17 research outputs found
High-dimensional limits of eigenvalue distributions for general Wishart process
In this article, we obtain an equation for the high-dimensional limit measure
of eigenvalues of generalized Wishart processes, and the results is extended to
random particle systems that generalize SDEs of eigenvalues. We also introduce
a new set of conditions on the coefficient matrices for the existence and
uniqueness of a strong solution for the SDEs of eigenvalues. The equation of
the limit measure is further discussed assuming self-similarity on the
eigenvalues.Comment: 28 page
Dynamic Covariance Models for Multivariate Financial Time Series
The accurate prediction of time-changing covariances is an important problem
in the modeling of multivariate financial data. However, some of the most
popular models suffer from a) overfitting problems and multiple local optima,
b) failure to capture shifts in market conditions and c) large computational
costs. To address these problems we introduce a novel dynamic model for
time-changing covariances. Over-fitting and local optima are avoided by
following a Bayesian approach instead of computing point estimates. Changes in
market conditions are captured by assuming a diffusion process in parameter
values, and finally computationally efficient and scalable inference is
performed using particle filters. Experiments with financial data show
excellent performance of the proposed method with respect to current standard
models
Gaussian Process Conditional Copulas with Applications to Financial Time Series
The estimation of dependencies between multiple variables is a central
problem in the analysis of financial time series. A common approach is to
express these dependencies in terms of a copula function. Typically the copula
function is assumed to be constant but this may be inaccurate when there are
covariates that could have a large influence on the dependence structure of the
data. To account for this, a Bayesian framework for the estimation of
conditional copulas is proposed. In this framework the parameters of a copula
are non-linearly related to some arbitrary conditioning variables. We evaluate
the ability of our method to predict time-varying dependencies on several
equities and currencies and observe consistent performance gains compared to
static copula models and other time-varying copula methods
A Mutually-Dependent Hadamard Kernel for Modelling Latent Variable Couplings
We introduce a novel kernel that models input-dependent couplings across
multiple latent processes. The pairwise joint kernel measures covariance along
inputs and across different latent signals in a mutually-dependent fashion. A
latent correlation Gaussian process (LCGP) model combines these non-stationary
latent components into multiple outputs by an input-dependent mixing matrix.
Probit classification and support for multiple observation sets are derived by
Variational Bayesian inference. Results on several datasets indicate that the
LCGP model can recover the correlations between latent signals while
simultaneously achieving state-of-the-art performance. We highlight the latent
covariances with an EEG classification dataset where latent brain processes and
their couplings simultaneously emerge from the model.Comment: 17 pages, 6 figures; accepted to ACML 201
Generalized Wishart processes for interpolation over diffusion tensor fields
Diffusion Magnetic Resonance Imaging (dMRI) is a non-invasive tool for watching the microstructure of fibrous nerve and muscle tissue. From dMRI, it is possible to estimate 2-rank diffusion tensors imaging (DTI) fields, that are widely used in clinical applications: tissue segmentation, fiber tractography, brain atlas construction, brain conductivity models, among others. Due to hardware limitations of MRI scanners, DTI has the difficult compromise between spatial resolution and signal noise ratio (SNR) during acquisition. For this reason, the data are often acquired with very low resolution. To enhance DTI data resolution, interpolation provides an interesting software solution. The aim of this work is to develop a methodology for DTI interpolation that enhance the spatial resolution of DTI fields. We assume that a DTI field follows a recently introduced stochastic process known as a generalized Wishart process (GWP), which we use as a prior over the diffusion tensor field. For posterior inference, we use Markov Chain Monte Carlo methods. We perform experiments in toy and real data. Results of GWP outperform other methods in the literature, when compared in different validation protocols
Nonparametric Bayesian Inference on Multivariate Exponential Families
We develop a model by choosing the maximum entropy distribution from the set of models satisfying certain smoothness and independence criteria; we show that inference on this model generalizes local kernel estimation to the context of Bayesian inference on stochastic processes. Our model enables Bayesian inference in contexts when standard techniques like Gaussian process inference are too expensive to apply. Exact inference on our model is possible for any likelihood function from the exponential family. Inference is then highly efficient, requiring only O (log N) time and O (N) space at run time. We demonstrate our algorithm on several problems and show quantifiable improvement in both speed and performance relative to models based on the Gaussian process.United States. Office of Naval Research (N00014-09-1-1052)United States. Office of Naval Research (N00014-10-1-0936
Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices
Modeling correlation (and covariance) matrices can be challenging due to the
positive-definiteness constraint and potential high-dimensionality. Our
approach is to decompose the covariance matrix into the correlation and
variance matrices and propose a novel Bayesian framework based on modeling the
correlations as products of unit vectors. By specifying a wide range of
distributions on a sphere (e.g. the squared-Dirichlet distribution), the
proposed approach induces flexible prior distributions for covariance matrices
(that go beyond the commonly used inverse-Wishart prior). For modeling
real-life spatio-temporal processes with complex dependence structures, we
extend our method to dynamic cases and introduce unit-vector Gaussian process
priors in order to capture the evolution of correlation among components of a
multivariate time series. To handle the intractability of the resulting
posterior, we introduce the adaptive -Spherical Hamiltonian Monte
Carlo. We demonstrate the validity and flexibility of our proposed framework in
a simulation study of periodic processes and an analysis of rat's local field
potential activity in a complex sequence memory task.Comment: 49 pages, 15 figure