Statistical Geometry Research Articles

Bayesian cluster analysis for registration and clustering homogeneous subgroups in multidimensional functional data

We introduce a new statistical model for clustering populations of functions and multidimensional curves where the domain is a real interval I. We consider that we are given a finite set of observations from a population of curves or functions with values in for a fixed and arguments in I. We are interested in the population background where the clustering is the process of grouping curves into homogeneous sub-populations. In particular, we define a distribution function for each sub-population and use the statistical geometry of the the space of smooth densities to explore a Bayesian model with a spherical Gaussian process prior. We also give the expression of the log-posterior distribution on coefficients resulting from the expansion. Finally, the practical interest of the proposed method is illustrated on simulated and real datasets of multidimensional curves.

Computer Test of a Modified Silicene/Graphite Anode for Lithium-Ion Batteries.

Despite the considerable efforts made to use silicon anodes and composites based on them in lithium-ion batteries, it is still not possible to overcome the difficulties associated with low conductivity, a decrease in the bulk energy density, and side reactions. In the present work, a new design of an electrochemical cell, whose anode is made in the form of silicene on a graphite substrate, is presented. The whole system was subjected to transmutation neutron doping. The molecular dynamics method was used to study the intercalation and deintercalation of lithium in a phosphorus-doped silicene channel. The maximum uniform filling of the channel with lithium is achieved at 3% and 6% P-doping of silicene. The high mobility of Li atoms in the channel creates the prerequisites for the fast charging of the battery. The method of statistical geometry revealed the irregular nature of the packing of lithium atoms in the channel. Stresses in the channel walls arising during its maximum filling with lithium are significantly inferior to the tensile strength even in the presence of polyvacancies in doped silicene. The proposed design of the electrochemical cell is safe to operate.

Moment maps, strict linear precision, and maximum likelihood degree one

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

A statistical geometry approach to length scales in phase field modelling of fracture and strength of porous microstructures

In phase field methods for fracture, versatility is acquired at the cost of the addition of a new parameter, a length scale parameter. The length scale parameter affects notch sensitivity, which in cellular materials is typically related to lengths of the material microstructure. Here, the relation between this length scale parameter and observable microstructural lengths of a cellular material is investigated numerically, specifically lengths derived using statistical geometry of random Voronoi tessellations.It is found that the fracture load of a homogeneous continuum model (i.e. a macroscopic model) coincides with that of a microstructured model if the length scale parameter is chosen to be the same in both models, while approximate macroscale stiffness and energy release rate are obtained by scaling the properties of the microstructured model with powers of the relative density. The correlation between the micro- and macroscale models is best when the length parameter is chosen as approximately two to three times the average cell size of the microstructure, depending on the relative density – which is also equal to approximately eight times a critical defect length of the Voronoi tessellation, regardless of relative density – as the microstructured material then behaves more like a continuum. If the length scale parameter needs to be smaller than twice the cell size or five times the critical length, the crack path is sensitive to features in the microstructure, and continuum modelling of the porous material cannot be advised.

Investigation of Phase-Locked Loop Statistics via Numerical Implementation of the Fokker–Planck Equation

The goal of this paper is to explore the effect of various parameters on the information geometric structure of the phase-locked loop (PLL) statistics, both transient and stationary. Comprehensive treatment on the behavior of PLL statistics will be given. The behavior of the phase-error statistics of the first-order PLL, in the presence of additive white Gaussian noise (WGN) is investigated through solving the differential equations known as the Fokker–Planck (FP) equation using the implicit Crank–Nicolson finite-difference method. The PLL is one of the most commonly used circuits in electrical engineering. A full knowledge of probability density functions (PDFs) of the phase-error statistics becomes essential in understanding the PLLs. Several illustrative examples are presented to provide profound insights on understanding the PLL statistics both qualitatively and quantitatively. Results covered include the transient and stationary statistics for the nonmodulo-2π probability density function, modulo-2π probability density function, and cycle slipping density function, of the phase error. Various numerical settings of PLL parameters are involved, including the detuning factor and signal-to-noise ratio (SNR). The results presented in this paper elucidate the link between various parameters and the information geometry of the phase-error statistics and form a basis for future investigation on PLL designs.

Predicting transport characteristics of hyperuniform porous media via rigorous microstructure-property relations

This paper is concerned with the estimation of the effective transport characteristics of fluid-saturated porous media via rigorous microstructure-property relations. We are particularly interested in predicting the formation factor F, mean survival time τ, principal NMR (diffusion) relaxation time T1, principal viscous relaxation time Θ1, and fluid permeability k. To do so, we employ rigorous methods to estimate the fluid permeability and these other transport properties of “hyperuniform” and nonhyperuniform models of porous media from microstructural information. Disordered hyperuniform materials are exotic amorphous states of matter that have attracted great attention in the physical, mathematical and biological science but little is known about their fluid transport characteristics. In carrying out this investigation, we not only draw from ideas and results of the emerging field of hyperuniformity, but from homogenization theory, statistical geometry, differential equations (spectrum of Laplace and Stokes operators), and the covering and quantizer problems of discrete geometry. Among other results, we derive a Fourier representation of a classic rigorous upper bound on the fluid permeability that depends on the spectral density to infer how the permeabilities of hyperuniform porous media perform relative to those of nonhyperuniform ones. We find that the velocity fields in nonhyperuniform porous media are generally much more localized over the pore space compared to those in their hyperuniform counterparts, which has implications for their permeabilities. Rigorous bounds on transport properties suggest a new approximate formula for the fluid permeability that provides reasonably accurate permeability predictions of a certain class of hyperuniform and nonhyperuniform porous media. These comparative studies shed new light on the microstructural characteristics that determine the transport properties of general porous media. Our findings also have implications for the design of porous materials with desirable transport properties.

Global Geometry of Bayesian Statistics.

In the previous work of the author, a non-trivial symmetry of the relative entropy in the information geometry of normal distributions was discovered. The same symmetry also appears in the symplectic/contact geometry of Hilbert modular cusps. Further, it was observed that a contact Hamiltonian flow presents a certain Bayesian inference on normal distributions. In this paper, we describe Bayesian statistics and the information geometry in the language of current geometry in order to spread our interest in statistics through general geometers and topologists. Then, we foliate the space of multivariate normal distributions by symplectic leaves to generalize the above result of the author. This foliation arises from the Cholesky decomposition of the covariance matrices.

Identifying the Latent Space Geometry of Network Models Through Analysis of Curvature

Statistically modeling networks, across numerous disciplines and contexts, is fundamentally challenging because of (often high-order) dependence between connections. A common approach assigns each person in the graph to a position on a low-dimensional manifold. Distance between individuals in this (latent) space is inversely proportional to the likelihood of forming a connection. The choice of the latent geometry (the manifold class, dimension, and curvature) has consequential impacts on the substantive conclusions of the model. More positive curvature in the manifold, for example, encourages more and tighter communities; negative curvature induces repulsion among nodes. Currently, however, the choice of the latent geometry is an a priori modeling assumption and there is limited guidance about how to make these choices in a data-driven way. In this work, we present a method to consistently estimate the manifold type, dimension, and curvature from an empirically relevant class of latent spaces: simply connected, complete Riemannian manifolds of constant curvature. Our core insight comes by representing the graph as a noisy distance matrix based on the ties between cliques. Leveraging results from statistical geometry, we develop hypothesis tests to determine whether the observed distances could plausibly be embedded isometrically in each of the candidate geometries. We explore the accuracy of our approach with simulations and then apply our approach to data-sets from economics and sociology as well as neuroscience.

Sustainable Wireless IoT Networks With RF Energy Charging Over Wi-Fi (CoWiFi)

Radio frequency (RF) energy harvesting is a promising technology that enables self-sustainable wireless Internet of Things (IoT) networks. In this article, we analyze the energy harvesting performance of a Wi-Fi-based IoT network, where a large number of IoT devices are connected via Wi-Fi for both data communication and energy transfer. Applying probability theory and statistical geometry, we first develop an analytical model to study the energy sustainability of wireless IoT devices with Wi-Fi charging, which operate in an active/charging mode and access the channel using carrier sensing multiple access with collision avoidance (CSMA/CA) protocol. Based on the analysis, we derive the necessary and sufficient conditions for the AP beaconing frequency and the charging period of IoT devices to ensure that a network with a general random topology is long-term energy sustainable. It is shown that transmission collisions due to random access result in too much energy consumption that makes it difficult to achieve energy sustainability of IoT devices. To maximize the total network throughput while ensuring long-term energy sustainability of Wi-Fi IoT devices, a distributed energy-sustainable throughput-optimal algorithm is proposed for user charging period selection. Extensive simulations using NS-3 validate the analysis and demonstrate the efficiency of the proposed algorithm.

Lie groupoids in information geometry

We demonstrate that the proper general setting for contrast (potential) functions in statistical and information geometry is the one provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. If the two-form is non-degenerate, it defines a ‘pseudo-Riemannian’ metric on the Lie algebroid and a family of Lie algebroid torsion-free connections, including the Levi-Civita connection of the metric. In this framework, the two-point functions are just functions on the pair groupoid with the ‘standard’ metric and affine connection on the Lie algebroid . We study also reductions of such systems and infinite-dimensional examples. In particular, we find a contrast function defining the Fubini-Study metric on the Hilbert projective space.

Pontos (in)comuns nos processos de formação de professores de matemática identificados ao longo de quase duas décadas (2000-2017)

Neste artigo foram contabilizados artigos publicados em quatro revistas eletrônicas com Qualis A-1 e B-1. Nessas revistas procurou-se quais seriam as questões (in)comuns que têm norteado os processos de formação de professores. Trata-se, assim, de uma pesquisa qualitativa com abordagem fenomenológica, realizada por meio de busca documental e bibliográfica. O método de análise utilizado foi a meta-análise. Para esse fim, selecionou-se um total de 40 artigos, tendo sido localizados, entre estes, sete artigos de universidades da Espanha e um de Honduras. Dado o número de trabalhos selecionados, foi possível determinar o quanto as publicações de diferentes revistas apontavam, naquele momento, para diferentes ações formativas. Conclui-se que foram constatadas 30 características nos 40 artigos analisados. Alguns temas foram identificados simultaneamente em diferentes revistas, o qual referia-se ao modo como o professor se comunica em sala de aula. Políticas públicas voltadas às ações dos professores estiveram presentes numa ocorrência de dois artigos. O uso de tecnologias pelos professores em formação foi uma questão abordada em quatro dos 40 artigos selecionados. O tema apresentado numa maior frequência entre as publicações dessas revistas foi a preocupação em desenvolver conteúdos matemáticos. Essa questão foi manifestada em seis artigos (estudo sobre funções, geometria, números decimais e conjuntos numéricos na educação básica; estatística, limites e continuidades no ensino superior). Pode-se afirmar, também, que a característica da formação de professores, realizada no período referente à década de 60, onde havia uma preocupação conteudista, é ainda a maior presença, devido aos dados levantados neste artigo.

Frontal crash simulations using parametric human models representing a diverse population

Objective: Analyses of crash data have shown that older, obese, and/or female occupants have a higher risk of injury in frontal crashes compared to the rest of the population. The objective of this study was to use parametric finite element (FE) human models to assess the increased injury risks and identify safety concerns for these vulnerable populations.Methods: We sampled 100 occupants based on age, sex, stature, and body mass index (BMI) to span a wide range of the U.S. adult population. The target anatomical geometry for each of the 100 models was predicted by the statistical geometry models for the rib cage, pelvis, femur, tibia, and external body surface developed previously. A regional landmark-based mesh morphing method was used to morph the Global Human Body Models Consortium (GHBMC) M50-OS model into the target geometries. The morphed human models were then positioned in a validated generic vehicle driver compartment model using a statistical driving posture model. Frontal crash simulations based on U.S. New Car Assessment Program (U.S. NCAP) were conducted. Body region injury risks were calculated based on the risk curves used in the US NCAP, except that scaling was used for the neck, chest, and knee–thigh–hip injury risk curves based on the sizes of the bony structures in the corresponding body regions. Age effects were also considered for predicting chest injury risk.Results: The simulations demonstrated that driver stature and body shape affect occupant interactions with the restraints and consequently affect occupant kinematics and injury risks in severe frontal crashes. U-shaped relations between occupant stature/weight and head injury risk were observed. Chest injury risk was strongly affected by age and sex, with older female occupants having the highest risk. A strong correlation was also observed between BMI and knee–thigh–hip injury risk, whereas none of the occupant parameters meaningfully affected neck injury risks.Conclusions: This study is the first to use a large set of diverse FE human models to investigate the combined effects of age, sex, stature, and BMI on injury risks in frontal crashes. The study demonstrated that parametric human models can effectively predict the injury trends for the population and may now be used to optimize restraint systems for people who are not similar in size and shape to the available anthropomorphic test devices (ATDs). New restraints that adapt to occupant age, sex, stature, and body shape may improve crash safety for all occupants.

Statistical geometry characterization of local structure of TMAO, TBA and urea aqueous solutions

Recently, we have revealed that trimethylamine-N-oxide (TMAO) molecules in aqueous solutions are distributed generally like random spheres in space (A.V. Anikeenko et al. J. Mol. Liq. 245 (2017) 35–41). We have shown that the variance of the distribution of Voronoi region volumes for the TMAO molecules and the hard random spheres are identical at the corresponding concentrations up to the mole fraction x ~ 0.1 (or packing fraction η ~ 0.2). In the current work, we study clusters in TMAO solutions and calculate the weighted mean cluster size and some topological cluster characteristics. For all these parameters, a great matching of TMAO and random spheres is also observed. The same analysis for tert-butyl alcohol (TBA) and urea aqueous solutions is carried out. These molecules have a tendency to cluster even at very low concentrations. However, basic geometrical features of the clusters are the same both for solutions and for the system of random spheres, indicating that geometrical laws governing the allocation of non-overlapping spheres in space should be taken into account when describing the structure of solutions even at rather low concentrations.

Nonperturbative thermodynamic geometry of nonextensive statistics

Following our earlier work on the perturbative thermodynamic geometry of nonextensive quantum and classical gases [H. Mohammadzadeh, F. Adli and S. Nouri, Phys. Rev. E 94 (2016) 062118], we study [Formula: see text]-generalized Bose–Einstein, Fermi–Dirac and classical statistics nonperturbatively. We define [Formula: see text]-generalized polylogarithm functions and evaluate thermodynamics quantities such as internal energy and particle number. We construct the thermodynamic geometry of nonextensive Bose (Fermi) ideal gas and show that the thermodynamic curvature is positive(negative) in full physical range as the same as ordinary statistics. Also, we show that the thermodynamic geometry of nonextensive ideal classical gas is flat, similar to the ordinary one. Therefore, the nonextensive parameter does not change the nature of intrinsic statistical interactions. We argue that the nonextensive boson gas might be more stable than the boson gas due to conjectural interpretation of thermodynamic curvature. In the following, we extract the singular points of thermodynamic curvature of nonextensive Bose gas and relate it to the condensation. We evaluate some thermodynamic quantities such as heat capacities, compressibility and [Formula: see text]-dependent phase transition temperature. We show that the heat capacity is not differentiable at critical temperature, [Formula: see text] which is reduced by increasing nonextensive parameter [Formula: see text]. Moreover, the critical temperature and possibility of condensation is investigated for different values of nonextensive parameter in various dimensions.

Multiscale computational homogenization of woven composites from microscale to mesoscale using data-driven self-consistent clustering analysis

Compared with the traditional phenomenological method, the multiscale simulation has significant advantages. This paper presents a methodology and computational homogenization framework to predict the macroscale behavior of woven composites based on fiber and matrix in microscale. The major challenge conducting multiscale analysis is the huge computational cost. To improve the efficiency, one of the reduced order models, which is called the data-driven self-consistent clustering analysis (SCA), is introduced and the multiscale framework is proposed by integrating two SCA solvers from different scales. The macroscale performance of 4-H satin weave carbon/carbon composites is investigated using the proposed framework. In order to reconstruct a real microstructure representative volume element (RVE), both microscale and mesoscale architectures are observed using scanning electron microscopy (SEM) and optical microscopy, and statistical geometry features are obtained. In addition, the SCA method is also verified by comparing the results with the finite element method (FEM). The uniaxial tension process is simulated using the multiscale approach, and strain/stress fields in both mesoscale and microscale can be captured simultaneously. Moreover, the uniaxial tensile experiments are also carried out to validate this framework, which shows high efficiency and great accuracy.

Shrinkage Estimation of Large Covariance Matrices: Keep it Simple, Statistician?

Under rotation-equivariant decision theory, sample covariance matrix eigenvalues can be optimally shrunk by recombining sample eigenvectors with a (potentially nonlinear) function of the unobservable population covariance matrix. The optimal shape of this function reflects the loss/risk that is to be minimized. We solve the problem of optimal covariance matrix estimation under a variety of loss functions motivated by statistical precedent, probability theory, and differential geometry. A key ingredient of our nonlinear shrinkage methodology is a new estimator of the angle between sample and population eigenvectors, without making strong assumptions on the population eigenvalues. We also introduce a broad family of covariance matrix estimators that can handle all regular functional transformations of the population covariance matrix under large-dimensional asymptotics. In addition, we compare via Monte Carlo simulations our methodology to two simpler ones from the literature, linear shrinkage and shrinkage based on the spiked covariance model.

Computational investigation of a promising Si-Cu anode material.

The lack of suitable anode materials is a limiting factor in the creation of a new generation of lithium-ion batteries. We use the molecular dynamics method to explore the processes of intercalation and deintercalation of lithium in the anode element, represented by two sheets of silicene, on a copper substrate. It is shown that the presence of vacancy-type defects in silicene increases the electrode capacitance, which becomes especially significant with bivacancies. However, the enlargement of defect sizes reduces the strength of the silicene channel during cycling and in the presence of hexavacancies it suffers a strong deformation and becomes impassable for Li+ ions during intercalation. The presence of a copper substrate greatly changes the electronic properties of silicene. The calculated DOS spectrum shows that silicon on a copper substrate acquires metallic properties. To analyze the structure we used the statistical geometry method. Lithium atoms in the channel are predominantly irregularly packed. However, part of the Li atoms are located above the hexagonal Si cells. The average stresses in silicene, calculated with limiting filling of the channel with lithium, are usually small. However, in the case of silicene with monovacancies, the tensile stress reaches 12.5% of the ultimate tensile stress. Evaluation of the dynamic stress observed in silicene during cycling shows that its value is less than 5% of the ultimate tensile stress.

The fraction of overlapping interphase around 2D and 3D polydisperse non-spherical particles: Theoretical and numerical models

The fraction of interphase is an important microstructure parameter in the prediction of macroscopic properties of particulate composites. Currently, some researchers have presented theoretical and numerical investigations on the interphase fraction for spherical particle systems, and even quantify the influence of interphase fraction on the overall elastic and transport properties of particulate composites. However, the overlapping interphase fraction in polydisperse non-spherical particle systems is still an open issue. In this work, a generic theoretical model is formulated to derive the overlapping interphase fraction for polydisperse 2D non-circular and 3D non-spherical particle systems by means of the statistical geometry of composites. In this model, the morphology of interphase coated onto the surface of non-spherical particles can be characterized by circularity and sphericity of particles that are important parameters to describe the shape of particles. On the other hand, numerical simulations for the one-point probability function of interphase are presented to verify the proposed theoretical framework. In the numerical simulations, a novel algorithm is developed by reducing the problem of identifying the precise location between an arbitrary spatial point and interphase to the basic issue of finding the distance from the point to the surface of particles. If the distance is less than the interphase thickness, the point definitely locates inside interphase. In addition, the algorithm can be further used to detect the overlap between adjacent ellipsoidal particles (2D ellipses). Moreover, a variety of particle shapes (such as regular polygons and ellipses with the same circularity, and regular polyhedrons and ellipsoids with the same sphericity) are taken into account to generate particle packing structures. Then, the fraction of overlapping interphase in each packing structure is statistically obtained. Results show that statistical values are consistent with their theoretical values under the same conditions. This validates the reliability of the theoretical framework. Finally, the effects of particle size distributions and interphase thicknesses on the interphase fraction are investigated. It can be found that the interphase fraction increases with the increase of particle volume fraction, particle fineness and interphase thickness.

Riemannian Geometry of Ising Model in the Bethe Approximation

A method combining statistical equilibrium theory and metric geometry is used to study thermodynamic scalar curvature in the neighborhood of the Curie critical temperature for an Ising model of ferromagnetism. Using a Bethe type free energy expression, a non-diagonal metric is introduced on the two-dimensional phase space of long-range and short-range order parameters. Based on the metric elements Christoffel symbols, curvature tensor and Ricci tensor are found. An expression (containing equilibrium order parameters) is derived for Riemann scalar curvature (R). Its behavior near the critical temperature is examined analytically. We find that R tends toward plus infinity while approaching the critical point. This result fits well with those in the exact one-dimensional chain and mean-field Ising model in the lowest order approximation.

How Gauss and Markov Met Pythagoras: Geometry in Econometrics

Pursuing abstractness and generality, a number of books and articles in econometrics rely heavily on standard algebraic proofs. However, most of the theorems proved in such a way have a strong geometric appeal. This paper demonstrates how shorter, less technical and even more beautiful the proofs can be when they are based on geometric theorems. We show that this technique can be extended to explain concepts in probability and statistics. Although most of the theorems and ideas are not completely new and there are even a few works on the geometry in econometrics and statistics, this paper introduces alternative explanations and more general results in some cases. Further, there are algebraic proofs provided parallel to geometric ones and illustrations which are our own work.