Analysing time series structure with hidden Markov models

This paper consides the problem of extracting the relationships between two time series in a non-linear non-stationary environment with Hidden Markov Models (HMMs). We describe an algorithm which is capable of identifying associations between variables. The method is applied both to synthetic data and real data. We show that HMMs are capable of modelling the oil drilling process and that they outperform existing methods.

Bayesian online algorithms for learning in discrete Hidden Markov Models

We propose and analyze two different Bayesian online algorithms for learning in discrete Hidden Markov Models and compare their performance with the already known Baldi-Chauvin Algorithm. Using the Kullback-Leibler divergence as a measure of generalization we draw learning curves in simplified situations for these algorithms and compare their performances.

Online learning in discrete hidden Markov models

We present and analyze three different online algorithms for learning in discrete Hidden Markov Models (HMMs) and compare their performance with the Baldi-Chauvin Algorithm. Using the Kullback-Leibler divergence as a measure of the generalization error we draw learning curves in simplified situations and compare the results. The performance for learning drifting concepts of one of the presented algorithms is analyzed and compared with the Baldi-Chauvin algorithm in the same situations. A brief discussion about learning and symmetry breaking based on our results is also presented. © 2006 American Institute of Physics.

Time delay estimation with hidden Markov models

Most traditional methods for extracting the relationships between two time series are based on cross-correlation. In a non-linear non-stationary environment, these techniques are not sufficient. We show in this paper how to use hidden Markov models to identify the lag (or delay) between different variables for such data. Adopting an information-theoretic approach, we develop a procedure for training HMMs to maximise the mutual information (MMI) between delayed time series. The method is used to model the oil drilling process. We show that cross-correlation gives no information and that the MMI approach outperforms maximum likelihood.

Hidden state models for time series

Amongst all the objectives in the study of time series, uncovering the dynamic law of its generation is probably the most important. When the underlying dynamics are not available, time series modelling consists of developing a model which best explains a sequence of observations. In this thesis, we consider hidden space models for analysing and describing time series. We first provide an introduction to the principal concepts of hidden state models and draw an analogy between hidden Markov models and state space models. Central ideas such as hidden state inference or parameter estimation are reviewed in detail. A key part of multivariate time series analysis is identifying the delay between different variables. We present a novel approach for time delay estimating in a non-stationary environment. The technique makes use of hidden Markov models and we demonstrate its application for estimating a crucial parameter in the oil industry. We then focus on hybrid models that we call dynamical local models. These models combine and generalise hidden Markov models and state space models. Probabilistic inference is unfortunately computationally intractable and we show how to make use of variational techniques for approximating the posterior distribution over the hidden state variables. Experimental simulations on synthetic and real-world data demonstrate the application of dynamical local models for segmenting a time series into regimes and providing predictive distributions.

Modelling data and voice traffic over IP networks using continuous-time Markov models

Common approaches to IP-traffic modelling have featured the use of stochastic models, based on the Markov property, which can be classified into black box and white box models based on the approach used for modelling traffic. White box models, are simple to understand, transparent and have a physical meaning attributed to each of the associated parameters. To exploit this key advantage, this thesis explores the use of simple classic continuous-time Markov models based on a white box approach, to model, not only the network traffic statistics but also the source behaviour with respect to the network and application. The thesis is divided into two parts: The first part focuses on the use of simple Markov and Semi-Markov traffic models, starting from the simplest two-state model moving upwards to n-state models with Poisson and non-Poisson statistics. The thesis then introduces the convenient to use, mathematically derived, Gaussian Markov models which are used to model the measured network IP traffic statistics. As one of the most significant contributions, the thesis establishes the significance of the second-order density statistics as it reveals that, in contrast to first-order density, they carry much more unique information on traffic sources and behaviour. The thesis then exploits the use of Gaussian Markov models to model these unique features and finally shows how the use of simple classic Markov models coupled with use of second-order density statistics provides an excellent tool for capturing maximum traffic detail, which in itself is the essence of good traffic modelling. The second part of the thesis, studies the ON-OFF characteristics of VoIP traffic with reference to accurate measurements of the ON and OFF periods, made from a large multi-lingual database of over 100 hours worth of VoIP call recordings. The impact of the language, prosodic structure and speech rate of the speaker on the statistics of the ON-OFF periods is analysed and relevant conclusions are presented. Finally, an ON-OFF VoIP source model with log-normal transitions is contributed as an ideal candidate to model VoIP traffic and the results of this model are compared with those of previously published work.

Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.

GTM through time

The standard GTM (generative topographic mapping) algorithm assumes that the data on which it is trained consists of independent, identically distributed (iid) vectors. For time series, however, the iid assumption is a poor approximation. In this paper we show how the GTM algorithm can be extended to model time series by incorporating it as the emission density in a hidden Markov model. Since GTM has discrete hidden states we are able to find a tractable EM algorithm, based on the forward-backward algorithm, to train the model. We illustrate the performance of GTM through time using flight recorder data from a helicopter.

Delay estimation for multivariate time series

Most traditional methods for extracting the relationships between two time series are based on cross-correlation. In a non-linear non-stationary environment, these techniques are not sufficient. We show in this paper how to use hidden Markov models (HMMs) to identify the lag (or delay) between different variables for such data. We first present a method using maximum likelihood estimation and propose a simple algorithm which is capable of identifying associations between variables. We also adopt an information-theoretic approach and develop a novel procedure for training HMMs to maximise the mutual information between delayed time series. Both methods are successfully applied to real data. We model the oil drilling process with HMMs and estimate a crucial parameter, namely the lag for return.

Dynamical local models for segmentation and prediction of financial time series

In the analysis and prediction of many real-world time series, the assumption of stationarity is not valid. A special form of non-stationarity, where the underlying generator switches between (approximately) stationary regimes, seems particularly appropriate for financial markets. We introduce a new model which combines a dynamic switching (controlled by a hidden Markov model) and a non-linear dynamical system. We show how to train this hybrid model in a maximum likelihood approach and evaluate its performance on both synthetic and financial data.

Bayesian analysis of the scatterometer wind retrieval inverse problem:Some new approaches

The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.

Discriminative training of the hidden vector state model for semantic parsing

In this paper, we discuss how discriminative training can be applied to the hidden vector state (HVS) model in different task domains. The HVS model is a discrete hidden Markov model (HMM) in which each HMM state represents the state of a push-down automaton with a finite stack size. In previous applications, maximum-likelihood estimation (MLE) is used to derive the parameters of the HVS model. However, MLE makes a number of assumptions and unfortunately some of these assumptions do not hold. Discriminative training, without making such assumptions, can improve the performance of the HVS model by discriminating the correct hypothesis from the competing hypotheses. Experiments have been conducted in two domains: the travel domain for the semantic parsing task using the DARPA Communicator data and the Air Travel Information Services (ATIS) data and the bioinformatics domain for the information extraction task using the GENIA corpus. The results demonstrate modest improvements of the performance of the HVS model using discriminative training. In the travel domain, discriminative training of the HVS model gives a relative error reduction rate of 31 percent in F-measure when compared with MLE on the DARPA Communicator data and 9 percent on the ATIS data. In the bioinformatics domain, a relative error reduction rate of 4 percent in F-measure is achieved on the GENIA corpus.

A hybrid generative/discriminative framework to train a semantic parser from an un-annotated corpus

We propose a hybrid generative/discriminative framework for semantic parsing which combines the hidden vector state (HVS) model and the hidden Markov support vector machines (HM-SVMs). The HVS model is an extension of the basic discrete Markov model in which context is encoded as a stack-oriented state vector. The HM-SVMs combine the advantages of the hidden Markov models and the support vector machines. By employing a modified K-means clustering method, a small set of most representative sentences can be automatically selected from an un-annotated corpus. These sentences together with their abstract annotations are used to train an HVS model which could be subsequently applied on the whole corpus to generate semantic parsing results. The most confident semantic parsing results are selected to generate a fully-annotated corpus which is used to train the HM-SVMs. The proposed framework has been tested on the DARPA Communicator Data. Experimental results show that an improvement over the baseline HVS parser has been observed using the hybrid framework. When compared with the HM-SVMs trained from the fully-annotated corpus, the hybrid framework gave a comparable performance with only a small set of lightly annotated sentences. © 2008. Licensed under the Creative Commons.

Generalized methods and solvers for noise removal from piecewise constant signals. II. New methods

Removing noise from signals which are piecewise constant (PWC) is a challenging signal processing problem that arises in many practical scientific and engineering contexts. In the first paper (part I) of this series of two, we presented background theory building on results from the image processing community to show that the majority of these algorithms, and more proposed in the wider literature, are each associated with a special case of a generalized functional, that, when minimized, solves the PWC denoising problem. It shows how the minimizer can be obtained by a range of computational solver algorithms. In this second paper (part II), using this understanding developed in part I, we introduce several novel PWC denoising methods, which, for example, combine the global behaviour of mean shift clustering with the local smoothing of total variation diffusion, and show example solver algorithms for these new methods. Comparisons between these methods are performed on synthetic and real signals, revealing that our new methods have a useful role to play. Finally, overlaps between the generalized methods of these two papers and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.

Training Bayesian networks for image segmentation

We are concerned with the problem of image segmentation in which each pixel is assigned to one of a predefined finite number of classes. In Bayesian image analysis, this requires fusing together local predictions for the class labels with a prior model of segmentations. Markov Random Fields (MRFs) have been used to incorporate some of this prior knowledge, but this not entirely satisfactory as inference in MRFs is NP-hard. The multiscale quadtree model of Bouman and Shapiro (1994) is an attractive alternative, as this is a tree-structured belief network in which inference can be carried out in linear time (Pearl 1988). It is an hierarchical model where the bottom-level nodes are pixels, and higher levels correspond to downsampled versions of the image. The conditional-probability tables (CPTs) in the belief network encode the knowledge of how the levels interact. In this paper we discuss two methods of learning the CPTs given training data, using (a) maximum likelihood and the EM algorithm and (b) emphconditional maximum likelihood (CML). Segmentations obtained using networks trained by CML show a statistically-significant improvement in performance on synthetic images. We also demonstrate the methods on a real-world outdoor-scene segmentation task.