2,101 research outputs found
An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data
We provide a probabilistic and infinitesimal view of how the principal
component analysis procedure (PCA) can be generalized to analysis of nonlinear
manifold valued data. Starting with the probabilistic PCA interpretation of the
Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an
intrinsic way that does not resort to linearization of the data space. The
underlying probability model is constructed by mapping a Euclidean stochastic
process to the manifold using stochastic development of Euclidean
semimartingales. The construction uses a connection and bundles of covariant
tensors to allow global transport of principal eigenvectors, and the model is
thereby an example of how principal fiber bundles can be used to handle the
lack of global coordinate system and orientations that characterizes manifold
valued statistics. We show how curvature implies non-integrability of the
equivalent of Euclidean principal subspaces, and how the stochastic flows
provide an alternative to explicit construction of such subspaces. We describe
estimation procedures for inference of parameters and prediction of principal
components, and we give examples of properties of the model on embedded
surfaces
Anisotropically weighted and nonholonomically constrained evolutions on manifolds
We present evolution equations for a family of paths that results from
anisotropically weighting curve energies in non-linear statistics of manifold
valued data. This situation arises when performing inference on data that have
non-trivial covariance and are anisotropic distributed. The family can be
interpreted as most probable paths for a driving semi-martingale that through
stochastic development is mapped to the manifold. We discuss how the paths are
projections of geodesics for a sub-Riemannian metric on the frame bundle of the
manifold, and how the curvature of the underlying connection appears in the
sub-Riemannian Hamilton-Jacobi equations. Evolution equations for both metric
and cometric formulations of the sub-Riemannian metric are derived. We
furthermore show how rank-deficient metrics can be mixed with an underlying
Riemannian metric, and we relate the paths to geodesics and polynomials in
Riemannian geometry. Examples from the family of paths are visualized on
embedded surfaces, and we explore computational representations on finite
dimensional landmark manifolds with geometry induced from right-invariant
metrics on diffeomorphism groups
A Meshfree Generalized Finite Difference Method for Surface PDEs
In this paper, we propose a novel meshfree Generalized Finite Difference
Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative
approximations for the same are done directly on the tangent space, in a manner
that mimics the procedure followed in volume-based meshfree GFDMs. As a result,
the proposed method not only does not require a mesh, it also does not require
an explicit reconstruction of the manifold. In contrast to existing methods, it
avoids the complexities of dealing with a manifold metric, while also avoiding
the need to solve a PDE in the embedding space. A major advantage of this
method is that all developments in usual volume-based numerical methods can be
directly ported over to surfaces using this framework. We propose
discretizations of the surface gradient operator, the surface Laplacian and
surface Diffusion operators. Possibilities to deal with anisotropic and
discontinous surface properties (with large jumps) are also introduced, and a
few practical applications are presented
Kinetics and Mechanism of Metal Nanoparticle Growth via Optical Extinction Spectroscopy and Computational Modeling: The Curious Case of Colloidal Gold
An overarching computational framework unifying several optical theories to
describe the temporal evolution of gold nanoparticles (GNPs) during a seeded
growth process is presented. To achieve this, we used the inexpensive and
widely available optical extinction spectroscopy, to obtain quantitative
kinetic data. In situ spectra collected over a wide set of experimental
conditions were regressed using the physical model, calculating light
extinction by ensembles of GNPs during the growth process. This model provides
temporal information on the size, shape, and concentration of the particles and
any electromagnetic interactions between them. Consequently, we were able to
describe the mechanism of GNP growth and divide the process into distinct
genesis periods. We provide explanations for several longstanding mysteries,
for example, the phenomena responsible for the purple-greyish hue during the
early stages of GNP growth, the complex interactions between nucleation,
growth, and aggregation events, and a clear distinction between agglomeration
and electromagnetic interactions. The presented theoretical formalism has been
developed in a generic fashion so that it can readily be adapted to other
nanoparticulate formation scenarios such as the genesis of various metal
nanoparticles.Comment: Main text and supplementary information (accompanying MATLAB codes
available on the journal webpage
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
Computational methods to predict and enhance decision-making with biomedical data.
The proposed research applies machine learning techniques to healthcare applications. The core ideas were using intelligent techniques to find automatic methods to analyze healthcare applications. Different classification and feature extraction techniques on various clinical datasets are applied. The datasets include: brain MR images, breathing curves from vessels around tumor cells during in time, breathing curves extracted from patients with successful or rejected lung transplants, and lung cancer patients diagnosed in US from in 2004-2009 extracted from SEER database. The novel idea on brain MR images segmentation is to develop a multi-scale technique to segment blood vessel tissues from similar tissues in the brain. By analyzing the vascularization of the cancer tissue during time and the behavior of vessels (arteries and veins provided in time), a new feature extraction technique developed and classification techniques was used to rank the vascularization of each tumor type. Lung transplantation is a critical surgery for which predicting the acceptance or rejection of the transplant would be very important. A review of classification techniques on the SEER database was developed to analyze the survival rates of lung cancer patients, and the best feature vector that can be used to predict the most similar patients are analyzed
Towards Precision LSST Weak-Lensing Measurement - I: Impacts of Atmospheric Turbulence and Optical Aberration
The weak-lensing science of the LSST project drives the need to carefully
model and separate the instrumental artifacts from the intrinsic lensing
signal. The dominant source of the systematics for all ground based telescopes
is the spatial correlation of the PSF modulated by both atmospheric turbulence
and optical aberrations. In this paper, we present a full FOV simulation of the
LSST images by modeling both the atmosphere and the telescope optics with the
most current data for the telescope specifications and the environment. To
simulate the effects of atmospheric turbulence, we generated six-layer phase
screens with the parameters estimated from the on-site measurements. For the
optics, we combined the ray-tracing tool ZEMAX and our simulated focal plane
data to introduce realistic aberrations and focal plane height fluctuations.
Although this expected flatness deviation for LSST is small compared with that
of other existing cameras, the fast f-ratio of the LSST optics makes this focal
plane flatness variation and the resulting PSF discontinuities across the CCD
boundaries significant challenges in our removal of the systematics. We resolve
this complication by performing PCA CCD-by-CCD, and interpolating the basis
functions using conventional polynomials. We demonstrate that this PSF
correction scheme reduces the residual PSF ellipticity correlation below 10^-7
over the cosmologically interesting scale. From a null test using HST/UDF
galaxy images without input shear, we verify that the amplitude of the galaxy
ellipticity correlation function, after the PSF correction, is consistent with
the shot noise set by the finite number of objects. Therefore, we conclude that
the current optical design and specification for the accuracy in the focal
plane assembly are sufficient to enable the control of the PSF systematics
required for weak-lensing science with the LSST.Comment: Accepted to PASP. High-resolution version is available at
http://dls.physics.ucdavis.edu/~mkjee/LSST_weak_lensing_simulation.pd
Virtual cardiac monolayers for electrical wave propagation
The complex structure of cardiac tissue is considered to be one of the main determinants of an arrhythmogenic substrate. This study is aimed at developing the first mathematical model to describe the formation of cardiac tissue, using a joint in silico-in vitro approach. First, we performed experiments under various conditions to carefully characterise the morphology of cardiac tissue in a culture of neonatal rat ventricular cells. We considered two cell types, namely, cardiomyocytes and fibroblasts. Next, we proposed a mathematical model, based on the Glazier-Graner-Hogeweg model, which is widely used in tissue growth studies. The resultant tissue morphology was coupled to the detailed electrophysiological Korhonen-Majumder model for neonatal rat ventricular cardiomyocytes, in order to study wave propagation. The simulated waves had the same anisotropy ratio and wavefront complexity as those in the experiment. Thus, we conclude that our approach allows us to reproduce the morphological and physiological properties of cardiac tissue
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