We consider the analysis of sets of categorical sequences consisting of piecewise homogeneous Markov segments. The sequences are assumed to be governed by a common underlying process with segments occurring in the same order for each sequence. Segments are defined by a set of unobserved changepoints where the positions and number of changepoints can vary from sequence to sequence. We propose a Bayesian framework for analyzing such data, placing priors on the locations of the changepoints and on the transition matrices and using Markov chain Monte Carlo (MCMC) techniques to obtain posterior samples given the data. Experimental results using simulated data illustrates how the methodology can be used for inference of posterior distributions for parameters and changepoints, as well as the ability to handle considerable variability in the locations of the changepoints across different sequences. We also investigate the application of the approach to sequential data from two applications involving monsoonal rainfall patterns and branching patterns in trees.
In my PhD thesis, we elaborate upon Bayesian changepoint analysis, whereby our focus is on three big topics: approximate sampling via MCMC, exact inference and uncertainty quantification. Besides, modeling matters are discussed in an ongoing fashion. Our findings are underpinned through several changepoint examples with a focus on a well-log drilling data.
Piecewise growth mixture models (PGMM) are a flexible and useful class of methods for analyzing segmented trends in individual growth trajectory over time, where the individuals come from a mixture of two or more latent classes. These models allow each segment of the overall developmental process within each class to have a different functional form; examples include two linear phases of growth, or a quadratic phase followed by a linear phase. The changepoint (knot) is the time of transition from one developmental phase (segment) to another. Inferring the location of the changepoint(s) is ofte…
Near real time change detection is important for a variety of Earth monitoring applications and remains a high priority for remote sensing science. Data sparsity, subtle changes, seasonal trends, and the presence of outliers make detecting actual landscape changes challenging. Adams and MacKay (2007) introduced Bayesian Online Changepoint Detection (BOCPD), a computationally efficient, exact Bayesian method for change detection. Incorporation of prior information allows for relaxed dependence on dense data and an extensive stable period, making this method applicable to relatively short time s…
By attaching auxiliary event times to the chronologically ordered observations, we formulate the Bayesian multiple changepoint problem of discrete-time observations into that of continuous-time ones. A version of forward-filtering backward-sampling (FFBS) algorithm is proposed for the simulation of changepoints within a collapsed Gibbs sampling scheme. Ideally, both the computational cost and memory cost of the FFBS algorithm can be quadratically scaled down to the number of changepoints, instead of the number of observations, which is otherwise prohibitive for a long sequence of observations.…
This paper proposes a new minimum description length procedure to detect multiple changepoints in time series data when some times are a priori thought more likely to be changepoints. This scenario arises with temperature time series homogenization pursuits, our focus here. Our Bayesian procedure constructs a natural prior distribution for the situation, and is shown to estimate the changepoint locations consistently, with an optimal convergence rate. Our methods substantially improve changepoint detection power when prior information is available. The methods are also tailored to bivariate da…
A Bayesian approach is developed to analyze change points in multivariate time series and space-time data. The methodology is used to assess the impact of extended inundation on the ecosystem of the Gulf Plains bioregion in northern Australia. The proposed approach can be implemented for dynamic mixture models that have a conditionally Gaussian state space representation. Details are given on how to efficiently implement the algorithm for a general class of multivariate time series and space-time models. This efficient implementation makes it feasible to analyze high dimensional, but of realis…
In regression models, predictor variables with inherent ordering, such as tumor staging ranging and ECOG performance status, are commonly seen in medical settings. Statistically, it may be difficult to determine the functional form of an ordinal predictor variable. Often, such a variable is dichotomized based on whether it is above or below a certain cutoff. Other methods conveniently treat the ordinal predictor as a continuous variable and assume a linear relationship with the outcome. However, arbitrarily choosing a method may lead to inaccurate inference and treatment. In this paper, we pro…
Copy number variations in cancer cells and volatility fluctuations in stock prices are commonly manifested as changepoints occurring at the same positions across related data sequences. We introduce a Bayesian modeling framework, BASIC, that employs a changepoint prior to capture the co-occurrence tendency in data of this type. We design efficient algorithms to sample from and maximize over the BASIC changepoint posterior and develop a Monte Carlo expectation-maximization procedure to select prior hyperparameters in an empirical Bayes fashion. We use the resulting BASIC framework to analyze DN…
Surprise Reduction and Nullification in Bayesian and Inverse Bayesian Inference under Ambiguous Prediction-Error Attribution
Tokyo Denki University（东京都立大学）；Osaka Metropolitan University（大阪公立大学）；The University of Tokyo（东京大学）；Utsunomiya University（宇都宫大学）；Waseda University（早稻田大学）
Changepoints are abrupt variations in the generative parameters of a data sequence. Online detection of changepoints is useful in modelling and prediction of time series in application areas such as finance, biometrics, and robotics. While frequentist methods have yielded online filtering and prediction techniques, most Bayesian papers have focused on the retrospective segmentation problem. Here we examine the case where the model parameters before and after the changepoint are independent and we derive an online algorithm for exact inference of the most recent changepoint. We compute the prob…
This paper presents a Markov chain Monte Carlo method to generate approximate posterior samples in retrospective multiple changepoint problems where the number of changes is not known in advance. The method uses conjugate models whereby the marginal likelihood for the data between consecutive changepoints is tractable. Inclusion of hyperpriors gives a near automatic algorithm providing a robust alternative to popular filtering recursions approaches in cases which may be sensitive to prior information. Three real examples are used to demonstrate the proposed approach.
Identifying changes in the generative process of sequential data, known as changepoint detection, has become an increasingly important topic for a wide variety of fields. A recently developed approach, which we call EXact Online Bayesian Changepoint Detection (EXO), has shown reasonable results with efficient computation for real time updates. The method is based on a \textit{forward} recursive message-passing algorithm. However, the detected changepoints from these methods are unstable. We propose a new algorithm called Lagged EXact Online Bayesian Changepoint Detection (LEXO) that improves t…
This paper proposes approaches for the analysis of multiple changepoint models when dependency in the data is modelled through a hierarchical Gaussian Markov random field. Integrated nested Laplace approximations are used to approximate data quantities, and an approximate filtering recursions approach is proposed for savings in compuational cost when detecting changepoints. All of these methods are simulation free. Analysis of real data demonstrates the usefulness of the approach in general. The new models which allow for data dependence are compared with conventional models where data within …
Princeton University（普林斯顿大学）；Arm ML Research Lab（Arm机器学习研究实验室）
Online algorithms for detecting changepoints, or abrupt shifts in the behavior of a time series, are often deployed with limited resources, e.g., to edge computing settings such as mobile phones or industrial sensors. In these scenarios it may be beneficial to trade the cost of collecting an environmental measurement against the quality or "fidelity" of this measurement and how the measurement affects changepoint estimation. For instance, one might decide between inertial measurements or GPS to determine changepoints for motion. A Bayesian approach to changepoint detection is particularly appe…
The aim of the present study is to detect abrupt trend changes in the mean of a multidimensional sequential signal. Directly inspired by papers of Fernhead and Liu ([4] and [5]), this work describes the signal in a hierarchical manner : the change dates of a time segmentation process trigger the renewal of a piece-wise constant emission law. Bayesian posterior information on the change dates and emission parameters is obtained. These estimations can be revised online, i.e. as new data arrive. This paper proposes explicit formulations corresponding to various emission laws, as well as a general…
We consider Bayesian analysis of a class of multiple changepoint models. While there are a variety of efficient ways to analyse these models if the parameters associated with each segment are independent, there are few general approaches for models where the parameters are dependent. Under the assumption that the dependence is Markov, we propose an efficient online algorithm for sampling from an approximation to the posterior distribution of the number and position of the changepoints. In a simulation study, we show that the approximation introduced is negligible. We illustrate the power of ou…
While there have been a lot of recent developments in the context of Bayesian model selection and variable selection for high dimensional linear models, there is not much work in the presence of change point in literature, unlike the frequentist counterpart. We consider a hierarchical Bayesian linear model where the active set of covariates that affects the observations through a mean model can vary between different time segments. Such structure may arise in social sciences/ economic sciences, such as sudden change of house price based on external economic factor, crime rate changes based on …
Recent findings suggest that abnormal operating conditions of equipment in the oil and gas supply chain represent a large fraction of anthropogenic methane emissions. Thus, effective mitigation of emissions necessitates rapid identification and repair of sources caused by faulty equipment. In addition to advances in sensing technology that allow for more frequent surveillance, prompt and cost-effective identification of sources requires computational frameworks that provide automatic fault detection. Here, we present a changepoint detection algorithm based on a recursive Bayesian scheme that a…
Material indentation studies, in which a probe is brought into controlled physical contact with an experimental sample, have long been a primary means by which scientists characterize the mechanical properties of materials. More recently, the advent of atomic force microscopy, which operates on the same fundamental principle, has in turn revolutionized the nanoscale analysis of soft biomaterials such as cells and tissues. This paper addresses the inferential problems associated with material indentation and atomic force microscopy, through a framework for the changepoint analysis of pre- and p…
We consider a sequential Bayesian changepoint detection problem for a general stochastic model, assuming that the observed data may be dependent and non-identically distributed and the prior distribution of the change point is arbitrary, not necessarily geometric. Tartakovsky and Veeravalli (2004) developed a general asymptotic theory of changepoint detection in the non-iid case and discrete time, and Baron and Tartakovsky (2006) in continuous time assuming certain stability of the log-likelihood ratio process. This stability property was formulated in terms of the r-quick convergence of the n…
Carnegie Mellon University（卡内基梅隆大学）；New York University（纽约大学）；Philips Research（飞利浦研究院）；Cornell University（康奈尔大学）
Identifying changes in model parameters is fundamental in machine learning and statistics. However, standard changepoint models are limited in expressiveness, often addressing unidimensional problems and assuming instantaneous changes. We introduce change surfaces as a multidimensional and highly expressive generalization of changepoints. We provide a model-agnostic formalization of change surfaces, illustrating how they can provide variable, heterogeneous, and non-monotonic rates of change across multiple dimensions. Additionally, we show how change surfaces can be used for counterfactual pre…
The Alan Turing Institute（艾伦·图灵研究所）；University of Warwick（华威大学）
We present the very first robust Bayesian Online Changepoint Detection algorithm through General Bayesian Inference (GBI) with $β$-divergences. The resulting inference procedure is doubly robust for both the parameter and the changepoint (CP) posterior, with linear time and constant space complexity. We provide a construction for exponential models and demonstrate it on the Bayesian Linear Regression model. In so doing, we make two additional contributions: Firstly, we make GBI scalable using Structural Variational approximations that are exact as $β\to 0$. Secondly, we give a principled way o…
University College London（伦敦大学学院）；University College Hospital（伦敦大学学院医院）；University of Southampton（南安普顿大学）
Numerous lung diseases, such as idiopathic pulmonary fibrosis (IPF), exhibit dilation of the airways. Accurate measurement of dilatation enables assessment of the progression of disease. Unfortunately the combination of image noise and airway bifurcations causes high variability in the profiles of cross-sectional areas, rendering the identification of affected regions very difficult. Here we introduce a noise-robust method for automatically detecting the location of progressive airway dilatation given two profiles of the same airway acquired at different time points. We propose a probabilistic…
We propose a new Bayesian Markov switching regression model for multidimensional arrays (tensors) of binary time series. We assume a zero-inflated logit regression with time-varying parameters and apply it to multilayer temporal networks. The original contribution is threefold. First, to avoid over-fitting we propose a parsimonious parametrization based on a low-rank decomposition of the tensor of regression coefficients. Second, we assume the parameters are driven by a hidden Markov chain, thus allowing for structural changes in the network topology. We follow a Bayesian approach to inference…
Within a Bayesian retrospective framework, we present a way of examining the distribution of \cps through a novel set estimator. For a given level, $α$, we aim at smallest sets that cover all \cps with a probability of at least $1-α$. These so-called smallest simultaneous credible regions, computed for certain values of $α$, provide parsimonious representations of the possible \cp locations. In addition, combining them for a range of different $α$'s enables very informative yet condensed visualisations. Therewith we allow for the evaluation of model choices and the analysis of \cp data to an u…
We consider the problem of flexible modeling of higher order hidden Markov models when the number of latent states and the nature of the serial dependence, including the true order, are unknown. We propose Bayesian nonparametric methodology based on tensor factorization techniques that can characterize any transition probability with a specified maximal order, allowing automated selection of the important lags and capturing higher order interactions among the lags. Theoretical results provide insights into identifiability of the emission distributions and asymptotic behavior of the posterior. …
We propose a Bayesian hierarchical model to simultaneously estimate mean based changepoints in spatially correlated functional time series. Unlike previous methods that assume a shared changepoint at all spatial locations or ignore spatial correlation, our method treats changepoints as a spatial process. This allows our model to respect spatial heterogeneity and exploit spatial correlations to improve estimation. Our method is derived from the ubiquitous cumulative sum (CUSUM) statistic that dominates changepoint detection in functional time series. However, instead of directly searching for t…
We consider the problem of flexible modeling of higher order Markov chains when an upper bound on the order of the chain is known but the true order and nature of the serial dependence are unknown. We propose Bayesian nonparametric methodology based on conditional tensor factorizations, which can characterize any transition probability with a specified maximal order. The methodology selects the important lags and captures higher order interactions among the lags, while also facilitating calculation of Bayes factors for a variety of hypotheses of interest. We design efficient Markov chain Monte…
In this article a flexible Bayesian non-parametric model is proposed for non-homogeneous hidden Markov models. The model is developed through the amalgamation of the ideas of hidden Markov models and predictor dependent stick-breaking processes. Computation is carried out using auxiliary variable representation of the model which enable us to perform exact MCMC sampling from the posterior. Furthermore, the model is extended to the situation when the predictors can simultaneously in influence the transition dynamics of the hidden states as well as the emission distribution. Estimates of few ste…
We propose a Bayesian inference approach for a class of latent Markov models. These models are widely used for the analysis of longitudinal categorical data, when the interest is in studying the evolution of an individual unobservable characteristic. We consider, in particular, the basic latent Markov, which does not account for individual covariates, and its version that includes such covariates in the measurement model. The proposed inferential approach is based on a system of priors formulated on a transformation of the initial and transition probabilities of the latent Markov chain. This s…
We propose a Bayesian hidden Markov model for analyzing time series and sequential data where a special structure of the transition probability matrix is embedded to model explicit-duration semi-Markovian dynamics. Our formulation allows for the development of highly flexible and interpretable models that can integrate available prior information on state durations while keeping a moderate computational cost to perform efficient posterior inference. We show the benefits of choosing a Bayesian approach for HSMM estimation over its frequentist counterpart, in terms of model selection and out-of-…
We develop two models for Bayesian estimation and selection in high-order, discrete-state Markov chains. Both are based on the mixture transition distribution, which constructs a transition probability tensor with additive mixing of probabilities from first-order transition matrices. We demonstrate two uses for the proposed models: parsimonious approximation of high-order dynamics by mixing lower-order transition models, and order/lag selection through over-specification and shrinkage via priors for sparse probability vectors. The priors further shrink all models to an identifiable and interpr…
Studying the neurological, genetic and evolutionary basis of human vocal communication mechanisms using animal vocalization models is an important field of neuroscience. The data sets typically comprise structured sequences of syllables or `songs' produced by animals from different genotypes under different social contexts. We develop a novel Bayesian semiparametric framework for inference in such data sets. Our approach is built on a novel class of mixed effects Markov transition models for the songs that accommodates exogenous influences of genotype and context as well as animal-specific het…
This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation. The holding times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning …
Scientists often must simultaneously localize and discover signals. For instance, in genetic fine-mapping, high correlations between nearby genetic variants make it hard to identify the exact locations of causal variants. So the statistical task is to output as many disjoint regions containing a signal as possible, each as small as possible, while controlling false positives. Similar problems arise in any application where signals cannot be perfectly localized, such as locating stars in astronomical surveys and changepoint detection in sequential data. Common Bayesian approaches to these probl…
In modern business modeling and analytics, data monitoring plays a critical role. Nowadays, sophisticated models often rely on hundreds or even thousands of input variables. Over time, structural changes such as abrupt level shifts or trend slope changes may occur among some of these variables, likely due to changes in economy or government policies. As a part of data monitoring, it is important to identify these changepoints, in terms of which variables exhibit such changes, and what time locations do the changepoints occur. Being alerted about the changepoints can help modelers decide if mod…
We discuss and extend methods for estimating Markovian-Regime-Switching (MRS) and trend models for wholesale electricity prices. We argue the existing methods of trend estimation used in the electricity price modelling literature either require an ambiguous definition of an extreme price, or lead to issues when implementing model selection [23]. The first main contribution of this paper is to design and infer a model which has a model-based definition of extreme prices and permits the use of model selection criteria. Due to the complexity of the MRS models inference is not straightforward. In …
The Markov chain random field (MCRF) model/theory provides a non-linear spatial Bayesian updating solution at the neighborhood nearest data level for simulating categorical spatial variables. In the MCRF solution, the spatial dependencies among nearest data and the central random variable is a probabilistic directed acyclic graph that conforms to a neighborhood-based Bayesian network on spatial data. By selecting different neighborhood sizes and structures, applying the spatial conditional independence assumption to nearest neighbors, or incorporating ancillary information, one may construct s…
Change detection involves segmenting sequential data such that observations in the same segment share some desired properties. Multivariate change detection continues to be a challenging problem due to the variety of ways change points can be correlated across channels and the potentially poor signal-to-noise ratio on individual channels. In this paper, we are interested in locating additive outliers (AO) and level shifts (LS) in the unsupervised setting. We propose ABACUS, Automatic BAyesian Changepoints Under Sparsity, a Bayesian source separation technique to recover latent signals while al…
This paper introduces a new model for panel data with Markov-switching GARCH effects. The model incorporates a series-specific hidden Markov chain process that drives the GARCH parameters. To cope with the high-dimensionality of the parameter space, the paper exploits the cross-sectional clustering of the series by first assuming a soft parameter pooling through a hierarchical prior distribution with two-step procedure, and then introducing clustering effects in the parameter space through a nonparametric prior distribution. The model and the proposed inference are evaluated through a simulati…
Graphical models express conditional independence relationships among variables. Although methods for vector-valued data are well established, functional data graphical models remain underdeveloped. We introduce a notion of conditional independence between random functions, and construct a framework for Bayesian inference of undirected, decomposable graphs in the multivariate functional data context. This framework is based on extending Markov distributions and hyper Markov laws from random variables to random processes, providing a principled alternative to naive application of multivariate m…
Stationary time series models built from parametric distributions are, in general, limited in scope due to the assumptions imposed on the residual distribution and autoregression relationship. We present a modeling approach for univariate time series data, which makes no assumptions of stationarity, and can accommodate complex dynamics and capture nonstandard distributions. The model for the transition density arises from the conditional distribution implied by a Bayesian nonparametric mixture of bivariate normals. This implies a flexible autoregressive form for the conditional transition dens…
University of Warwick（华威大学）；The Alan Turing Institute for Data Science & AI（艾伦·图灵数据科学与人工智能研究所）
Bayesian On-line Changepoint Detection is extended to on-line model selection and non-stationary spatio-temporal processes. We propose spatially structured Vector Autoregressions (VARs) for modelling the process between changepoints (CPs) and give an upper bound on the approximation error of such models. The resulting algorithm performs prediction, model selection and CP detection on-line. Its time complexity is linear and its space complexity constant, and thus it is two orders of magnitudes faster than its closest competitor. In addition, it outperforms the state of the art for multivariate …
When analysing multiple time series that may be subject to changepoints, it is sometimes possible to specify a priori, by means of a graph, which pairs of time series are likely to be impacted by simultaneous changepoints. This article proposes an informative prior for changepoints which encodes the information contained in the graph, inducing a changepoint model for multiple time series that borrows strength across clusters of connected time series to detect weak signals for synchronous changepoints. The graphical model for changepoints is further extended to allow dependence between nearby b…
The literature on Bayesian methods for the analysis of discrete-time semi-Markov processes is sparse. In this paper, we introduce the semi-Markov beta-Stacy process, a stochastic process useful for the Bayesian non-parametric analysis of semi-Markov processes. The semi-Markov beta-Stacy process is conjugate with respect to data generated by a semi-Markov process, a property which makes it easy to obtain probabilistic forecasts. Its predictive distributions are characterized by a reinforced random walk on a system of urns.
Changepoint analysis (also known as segmentation analysis) aims at analyzing an ordered, one-dimensional vector, in order to find locations where some characteristic of the data changes. Many models and algorithms have been studied under this theme, including models for changes in mean and / or variance, changes in linear regression parameters, etc. In this work, we are interested in an algorithm for the segmentation of long duration acoustic signals; the segmentation is based on the change of the RMS power of the signal. We investigate a Bayesian model with two possible parameterizations, and…
Changepoint models typically assume the data within each segment are independent and identically distributed conditional on some parameters which change across segments. This construction may be inadequate when data are subject to local correlation patterns, often resulting in many more changepoints fitted than preferable. This article proposes a Bayesian changepoint model which relaxes the assumption of exchangeability within segments. The proposed model supposes data within a segment are $m$-dependent for some unkown $m \geqslant0$ which may vary between segments, resulting in a model suitab…
There is much interest in the Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) as a natural Bayesian nonparametric extension of the ubiquitous Hidden Markov Model for learning from sequential and time-series data. However, in many settings the HDP-HMM's strict Markovian constraints are undesirable, particularly if we wish to learn or encode non-geometric state durations. We can extend the HDP-HMM to capture such structure by drawing upon explicit-duration semi-Markovianity, which has been developed mainly in the parametric frequentist setting, to allow construction of highly interp…
Bayesian inference for Markov jump processes (MJPs) where available observations relate to either system states or jumps typically relies on data-augmentation Markov Chain Monte Carlo. State-of-the-art developments involve representing MJP paths with auxiliary candidate jump times that are later thinned. However, these algorithms are i) unfeasible in situations involving large or infinite capacity systems and ii) not amenable for all observation types. In this paper we establish and present a general data-augmentation framework for population MJPs based on uniformized representations of the un…
