3,038 research outputs found
Implicit Copulas from Bayesian Regularized Regression Smoothers
We show how to extract the implicit copula of a response vector from a
Bayesian regularized regression smoother with Gaussian disturbances. The copula
can be used to compare smoothers that employ different shrinkage priors and
function bases. We illustrate with three popular choices of shrinkage priors
--- a pairwise prior, the horseshoe prior and a g prior augmented with a point
mass as employed for Bayesian variable selection --- and both univariate and
multivariate function bases. The implicit copulas are high-dimensional, have
flexible dependence structures that are far from that of a Gaussian copula, and
are unavailable in closed form. However, we show how they can be evaluated by
first constructing a Gaussian copula conditional on the regularization
parameters, and then integrating over these. Combined with non-parametric
margins the regularized smoothers can be used to model the distribution of
non-Gaussian univariate responses conditional on the covariates. Efficient
Markov chain Monte Carlo schemes for evaluating the copula are given for this
case. Using both simulated and real data, we show how such copula smoothing
models can improve the quality of resulting function estimates and predictive
distributions
Variational Bayes Estimation of Discrete-Margined Copula Models with Application to Time Series
We propose a new variational Bayes estimator for high-dimensional copulas
with discrete, or a combination of discrete and continuous, margins. The method
is based on a variational approximation to a tractable augmented posterior, and
is faster than previous likelihood-based approaches. We use it to estimate
drawable vine copulas for univariate and multivariate Markov ordinal and mixed
time series. These have dimension , where is the number of observations
and is the number of series, and are difficult to estimate using previous
methods. The vine pair-copulas are carefully selected to allow for
heteroskedasticity, which is a feature of most ordinal time series data. When
combined with flexible margins, the resulting time series models also allow for
other common features of ordinal data, such as zero inflation, multiple modes
and under- or over-dispersion. Using six example series, we illustrate both the
flexibility of the time series copula models, and the efficacy of the
variational Bayes estimator for copulas of up to 792 dimensions and 60
parameters. This far exceeds the size and complexity of copula models for
discrete data that can be estimated using previous methods
Bayesian Approaches to Copula Modelling
Copula models have become one of the most widely used tools in the applied
modelling of multivariate data. Similarly, Bayesian methods are increasingly
used to obtain efficient likelihood-based inference. However, to date, there
has been only limited use of Bayesian approaches in the formulation and
estimation of copula models. This article aims to address this shortcoming in
two ways. First, to introduce copula models and aspects of copula theory that
are especially relevant for a Bayesian analysis. Second, to outline Bayesian
approaches to formulating and estimating copula models, and their advantages
over alternative methods. Copulas covered include Archimedean, copulas
constructed by inversion, and vine copulas; along with their interpretation as
transformations. A number of parameterisations of a correlation matrix of a
Gaussian copula are considered, along with hierarchical priors that allow for
Bayesian selection and model averaging for each parameterisation. Markov chain
Monte Carlo sampling schemes for fitting Gaussian and D-vine copulas, with and
without selection, are given in detail. The relationship between the prior for
the parameters of a D-vine, and the prior for a correlation matrix of a
Gaussian copula, is discussed. Last, it is shown how to compute Bayesian
inference when the data are discrete-valued using data augmentation. This
approach generalises popular Bayesian methods for the estimation of models for
multivariate binary and other ordinal data to more general copula models.
Bayesian data augmentation has substantial advantages over other methods of
estimation for this class of models
Bayesian Variable Selection for Non-Gaussian Responses: A Marginally Calibrated Copula Approach
We propose a new highly flexible and tractable Bayesian approach to undertake
variable selection in non-Gaussian regression models. It uses a copula
decomposition for the joint distribution of observations on the dependent
variable. This allows the marginal distribution of the dependent variable to be
calibrated accurately using a nonparametric or other estimator. The family of
copulas employed are `implicit copulas' that are constructed from existing
hierarchical Bayesian models widely used for variable selection, and we
establish some of their properties. Even though the copulas are
high-dimensional, they can be estimated efficiently and quickly using Markov
chain Monte Carlo (MCMC). A simulation study shows that when the responses are
non-Gaussian the approach selects variables more accurately than contemporary
benchmarks. A real data example in the Web Appendix illustrates that accounting
for even mild deviations from normality can lead to a substantial increase in
accuracy. To illustrate the full potential of our approach we extend it to
spatial variable selection for fMRI. Using real data, we show our method allows
for voxel-specific marginal calibration of the magnetic resonance signal at
over 6,000 voxels, leading to an increase in the quality of the activation
maps
Deep Distributional Time Series Models and the Probabilistic Forecasting of Intraday Electricity Prices
Recurrent neural networks (RNNs) with rich feature vectors of past values can
provide accurate point forecasts for series that exhibit complex serial
dependence. We propose two approaches to constructing deep time series
probabilistic models based on a variant of RNN called an echo state network
(ESN). The first is where the output layer of the ESN has stochastic
disturbances and a shrinkage prior for additional regularization. The second
approach employs the implicit copula of an ESN with Gaussian disturbances,
which is a deep copula process on the feature space. Combining this copula with
a non-parametrically estimated marginal distribution produces a deep
distributional time series model. The resulting probabilistic forecasts are
deep functions of the feature vector and also marginally calibrated. In both
approaches, Bayesian Markov chain Monte Carlo methods are used to estimate the
models and compute forecasts. The proposed deep time series models are suitable
for the complex task of forecasting intraday electricity prices. Using data
from the Australian National Electricity Market, we show that our models
provide accurate probabilistic price forecasts. Moreover, the models provide a
flexible framework for incorporating probabilistic forecasts of electricity
demand as additional features. We demonstrate that doing so in the deep
distributional time series model in particular, increases price forecast
accuracy substantially
Automated potentiometric techniques for the on-site monitoring of anion concentrations in water
Students supported: 2 Grad Students, 2 Undergraduate StudentsCharacteristics of several of the new non-glass ion-selective electrodes and of several reference electrodes were studied in detail. The most significant finding of the research was the development of a new high accuracy standard addition technique for the potentiometric determination of nitrate ion in a cell without liquid junction using the fluoride ion-selective electrode as a reference. This method eliminates the liquid junction and dilution and activity coefficient changes cancel. The standard addition technique with the fluoride electrode as a reference is applicable to all monovalent anions for which an electrode is available and enables the determination of anion concentrations to within 1 percent relative accuracy. Low temperature potentiometric titrations of perchlorate and tetrafluoroborate ions were investigated and found to give very high accuracy and reasonably good sensitivity for these ions. Tetraphenylarsonium chloride was used as a titrant for perchlorate and tetrafluoroborate. The application of the nitrate ion-selective electrode to following nitrate ion concentration in microbial cultures was investigated. It was found to be useful for following nitrate ion reduction in situ.Project # B-023-MO Agreement # 14-01-0001-191
- …