A Bayesian Nonparametric Markovian Model for Nonstationary Time Series

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 density, defining a time-homogeneous, nonstationary, Markovian model for real-valued data indexed in discrete-time. To obtain a more computationally tractable algorithm for posterior inference, we utilize a square-root-free Cholesky decomposition of the mixture kernel covariance matrix. Results from simulated data suggest the model is able to recover challenging transition and predictive densities. We also illustrate the model on time intervals between eruptions of the Old Faithful geyser. Extensions to accommodate higher order structure and to develop a state-space model are also discussed

EMMIX-uskew: An R Package for Fitting Mixtures of Multivariate Skew t-distributions via the EM Algorithm

This paper describes an algorithm for fitting finite mixtures of unrestricted Multivariate Skew t (FM-uMST) distributions. The package EMMIX-uskew implements a closed-form expectation-maximization (EM) algorithm for computing the maximum likelihood (ML) estimates of the parameters for the (unrestricted) FM-MST model in R. EMMIX-uskew also supports visualization of fitted contours in two and three dimensions, and random sample generation from a specified FM-uMST distribution. Finite mixtures of skew t-distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour, for example, datasets from flow cytometry. In recent years, various versions of mixtures with multivariate skew t (MST) distributions have been proposed. However, these models adopted some restricted characterizations of the component MST distributions so that the E-step of the EM algorithm can be evaluated in closed form. This paper focuses on mixtures with unrestricted MST components, and describes an iterative algorithm for the computation of the ML estimates of its model parameters. The usefulness of the proposed algorithm is demonstrated in three applications to real data sets. The first example illustrates the use of the main function fmmst in the package by fitting a MST distribution to a bivariate unimodal flow cytometric sample. The second example fits a mixture of MST distributions to the Australian Institute of Sport (AIS) data, and demonstrate that EMMIX-uskew can provide better clustering results than mixtures with restricted MST components. In the third example, EMMIX-uskew is applied to classify cells in a trivariate flow cytometric dataset. Comparisons with other available methods suggests that the EMMIX-uskew result achieved a lower misclassification rate with respect to the labels given by benchmark gating analysis

Estimating Local Function Complexity via Mixture of Gaussian Processes

Real world data often exhibit inhomogeneity, e.g., the noise level, the sampling distribution or the complexity of the target function may change over the input space. In this paper, we try to isolate local function complexity in a practical, robust way. This is achieved by first estimating the locally optimal kernel bandwidth as a functional relationship. Specifically, we propose Spatially Adaptive Bandwidth Estimation in Regression (SABER), which employs the mixture of experts consisting of multinomial kernel logistic regression as a gate and Gaussian process regression models as experts. Using the locally optimal kernel bandwidths, we deduce an estimate to the local function complexity by drawing parallels to the theory of locally linear smoothing. We demonstrate the usefulness of local function complexity for model interpretation and active learning in quantum chemistry experiments and fluid dynamics simulations.Comment: 19 pages, 16 figure

Flexible modelling in statistics: past, present and future

In times where more and more data become available and where the data exhibit rather complex structures (significant departure from symmetry, heavy or light tails), flexible modelling has become an essential task for statisticians as well as researchers and practitioners from domains such as economics, finance or environmental sciences. This is reflected by the wealth of existing proposals for flexible distributions; well-known examples are Azzalini's skew-normal, Tukey's $g$-and-$h$, mixture and two-piece distributions, to cite but these. My aim in the present paper is to provide an introduction to this research field, intended to be useful both for novices and professionals of the domain. After a description of the research stream itself, I will narrate the gripping history of flexible modelling, starring emblematic heroes from the past such as Edgeworth and Pearson, then depict three of the most used flexible families of distributions, and finally provide an outlook on future flexible modelling research by posing challenging open questions.Comment: 27 pages, 4 figure

Joint Modeling and Registration of Cell Populations in Cohorts of High-Dimensional Flow Cytometric Data

In systems biomedicine, an experimenter encounters different potential sources of variation in data such as individual samples, multiple experimental conditions, and multi-variable network-level responses. In multiparametric cytometry, which is often used for analyzing patient samples, such issues are critical. While computational methods can identify cell populations in individual samples, without the ability to automatically match them across samples, it is difficult to compare and characterize the populations in typical experiments, such as those responding to various stimulations or distinctive of particular patients or time-points, especially when there are many samples. Joint Clustering and Matching (JCM) is a multi-level framework for simultaneous modeling and registration of populations across a cohort. JCM models every population with a robust multivariate probability distribution. Simultaneously, JCM fits a random-effects model to construct an overall batch template -- used for registering populations across samples, and classifying new samples. By tackling systems-level variation, JCM supports practical biomedical applications involving large cohorts