72,234 research outputs found
Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres
Functional data analysis on nonlinear manifolds has drawn recent interest.
Sphere-valued functional data, which are encountered for example as movement
trajectories on the surface of the earth, are an important special case. We
consider an intrinsic principal component analysis for smooth Riemannian
manifold-valued functional data and study its asymptotic properties. Riemannian
functional principal component analysis (RFPCA) is carried out by first mapping
the manifold-valued data through Riemannian logarithm maps to tangent spaces
around the time-varying Fr\'echet mean function, and then performing a
classical multivariate functional principal component analysis on the linear
tangent spaces. Representations of the Riemannian manifold-valued functions and
the eigenfunctions on the original manifold are then obtained with exponential
maps. The tangent-space approximation through functional principal component
analysis is shown to be well-behaved in terms of controlling the residual
variation if the Riemannian manifold has nonnegative curvature. Specifically,
we derive a central limit theorem for the mean function, as well as root-
uniform convergence rates for other model components, including the covariance
function, eigenfunctions, and functional principal component scores. Our
applications include a novel framework for the analysis of longitudinal
compositional data, achieved by mapping longitudinal compositional data to
trajectories on the sphere, illustrated with longitudinal fruit fly behavior
patterns. RFPCA is shown to be superior in terms of trajectory recovery in
comparison to an unrestricted functional principal component analysis in
applications and simulations and is also found to produce principal component
scores that are better predictors for classification compared to traditional
functional functional principal component scores
Statistical mechanics characterization of spatio-compositional inhomogeneity
On the basis of a model system of pillars built of unit cubes, a
two-component entropic measure for the multiscale analysis of
spatio-compositional inhomogeneity is proposed. It quantifies the statistical
dissimilarity per cell of the actual configurational macrostate and the
theoretical reference one that maximizes entropy. Two kinds of disorder
compete: i) the spatial one connected with possible positions of pillars inside
a cell (the first component of the measure), ii) the compositional one linked
to compositions of each local sum of their integer heights into a number of
pillars occupying the cell (the second component). As both the number of
pillars and sum of their heights are conserved, the upper limit for a pillar
height hmax occurs. If due to a further constraint there is the more demanding
limit h <= h* < hmax, the exact number of restricted compositions can be then
obtained only through the generating function. However, at least for systems
with exclusively composition degrees of freedom, we show that the neglecting of
the h* is not destructive yet for a nice correlation of the h*-constrained
entropic measure and its less demanding counterpart, which is much easier to
compute. Given examples illustrate a broad applicability of the measure and its
ability to quantify some of the subtleties of a fractional Brownian motion,
time evolution of a quasipattern [28,29] and reconstruction of a laser-speckle
pattern [2], which are hardly to discern or even missed.Comment: 17 pages, 5 figure
On the relative importance of thermal and chemical buoyancy in regular and impact-induced melting in a Mars-like planet
We ran several series of two-dimensional numerical mantle convection
simulations representing in idealized form the thermochemical evolution of a
Mars-like planet. In order to study the importance of compositional buoyancy of
melting mantle, the models were set up in pairs of one including all thermal
and compositional contributions to buoyancy and one accounting only for the
thermal contributions. In several of the model pairs, single large impacts were
introduced as causes of additional strong local anomalies, and their evolution
in the framework of the convecting mantle was tracked. The models confirm that
the additional buoyancy provided by the depletion of the mantle by regular
melting can establish a global stable stratification of the convecting mantle
and throttle crust production. Furthermore, the compositional buoyancy is
essential in the stabilization and preservation of local compositional
anomalies directly beneath the lithosphere and offers a possible explanation
for the existence of distinct, long-lived reservoirs in the martian mantle. The
detection of such anomalies by geophysical means is probably difficult,
however; they are expected to be detected by gravimetry rather than by seismic
or heat flow measurements. The results further suggest that the crustal
thickness can be locally overestimated by up to ~20 km if impact-induced
density anomalies in the mantle are neglected.Comment: 29 pages, 10 figure
A Unified Framework for Compositional Fitting of Active Appearance Models
Active Appearance Models (AAMs) are one of the most popular and
well-established techniques for modeling deformable objects in computer vision.
In this paper, we study the problem of fitting AAMs using Compositional
Gradient Descent (CGD) algorithms. We present a unified and complete view of
these algorithms and classify them with respect to three main characteristics:
i) cost function; ii) type of composition; and iii) optimization method.
Furthermore, we extend the previous view by: a) proposing a novel Bayesian cost
function that can be interpreted as a general probabilistic formulation of the
well-known project-out loss; b) introducing two new types of composition,
asymmetric and bidirectional, that combine the gradients of both image and
appearance model to derive better conver- gent and more robust CGD algorithms;
and c) providing new valuable insights into existent CGD algorithms by
reinterpreting them as direct applications of the Schur complement and the
Wiberg method. Finally, in order to encourage open research and facilitate
future comparisons with our work, we make the implementa- tion of the
algorithms studied in this paper publicly available as part of the Menpo
Project.Comment: 39 page
Unifying Amplitude and Phase Analysis: A Compositional Data Approach to Functional Multivariate Mixed-Effects Modeling of Mandarin Chinese
Mandarin Chinese is characterized by being a tonal language; the pitch (or
) of its utterances carries considerable linguistic information. However,
speech samples from different individuals are subject to changes in amplitude
and phase which must be accounted for in any analysis which attempts to provide
a linguistically meaningful description of the language. A joint model for
amplitude, phase and duration is presented which combines elements from
Functional Data Analysis, Compositional Data Analysis and Linear Mixed Effects
Models. By decomposing functions via a functional principal component analysis,
and connecting registration functions to compositional data analysis, a joint
multivariate mixed effect model can be formulated which gives insights into the
relationship between the different modes of variation as well as their
dependence on linguistic and non-linguistic covariates. The model is applied to
the COSPRO-1 data set, a comprehensive database of spoken Taiwanese Mandarin,
containing approximately 50 thousand phonetically diverse sample contours
(syllables), and reveals that phonetic information is jointly carried by both
amplitude and phase variation.Comment: 49 pages, 13 figures, small changes to discussio
Convection in the Earth's core driven by lateral variations in the core-mantle boundary heat flux
Moving core fluid maintains an isothermal core-mantle boundary (CMB), so lateral variations in the CMB heat flow result from mantle convection. Such variations will
drive thermal winds, even if the top of the core is stably stratified. These flows may contribute to the magnetic secular variation and are investigated here using a simple,
non-magnetic numerical model of the core. The results depend on the equatorial symmetry of the boundary heat flux variation. Large-scale equatorially symmetric
(ES) heat flux variations at the outer surface of a rapidly rotating spherical shell drive
deeply penetrating flows that are strongly suppressed in stratified fluid. Smaller-scale
ES heat flux variations drive flows less dominated by rotation and so less inhibited
by stratification. Equatorially anti-symmetric flux variations drive flows an order of
magnitude less energetic than those driven by ES patterns but, due to the nature of the Coriolis force, are less suppressed by stratification. The response of the rotating core fluid to a general CMB heat flow pattern will then depend strongly on the subadiabatic temperature profile. Imposing a lateral heat flux variation linearly related to a model of seismic tomography in the lowermost mantle drives flow in a density stratified fluid that
reproduces some features found in flows inverted from geomagnetic data
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