5,097 research outputs found
Nonlinear Models Using Dirichlet Process Mixtures
We introduce a new nonlinear model for classification, in which we model the
joint distribution of response variable, y, and covariates, x,
non-parametrically using Dirichlet process mixtures. We keep the relationship
between y and x linear within each component of the mixture. The overall
relationship becomes nonlinear if the mixture contains more than one component.
We use simulated data to compare the performance of this new approach to a
simple multinomial logit (MNL) model, an MNL model with quadratic terms, and a
decision tree model. We also evaluate our approach on a protein fold
classification problem, and find that our model provides substantial
improvement over previous methods, which were based on Neural Networks (NN) and
Support Vector Machines (SVM). Folding classes of protein have a hierarchical
structure. We extend our method to classification problems where a class
hierarchy is available. We find that using the prior information regarding the
hierarchical structure of protein folds can result in higher predictive
accuracy
Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches
Imaging spectrometers measure electromagnetic energy scattered in their
instantaneous field view in hundreds or thousands of spectral channels with
higher spectral resolution than multispectral cameras. Imaging spectrometers
are therefore often referred to as hyperspectral cameras (HSCs). Higher
spectral resolution enables material identification via spectroscopic analysis,
which facilitates countless applications that require identifying materials in
scenarios unsuitable for classical spectroscopic analysis. Due to low spatial
resolution of HSCs, microscopic material mixing, and multiple scattering,
spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus,
accurate estimation requires unmixing. Pixels are assumed to be mixtures of a
few materials, called endmembers. Unmixing involves estimating all or some of:
the number of endmembers, their spectral signatures, and their abundances at
each pixel. Unmixing is a challenging, ill-posed inverse problem because of
model inaccuracies, observation noise, environmental conditions, endmember
variability, and data set size. Researchers have devised and investigated many
models searching for robust, stable, tractable, and accurate unmixing
algorithms. This paper presents an overview of unmixing methods from the time
of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models
are first discussed. Signal-subspace, geometrical, statistical, sparsity-based,
and spatial-contextual unmixing algorithms are described. Mathematical problems
and potential solutions are described. Algorithm characteristics are
illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of
Selected Topics in Applied Earth Observations and Remote Sensin
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
The supervised hierarchical Dirichlet process
We propose the supervised hierarchical Dirichlet process (sHDP), a
nonparametric generative model for the joint distribution of a group of
observations and a response variable directly associated with that whole group.
We compare the sHDP with another leading method for regression on grouped data,
the supervised latent Dirichlet allocation (sLDA) model. We evaluate our method
on two real-world classification problems and two real-world regression
problems. Bayesian nonparametric regression models based on the Dirichlet
process, such as the Dirichlet process-generalised linear models (DP-GLM) have
previously been explored; these models allow flexibility in modelling nonlinear
relationships. However, until now, Hierarchical Dirichlet Process (HDP)
mixtures have not seen significant use in supervised problems with grouped data
since a straightforward application of the HDP on the grouped data results in
learnt clusters that are not predictive of the responses. The sHDP solves this
problem by allowing for clusters to be learnt jointly from the group structure
and from the label assigned to each group.Comment: 14 page
Sparse covariance estimation in heterogeneous samples
Standard Gaussian graphical models (GGMs) implicitly assume that the
conditional independence among variables is common to all observations in the
sample. However, in practice, observations are usually collected form
heterogeneous populations where such assumption is not satisfied, leading in
turn to nonlinear relationships among variables. To tackle these problems we
explore mixtures of GGMs; in particular, we consider both infinite mixture
models of GGMs and infinite hidden Markov models with GGM emission
distributions. Such models allow us to divide a heterogeneous population into
homogenous groups, with each cluster having its own conditional independence
structure. The main advantage of considering infinite mixtures is that they
allow us easily to estimate the number of number of subpopulations in the
sample. As an illustration, we study the trends in exchange rate fluctuations
in the pre-Euro era. This example demonstrates that the models are very
flexible while providing extremely interesting interesting insights into
real-life applications
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