Combining Functional Data Registration and Factor Analysis

We extend the definition of functional data registration to encompass a larger class of registration models. In contrast to traditional registration models, we allow for registered functions that have more than one primary direction of variation. The proposed Bayesian hierarchical model simultaneously registers the observed functions and estimates the two primary factors that characterize variation in the registered functions. Each registered function is assumed to be predominantly composed of a linear combination of these two primary factors, and the function-specific weights for each observation are estimated within the registration model. We show how these estimated weights can easily be used to classify functions after registration using both simulated data and a juggling dataset. Supplementary materials for this article are available online.     

The Momentum Map Representation of Images

This paper discusses the mathematical framework for designing methods of Large Deformation Diffeomorphic Matching (LDM) for image registration in computational anatomy. After reviewing the geometrical framework of LDM image registration methods, we prove a theorem showing that these methods may be designed by using the actions of diffeomorphisms on the image data structure to define their associated momentum representations as (cotangent-lift) momentum maps. To illustrate its use, the momentum map theorem is shown to recover the known algorithms for matching landmarks, scalar images, and vector fields. After briefly discussing the use of this approach for diffusion tensor (DT) images, we explain how to use momentum maps in the design of registration algorithms for more general data structures. For example, we extend our methods to determine the corresponding momentum map for registration using semidirect product groups, for the purpose of matching images at two different length scales. Finally, we discuss the use of momentum maps in the design of image registration algorithms when the image data is defined on manifolds instead of vector spaces.     

A COMPOUND POISSON MODEL FOR LEARNING DISCRETE BAYESIAN NETWORKS

We introduce here the concept of Bayesian networks, in compound Poisson model, which provides a graphical modeling framework that encodes the joint probability distribution for a set of random variables within a directed acyclic graph. We suggest an approach proposal which offers a new mixed implicit estimator. We show that the implicit approach applied in compound Poisson model is very attractive for its ability to understand data and does not require any prior information. A comparative study between learned estimates given by implicit and by standard Bayesian approaches is established. Under some conditions and based on minimal squared error calculations, we show that the mixed implicit estimator is better than the standard Bayesian and the maximum likelihood estimators. We illustrate our approach by considering a simulation study in the context of mobile communication networks.     

Modeling first bid in retail secondary market online auctions: A Bayesian approach

We propose a Bayesian framework to model bid placement time in retail secondary market online business‐to‐business auctions. In doing so, we propose a Bayesian beta regression model to predict the first bidder and time to first bid, and a dynamic probit model to analyze participation. In our development, we consider both auction‐specific and bidder‐specific explanatory variables. While we primarily focus on the predictive performance of the models, we also discuss how auction features and bidders' heterogeneity could affect the bid timings, as well as auction participation. We illustrate the implementation of our models by applying to actual auction data and discuss additional insights provided by the Bayesian approach, which can benefit auctioneers.     

On‐site surrogates for large‐scale calibration

Motivated by a computer model calibration problem from the oil and gas industry, involving the design of a honeycomb seal, we develop a new Bayesian methodology to cope with limitations in the canonical apparatus stemming from several factors. We propose a new strategy of on‐site design and surrogate modeling for a computer simulator acting on a high‐dimensional input space that, although relatively speedy, is prone to numerical instabilities, missing data, and nonstationary dynamics. Our aim is to strike a balance between data‐faithful modeling and computational tractability in a calibration framework—tailoring the computer model to a limited field experiment. Situating our on‐site surrogates within the canonical calibration apparatus requires updates to that framework. We describe a novel yet intuitive Bayesian setup that carefully decomposes otherwise prohibitively large matrices by exploiting the sparse blockwise structure. Empirical illustrations demonstrate that this approach performs well on toy data and our motivating honeycomb example.     

Semiparametric Bayesian optimal replacement policies: application to railroad tracks

We present a Bayesian decision theoretic approach for developing replacement strategies. In so doing, we consider a semiparametric model to describe the failure characteristics of systems by specifying a nonparametric form for cumulative intensity function and by taking into account effect of covariates by a parametric form. Use of a gamma process prior for the cumulative intensity function complicates the Bayesian analysis when the updating is based on failure count data. We develop a Bayesian analysis of the model using Markov chain Monte Carlo methods and determine replacement strategies. Adoption of Markov chain Monte Carlo methods involves a data augmentation algorithm. We show the implementation of our approach using actual data from railroad tracks. Copyright © 2016 John Wiley & Sons, Ltd.     

A General Framework for Mining Massive Data Streams

In many domains, data now arrive faster than we are able to mine it. To avoid wasting these data, we must switch from the traditional “one-shot” data mining approach to systems that are able to mine continuous, high-volume, open-ended data streams as they arrive. In this article we identify some desiderata for such systems, and outline our framework for realizing them. A key property of our approach is that it minimizes the time required to build a model on a stream while guaranteeing (as long as the data are iid) that the model learned is effectively indistinguishable from the one that would be obtained using infinite data. Using this framework, we have successfully adapted several learning algorithms to massive data streams, including decision tree induction, Bayesian network learning, k-means clustering, and the EM algorithm for mixtures of Gaussians. These algorithms are able to process on the order of billions of examples per day using off-the-shelf hardware. Building on this, we are currently developing software primitives for scaling arbitrary learning algorithms to massive data streams with minimal effort.     

Adaptive Rejection Metropolis Simulated Annealing for Detecting Global Maximum Regions

A finite mixture model has been used to fit the data from heterogeneous populations to many applications. An Expectation Maximization (EM) algorithm is the most popular method to estimate parameters in a finite mixture model. A Bayesian approach is another method for fitting a mixture model. However, the EM algorithm often converges to the local maximum regions, and it is sensitive to the choice of starting points. In the Bayesian approach, the Markov Chain Monte Carlo (MCMC) sometimes converges to the local mode and is difficult to move to another mode. Hence, in this paper we propose a new method to improve the limitation of EM algorithm so that the EM can estimate the parameters at the global maximum region and to develop a more effective Bayesian approach so that the MCMC chain moves from one mode to another more easily in the mixture model. Our approach is developed by using both simulated annealing (SA) and adaptive rejection metropolis sampling (ARMS). Although SA is a well-known approach for detecting distinct modes, the limitation of SA is the difficulty in choosing sequences of proper proposal distributions for a target distribution. Since ARMS uses a piecewise linear envelope function for a proposal distribution, we incorporate ARMS into an SA approach so that we can start a more proper proposal distribution and detect separate modes. As a result, we can detect the maximum region and estimate parameters for this global region. We refer to this approach as ARMS annealing. By putting together ARMS annealing with the EM algorithm and with the Bayesian approach, respectively, we have proposed two approaches: an EM-ARMS annealing algorithm and a Bayesian-ARMS annealing approach. We compare our two approaches with traditional EM algorithm alone and Bayesian approach alone using simulation, showing that our two approaches are comparable to each other but perform better than EM algorithm alone and Bayesian approach alone. Our two approaches detect the global maximum region well and estimate the parameters in this region. We demonstrate the advantage of our approaches using an example of the mixture of two Poisson regression models. This mixture model is used to analyze a survey data on the number of charitable donations.     

A hierarchical model for the joint mortality analysis of pension scheme data with missing covariates

A hierarchical model is developed for the joint mortality analysis of pension scheme datasets. The proposed model allows for a rigorous statistical treatment of missing data. While our approach works for any missing data pattern, we are particularly interested in a scenario where some covariates are observed for members of one pension scheme but not the other. Therefore, our approach allows for the joint modelling of datasets which contain different information about individual lives. The proposed model generalizes the specification of parametric models when accounting for covariates. We consider parameter uncertainty using Bayesian techniques. Model parametrization is analysed in order to obtain an efficient MCMC sampler, and address model selection. The inferential framework described here accommodates any missing-data pattern, and turns out to be useful to analyse statistical relationships among covariates. Finally, we assess the financial impact of using the covariates, and of the optimal use of the whole available sample when combining data from different mortality experiences.     

Bayesian networks for mathematical models: Techniques for automatic construction and efficient inference

Expert knowledge in the form of mathematical models can be considered sufficient statistics of all prior experimentation in the domain, embodying generic or abstract knowledge of it. When used in a probabilistic framework, such models provide a sound foundation for data mining, inference, and decision making under uncertainty.We describe a methodology for encapsulating knowledge in the form of ordinary differential equations (ODEs) in dynamic Bayesian networks (DBNs). The resulting DBN framework can handle both data and model uncertainty in a principled manner, can be used for temporal data mining with noisy and missing data, and can be used to re-estimate model parameters automatically using data streams. A standard assumption when performing inference in DBNs is that time steps are fixed. Generally, the time step chosen is small enough to capture the dynamics of the most rapidly changing variable. This can result in DBNs having a natural time step that is very short, leading to inefficient inference; this is particularly an issue for DBNs derived from ODEs and for systems where the dynamics are not uniform over time.We propose an alternative to the fixed time step inference used in standard DBNs. In our algorithm, the DBN automatically adapts the time step lengths to suit the dynamics in each step. The resulting system allows us to efficiently infer probable values of hidden variables using multiple time series of evidence, some of which may be sparse, noisy or incomplete.We evaluate our approach with a DBN based on a variant of the van der Pol oscillator, and demonstrate an example where it gives more accurate results than the standard approach, but using only one tenth the number of time steps.We also apply our approach to a real-world example in critical care medicine. By incorporating knowledge in the form of an existing ODE model, we have built a DBN framework for efficiently predicting individualised patient responses using the available bedside and lab data.     

Bayesian adaptive Lasso

We propose the Bayesian adaptive Lasso (BaLasso) for variable selection and coefficient estimation in linear regression. The BaLasso is adaptive to the signal level by adopting different shrinkage for different coefficients. Furthermore, we provide a model selection machinery for the BaLasso by assessing the posterior conditional mode estimates, motivated by the hierarchical Bayesian interpretation of the Lasso. Our formulation also permits prediction using a model averaging strategy. We discuss other variants of this new approach and provide a unified framework for variable selection using flexible penalties. Empirical evidence of the attractiveness of the method is demonstrated via extensive simulation studies and data analysis.     

Behavioral modeling in weight loss interventions

Designing systems with human agents is difficult because it often requires models that characterize agents’ responses to changes in the system’s states and inputs. An example of this scenario occurs when designing treatments for obesity. While weight loss interventions through increasing physical activity and modifying diet have found success in reducing individuals’ weight, such programs are difficult to maintain over long periods of time due to lack of patient adherence. A promising approach to increase adherence is through the personalization of treatments to each patient. In this paper, we make a contribution toward treatment personalization by developing a framework for predictive modeling using utility functions that depend upon both time-varying system states and motivational states evolving according to some modeled process corresponding to qualitative social science models of behavior change. Computing the predictive model requires solving a bilevel program, which we reformulate as a mixed-integer linear program (MILP). This reformulation provides the first (to our knowledge) formulation for Bayesian inference that uses empirical histograms as prior distributions. We study the predictive ability of our framework using a data set from a weight loss intervention, and our predictive model is validated by comparison to standard machine learning approaches. We conclude by describing how our predictive model could be used for optimization, unlike standard machine learning approaches that cannot.     

Forecasting seasonal time series data: a Bayesian model averaging approach

A flexible Bayesian periodic autoregressive model is used for the prediction of quarterly and monthly time series data. As the unknown autoregressive lag order, the occurrence of structural breaks and their respective break dates are common sources of uncertainty these are treated as random quantities within the Bayesian framework. Since no analytical expressions for the corresponding marginal posterior predictive distributions exist a Markov Chain Monte Carlo approach based on data augmentation is proposed. Its performance is demonstrated in Monte Carlo experiments. Instead of resorting to a model selection approach by choosing a particular candidate model for prediction, a forecasting approach based on Bayesian model averaging is used in order to account for model uncertainty and to improve forecasting accuracy. For model diagnosis a Bayesian sign test is introduced to compare the predictive accuracy of different forecasting models in terms of statistical significance. In an empirical application, using monthly unemployment rates of Germany, the performance of the model averaging prediction approach is compared to those of model selected Bayesian and classical (non)periodic time series models.     

Bayesian approach for mixture models with grouped data

Finite mixture modeling approach is widely used for the analysis of bimodal or multimodal data that are individually observed in many situations. However, in some applications, the analysis becomes substantially challenging as the available data are grouped into categories. In this work, we assume that the observed data are grouped into distinct non-overlapping intervals and follow a finite mixture of normal distributions. For the inference of the model parameters, we propose a parametric approach that accounts for the categorical features of the data. The main idea of our method is to impute the missing information of the original data through the Bayesian framework using the Gibbs sampling techniques. The proposed method was compared with the maximum likelihood approach, which uses the Expectation-Maximization algorithm for the estimation of the model parameters. It was also illustrated with an application to the Old Faithful geyser data.     

An efficient sampling scheme for dynamic generalized models

A multimove sampling scheme for the state parameters of non-Gaussian and nonlinear dynamic models for univariate time series is proposed. This procedure follows the Bayesian framework, within a Gibbs sampling algorithm with steps of the Metropolis–Hastings algorithm. This sampling scheme combines the conjugate updating approach for generalized dynamic linear models, with the backward sampling of the state parameters used in normal dynamic linear models. A quite extensive Monte Carlo study is conducted in order to compare the results obtained using our proposed method, conjugate updating backward sampling (CUBS), with those obtained using some algorithms previously proposed in the Bayesian literature. We compare the performance of CUBS with other sampling schemes using two real datasets. Then we apply our algorithm in a stochastic volatility model. CUBS significantly reduces the computing time needed to attain convergence of the chains, and is relatively simple to implement.     

A general approach to Bayesian portfolio optimization

We develop a general approach to portfolio optimization taking account of estimation risk and stylized facts of empirical finance. This is done within a Bayesian framework. The approximation of the posterior distribution of the unknown model parameters is based on a parallel tempering algorithm. The portfolio optimization is done using the first two moments of the predictive discrete asset return distribution. For illustration purposes we apply our method to empirical stock market data where daily asset log-returns are assumed to follow an orthogonal MGARCH process with t-distributed perturbations. Our results are compared with other portfolios suggested by popular optimization strategies.     

Identifying Mixtures of Mixtures Using Bayesian Estimation

The use of a finite mixture of normal distributions in model-based clustering allows us to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework, we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior, where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition, this prior allows us to estimate the model using standard MCMC sampling methods. In combination with a post-processing approach which resolves the label switching issue and results in an identified model, our approach allows us to simultaneously (1) determine the number of clusters, (2) flexibly approximate the cluster distributions in a semiparametric way using finite mixtures of normals and (3) identify cluster-specific parameters and classify observations. The proposed approach is illustrated in two simulation studies and on benchmark datasets. Supplementary materials for this article are available online.     

Markov Chain Monte Carlo With Mixtures of Mutually Singular Distributions

Markov chain Monte Carlo (MCMC) methods for Bayesian computation are mostly used when the dominating measure is the Lebesgue measure, the counting measure, or a product of these. Many Bayesian problems give rise to distributions that are not dominated by the Lebesgue measure or the counting measure alone. In this article we introduce a simple framework for using MCMC algorithms in Bayesian computation with mixtures of mutually singular distributions. The idea is to find a common dominating measure that allows the use of traditional Metropolis-Hastings algorithms. In particular, using our formulation, the Gibbs sampler can be used whenever the full conditionals are available. We compare our formulation with the reversible jump approach and show that the two are closely related. We give results for three examples, involving testing a normal mean, variable selection in regression, and hypothesis testing for differential gene expression under multiple conditions. This allows us to compare the three methods considered: Metropolis-Hastings with mutually singular distributions, Gibbs sampler with mutually singular distributions, and reversible jump. In our examples, we found the Gibbs sampler to be more precise and to need considerably less computer time than the other methods. In addition, the full conditionals used in the Gibbs sampler can be used to further improve the estimates of the model posterior probabilities via Rao-Blackwellization, at no extra cost.     

Functional Additive Mixed Models

We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, for example, spatial, temporal, or longitudinal functional data. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. It accommodates densely or sparsely observed functional responses and predictors which may be observed with additional error and includes both spline-based and functional principal component-based terms. Estimation and inference in this framework is based on standard additive mixed models, allowing us to take advantage of established methods and robust, flexible algorithms. We provide easy-to-use open source software in the pffr() function for the R package refund. Simulations show that the proposed method recovers relevant effects reliably, handles small sample sizes well, and also scales to larger datasets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.     

Nonparametric spatial models for extremes: application to extreme temperature data

Estimating the probability of extreme temperature events is difficult because of limited records across time and the need to extrapolate the distributions of these events, as opposed to just the mean, to locations where observations are not available. Another related issue is the need to characterize the uncertainty in the estimated probability of extreme events at different locations. Although the tools for statistical modeling of univariate extremes are well-developed, extending these tools to model spatial extreme data is an active area of research. In this paper, in order to make inference about spatial extreme events, we introduce a new nonparametric model for extremes. We present a Dirichlet-based copula model that is a flexible alternative to parametric copula models such as the normal and t-copula. The proposed modelling approach is fitted using a Bayesian framework that allow us to take into account different sources of uncertainty in the data and models. We apply our methods to annual maximum temperature values in the east-south-central United States.