Multidimensional Distribution Research Articles

Stochastic Modeling of Multidimensional Particle Properties Using Parametric Copulas.

In this paper, prediction models are proposed which allow the mineralogical characterization of particle systems observed by X-ray micro tomography (XMT). The models are calibrated using 2D image data obtained by a combination of scanning electron microscopy and energy dispersive X-ray spectroscopy in a planar cross-section of the XMT data. To reliably distinguish between different minerals the models are based on multidimensional distributions of certain particle characteristics describing, for example, their size, shape, and texture. These multidimensional distributions are modeled using parametric Archimedean copulas which are able to describe the correlation structure of complex multidimensional distributions with only a few parameters. Furthermore, dimension reduction of the multidimensional vectors of particle characteristics is utilized to make non-parametric approaches such as the computation of distributions via kernel density estimation viable. With the help of such distributions the proposed prediction models are able to distinguish between different types of particles among the entire XMT image.

Approximate Bayesian Estimation Based on Fuzzy Interval Data from Multi-Dimensional Normal Distribution

Approximate Bayesian Estimation Based on Fuzzy Interval Data from Multi-Dimensional Normal Distribution

Diffusion tensor imaging reveals diffuse white matter injuries in locked-in syndrome patients.

Locked-in syndrome (LIS) is a state of quadriplegia and anarthria with preserved consciousness, which is generally triggered by a disruption of specific white matter fiber tracts, following a lesion in the ventral part of the pons. However, the impact of focal lesions on the whole brain white matter microstructure and structural connectivity pathways remains unknown. We used diffusion tensor magnetic resonance imaging (DT-MRI) and tract-based statistics to characterise the whole white matter tracts in seven consecutive LIS patients, with ventral pontine injuries but no significant supratentorial lesions detected with morphological MRI. The imaging was performed in the acute phase of the disease (26 ± 13 days after the accident). DT-MRI-derived metrics were used to quantitatively assess global white matter alterations. All diffusion coefficient Z-scores were decreased for almost all fiber tracts in all LIS patients, with diffuse white matter alterations in both infratentorial and supratentorial areas. A mixture model of two multidimensional Gaussian distributions was fitted to cluster the white matter fiber tracts studied in two groups: the least (group 1) and most injured white matter fiber tracts (group 2). The greatest injuries were revealed along pathways crossing the lesion responsible for the LIS: left and right medial lemniscus (98.4% and 97.9% probability of belonging to group 2, respectively), left and right superior cerebellar peduncles (69.3% and 45.7% probability) and left and right corticospinal tract (20.6% and 46.5% probability). This approach demonstrated globally compromised white matter tracts in the acute phase of LIS, potentially underlying cognitive deficits.

Modeling of Mathematical Processing of Physics Experimental Data in the Form of a Non-Markovian Multi-Resource Queuing System

The paper presents a mathematical model for processing of physics experimental data in the form of a non-Markovian infinite-server multi-resource queuing system with Markov modulated Poisson process arrivals and arbitrary service time. It is proved that the joint steady-state probability distribution of the total volume of the occupied resource of each type converges to a multidimensional Gaussian distribution under the asymptotic condition of the growing intensity of the arrival process. The parameters of this asymptotic distribution are derived.

Effect of correlation between different ply failures on reliability of laminate composites

The present study focuses on the correlation between different ply failures in laminate composites and its effect on the laminate reliability. First order reliability method (FORM) is employed to compute the failure probability of each ply, and the the correlation between different ply failures is derived from the linearized limit state function of each ply. Significant correlation between different ply failures is observed, and the magnitude of the correlation depends on the uncertainty level of stochastic variables. The correlation is demonstrated laying heavy influence on the laminate reliability. It is clarified that the max algorithm (adopted by previous studies) may provide a heavy underestimation on the laminate failure probability, depending on the magnitude of the correlation coefficient between different ply failures. On the contrary, the sum algorithm (adopted by previous studies) is found providing huge overestimation of the laminate failure probability due to its complete neglect of the correlation between different ply failures. To well account for the correlation between different ply failures in composite re liability evaluation, the integration of multidimensional normal distribution (IMND) is recommended to be employed.

Pyramidal Distribution: An Alternative Distribution for Bivariate Uniformity Test Power Comparison

The purpose of the multi-dimensional uniformity test is to check whether the underlying probability distribution, from which a random sample is drawn, differs from the multi-dimensional uniform distribution. The multi-dimensional uniformity test has applications in various fields such as biology, astronomy and computer science. Various statistical tests have been proposed for the multivariate uniformity. As a special case, the bivariate statistic test is discussed in this paper. To evaluate the performance of the uniformity tests, power comparison is the technique for selecting appropriate statistical test. The Monte Carlo simulations usually used to compare the power of the tests. Berrendero, Cuevas and Vazquez-Grande proposed a distance-to-boundary test, which was one of the latest published statistical tests for multi-dimensional uniformity [J. R. Berrendero, A. Cuevas and F. Vazquez-Grande, Testing multivariate uniformity: The distance-to-boundary method, Canad. J. Stat. 34 (2006) 693–707]. Chen and Hu proposed another test for multivariate uniformity [Z. Chen and T. Hu, Statistical test for bivariate uniformity, Adv. Stat. 2014 (2014) 740831]. Power comparison was conducted to compare these tests. In order to get more convincing power comparison results, more alternative distributions should be used when power study is conducted. This paper proposes a new bivariate distribution, named the pyramidal distribution, with support set [Formula: see text]. This distribution is quite flexible so that it can be used to produce different shapes of bivariate distributions. Because of that, the proposed distribution can be used as an alternative distribution in power comparison for bivariate uniformity test.

Towards end-to-end likelihood-free inference with convolutional neural networks.

Complex simulator-based models with non-standard sampling distributions require sophisticated design choices for reliable approximate parameter inference. We introduce a fast, end-to-end approach for approximate Bayesian computation (ABC) based on fully convolutional neural networks. The method enables users of ABC to derive simultaneously the posterior mean and variance of multidimensional posterior distributions directly from raw simulated data. Once trained on simulated data, the convolutional neural network is able to map real data samples of variable size to the first two posterior moments of the relevant parameter's distributions. Thus, in contrast to other machine learning approaches to ABC, our approach allows us to generate reusable models that can be applied by different researchers employing the same model. We verify the utility of our method on two common statistical models (i.e., a multivariate normal distribution and a multiple regression scenario), for which the posterior parameter distributions can be derived analytically. We then apply our method to recover the parameters of the leaky competing accumulator (LCA) model and we reference our results to the current state-of-the-art technique, which is the probability density estimation (PDA). Results show that our method exhibits a lower approximation error compared with other machine learning approaches to ABC. It also performs similarly to PDA in recovering the parameters of the LCA model.

VERIFICATION OF NONSTATIONARY POLYNOMIAL TRAFFIC MODELS

In this paper, we perform an experimental study of models for the optimal estimation of polynomial traffic in heterogeneous computer networks. To achieve the goal of the study, the tasks of verification of traffic models with different dimensions of polynomial traffic are set and solved. The verification procedure is understood as checking the adequacy of traffic models to reference models using selected criteria and adequacy indicators, in the form of which target functionals are used, which determine the accuracy and reliability of traffic modeling. The main criteria are point and interval estimates of concordance (consistency) among themselves of model and reference values of moments of polynomial traffic. Models of non-stationary traffic in heterogeneous computer networks are built using the quantization and Markov approximation methods and the least squares method. The multidimensional normal distribution of all values of aggregated traffic in the nonstationary interval is assumed to be Gaussian. To assess the accuracy of the approximation of traffic, the concept of reference traffic is introduced and a comparison of the first two moment characteristics of the bypass model and reference traffic is used. This allows you to compare multidimensional Gaussian distributions of reference and model unsteady traffic. An increase in the dimensionality of the models is accompanied by an increase in the accuracy of approximation of non-stationary traffic. The proposed system of indicators and methods of verification of models of non-stationary traffic with a different number of traffic states allow you to verify models with the necessary degree of completeness. The calculations of the verification criteria for models of nonstationary polynomial traffic of dimension n = 3 and n = 4 in heterogeneous computer networks in the discrete case using the Euclidean and Hilbert space metrics in the continuous case. The obtained results make it possible to identify general patterns of growth in the accuracy of traffic approximation (degree of model adequacy) depending on the increase in the dimension of the models and sample sizes.

Diffusion tensor distribution imaging.

Conventional diffusion MRI yields voxel‐averaged parameters that suffer from ambiguities for heterogeneous anisotropic materials such as brain tissue. Using principles from solid‐state NMR spectroscopy, we have previously introduced the shape of the diffusion encoding tensor as a separate acquisition dimension that disentangles isotropic and anisotropic contributions to the observed diffusivities, thereby allowing for unconstrained data inversion into diffusion tensor distributions with “size,” “shape,” and orientation dimensions. Here we combine our recent non‐parametric data inversion algorithm and data acquisition protocol with an imaging pulse sequence to demonstrate spatial mapping of diffusion tensor distributions using a previously developed composite phantom with multiple isotropic and anisotropic components. We propose a compact format for visualizing two‐dimensional arrays of the distributions, new scalar parameters quantifying intra‐voxel heterogeneity, and a binning procedure giving maps of all relevant parameters for each of the components resolved in the multidimensional distribution space.

Endogenous fluorescence can differentiate the keratoconic cornea

The purpose of the study was to investigate the endogenous fluorescence of the keratoconic cornea in order to analyze changes in the spectra due to the keratoconic stroma abnormalities.Twenty-two corneal buttons obtained from patients with keratoconus (KC, N = 22) at the time of penetrating keratoplasty were used. As a reference, twelve normal corneas (N = 12): ten from the Eye Bank and two from enucleated eyes due to choroidal melanoma were used. The fluorescence excitation/emission matrices (EEM) in the ranges of 250–400/260–600 nm were recorded.Healthy cornea, keratoconic cornea and sclera showed three main EEM bands, which correspond to the following fluorophores: tryptophan residues in the proteoglycan fraction of corneal/scleral stromas, naturally occurring collagen cross-links and the NAD(P)H fraction present in the metabolically active cells. Relative intensity factors S1, S2 and S3 describing the contribution of each kind of fluorophore to the total fluorescence of the tissue were calculated. Normal and keratoconic corneas show qualitatively similar fluorescence matrices, but the statistically significant differences in the mean values of the S1, S2 and S3 parameters for the KC and normal corneas were observed indicating changes in contribution of different fluorophores to the whole fluorescence of the tissue. Moreover, differences between multidimensional distribution of the relative intensity factors S1, S2 and S3 between these groups were demonstrated (p < 0.001).In conclusions: Differences in the relative intensity factors calculated on a basis of the fluorescence spectra can correspond to the changes found in the KC stroma regarding natural collagen cross-links and the proteoglycan fraction. These parameters well differentiate the KC and normal corneas that could serve as an additional tool for the keratoconus characterization.

Failure probability of corroded pipeline considering the correlation of random variables

In view of the shortcomings of independence assumption for random variables in the reliability analysis of corroded pipeline, an analytical method is put forward for the reliability of corroded pipeline considering the correlation among random variables. The models of corrosion perforation, local burst and rupture as well as their composite failure probability are also established. Based on JC method and orthogonal transformation, the multidimensional normal distribution function is used to propose a computational method for the multi-mode failure probability of corroded pipeline considering correlated random variables. This paper elaborates on the correlation of pipeline reliability analysis. Meanwhile, based on case studies, a study is conducted on how the pipeline failure probability is affected by the correlation between four pairs of random variables, including pipe diameter and wall thickness, the depth and length of the defect, the radial rate and axial rate of corrosion, as well as yield strength and tensile strength. The analysis results suggest that the correlation between random variables has no influence on the failure probability of pipeline corrosion perforation. The probability of local burst, rupture and composite failure increases with the increase of correlation coefficient between the defect depth and length as well as that between the radial and axial corrosion rate, and decreases with the increase of correlation coefficient between the pipe diameter and wall thickness. The larger the correlation coefficient between yield strength and tensile strength is, the greater the probability of local burst and rupture will be, while the composite failure probability remains the same. Besides, the impact of variables correlation on the failure probability of corroded pipeline decreases as the corrosion continues. The failure probability of corroded pipeline is most sensitive to the correlation coefficient between the radial and axial corrosion rate, and least sensitive to the correlation coefficient between the yield strength and tensile strength.

Sheet Metal Profiles with Variable Height: Numerical Analyses on Flexible Roller Beading

In the spirit of flexible manufacturing, the novel forming process “flexible roller beading” was developed, which allows the incremental production of height-variable sheet metal profiles. After designing the process and realizing a test facility for flexible roller beading, the feasibility was experimentally shown. The following step addresses the expansion of the process limits. With this aim, the mechanical behavior of the sheet metal during the process was investigated by means of FEA. Due to the variable cross-section development of the sheet metal profile, a multidimensional stress distribution was identified. Based on the present state of stress and strain, conclusions about the origin of appearing defect formations were drawn. Observed defects were sheet wrinkles as a result of compressive stresses in the profile flange and material thinning in the profile legs and bottom due to unintendedly exceeding tensile stresses. The influences of the forming strategy as well as tool- and workpiece-side variations on the quality of the manufacturing result were investigated. From the results of the analyses, measures to avoid component failure were derived. Given all the findings, guidelines were concluded that are to be considered in designing the forming sequence. With the insights into the occurring processes and the mastery of this novel forming process, important contributions are made to its industrial suitability. The approach of lightweight and load-oriented component design can be extended by realizing new families of sheet metal profiles. With respect to Industry 4.0, on-demand manufacturing is increasingly required, which is why flexible roller beading is of substantial relevance for the industrial sheet metal production.

Seismic risk analysis for reinforced concrete structures with both random and parallelepiped convex variables

This article develops an improved seismic risk assessment formulation exhibiting both random and bounded uncertainties using a probability and parallelepiped convex set mixed model. Limit thresholds for different types of components are described via a probabilistic model. The distribution parameters of limit thresholds are originally treated by employing a multidimensional parallelepiped convex model, in which marginal intervals are utilised to represent scattering levels for the distribution parameters, while relevant angle are employed to express the correlation between uncertain distribution parameters. The structural responses, i.e., engineering demand parameters (EDPs), are considered as correlated random variables and are assumed to follow a multidimensional lognormal distribution. A performance limit state function, which allows considering the relationship between the EDPs and the corresponding limit thresholds, is employed to reflect the coexistence of both random and parallelepiped convex variables. The limit state function is mapped into the standard parameter space via a transformation technique. Then, the improved seismic risk formulation, characterised through a probability and parallelepiped convex mixed variables, can be derived with the combination of the seismic fragility function and the ground motion hazard curve. The main purpose is to illustrate that the performance limit states should be properly modeled as random and parallelepiped convex mixed variables rather than only random or deterministic quantities. A six-story reinforced concrete building designed according to Chinese codes are used to illustrate the proposed approach for constructing hazard curves. The interstory drift and the peak floor acceleration are the selected EDPs, calculated through incremental dynamic analysis. The results demonstrate that the calculated failure probabilities for different limit states in 50 years are found capable of meeting the requirements of Chinese seismic norms after the proposed seismic risk formulation is adopted.

A New Generalized Inequality for Covariance in N Dimensions

Inspired by the work of Zhefei He and Mingjin Wang which was published in the Journal of Inequalities and Applications in 2015, this paper further generalizes some related results to the case of multidimensional random variables. The resulting inequality for covariance is then applied to different multidimensional statistical distributions (multiuniform, multinomial, and multinormal). Coordinate dependence of the inequality is also examined. The obtained formulas could be useful for making estimates in multivariate statistics.

Contrasting latitudinal diversity and co-occurrence patterns of soil fungi and plants in forest ecosystems

The diversity patterns of plants at large scale are broadly documented, but that of soil fungi remains elusive. Limited reports on soil fungal biogeography mainly are based on species richness. We carried out a large-scale field investigation on soil fungi originating from 365 forest plots covering five climate zones in China. We tested whether and to what degree a range of different fungal diversity estimates are subject to latitudinal constraints and compared co-occurrence patterns of plants and soil fungi. We found that alpha diversity of plants increased towards lower latitude, while that of soil fungi only tended to increase. Abiotic but not biotic factors controlled fungal alpha diversity. Multiple-site and pairwise beta diversity of soil fungi were closely linked to plant communities. Null model revealed an increasing interaction strength between plant species, while there was a hump-shaped pattern of interaction strength between fungal species from high-latitude towards low-latitude. Our results allow a deeper insight into the multidimensional distribution of soil fungal diversity and reveal contrasting co-occurrence patterns between soil fungi and plants along a latitude gradient, highlighting the need for more inclusive theories to explain these disparities between micro- and macroorganism communities.

A Novel Seismic Risk Analysis Method for Structures with Both Random and Convex Set Mixed Variables: Case Study of a RC Bridge

Assessing the demand hazards of structures is requested in the framework of performance-based earthquake engineering. An efficient method for estimating the seismic risk of structures is proposed in this paper. The relationship between multiple limit capacities and corresponding response parameters is denoted by using a generalized multidimensional limit state equation. The limit states of different components are described as random and convex mixed variables. The seismic responses of different components are considered dependent and follow a multidimensional lognormal distribution. The mathematical formula of multidimensional demand hazards of structures is then derived through combining the seismic fragility function and the seismic hazard curve. The proposed method is used to perform the demand hazard analysis and the parameter sensitivity analysis of a multispan continuous concrete girder bridge, selecting column ductility and bearing displacement as the two-dimensional seismic response parameters obtained by Incremental Dynamic Analysis. The results demonstrate that the coefficient of variation and correlation coefficient N, which are involved in the limit state equation, have an impact on the evaluation of the demand hazards.

基于多维混合柯西分布的点云配准

基于多维混合柯西分布的点云配准

Multi-dimensional analysis of wetness-dryness encountering of streamflow based on the Copula function in Lake Poyang Basin

鄱阳湖流域5大水系来水变化与湖区水文极值事件有密切关系，研究径流变化特征与丰枯遭遇规律对区域防洪抗旱有重要意义.本文运用Copula函数构建了鄱阳湖水系多维径流联合分布模型，采用特枯、偏枯、平水、偏丰和特丰的径流丰枯分类，定量研究了鄱阳湖5大水系丰枯遭遇的问题，探讨了多维丰枯遭遇同步联合概率的变化特征.结果表明：鄱阳湖水系河流之间的径流具有较高的相关性，Gaussian Copula函数能较好地模拟二维至五维的径流联合分布.多条河流的丰枯遭遇随着维数的增加，丰枯组合增加，丰枯同步的联合概率明显下降，且丰枯同步的最大联合概率趋向于丰枯两端.对于相同的概率区间，非汛期径流的丰枯同步联合概率明显大于年径流和汛期径流，而年径流和汛期径流之间的丰枯同步联合概率差别较小.同处于流域北部或南部或相邻的河流之间的组合，其同步联合概率相较其他组合大，而南、北河流组合的同步联合概率相对较小.该研究可为流域水资源管理及水旱灾害预防提供科学依据.;According to previous research, there was a close relationship between the change of streamflow and the hydrological extreme events in Lake Poyang Basin. Based on the Copula function, the multi-dimensional joint distributions of streamflow in Lake Poyang Basin was built. Streamflow was classified into extra dryness, ordinary dryness, normal water, ordinary wetness and extra wetness, the problem of wetness-dryness encountering of streamflow was discussed quantitatively. Multi-dimensional joint frequencies of identical probability interval of streamflow were analysed. Results showed that there were high dependences between two rivers of Lake Poyang Basin. The Gaussian Copula fitted well the joint distributions of streamflow from two-dimension to five-dimension. As the dimension of joint frequency of multi-streamflow increased, the number of combination of wetness and dryness encountering increased, yet the joint frequency of identical probability interval decreased significantly, and then the maximum joint frequency of identical probability intervals tended to occur in the upper or down interval of the distribution of streamflow. The joint frequencies of identical probability interval of streamflow in dry period were greater than those of annual streamflow and streamflow in flood period, and those between annual streamflow and streamflow in flood were slightly different. The joint frequencies of identical probability interval of streamflow between northern rivers, southern rivers, or adjacent rivers were generally greater than those between northern and southern rivers. This study provides scientific basis for river basin resources management and drought event prevention.

Real-Time Characterization of Crystal Shape and Size Distribution Based on Moving Window and 3D Imaging in a Stirred Tank

Crystal shape distribution, i.e. the multidimensional size distribution of crystals, is of great importance to their down-stream processing such as in filtration as well as to the end-use properties including the dissolution rate and bioavailability for crystalline pharmaceuticals. Engineering crystal shape and shape distribution requires knowledge about the growth behavior of different crystal facets under varied operational conditions e.g. supersaturations. Measurement of the facet growth rates and growth kinetics of static crystals in a crystallizer without stirring has been reported previously. Here attention is given to study on real-time characterization of the 3D facet growth behavior of crystals in a stirred tank where crystals are constantly moving and rotating. The measurement technique is stereo imaging and the crystal shape reconstruction is based on a stereo imaging camera model. By reference to a case study on potash alum crystallization, it is demonstrated that the crystal size and shape distributions (CSSD) of moving and rotating potash alum crystals in the solution can be reconstructed. The moving window approach was used to correlate 3D face growth kinetics with supersaturation (in the range 0.04 - 0.12) given by an ATR FTIR probe. It revealed that {100} is the fastest growing face, leading to a rapid reduction of its area, while the {111} face has the slowest growth rate, reflected in its area continuously getting larger.

Applications of Machine Learning at BESIII

BESIII is an experiment at the high precision frontier of hadron physics in τ-charm region. Machine learning techniques have been used to improve the performance of BESIII software. In this proceeding, we present novel approaches with XGBoost for multi-dimensional distribution reweighting, muon identification and cluster reconstruction for CGEM (Cylindrical Gas Electron Multiplier) inner tracker.