142 research outputs found
An Additive Bivariate Hierarchical Model for Functional Data and Related Computations
The work presented in this dissertation centers on the theme of regression and
computation methodology. Functional data is an important class of longitudinal
data, and principal component analysis is an important approach to regression with
this type of data. Here we present an additive hierarchical bivariate functional data
model employing principal components to identify random e ects. This additive
model extends the univariate functional principal component model. These models
are implemented in the pfda package for R. To t the curves from this class of models
orthogonalized spline basis are used to reduce the dimensionality of the t, but retain
exibility. Methods for handing spline basis functions in a purely analytical manner,
including the orthogonalizing process and computing of penalty matrices used to t
the principal component models are presented. The methods are implemented in the
R package orthogonalsplinebasis.
The projects discussed involve complicated coding for the implementations in R.
To facilitate this I created the NppToR utility to add R functionality to the popular
windows code editor Notepad . A brief overview of the use of the utility is also
included
Clustering of longitudinal curves via a penalized method and EM algorithm
In this article, the subgroup analysis is considered for longitudinal curves
under the framework of functional principal component analysis. The mean
functions of different curves are assumed to be in different groups but share
the same covariance structure. The mean functions are written as B-spline
functions and the subgroups are found through a concave pairwise fusion method.
The EM algorithm and the alternating direction method of multiplier algorithm
(ADMM) are combined to estimate the group structure, mean functions and
covariance function simultaneously. In the simulation study, the performance of
the proposed method is compared with the existing subgrouping method, which
ignores the covariance structure, in terms of the accuracy for estimating the
number of subgroups and mean functions. The results suggest that ignoring
covariance structure will have a great effect on the performance of estimating
the number of groups and estimating accuracy. Including pairwise weights in the
pairwise penalty functions is also explored in a spatial lattice setting to
take consideration of the spatial information. The results show that
incorporating spatial weights will improve the performance
Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction
In this paper, we study the inverse boundary value problem for the wave
equation with a view towards an explicit reconstruction procedure. We consider
both the anisotropic problem where the unknown is a general Riemannian metric
smoothly varying in a domain, and the isotropic problem where the metric is
conformal to the Euclidean metric. Our objective in both cases is to construct
the metric, using either the Neumann-to-Dirichlet (N-to-D) map or
Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we
construct the metric in the boundary normal (or semi-geodesic) coordinates via
reconstruction of the wave field in the interior of the domain. In the
isotropic case we can go further and construct the wave speed in the Euclidean
coordinates via reconstruction of the coordinate transformation from the
boundary normal coordinates to the Euclidean coordinates. Both cases utilize a
variant of the Boundary Control method, and work by probing the interior using
special boundary sources. We provide a computational experiment to demonstrate
our procedure in the isotropic case with N-to-D data.Comment: 24 pages, 6 figure
VADER: A Flexible, Robust, Open-Source Code for Simulating Viscous Thin Accretion Disks
The evolution of thin axisymmetric viscous accretion disks is a classic
problem in astrophysics. While models based on this simplified geometry provide
only approximations to the true processes of instability-driven mass and
angular momentum transport, their simplicity makes them invaluable tools for
both semi-analytic modeling and simulations of long-term evolution where two-
or three-dimensional calculations are too computationally costly. Despite the
utility of these models, the only publicly-available frameworks for simulating
them are rather specialized and non-general. Here we describe a highly
flexible, general numerical method for simulating viscous thin disks with
arbitrary rotation curves, viscosities, boundary conditions, grid spacings,
equations of state, and rates of gain or loss of mass (e.g., through winds) and
energy (e.g., through radiation). Our method is based on a conservative,
finite-volume, second-order accurate discretization of the equations, which we
solve using an unconditionally-stable implicit scheme. We implement Anderson
acceleration to speed convergence of the scheme, and show that this leads to
factor of speed gains over non-accelerated methods in realistic
problems, though the amount of speedup is highly problem-dependent. We have
implemented our method in the new code Viscous Accretion Disk Evolution
Resource (VADER), which is freely available for download from
https://bitbucket.org/krumholz/vader/ under the terms of the GNU General Public
License.Comment: 58 pages, 13 figures, accepted to Astronomy & Computing; this version
includes more discussion, but no other changes; code is available for
download from https://bitbucket.org/krumholz/vader
Efficient implementation of atom-density representations
Physically motivated and mathematically robust atom-centered representations of molecular structures are key to the success of modern atomistic machine learning. They lie at the foundation of a wide range of methods to predict the properties of both materials and molecules and to explore and visualize their chemical structures and compositions. Recently, it has become clear that many of the most effective representations share a fundamental formal connection. They can all be expressed as a discretization of n-body correlation functions of the local atom density, suggesting the opportunity of standardizing and, more importantly, optimizing their evaluation. We present an implementation, named librascal, whose modular design lends itself both to developing refinements to the density-based formalism and to rapid prototyping for new developments of rotationally equivariant atomistic representations. As an example, we discuss smooth overlap of atomic position (SOAP) features, perhaps the most widely used member of this family of representations, to show how the expansion of the local density can be optimized for any choice of radial basis sets. We discuss the representation in the context of a kernel ridge regression model, commonly used with SOAP features, and analyze how the computational effort scales for each of the individual steps of the calculation. By applying data reduction techniques in feature space, we show how to reduce the total computational cost by a factor of up to 4 without affecting the model’s symmetry properties and without significantly impacting its accuracy
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