Geometry Of Elements Research Articles

Static analysis of masonry polygonal domes with membrane forces

Masonry polygonal domes are parts of several historic monuments and masonry buildings. The analysis of polygonal domes with a static approach is here presented, considering the structure made of vault cells and ribs. The vaults are portions of a cylindrical surface with single curvature in the meridian direction, and translation of the generatrix curve in the direction of the cylinder axis. The system is divided into rigid elements with interfaces. The first part of the work regards the geometry of the rigid elements, in particular the angles that characterize the shape of these. Then the equilibrium equations applied to every element are presented, with a procedure to calculate the membrane response for the self-weight in a non-redundant system, under symmetry conditions. The sum of the solution for the vaults is combined with effects of the weight of ribs and radial walls. The use of the equilibrium solution for pointed domes, domes with a lantern and oculus or additional load to the dome self-weight is shown. A limit analysis static approach is formulated calculating the maximum load relative to the equilibrium of the rigid elements considering the interfaces tensile strength and limited cohesion and friction for the shear resistance.

Analysis of the Dispersive and Distributive Mixing Effect of Screw Elements on the Co-Rotating Twin-Screw Extruder with Particle Tracking.

Compounding is an important step in processing base polymers and is used to incorporate various additives into a polymer. For this purpose, different screw elements are used for dispersive and distributive mixing on a co-rotating twin-screw extruder. Optimising the screw configuration requires precise knowledge of the screw elements' mixing properties, which have not been thoroughly investigated. This study analyses the mixing behaviour of individual screw elements regarding dispersive and distributive mixing using 3D CFD flow simulations with subsequent particle tracking. For distributive mixing, the particle distribution behind the screw elements in the XY plane is analysed and the mixing index MQ, which relates the standard deviation and the mean value of the triangular areas between the particles, is calculated. For dispersive mixing, the maximum shear stress on the particle path and the integral of the shear stress over the residence time of each individual particle are determined. The results show that screw element geometry and rotation speed have a significant influence on dispersive and distributive mixing. In addition, better dispersive mixing is achievable with highly viscous materials. These findings enable the optimisation of the mixing zone of a co-rotating twin-screw extruder for the efficient mixing of mineral fillers.

Seismic design of steel moment‐resisting knee‐braced frame system by failure mode control

AbstractThis paper presents the seismic design of a steel moment‐resisting knee‐braced frame (MKF) using the theory of plastic mechanism control (TPMC) within the capacity‐based design framework. The MKF is an alternative system to MRFs, wherein knee elements are utilized to provide rigid connections and enhance lateral stiffness. Capacity‐based design, the predominant approach in current seismic provisions, relies on two key principles: (1) selecting specific structural components as fuses with sufficient ductility to dissipate seismic energy, and (2) ensuring non‐fuse elements can resist the maximum probable reactions from these fuses. The ultimate goal is to achieve a global mechanism where yielding occurs in all structural fuses and at the base of first‐story columns. However, existing seismic design provisions often struggle to fully satisfy the second principle due to the lack of a method for controlling failure modes. TPMC addresses this challenge by ensuring compliance with the second principle, grounding its approach in the kinematic method and the mechanism equilibrium curve within the rigid‐plastic analysis framework. By considering all potential story‐based undesirable mechanisms and calculating the required plastic moment of columns up to a target design displacement, TPMC ensures adherence to the second principle of the capacity‐based design approach, leading to the achievement of a global collapse mechanism. In this paper, an iterative method is proposed for designing beams and knee elements by considering plastic hinges at both ends of the beams, followed by a TPMC‐based methodology for designing columns to ensure a global mechanism. A parametric analysis of a single‐story single‐span MKF explores the effects of knee element geometry ( and ) on component demands. The results indicate that optimal parameter ranges of and can minimize the demands for MKF components. Practical design examples are illustrated using three steel MKFs, each consisting of four, seven, and ten stories with five spans. Pushover analysis and nonlinear dynamic analyses were performed to demonstrate the effectiveness of the proposed design procedure in ensuring the attainment of a global mechanism and excellent seismic performance under real ground motions.

Electrical damage mechanism analysis of traction motor bearings on high-speed trains considering overvoltage impulse

ABSTRACT As the critical rotating knuckle, traction motor bearings play a vital role in transmitting torque from motors to wheels for high-speed trains, while some electrochemical corrosion signs appear in traction motor bearings. To maintain the safety of the traction system, it is essential to explore the electrical damage mechanism of traction motor bearings considering the impact brought from current or overvoltage impulses. Herein, via analysing the electrochemical corrosion evolutionary process of bearings, main influencing factors are determined that both transient overvoltage impulse and long-term grounding current bring dramatic impact on bearings. To assess overvoltage distribution on different carriages, a ‘catenary-train-substation’ traction power supply model is launched based on practically collected impedances. A ‘carriage-bogie-motor’ circuit model is built to evaluate the overvoltage propagation path, meanwhile, the 3D finite element geometry of the bogie with bearings is launched to observe the overvoltage distribution in the bearings for evaluating breakdown risk.

Eksplorasi Etnomatematika Geometri Bangun Datar Segitiga pada Pakaian Tradisonal Sortopi Khas Suku Batak Toba

Exploration of ethnomathematics in the context of flat geometry, especially triangles, in traditional Batak Toba clothing, is an interesting and important study. This study aims to reveal the relationship between mathematical concepts and local culture, and how geometric elements can be found in the design of Sortopi traditional clothing. This research method uses a qualitative approach with data sources, namely library studies, from several literature reviews according to the topic and purpose of the study. The results of the study show that Sortopi is a complement to traditional Batak clothing used by men as a crown or headband. The shape of Sortopi contains elements of isosceles triangle flat geometry, thus showing that cultural elements are inseparable from mathematics.

An ultra-broadband low profile modified chessboard metasurface with improved backscattering reduction

Building upon the concepts of polarization conversion and the mechanism of destructive interference, a single-layered ultra-broadband metasurface for radar cross-section (RCS) or backscattering reduction is proposed in this work. A diffusion-assisted checkerboard metasurface with re-tailored unit cells at the center leads to a significant improvement in the stealth characteristics. For this, a low profile and less complex unit cell is proposed, comprising a long metallic strip along the diagonal and two thin horizontal stubs at the edges. This unique arrangement exhibits more than 90% polarization conversion efficiency from 13.2 to 38.2 GHz. Modifying the geometry of central elements in a conventional checkerboard metasurface achieves a minimum of 15 dB RCS reduction from 10 to 38 GHz. Also, as we ascend the frequency spectrum, the beams are scattered in unintended directions with reduced signal strength. Notably, there is no beam in the boresight direction, as opposed to the conventional chessboard configurations. For off-normal incidences, the metasurface exhibits good angular stability, and a minimum of 10 dB backscattering reduction is maintained up to an incidence angle variation of 60°. The final optimized fabricated prototype exhibits measurement results that agree with the simulation results for normal and wide-angle incidences.

Analytical and geometric interpretation of the flat arrangement of points by means of metric geometry in the study of metric spaces

Abstract When studying metric spaces, students of higher education often have difficulties with understanding the basic concepts and properties of these spaces. This, to a large extent, is a consequence of the significant level of formalization of such concepts on the one hand, and the preservation of the corresponding formulations and names familiar to students from a school mathematics course. To overcome these difficulties, it is advisable to use methods of geometric interpretation and visualization of these properties. At the same time, it is appropriate to use elements of metric geometry. Its methods make it possible to interpret the geometric features of the mutual placement of points of metric space in Cartesian (rectangular) coordinate systems, which are familiar to students of higher education. Moreover, it becomes possible to visualize these features with the help of graphic editors, since they, as a rule, use numerical values of the coordinates of points to visualize them. Based on the definition of an angle as an ordered trio of points of an arbitrary metric space, and the angular characteristic of this angle, the fact of the flat placement of four points of a non-Euclidean metric space is established, and examples of digital visualization of this arrangement using the dynamic geometric environment GeoGebra 3D are given.

Numerical study of influencing geometry of separator elements for cleaning dust of food production on its efficiency

At modern food enterprises, the task of providing high-quality air cleaning from dust is relevant. The paper proposes an original dust separator, the peculiarity of which is the presence of structural elements of different shapes, arranged in a chess order. This design solution ensures the formation of a wave-like structure of the gas-dust flow within the device. The separation of solids from air is due to inertial and centrifugal forces. The aim of the work is to evaluate the fractional efficiency of this separator with structural elements of different geometric shape during the dust capture process. The paper considers I-shaped, P-shaped, arc-shaped and V-shaped structural elements. The study is carried out by numerical simulation at a gas dust flow rate of 0.5 to 3 m/s and a particle size of 10 to 200 ?m. During the work it has been found that the shape of the structural elements largely determines the efficiency of separation of dust material from the gas stream. The V-shaped design elements have shown the highest average separation efficiency compared to other shapes under other equal conditions. This is due to the fact that in such a separator the particles are directed to their apex, where the velocity of the particles decreases contributing to their subsequent settling into the bunker. The maximum efficiency for V-shaped elements is on average 80.1 % at a gas dust inlet velocity of 0.5 m/s. The lowest efficiency is observed for I-shaped elements, as particles return to the air stream after rebounding from the walls.

Методы монохроматизации синхротронного излучения (обзор исследований)

The paper presents an analysis of studies related to the monochromatization of X-ray radiation (XR) at synchrotron radiation sources. A review of monochromators based on of X-ray diffraction on crystals is given, and the peculiarities of their technical realization are considered. The ideas about monochromators which include multilayer structures are examined. The authors also study technical problems arising during designing devices and its possible solutions. Introduction. The possibilities of using X-rays in scientific research are described. The high efficiency of synchrotron radiation sources is noted, and its characterization is given. Elementary information about diffraction of X-rays. The paper describes the properties of X-ray radiation and the possibilities of its using while studying various materials. Degree of monochromaticity. The degree of monochromaticity is an important characteristic of the synchrotron radiation (SR). Depending on the width of the wavelength band, “white”, “pink” and monochromatic beams are distinguished. Monochromators based on multilayer structures are used to obtain “pink” beams. Monochromatic radiation is formed using monocrystals. When conducting experiments with “white” beams, the monochromator is not used. The authors also describe the factors that violate the ideal fulfillment of the Wolf-Bragg condition and affect the degree of monochromaticity (heat, vibration). The reflectivity values at different beam grazing angles are noted to have different widths. Monochromators based on multilayer structures. Periodic structures combining thin layers of two heterogeneous materials make it possible to obtain “pink” beams. The wavelength bandwidth of such devices is one or two orders of magnitude greater than that of monochromators using crystals as optical elements. Configurations and geometry of optical elements. There are two types of X-ray diffraction on a crystal: Bragg and Laue diffraction. Bragg diffraction refers to reflective geometry, Laue diffraction is based on the passage of beams through the crystal. The section provides examples of monochromators with different configurations of crystals and X-ray mirrors. The arrangement of optical elements in a monochromator plays an important role in the geometry of the beam path. When designing monochromators, it is necessary to take into account the methods of fixation and orientation of the rotation axes of optical elements. Examples of monochromators with different configurations of crystals and X-ray mirrors are given. Focusing monochromators. It is possible to provide sagittal and meridional types of deformation by bending the optical element of the monochromator. Due to the curved crystal surface the beam is not only monochromatized but also subjected to focusing. Modern focusing monochromators are equipped with adaptivity elements allowing it to change the radius of curvature of the optical element. Examples of practical realization of such monochromators are presented. Thermal load of SR on optical elements. The SR is characterized by high brightness and a wide spectrum of emitted wavelengths. While operating optical elements of SR stations absorb a large amount of thermal power. The problems of heat dissipation have a fundamental influence on the quality of synchrotron radiation monochromatization. Additional information about monochromators. Examples of special design solutions for monochromators are given. Conclusion. The design of monochromators is relevant to the synchrotron radiation source 4+ “SKIF” under construction in Novosibirsk.

Deep learning-driven optimization of real-time ray tracing for enhanced immersive virtual reality

Photorealistic rendering is essential for immersive virtual reality and real-time ray tracing is required for this, yet the necessary computational load has so far hindered its use in scenarios where a fast response and high frame rates are paramount. To overcome this challenge, we present a deep learning-based model that optimises the computational load of ray tracing through utilising convolutional neural networks (CNNs) to predict lightscene interactions. We achieve this by training CNNs from datasets with offline ray-traced images. Using this learning process, we approximate the contents of light transport simulations from scene point sampling. The use of this optimised algorithm in virtual reality scenes, thus, enables renderings of more complex scenes, with dynamic lighting, complex geometry and interactive elements under high frame rates, keeping users immersed and responsive to events. Our results show that this method achieves visual similarities to traditional ray tracing, which is photorealistic rendering, but can be performed in real-time in virtual reality. We anticipate this work leads to many potential applications of ray tracing in many scenarios in virtual reality, such as gaming, architectural rendering and training.

Cosmography of the local Universe by multipole analysis of the expansion rate fluctuation field

We establish a relationship between the multipoles of the expansion rate fluctuation field η, which capture in an accurate way deviations from isotropy in the redshift-distance relation, and the multipoles of the covariant cosmographic parameters—Hubble ℍ o , deceleration ℚ o , jerk 𝕁 o , and curvature ℝ o . These parameters, derived from the third-order expansion of the luminosity distance with respect to redshift in a generic spacetime, provide model-independent insight into the geometry and symmetries of the cosmic line element without requiring the Cosmological Principle. Moreover, we demonstrate that although this approach is fully nonperturbative and does not rely on concepts such as peculiar velocities, it has the potential to constrain the motion of the matter frame at the observer's position relative to the cosmic microwave background.We use two analytical axisymmetric models, motivated by observational evidence, to test the formalism and the effectiveness of the third-order expansion of the luminosity distance. These models also help to predict the precision with which future surveys, like the Zwicky Transient Facility, will constrain the covariant cosmographic parameters in the local (z < 0.1) Universe.

Out-of-plane cross-angle bending stiffness of cross-laminated timber

The increasing use of cross-laminated timber (CLT) in construction worldwide can be attributed to its combination of sustainability, structural efficiency, biaxial strength and stiffness, short construction times, and architectural flexibility. However, when applied in free-form structures, CLT panels may exhibit irregular element geometries, resulting in deviations from their major or minor strength axes. The required out-of-plane cross-angle bending stiffness is not addressed by commonly used design methods. This study addresses this gap by examining the cross-angle bending stiffness of CLT panel segments using experimental, analytical, and numerical methods. Three different lay-ups (3-ply, 4-ply, and 5-ply) and five cross-angles (ranging from 0° to 90°) were tested under 3-point bending, demonstrating the nonlinear relation between cross-angle and out-of-plane bending stiffness. The lowest stiffness was observed for intermediate cross-angles and not when loaded along the minor strength direction. A proposed analytical model that combines the shear analogy with Hankinson's equation yielded adequate approximations. The model was verified by numerical simulations and then applied to predict the out-of-plane cross-angle bending stiffness of a wide range of CLT layups. This research provides valuable insights for working with CLT in complex geometrical designs, enhancing preliminary design.

Evolution of the transtensional Barreirinhas pull-apart system in the Brazilian Equatorial margin and its correlation with the African conjugate counterpart

The Barreirinhas pull-apart system encompasses marginal basins in divergent and transform margin segments in the central sector of the Brazilian Equatorial Margin and its African conjugate counterpart. This ancient pull-apart system evolved through transtensional strike-slip motion within a highly heterogeneous crystalline basement affected by multiple rift phases. The geometry and development of pull-apart structural elements during the final rifting phase before continental breakup and the mechanisms and extent to which they were influenced by preexisting crustal heterogeneities are comprehensively addressed using an extensive database of potential field (magnetic and gravity) and 2D seismic reflection data. We also assess the lithospheric thermomechanical conditions and their influence on transtensional extension throughout Curie Point Depth, Heat Flow, and Moho depth, derived from potential field data and published seismological models. Plate reconstruction of Brazilian and African equatorial margins based on gravity patterns and comparison with sandbox analog models allow a 3D synoptic model to reveal the Barreirinhas pull-apart system evolution during the Equatorial Atlantic opening. During the rift phase I, the location of major grabens was controlled by favorably oriented Neoproterozoic shear zones, while the cooler, stronger, and thicker crust beneath cratonic areas formed the western barrier to strike-slip rift activity during rift phase II. This same geological domain anchored the onset of the pull-apart system in the last rift phase III, whose principal displacement zones developed along the extensive oceanic fracture zones linked by sigmoidal fault systems. Toward the end of the rift phase, a large asymmetric lozangle to a lazy-Z-shaped, pull-apart basin developed above low overlapping ∼90°, releasing stepover in oblique transtensional strike-slip motion.

Probing the Depths of Electrocatalysis Systems: X-Ray Absorption Spectroscopy Reveals Catalyst Behavior

X-ray absorption spectroscopy (XAS) is a powerful experimental technique used to study the electronic structure and local environment of atoms in a material. It provides valuable insights into the chemical bonding, oxidation states, and coordination geometry of elements. XAS has been widely applied in various fields of electrochemistry, including popular topics like the hydrogen evolution reaction (HER), oxygen reduction reaction (ORR), and CO2 reduction reaction (CO2RR).Despite the growing recognition of the practical importance of XAS techniques, correlating experimental spectroscopy with the working mechanisms and deciphering the structure of the electrochemical interface under reaction conditions remains challenging. In the case of CO2RR, we utilized operando XAS to investigate multiple Cu-based catalysts in different cell configurations. Our findings revealed that altering the reaction environment can induce an alternative pathway for catalyst structural changes. Moreover, operando XAS demonstrated that the reduction of Cu oxide catalysts is not solely governed by applied potentials but is also influenced by mass transport and other parameters. For ORR, in situ XAS not only provides information about the local environment changes of Pt-based catalysts but also unveils surface properties through △μ-X-ray adsorption near-edge spectroscopy (△μ-XANES). This further aids in the discovery of degradation mechanisms in proton exchange membrane fuel cells (PEMFCs). In the case of HER, in situ XAS enables characterization beyond catalysts and extends to the electrochemical interface.Overall, XAS techniques play a crucial role in understanding the electronic structure and local environment of atoms in electrochemical systems, shedding light on reaction mechanisms and catalyst performance under realistic conditions Acknowledgement The authors thank the funding source from Liquid Sunlight Alliance under the U.S. DOE, Office of Science, Office of Basic Energy Sciences, Fuels Award Number DE-SC0021266 and the Laboratory Directed Research and Development Program of the Lawrence Berkeley National Laboratory under the U.S. DOE Contract No. DE-AC02-05CH11231.

On setting boundary conditions for calculations of turbomachines in 3-d computer packages

Unfortunately, in a growing number of publications of the results of the study of turbomachines in ANSYS and other packages, insufficient attention is paid to the choice of the flow model (absolute or relative) and the setting of boundary conditions in the areas adjacent to the rotating impeller: shroud seal, diaphragm (end) seals, gap over shroudless blades, steam balance holes, etc.As a rule, these areas contain such elements of geometry as a “ledge”, “projection”, “crest”, the flow around which, and, consequently, their resistance depends on the angle of incoming flow relative to the barrier. In the stages of turbomachines in sections behind fixed blade units, angle α1 between the flow velocity vector (in absolute motion) and the circumferential velocity is 10°÷25° (for steam turbine stages), 20°÷35° (for gas turbine stages) and 40°÷75° ((for axial compressor stages). It is shown (using the example of viscous flow around the “ledge + projection” system) that a change in the angle in the range of 15°÷90° changes the resistance by almost 2 times.The choice of a particular type of flow model (absolute or relative) for the domain primarily determines the value of the angle of interaction of the flow lines with an obstacle: it remains close to α1, or close to β1 = α1 + 20°÷70°, which significantly changes the qualitative picture of the viscous flow around the characteristic elements of geometry (“ledge”, “projection”, “crest”), and the values of integral parameters (flow rate, power, efficiency). The lack of recommendations on the correct choice of boundary conditions will inevitably lead not only to the inability to compare the results of various published studies, but also to their objective value. Therefore, the need to substantiate proposals for the choice of flow models in the areas of labyrinth seals, steam balance holes, gaps over shroudless blades, etc. is very relevant.

Исследование процессов тепломассопереноса в нефтяной скважине с~призабойным нагревателем различной длины

This paper considers a three-dimensional mathematical model of heat and mass transfer in an oil well with a bottom-hole heater. The vertical section of an oil well with an electric bottom-hole heater device for wells with highly viscous and paraffinic oil was investigated. The bottomhole heater is a monolithic cylinder located in the lower part of the tubing. The tubing contains perforations through which the oil fluid enters the tubing. The use of a local heater in the near-wellbore area makes it possible to increase the temperature of the fluid flowing through it, thereby reducing the viscosity of the oil and the load on the electric centrifugal pump, for which the critical value of the viscosity of the pumped medium is determined. The implementation of this model was carried out by the finite volume method using the ANSYS engineering simulation software. The geometry of finite elements and the finite volume mesh generated, an analogue of a continuous computational domain, were constructed by the MESH preprocessor. The finite volume mesh consists of polyhedral elements. Using the model, the fields of temperature, velocity and viscosity were obtained throughout the entire volume of the area under study. The curves showing the dependence of the average temperature of the cross section on the longitudinal coordinate of the tubing and the dependence of the average temperature at the inlet to an electric centrifugal pump on the heater power are presented. The viscosity value at the maximum permissible temperature of the heater, as well as the value of sufficient heater power to ensure uninterrupted pumping of petroleum liquid, were determined. The results obtained can be used to significantly improve the operating efficiency of an electric centrifugal pump, to increase the turnaround time and to reduce material costs during field development.

Towards the prediction of barrel fill level in twin-screw wet granulation

The barrel fill level is defined as the fraction of the free available volume for a given screw configuration that is occupied by the wet material and is an interplay of the material throughput, screw speed, screw setup, barrel length of the twin-screw granulator used and the properties of the starting material. The fill level has a major impact on mixing and densification of the wetted mass and thus on the granules produced. It influences the twin-screw granulation process accordingly.In the current study, a model has been developed which is predictive in terms of material hold-ups in the barrel at various process settings by considering the geometries of the different screw elements in a configuration and the conveying velocity of the wet mass through the barrel. The model was checked on two granulators of different dimensions with various screw configurations, different materials and at different process settings.The model represents a step forward in predicting the barrel fill level but further research with a broader spectrum of materials, screw configurations and process settings is still needed and additional twin-screw granulators of other dimensions must be investigated.

Injection Moulded Part Analysis by Alignment of Simulated and Referent Part Geometry

Warpage of the injection moulded parts is one of the most common defects after demoulding, which is intensively present in thin-walled moulded parts or parts with non-uniform wall thickness. The level of the warpage can be reduced by optimizing the mould tempering system in the phase of mould design, by optimizing the injection moulding parameters and recently by optimizing the shape of the mould cavity elements geometry – so-called inverse contouring. Computer simulation of injection moulding process can show the level of moulded part warpage after ejection from the mould, but it will not allow detailed measuring of critical moulded part measures deviations. Detailed analysis and prediction of the critical moulded part dimensions can be performed by aligning of the deformed moulded part design obtained with computer simulation, with moulded part reference CAD geometry. This paper shows the main steps and an example of the geometry aligning process for a specific thermoplastic part.

The influence of longitudinal bending stiffness on running economy and biomechanics in male and female runners

The purpose of this study was to evaluate the influence of different longitudinal bending stiffnesses (LBS) on running economy and lower extremity joint biomechanics in male and female runners. Thirty participants (15 F;15M) performed a treadmill protocol of five-minute randomised aerobic runs with four different footwear conditions varying in LBS: 16.1 N/mm; 32.7 N/mm; 46.1 N/mm; 90.1 N/mm. Biomechanical data was collected at the metatarsophalangeal (MTP) and ankle joints. Running economy was quantified via oxygen consumption data. Oxygen consumption observed no significant differences between footwear conditions (p = 0.960) or significant interaction between footwear and sex (p = 0.126). Stance time observed a significant difference between footwear conditions (p < 0.001), and a significant interaction of footwear and sex (p = 0.008), increasing in females as stiffness increased. At the MTP joint, a shoe effect was present for peak bending angle (p < 0.001), peak extension (p = 0.040) and flexion (p < 0.001) angular velocity, and energy generated at the joint (p = 0.010). There was also an interaction effect of peak MTP extension angular velocity (p = 0.044), with females slightly decreasing as stiffness increased, with no reaction from males. At the ankle joint, a shoe effect was present for peak dorsiflexion (p < 0.001) and plantarflexion (p < 0.001) angular velocity, peak negative power (p < 0.001) and energy absorbed at the joint (p < 0.001). Comparing the interaction between shoe condition and male/female runners, male and female runners reacted similarly to the increased LBS with no effect on running economy and subtle changes in the stance time and MTP angular velocity for female runners. While further research is needed in this area investigating aspects related to sex-specific optimal stiffness element location and geometry, it does not appear that the stiffness magnitude has a sex dependence.

Kernel Angle Dependence Measures in Metric Spaces

Measuring and testing dependence between data in separable metric spaces is of great importance in modern statistics. Most existing work relied on the distance between random variables, which inevitably required the moment conditions to guarantee the distance is well-defined. Based on the geometry element “angle”, we develop a novel class of nonlinear dependence measures for data in metric space that can avoid such conditions. Specifically, by making use of the reproducing kernel Hilbert space equipped with Gaussian measure, we introduce kernel angle covariances that can be applied to various types of data, including low dimensional vector, high dimensional vectors, non-Euclidean data like symmetric positive definite matrices, and compositional data. We estimate kernel angle covariances based on U-statistic and establish the corresponding independence tests via gamma approximation. Our kernel angle independence tests, imposing no-moment conditions on kernels, are robust with heavy-tailed random variables. We conduct comprehensive simulation studies and apply our proposed methods to a facial recognition task. Our kernel angle covariances-based tests show remarkable performances in dealing with image data. All the codes and proofs are included in the supplementary materials.