71,057 research outputs found
Toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data
pre-printThis paper proposes an original approach for the statistical analysis of longitudinal shape data. The proposed method allows the characterization of typical growth patterns and subject-specific shape changes in repeated time-series observations of several subjects. This can be seen as the extension of usual longitudinal statistics of scalar measurements to high-dimensional shape or image data. The method is based on the estimation of continuous subject-specific growth trajectories and the comparison of such temporal shape changes across subjects. Differences between growth trajectories are decomposed into morphological deformations, which account for shape changes independent of the time, and time warps, which account for different rates of shape changes over time. Given a longitudinal shape data set, we estimate a mean growth scenario representative of the population, and the variations of this scenario both in terms of shape changes and in terms of change in growth speed. Then, intrinsic statistics are derived in the space of spatiotemporal deformations, which characterize the typical variations in shape and in growth speed within the studied population. They can be used to detect systematic developmental delays across subjects. In the context of neuroscience, we apply this method to analyze the differences in the growth of the hippocampus in children diagnosed with autism, developmental delays and in controls. Result suggest that group differences may be better characterized by a different speed of maturation rather than shape differences at a given age. In the context of anthropology, we assess the differences in the typical growth of the endocranium between chimpanzees and bonobos. We take advantage of this study to show the robustness of the method with respect to change of parameters and perturbation of the age estimates
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 Lyapunov theory based on bifurcation analysis of energy cascade in isotropic homogeneous turbulence: a physical -- mathematical review
This work presents a review of previous articles dealing with an original
turbulence theory proposed by the author, and provides new theoretical insights
into some related issues. The new theoretical procedures and methodological
approaches confirm and corroborate the previous results. These articles study
the regime of homogeneous isotropic turbulence for incompressible fluids and
propose theoretical approaches based on a specific Lyapunov theory for
determining the closures of the von K\'arm\'an-Howarth and Corrsin equations,
and the statistics of velocity and temperature difference. Furthermore, novel
theoretical issues are here presented among which we can mention the following
ones. The bifurcation rate of the velocity gradient, calculated along fluid
particles trajectories, is shown to be much larger than the corresponding
maximal Lyapunov exponent. On that basis, an interpretation of the energy
cascade phenomenon is given and the statistics of finite time Lyapunov exponent
of the velocity gradient is shown to be represented by normal distribution
functions. Next, the self--similarity produced by the proposed closures is
analyzed, and a proper bifurcation analysis of the closed von
K\'arm\'an--Howarth equation is performed. This latter investigates the route
from developed turbulence toward the non--chaotic regimes, leading to an
estimate of the critical Taylor scale Reynolds number. A proper statistical
decomposition based on extended distribution functions and on the
Navier--Stokes equations is presented, which leads to the statistics of
velocity and temperature difference.Comment: physical--mathematical review of previous works and new theoretical
insights into some relates issue
Statistics of finite scale local Lyapunov exponents in fully developed homogeneous isotropic turbulence
The present work analyzes the statistics of finite scale local Lyapunov
exponents of pairs of fluid particles trajectories in fully developed
incompressible homogeneous isotropic turbulence. According to the hypothesis of
fully developed chaos, this statistics is here analyzed assuming that the
entropy associated to the fluid kinematic state is maximum. The distribution of
the local Lyapunov exponents results to be an unsymmetrical uniform function in
a proper interval of variation. From this PDF, we determine the relationship
between average and maximum Lyapunov exponents, and the longitudinal velocity
correlation function. This link, which in turn leads to the closure of von
K\`arm\`an-Howarth and Corrsin equations, agrees with results of previous
works, supporting the proposed PDF calculation, at least for the purposes of
the energy cascade main effect estimation. Furthermore, through the property
that the Lyapunov vectors tend to align the direction of the maximum growth
rate of trajectories distance, we obtain the link between maximum and average
Lyapunov exponents in line with the previous results. To validate the proposed
theoretical results, we present different numerical simulations whose results
justify the hypotheses of the present analysis.Comment: Research article. arXiv admin note: text overlap with
arXiv:1706.0097
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