The Multivariate Generalised von Mises distribution: Inference and applications

Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First we introduce a new multivariate distribution over circular variables, called the multivariate Generalised von Mises (mGvM) distribution. This distribution can be constructed by restricting and renormalising a general multivariate Gaussian distribution to the unit hyper-torus. Previously proposed multivariate circular distributions are shown to be special cases of this construction. Second, we introduce a new probabilistic model for circular regression, that is inspired by Gaussian Processes, and a method for probabilistic principal component analysis with circular hidden variables. These models can leverage standard modelling tools (e.g. covariance functions and methods for automatic relevance determination). Third, we show that the posterior distribution in these models is a mGvM distribution which enables development of an efficient variational free-energy scheme for performing approximate inference and approximate maximum-likelihood learning.AKWN thanks CAPES grant BEX 9407-11-1. JF thanks the Danish Council for Independent Research grant 0602- 02909B. RET thanks EPSRC grants EP/L000776/1 and EP/M026957/1

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

A Bayesian approach to the semi-analytic model of galaxy formation: methodology

We believe that a wide range of physical processes conspire to shape the observed galaxy population but we remain unsure of their detailed interactions. The semi-analytic model (SAM) of galaxy formation uses multi-dimensional parameterisations of the physical processes of galaxy formation and provides a tool to constrain these underlying physical interactions. Because of the high dimensionality, the parametric problem of galaxy formation may be profitably tackled with a Bayesian-inference based approach, which allows one to constrain theory with data in a statistically rigorous way. In this paper we develop a SAM in the framework of Bayesian inference. We show that, with a parallel implementation of an advanced Markov-Chain Monte-Carlo algorithm, it is now possible to rigorously sample the posterior distribution of the high-dimensional parameter space of typical SAMs. As an example, we characterise galaxy formation in the current $\Lambda$CDM cosmology using the stellar mass function of galaxies as an observational constraint. We find that the posterior probability distribution is both topologically complex and degenerate in some important model parameters, suggesting that thorough explorations of the parameter space are needed to understand the models. We also demonstrate that because of the model degeneracy, adopting a narrow prior strongly restricts the model. Therefore, the inferences based on SAMs are conditional to the model adopted. Using synthetic data to mimic systematic errors in the stellar mass function, we demonstrate that an accurate observational error model is essential to meaningful inference.Comment: revised version to match published article published in MNRA

The joint projected normal and skew-normal: a distribution for poly-cylindrical data

The contribution of this work is the introduction of a multivariate circular-linear (or poly- cylindrical) distribution obtained by combining the projected and the skew-normal. We show the flexibility of our proposal, its property of closure under marginalization and how to quantify multivariate dependence. Due to a non-identifiability issue that our proposal inherits from the projected normal, a compu- tational problem arises. We overcome it in a Bayesian framework, adding suitable latent variables and showing that posterior samples can be obtained with a post-processing of the estimation algo- rithm output. Under specific prior choices, this approach enables us to implement a Markov chain Monte Carlo algorithm relying only on Gibbs steps, where the updates of the parameters are done as if we were working with a multivariate normal likelihood. The proposed approach can be also used with the projected normal. As a proof of concept, on simulated examples we show the ability of our algorithm in recovering the parameters values and to solve the identification problem. Then the proposal is used in a real data example, where the turning-angles (circular variables) and the logarithm of the step-lengths (linear variables) of four zebras are jointly modelled

Nonparametric Dynamic State Space Modeling of Observed Circular Time Series with Circular Latent States: A Bayesian Perspective

Circular time series has received relatively little attention in statistics and modeling complex circular time series using the state space approach is non-existent in the literature. In this article we introduce a flexible Bayesian nonparametric approach to state space modeling of observed circular time series where even the latent states are circular random variables. Crucially, we assume that the forms of both observational and evolutionary functions, both of which are circular in nature, are unknown and time-varying. We model these unknown circular functions by appropriate wrapped Gaussian processes having desirable properties. We develop an effective Markov chain Monte Carlo strategy for implementing our Bayesian model, by judiciously combining Gibbs sampling and Metropolis-Hastings methods. Validation of our ideas with a simulation study and two real bivariate circular time series data sets, where we assume one of the variables to be unobserved, revealed very encouraging performance of our model and methods. We finally analyse a data consisting of directions of whale migration, considering the unobserved ocean current direction as the latent circular process of interest. The results that we obtain are encouraging, and the posterior predictive distribution of the observed process correctly predicts the observed whale movement.Comment: This significantly updated version will appear in Journal of Statistical Theory and Practic

Bayesian inferences of galaxy formation from the K-band luminosity and HI mass functions of galaxies: constraining star formation and feedback

We infer mechanisms of galaxy formation for a broad family of semi-analytic models (SAMs) constrained by the K-band luminosity function and HI mass function of local galaxies using tools of Bayesian analysis. Even with a broad search in parameter space the whole model family fails to match to constraining data. In the best fitting models, the star formation and feedback parameters in low-mass haloes are tightly constrained by the two data sets, and the analysis reveals several generic failures of models that similarly apply to other existing SAMs. First, based on the assumption that baryon accretion follows the dark matter accretion, large mass-loading factors are required for haloes with circular velocities lower than 200 km/s, and most of the wind mass must be expelled from the haloes. Second, assuming that the feedback is powered by Type-II supernovae with a Chabrier IMF, the outflow requires more than 25% of the available SN kinetic energy. Finally, the posterior predictive distributions for the star formation history are dramatically inconsistent with observations for masses similar to or smaller than the Milky-Way mass. The inferences suggest that the current model family is still missing some key physical processes that regulate the gas accretion and star formation in galaxies with masses below that of the Milky Way.Comment: 17 pages, 9 figures, 1 table, accepted for publication in MNRA

The Mass of the Black Hole in Arp 151 from Bayesian Modeling of Reverberation Mapping Data

Supermassive black holes are believed to be ubiquitous at the centers of galaxies. Measuring their masses is extremely challenging yet essential for understanding their role in the formation and evolution of cosmic structure. We present a direct measurement of the mass of a black hole in an active galactic nucleus (Arp 151) based on the motion of the gas responsible for the broad emission lines. By analyzing and modeling spectroscopic and photometric time series, we find that the gas is well described by a disk or torus with an average radius of 3.99 +- 1.25 light days and an opening angle of 68.9 (+21.4, -17.2) degrees, viewed at an inclination angle of 67.8 +- 7.8 degrees (that is, closer to face-on than edge-on). The black hole mass is inferred to be 10^(6.51 +- 0.28) solar masses. The method is fully general and can be used to determine the masses of black holes at arbitrary distances, enabling studies of their evolution over cosmic time.Comment: Accepted for publication in ApJ Letter