5,537 research outputs found
Linear Mixed Models with Marginally Symmetric Nonparametric Random Effects
Linear mixed models (LMMs) are used as an important tool in the data analysis
of repeated measures and longitudinal studies. The most common form of LMMs
utilize a normal distribution to model the random effects. Such assumptions can
often lead to misspecification errors when the random effects are not normal.
One approach to remedy the misspecification errors is to utilize a point-mass
distribution to model the random effects; this is known as the nonparametric
maximum likelihood-fitted (NPML) model. The NPML model is flexible but requires
a large number of parameters to characterize the random-effects distribution.
It is often natural to assume that the random-effects distribution be at least
marginally symmetric. The marginally symmetric NPML (MSNPML) random-effects
model is introduced, which assumes a marginally symmetric point-mass
distribution for the random effects. Under the symmetry assumption, the MSNPML
model utilizes half the number of parameters to characterize the same number of
point masses as the NPML model; thus the model confers an advantage in economy
and parsimony. An EM-type algorithm is presented for the maximum likelihood
(ML) estimation of LMMs with MSNPML random effects; the algorithm is shown to
monotonically increase the log-likelihood and is proven to be convergent to a
stationary point of the log-likelihood function in the case of convergence.
Furthermore, it is shown that the ML estimator is consistent and asymptotically
normal under certain conditions, and the estimation of quantities such as the
random-effects covariance matrix and individual a posteriori expectations is
demonstrated
Iteratively-Reweighted Least-Squares Fitting of Support Vector Machines: A Majorization--Minimization Algorithm Approach
Support vector machines (SVMs) are an important tool in modern data analysis.
Traditionally, support vector machines have been fitted via quadratic
programming, either using purpose-built or off-the-shelf algorithms. We present
an alternative approach to SVM fitting via the majorization--minimization (MM)
paradigm. Algorithms that are derived via MM algorithm constructions can be
shown to monotonically decrease their objectives at each iteration, as well as
be globally convergent to stationary points. We demonstrate the construction of
iteratively-reweighted least-squares (IRLS) algorithms, via the MM paradigm,
for SVM risk minimization problems involving the hinge, least-square,
squared-hinge, and logistic losses, and 1-norm, 2-norm, and elastic net
penalizations. Successful implementations of our algorithms are presented via
some numerical examples
Maximum Likelihood Estimation of Triangular and Polygonal Distributions
Triangular distributions are a well-known class of distributions that are
often used as elementary example of a probability model. In the past,
enumeration and order statistic-based methods have been suggested for the
maximum likelihood (ML) estimation of such distributions. A novel
parametrization of triangular distributions is presented. The parametrization
allows for the construction of an MM (minorization--maximization) algorithm for
the ML estimation of triangular distributions. The algorithm is shown to both
monotonically increase the likelihood evaluations, and be globally convergent.
Using the parametrization is then applied to construct an MM algorithm for the
ML estimation of polygonal distributions. This algorithm is shown to have the
same numerical properties as that of the triangular distribution. Numerical
simulation are provided to demonstrate the performances of the new algorithms
against established enumeration and order statistics-based methods
Higher Order Effects in the Dielectric Constant of Percolative Metal-Insulator Systems above the Critical Point
The dielectric constant of a conductor-insulator mixture shows a pronounced
maximum above the critical volume concentration. Further experimental evidence
is presented as well as a theoretical consideration based on a phenomenological
equation. Explicit expressions are given for the position of the maximum in
terms of scaling parameters and the (complex) conductances of the conductor and
insulator. In order to fit some of the data, a volume fraction dependent
expression for the conductivity of the more highly conductive component is
introduced.Comment: 4 pages, Latex, 4 postscript (*.epsi) files submitted to Phys Rev.
Mixtures of Spatial Spline Regressions
We present an extension of the functional data analysis framework for
univariate functions to the analysis of surfaces: functions of two variables.
The spatial spline regression (SSR) approach developed can be used to model
surfaces that are sampled over a rectangular domain. Furthermore, combining SSR
with linear mixed effects models (LMM) allows for the analysis of populations
of surfaces, and combining the joint SSR-LMM method with finite mixture models
allows for the analysis of populations of surfaces with sub-family structures.
Through the mixtures of spatial splines regressions (MSSR) approach developed,
we present methodologies for clustering surfaces into sub-families, and for
performing surface-based discriminant analysis. The effectiveness of our
methodologies, as well as the modeling capabilities of the SSR model are
assessed through an application to handwritten character recognition
A Block Minorization--Maximization Algorithm for Heteroscedastic Regression
The computation of the maximum likelihood (ML) estimator for heteroscedastic
regression models is considered. The traditional Newton algorithms for the
problem require matrix multiplications and inversions, which are bottlenecks in
modern Big Data contexts. A new Big Data-appropriate minorization--maximization
(MM) algorithm is considered for the computation of the ML estimator. The MM
algorithm is proved to generate monotonically increasing sequences of
likelihood values and to be convergent to a stationary point of the
log-likelihood function. A distributed and parallel implementation of the MM
algorithm is presented and the MM algorithm is shown to have differing time
complexity to the Newton algorithm. Simulation studies demonstrate that the MM
algorithm improves upon the computation time of the Newton algorithm in some
practical scenarios where the number of observations is large
Approximation by finite mixtures of continuous density functions that vanish at infinity
Given sufficiently many components, it is often cited that finite mixture
models can approximate any other probability density function (pdf) to an
arbitrary degree of accuracy. Unfortunately, the nature of this approximation
result is often left unclear. We prove that finite mixture models constructed
from pdfs in can be used to conduct approximation of various
classes of approximands in a number of different modes. That is, we prove
approximands in can be uniformly approximated, approximands
in can be uniformly approximated on compact sets, and
approximands in can be approximated with respect to the
, for . Furthermore, we also prove
that measurable functions can be approximated, almost everywhere
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