66 research outputs found
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
A Fully Nonparametric Modelling Approach to Binary Regression
We propose a general nonparametric Bayesian framework for binary regression,
which is built from modeling for the joint response-covariate distribution. The
observed binary responses are assumed to arise from underlying continuous
random variables through discretization, and we model the joint distribution of
these latent responses and the covariates using a Dirichlet process mixture of
multivariate normals. We show that the kernel of the induced mixture model for
the observed data is identifiable upon a restriction on the latent variables.
To allow for appropriate dependence structure while facilitating
identifiability, we use a square-root-free Cholesky decomposition of the
covariance matrix in the normal mixture kernel. In addition to allowing for the
necessary restriction, this modeling strategy provides substantial
simplifications in implementation of Markov chain Monte Carlo posterior
simulation. We present two data examples taken from areas for which the
methodology is especially well suited. In particular, the first example
involves estimation of relationships between environmental variables, and the
second develops inference for natural selection surfaces in evolutionary
biology. Finally, we discuss extensions to regression settings with
multivariate ordinal responses
Bayesian semiparametric modeling for stochastic precedence, with applications in epidemiology and survival analysis
Bayesian spectral modeling for multiple time series
We develop a novel Bayesian modeling approach to spectral density estimation for multiple time series. The log-periodogram distribution for each series is modeled as a mixture of Gaussian distributions with frequency-dependent weights and mean functions. The implied model for the log-spectral density is a mixture of linear mean functions with frequency-dependent weights. The mixture weights are built through successive differences of a logit-normal distribution function with frequency-dependent parameters. Building from the construction for a single spectral density, we develop a hierarchical extension for multiple time series. Specifically, we set the mean functions to be common to all spectral densities and make the weights specific to the time series through the parameters of the logit-normal distribution. In addition to accommodating flexible spectral density shapes, a practically important feature of the proposed formulation is that it allows for ready posterior simulation through a Gibbs sampler with closed form full conditional distributions for all model parameters. The modeling approach is illustrated with simulated datasets, and used for spectral analysis of multichannel electroencephalographic recordings (EEGs), which provides a key motivating application for the proposed methodology
Modeling for seasonal marked point processes: An analysis of evolving hurricane occurrences
Seasonal point processes refer to stochastic models for random events which
are only observed in a given season. We develop nonparametric Bayesian
methodology to study the dynamic evolution of a seasonal marked point process
intensity. We assume the point process is a nonhomogeneous Poisson process and
propose a nonparametric mixture of beta densities to model dynamically evolving
temporal Poisson process intensities. Dependence structure is built through a
dependent Dirichlet process prior for the seasonally-varying mixing
distributions. We extend the nonparametric model to incorporate time-varying
marks, resulting in flexible inference for both the seasonal point process
intensity and for the conditional mark distribution. The motivating application
involves the analysis of hurricane landfalls with reported damages along the
U.S. Gulf and Atlantic coasts from 1900 to 2010. We focus on studying the
evolution of the intensity of the process of hurricane landfall occurrences,
and the respective maximum wind speed and associated damages. Our results
indicate an increase in the number of hurricane landfall occurrences and a
decrease in the median maximum wind speed at the peak of the season.
Introducing standardized damage as a mark, such that reported damages are
comparable both in time and space, we find that there is no significant rising
trend in hurricane damages over time.Comment: Published at http://dx.doi.org/10.1214/14-AOAS796 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bayesian Nonparametric Density Autoregression with Lag Selection
We develop a Bayesian nonparametric autoregressive model applied to flexibly
estimate general transition densities exhibiting nonlinear lag dependence. Our
approach is related to Bayesian density regression using Dirichlet process
mixtures, with the Markovian likelihood defined through the conditional
distribution obtained from the mixture. This results in a Bayesian
nonparametric extension of a mixtures-of-experts model formulation. We address
computational challenges to posterior sampling that arise from the Markovian
structure in the likelihood. The base model is illustrated with synthetic data
from a classical model for population dynamics, as well as a series of waiting
times between eruptions of Old Faithful Geyser. We study inferences available
through the base model before extending the methodology to include automatic
relevance detection among a pre-specified set of lags. Inference for global and
local lag selection is explored with additional simulation studies, and the
methods are illustrated through analysis of an annual time series of pink
salmon abundance in a stream in Alaska. We further explore and compare
transition density estimation performance for alternative configurations of the
proposed model
Structured Mixture of Continuation-ratio Logits Models for Ordinal Regression
We develop a nonparametric Bayesian modeling approach to ordinal regression
based on priors placed directly on the discrete distribution of the ordinal
responses. The prior probability models are built from a structured mixture of
multinomial distributions. We leverage a continuation-ratio logits
representation to formulate the mixture kernel, with mixture weights defined
through the logit stick-breaking process that incorporates the covariates
through a linear function. The implied regression functions for the response
probabilities can be expressed as weighted sums of parametric regression
functions, with covariate-dependent weights. Thus, the modeling approach
achieves flexible ordinal regression relationships, avoiding linearity or
additivity assumptions in the covariate effects. A key model feature is that
the parameters for both the mixture kernel and the mixture weights can be
associated with a continuation-ratio logits regression structure. Hence, an
efficient and relatively easy to implement posterior simulation method can be
designed, using P\'olya-Gamma data augmentation. Moreover, the model is built
from a conditional independence structure for category-specific parameters,
which results in additional computational efficiency gains through partial
parallel sampling. In addition to the general mixture structure, we study
simplified model versions that incorporate covariate dependence only in the
mixture kernel parameters or only in the mixture weights. For all proposed
models, we discuss approaches to prior specification and develop Markov chain
Monte Carlo methods for posterior simulation. The methodology is illustrated
with several synthetic and real data examples
- …