MultiLayer Perceptron neural network with error back-propagation learning algorithm, and Adaptive Network based Fuzzy Inference System. Particularly, cross validation techniques based on the evaluation of error indices on multiple validation sets is utilized to optimize the number of neurons and to prevent over fitting in the incremental learning algorithms. To make a fair comparison between neural and neurofuzzy models, they are compared at their best structure based on their prediction accuracy, generalization, and computational complexity. The experiments are basically designed to analyze the generalization capability and accuracy of the learning techniques when dealing with limited number of training samples from deterministic chaotic time series, but the effect of noise on the performance of the techniques is also considered. Various chaotic systems and time series including Lorenz system, Mackey-Glass chaotic equation, Henon map, AE geomagnetic activity index, and sunspot numbers are examined as case studies. The obtained results indicate the superior performance of incremental learning algorithms and their respective networks, such as, OLS for RBF network and LoLiMoT for locally linear neurofuzzy model.     

Stochastic Fragments: A Framework for the Exact Reduction of the Stochastic Semantics of Rule-Based Models

In this paper, we propose an abstract interpretation-based framework for reducing the state space of stochastic semantics for protein-protein interaction networks. Our approach consists in quotienting the state space of networks. Yet interestingly, we do not apply the widely-used strong lumpability criterion which imposes that two equivalent states behave similarly with respect to the quotient, but a weak version of it. More precisely, our framework detects and proves some invariants about the dynamics of the system: indeed the quotient of the state space is such that the probability of being in a given state knowing that this state is in a given equivalence class, is an invariant of the semantics. Then we introduce an individual-based stochastic semantics (where each agent is identified by a unique identifier) for the programs of a rule-based language (namely Kappa) and we use our abstraction framework for deriving a sound population-based semantics and a sound fragments-based semantics, which give the distribution of the traces respectively for the number of instances of molecular species and for the number of instances of partially defined molecular species. These partially defined species are chosen automatically thanks to a dependency analysis which is also described in the paper.     

Filtering of discrete‐time Markov jump linear systems with uncertain transition probabilities

This article addresses the filtering design problem for discrete‐time Markov jump linear systems (MJLS) under the assumption that the transition probabilities are not completely known. We present the methods to determine ??₂‐ and ??_∞‐norm bounded filters for MJLS whose transition probability matrices have uncertainties in a convex polytope and establish an equivalence with the ones with partly unknown elements. The proposed design, based on linear matrix inequalities, allows different assumptions on Markov mode availability to the filter and on system parameter uncertainties to be taken into account. Under mode‐dependent assumption and internal model knowledge, observer‐based filters can be obtained and it is shown theoretically that our method outperforms some available ones in the literature to date. Numerical examples illustrate this claim. Copyright © 2010 John Wiley & Sons, Ltd.     

一类非线性系统的基于时间序列的多模型自适应控制

针对一类非线性离散时间系统,提出了一种基于时间序列的多模型自适应控制器(Multiple models adaptive controller, MMAC). 该控制器首先利用聚类方法建立多个线性固定模型,然后,利用系统的时间序列和方向导数建立一个 反映工作点变化趋势的局部加权模型,在此基础上增加了一个全局自适应模型和一个可重新赋值的 自适应模型,并设计了一个切换机构选择最优模型实现控制.仿真结果表明该控制器不但具有良好 的暂态性能、较快的控制速度,而且在相似的控制效果下,可以极大地减少模型的数量.     

Stochastic stability and robust stabilization of semi‐Markov jump linear systems

The semi‐Markov jump linear system (S‐MJLS) is more general than the Markov jump linear system (MJLS) in modeling some practical systems. Unlike the constant transition rates in the MJLS, the transition rates of the S‐MJLS are time varying. This paper focuses on the robust stochastic stability condition and the robust control design problem for the S‐MJLS with norm‐bounded uncertainties. The infinitesimal generator for the constructed Lyapunov function is first derived. Numerically solvable sufficient conditions for the stochastic stability of S‐MJLSs are then established in terms of linear matrix inequalities. To reduce the conservativeness of the stability conditions, we propose to incorporate the upper and lower bounds of the transition rate and meanwhile apply a new partition scheme. The robust state feedback controller is accordingly developed. Simulation studies and comparisons demonstrate the effectiveness and advantages of the proposed methods. Copyright © 2012 John Wiley & Sons, Ltd.