Parameterization Of Distribution Research Articles

A New Look at the Dirichlet Distribution: Robustness, Clustering, and Both Together

AbstractCompositional data have peculiar characteristics that pose significant challenges to traditional statistical methods and models. Within this framework, we use a convenient mode parametrized Dirichlet distribution across multiple fields of statistics. In particular, we propose finite mixtures of unimodal Dirichlet (UD) distributions for model-based clustering and classification. Then, we introduce the contaminated UD (CUD) distribution, a heavy-tailed generalization of the UD distribution that allows for a more flexible tail behavior in the presence of atypical observations. Thirdly, we propose finite mixtures of CUD distributions to jointly account for the presence of clusters and atypical points in the data. Parameter estimation is carried out by directly maximizing the maximum likelihood or by using an expectation-maximization (EM) algorithm. Two analyses are conducted on simulated data to illustrate the effects of atypical observations on parameter estimation and data classification, and how our proposals address both aspects. Furthermore, two real datasets are investigated and the results obtained via our models are discussed.

Compton rocket effect due to the action of radiation reaction force in degenerate plasma

A closed set of fluid equations with radiation reaction force (RRF) are constructed from the moments of the appropriate single particle kinetic equation describing a relativistic degenerate (high density) electron plasma. The closure, in analogy with the Maxwellian closure for non-degenerate plasmas, is affected via a parametrized Fermi-Dirac distribution. It is shown that the degeneracy increases RRF just as will be predicted from the so-called “Compton Rocket” effect.

TOWARDS THEORETICAL MODELLING OF DISTRIBUTION OF MATTER IN UNIVERSE

The description of the distribution of matter in the universe requires not only accurate observations but also adequate approaches to their theoretical interpretation. This paper proposes a method of parameterization of distributions with a morphologically complex topology using structural invariants (for example, Euler), based on which it is possible to distinguish clusters with different topologies (in the observation plane). Based on the introduced classification and appropriate scaling, it becomes possible to estimate the distribution of matter, for example, within the framework of the mean-field model. To study the kinetics of the evolution of matter distribution, it is proposed to introduce the appropriate ordering parameter, which is built based on the calibrating and current values of the Euler-Poincaré invariants. This approach, by constructing phase diagrams for the ordering parameter, allows you to track the details of the kinetics of the evolution of matter distributions, studying, in particular, the temporal hierarchy of relaxation times of intermediate states. The temporal kinetics of this approach are described using simple kinetic equations that describe the relaxation of the ordering parameter field, and the values of invariants determined with the help of appropriate measurements (observations) appear as initial conditions. This approach can be seen as an alternative to other approaches to parameterization of structurally complex systems, such as Voronoi methods, graphs, etc. A comparative analysis of the results of various alternative approaches to the parameterization of topologically complex distributions of matter, which will be conducted in the future, should contribute to the deepening of existing ideas about the nature of the distribution of matter in the Universe.

Battery defect detection for real world vehicles based on Gaussian distribution parameterization developed LCSS

Accurate detection of early faults in lithium-ion (Li-ion) battery packs plays an important role in preventing safety accidents and reducing property damage. At present, fault diagnosis research based on actual vehicle single-cell voltage consistency has become a hot topic, but the consistency evolution law is not uniform due to complex environmental conditions and operating conditions, so the generalization of current methods is poor. To this end, this paper proposes a multi-layer fault diagnosis framework, which first considers the evolution law of single-unit voltage difference to determine whether there is a potential risk, and proposes a flexible screening method for state of charge (SOC) interval segments, which improves the accuracy and efficiency of inspection. Next, the thresholds required by the developed longest common subsequence (DLCSS) algorithm are parameterized based on Gaussian distribution and combined with z-score to construct fault diagnosis indicators. The parameters of the framework are calculated from different segment data statistics, enhancing the objectivity and interpretability of the parameters. The results are validated using actual vehicle data and show that the proposed framework can greatly reduce the required time and identify faulty batteries more accurately with a 96.2 % precision and a 72.4 % detection rate, which is better than other algorithms and has better robustness.

Considering the heavy-quark distribution functions at low x

In this paper, we investigate the behavior of charm and bottom distribution functions at low x with respect to the parametrization of the gluon distribution function in the framework of the nonlinear Gribov–Levin–Ryskin–Mueller–Qiu, Zhu-Ruan-Shen (GLR-MQ-ZRS) evolution equation. Also, we solve the GLR-MQ-ZRS equation using the parametrization behavior of the gluon distribution function. The computed results are compared with the NNPDF3.0, CT14 and GRV92 collaborations.

Parametrization of drop size distribution with rain rate for microwave and millimeter wave applications in Central Africa

Parametrization of drop size distribution with rain rate for microwave and millimeter wave applications in Central Africa

Optimization framework for patient-specific modeling under uncertainty.

Estimating a patient-specific computational model's parameters relies on data that is often unreliable and ill-suited for a deterministic approach. We develop an optimization-based uncertainty quantification framework for probabilistic model tuning that discovers model inputs distributions that generate target output distributions. Probabilistic sampling is performed using a surrogate model for computational efficiency, and a general distribution parameterization is used to describe each input. The approach is tested on seven patient-specific modeling examples using CircAdapt, a cardiovascular circulatory model. Six examples are synthetic, aiming to match the output distributions generated using known reference input data distributions, while the seventh example uses real-world patient data for the output distributions. Our results demonstrate the accurate reproduction of the target output distributions, with a correct recreation of the reference inputs for the six synthetic examples. Our proposed approach is suitable for determining the parameter distributions of patient-specific models with uncertain data and can be used to gain insights into the sensitivity of the model parameters to the measured data.

Connected and disconnected sea partons from the CT18 parametrization of PDFs

The separation of the connected and disconnected sea partons, which were uncovered in the Euclidean path-integral formulation of the hadronic tensor, is accommodated with an alternative parametrization of the non-perturbative parton distribution functions in the CT18 global analysis. This is achieved with the help of the distinct small $x$ behaviours of these two sea partons and the constraint from the lattice calculation of the ratio of the strange momentum fraction to that of the $\bar u$ or $\bar d$ in the disconnected insertion. The whole dataset of CT18 is used in this CT18CS fit. The impact of the recent SeaQuest data on the $\bar{d}(x)-\bar{u}(x)$ distribution of CT18CS is also discussed. The separate momentum fractions for the valence, the connected sea and disconnected sea of $u$ and $d$, the strange and the gluon partons are presented at $\mu =1.3$ GeV for the first time. They can be compared term-by-term with systematic error controlled lattice calculations.

Demographics of z ∼ 6 quasars in the black hole mass–luminosity plane

ABSTRACT We study the demographics of z ∼ 6 broad-line quasars in the black hole (BH) mass–luminosity plane using a sample of more than 100 quasars at 5.7 < z < 6.5. These quasars have well-quantified selection functions and nearly one-third of them also have virial BH masses estimated from near-IR spectroscopy. We use forward modelling of parametrized intrinsic distributions of BH masses and Eddington ratios, and account for the sample flux limits and measurement uncertainties of the BH masses and luminosities. We find significant differences between the intrinsic and observed distributions of the quantities due to measurement uncertainties and sample flux limits. There is also marginal evidence that the virial BH masses are susceptible to a positive luminosity-dependent bias (BH mass is overestimated when luminosity is above the average), and that the mean Eddington ratio increases with BH mass. Our models provide reliable constraints on the z ∼ 6 BH mass function at $M_{\rm BH}\gt 10^{8.5}\, M_\odot$, with a median 1σ uncertainty of ∼0.5 dex in abundance. The intrinsic Eddington ratio distribution of $M_{\rm BH}\gt 10^{8.5}\, M_\odot$ quasars can be approximated by a mass-dependent Schechter model, with a broad peak around log (Lbol/LEdd) ∼ −0.9. We also find that, at 4.5 ≲ z ≲ 6, the number densities of more massive BHs tend to decline more rapidly with increasing redshift, contrary to the trend at 2.5 ≲ z ≲ 4.5 reported previously.

Bubble-Turbulence Dynamics and Dissipation Beneath Laboratory Breaking Waves

Abstract Bubbles directly link sea surface structure to the dissipation rate of turbulence in the ocean surface layer through wave breaking, and they are an important vehicle for air–sea transfer of heat and gases and important for understanding both hurricanes and global climate. Adequate parameterization of bubble dynamics, especially in high winds, requires simultaneous measurements of surface waves and breaking-induced turbulence; collection of such data would be hazardous in the field, and they are largely absent from laboratory studies to date. We therefore present data from a series of laboratory wind-wave tank experiments designed to observe bubble size distributions in natural seawater beneath hurricane conditions and connect them to surface wave statistics and subsurface turbulence. A shadowgraph imager was used to observe bubbles in three different water temperature conditions. We used these controlled conditions to examine the role of stability, surface tension, and water temperature on bubble distributions. Turbulent kinetic energy dissipation rates were determined from subsurface ADCP data using a robust inertial-subrange identification algorithm and related to wind input via wave-dependent scaling. Bubble distributions shift from narrow to broadbanded and toward smaller radius with increased wind input and wave steepness. TKE dissipation rate and shear were shown to increase with wave steepness; this behavior is associated with a larger number of small bubbles in the distributions, suggesting shear is dominant in forcing bubbles in hurricane wind-wave conditions. These results have important implications for bubble-facilitated air–sea exchanges, near-surface ocean mixing, and the distribution of turbulence beneath the air–sea interface in hurricanes. Significance Statement Bubbles are a vehicle for the flux of heat, momentum, and gases between the atmosphere and ocean. These fluxes contribute to the energy budgets of hurricanes, climate, and upper-ocean biology. Few to no simultaneous measurements of surface waves, bubbles, and turbulence have been made in hurricane conditions. To improve numerical model representation of bubbles, we performed laboratory experiments to parameterize bubble size distributions using physical variables including wind and waves. Bubble distributions were found to become broadbanded and shift toward smaller radius with increased wind stress and wave steepness. Turbulence dissipation rate and shear were shown to increase with wave steepness. Our results give the first physically based bubble distribution parameterization from naturally breaking waves in hurricane-force conditions.

Simulation of in-ice cosmic ray air shower induced particle cascades

We present simulations of the development of high-energy cosmic-ray air showers penetrating high-altitude ice layers that can be found at the polar regions. We use a combination of the corsika Monte Carlo code and the geant4 simulation toolkit, and focus on the particle cascade that develops in the ice to describe its most prominent features. We discuss the impact of the ice layer on the total number of particles in function of depth of the air shower, and we give a general parametrization of the charge distribution in the cascade front in function of ${X}_{\mathrm{max}}$ of the cosmic ray air shower, which can be used for analytical and semianalytical calculations of the expected Askaryan radio emission of the in-ice particle cascade. We show that the core of the cosmic ray air shower dominates during the propagation in ice, therefore creating an in-ice particle cascade strongly resembling a neutrino-induced particle cascade. Finally, we present the results of simulations of the Askaryan radio emission of the in-ice particle cascade, showing that the emission is dominated by the shower core, and discuss the feasibility of detecting the plasma created by the particle cascade in the ice using RADAR echo techniques.

Bayesian Stokes inversion with normalizing flows

Stokes inversion techniques are very powerful methods for obtaining information on the thermodynamic and magnetic properties of solar and stellar atmospheres. In recent years, highly sophisticated inversion codes have been developed that are now routinely applied to spectro-polarimetric observations. Most of these inversion codes are designed to find an optimum solution to the nonlinear inverse problem. However, to obtain the location of potentially multimodal cases (ambiguities), the degeneracies and the uncertainties of each parameter inferred from the inversions algorithms – such as Markov chain Monte Carlo (MCMC) – require evaluation of the likelihood of the model thousand of times and are computationally costly. Variational methods are a quick alternative to Monte Carlo methods, and approximate the posterior distribution by a parametrized distribution. In this study, we introduce a highly flexible variational inference method to perform fast Bayesian inference, known as normalizing flows. Normalizing flows are a set of invertible, differentiable, and parametric transformations that convert a simple distribution into an approximation of any other complex distribution. If the transformations are conditioned on observations, the normalizing flows can be trained to return Bayesian posterior probability estimates for any observation. We illustrate the ability of the method using a simple Milne-Eddington model and a complex non-local thermodynamic equilibrium (NLTE) inversion. The method is extremely general and other more complex forward models can be applied. The training procedure need only be performed once for a given prior parameter space and the resulting network can then generate samples describing the posterior distribution several orders of magnitude faster than existing techniques.

Combined parameterization of material distribution and surface mesh for stiffener layout optimization of complex surfaces

Stiffener layout optimization of complex surfaces is fulfilled within the framework of topology optimization. A combined parameterization method is developed in two aspects. One is to parameterize the material distribution of the stiffener layout by means of B-spline. The other is to build the mapping relationship from the known 3D surface mesh of the thin-walled structure to its parametric domain by means of mesh parameterization. The influence of mesh parameterization upon the stiffener layout is discussed to reveal the matching issue of the combined parameterization. 3D complex surfaces represented by the triangular mesh can be dealt with even though analytical parametric equations are not available. Some numerical examples are solved to demonstrate the direct advantage and effectiveness of the proposed method.

Improved parametrized multiple window spectrogram with application in ship navigation systems

In analyzing non-stationary noisy signals with time-varying frequency content, it's convenient to use distribution methods in joint, time and frequency, domains. Besides different adaptive data-driven time-frequency (TF) representations, the approach with multiple orthogonal and optimally concentrated Hermite window functions is an effective solution to achieve a good trade-off between low variance and minimized stable bias estimates. In this paper, we propose a novel spectrogram method with multiple optimally parameterized Hermite window functions, with parameterization which includes a pair of free parameters to regulate the shape of the window functions. The computation is performed in the optimization process to minimize the variable projection (VP) functional problem. The proposed parametrized distribution method improves TF concentration and instantaneous frequency (IF) estimation accuracy, as shown in experimental results for synthetic signals and real-life ship motion response signals. With the optimization of nonlinear least-squares approximation of the ship response signals, the Hermite spectra are centralized, and only up to 15 basis functions are sufficient for concentration improvement in the TF domain.

Characteristics of Particles Emitted from Waste Fires-A Construction Materials Case Study.

This study aimed to determine the relative densities of populations of particles emitted in fire experiments of selected materials through direct measurement and parametrization of size distribution as number (NSD), volume (VSD), and mass (MSD). As objects of investigation, four typical materials used in construction and furniture were chosen: pinewood (PINE), laminated particle board (LPB), polyurethane (PUR), and poly(methyl methacrylate) (PMMA). The NSD and VSD were measured using an electric low-pressure impactor, while MSD was measured by weighing filters from the impactor using a microbalance. The parametrization of distributions was made assuming that each distribution can be expressed as the sum of an arbitrary number of log-normal distributions. In all materials, except PINE, the distributions of the particles emitted in fire experiments were the sum of two log-normal distributions; in PINE, the distribution was accounted for by only one log-normal distribution. The parametrization facilitated the determination of volume and mass abundances, and therefore, the relative density. The VSDs of particles generated in PINE, LPB, and PUR fires have similar location parameters, with a median volume diameter of 0.2–0.3 µm, whereas that of particles generated during PMMA burning is 0.7 µm. To validate the presented method, we burned samples made of the four materials in similar proportions and compared the measured VSD with the VSD predicted based on the weighted sum of VSD of raw materials. The measured VSD shifted toward smaller diameters than the predicted ones due to thermal decomposition at higher temperatures.

Variational Bayesian Kalman filter using natural gradient

We propose a technique based on the natural gradient method for variational lower bound maximization for a variational Bayesian Kalman filter. The natural gradient approach is applied to the Kullback-Leibler divergence between the parameterized variational distribution and the posterior density of interest. Using a Gaussian assumption for the parametrized variational distribution, we obtain a closed-form iterative procedure for the Kullback-Leibler divergence minimization, producing estimates of the variational hyper-parameters of state estimation and the associated error covariance. Simulation results in both a Doppler radar tracking scenario and a bearing-only tracking scenario are presented, showing that the proposed natural gradient method outperforms existing methods which are based on other linearization techniques in terms of tracking accuracy.

On the consistency of Bayes estimates for the infinite continuous mixture of Dirichlet distributions

In this paper, we introduced the infinite continuous mixture of Dirichlet distributions as a generalization of the infinite mixture of Dirichlet ones, in order to avoid the limitation of choosing the a priori sample size for the expectation \textit{a posteriori} estimator. Monte-Carlo sampling was used in order to obtain the \textit{posterior} distributions mixture, since this mixture is difficult to get analytically. A new parametrization of this proposed distribution was achieved. Then, we suggested a mixture expectation \textit{a posteriori} estimator of the unknown parameters. The proposed estimator solves the problem of how to construct a Bayesian estimation of proportions without specifying particular parameters and sample size of the prior knowledge. Some asymptotic properties of this estimator were derived, specifically, its bias and variance. The consistency and asymptotic normality of the estimator were also established when the sample size tends to infinity and its credible interval was determined. The performance of the proposed estimator was illustrated theoretically and by means of a simulation study. Ultimately, a comparative simulation study between the learned estimates, the proposed mixture expectation \textit{a posteriori}, standard Bayesian estimator, maximum likelihood and Jeffreys estimator, was established. According to this simulation, we were able to conclude that the prior infinite mixture of Dirichlet distributions offers higher accuracy and flexibility for modeling and learning data.

Improved Inference of Gaussian Mixture Copula Model for Clustering and Reproducibility Analysis using Automatic Differentiation

Copulas provide a modular parameterization of multivariate distributions that decouples the modeling of marginals from the dependencies between them. The Gaussian Mixture Copula Model (GMCM) is a highly flexible copula that can model many kinds of multi-modal dependencies, as well as asymmetric and tail dependencies. They have been effectively used in clustering non-Gaussian data and in Reproducibility Analysis, a meta-analysis method designed to verify the reliability and consistency of multiple high-throughput genomic experiments. Parameter estimation for GMCM is challenging due to its intractable likelihood. The best previous methods maximize a proxy-likelihood through a Pseudo Expectation Maximization (PEM) algorithm. No guarantees of convergence or convergence to the correct parameters are provided by those methods. Using Automatic Differentiation (AD), a method, called AD-GMCM, is developed that can maximize the exact GMCM likelihood. Simulation studies and experiments on real data show that AD-GMCM finds more accurate parameter estimates than PEM and yields better performance in clustering and reproducibility analysis. The advantages of an AD-based approach to address problems related to monotonic increase of likelihood and parameter identifiability in GMCM are discussed. The two well-known cases of degeneracy of maximum likelihood in GMM that can lead to spurious clustering solutions are analyzed for GMCM as well. The analysis reveals that, unlike GMM, GMCM is not affected in one of the cases.

On Matter and Pressure Distribution in Nucleons

Matter and pressure distribution in hadrons are studied in a dual analytic model of generalized parton distributions with complex Regge trajectories. An original parametrization of the pressure distribution in the nucleon is proposed, ensuring its stability and compatible with the experimental data from the JLab accelerator.

Cosmic rays and non-thermal emission in simulated galaxies − I. Electron and proton spectra compared to Voyager-1 data

ABSTRACT Current-day cosmic ray (CR) propagation studies use static Milky Way models and fit parametrized source distributions to data. Instead, we use three-dimensional magnetohydrodynamic (MHD) simulations of isolated galaxies with the moving-mesh code arepo that self-consistently accounts for hydrodynamic effects of CR protons. In post-processing, we calculate their steady-state spectra, taking into account all relevant loss processes. We show that this steady-state assumption is well justified in the disc and generally for regions that emit non-thermal radio and gamma rays. Additionally, we model the spectra of primary electrons, accelerated by supernova remnants, and secondary electrons and positrons produced in hadronic CR proton interactions with the gas. We find that proton spectra above 10 GeV only weakly depend on galactic radius, while they acquire a radial dependence at lower energies due to Coulomb interactions. Radiative losses steepen the spectra of primary CR electrons in the central galactic regions, while diffusive losses dominate in the outskirts. Secondary electrons exhibit a steeper spectrum than primaries because they originate from the transported steeper CR proton spectra. Consistent with Voyager-1 and AMS-02 data, our models (i) show a turnover of proton spectra below GeV energies due to Coulomb interactions so that electrons start to dominate the total particle spectra and (ii) match the shape of the positron fraction up to 10 GeV. We conclude that our steady-state CR modelling in MHD CR galaxy simulations is sufficiently realistic to capture the dominant transport effects shaping their spectra, arguing for a full MHD treatment to accurately model CR transport in the future.