Normalization Constant Research Articles

Refined saddlepoint approximations for the normalizing constant of the complex Bingham quart

The complex Bingham quartic (CBQ) distribution is defined on the unit complex sphere in and it is relevant for the statistical shape analysis of a -point landmark data in 2D. This extended the Fisher distribution on the unit spherical shape space 2 (1/2). The complex Bingham quartic (CBQ) distribution provides suitable shape parameters to comprise anisotropy. Under high concentrations, it looks like a multivariate Gaussian normal distribution but the main drawback of this planar shape distribution is that its normalizing constant does not have a simple closed explicit form representation. The present paper provides a modified approximation procedure for the indeterminate normalizing constant of the CBQ distribution based on saddlepoint approximations with a change of variable scheme. The modified saddlepoint approximations under a change of variable seem more precise as compared with the saddlepoint approximation without a change of variable approach.

Improving bridge estimators via f-GAN

Bridge sampling is a powerful Monte Carlo method for estimating ratios of normalizing constants. Various methods have been introduced to improve its efficiency. These methods aim to increase the overlap between the densities by applying appropriate transformations to them without changing their normalizing constants. In this paper, we first give a new estimator of the asymptotic relative mean square error (RMSE) of the optimal Bridge estimator by equivalently estimating an f-divergence between the two densities. We then utilize this framework and propose f-GAN-Bridge estimator (f-GB) based on a bijective transformation that maps one density to the other and minimizes the asymptotic RMSE of the optimal Bridge estimator with respect to the densities. This transformation is chosen by minimizing a specific f-divergence between the densities. We show f-GB is optimal in the sense that within any given set of candidate transformations, the f-GB estimator can asymptotically achieve an RMSE lower than or equal to that achieved by Bridge estimators based on any other transformed densities. Numerical experiments show that f-GB outperforms existing methods in simulated and real-world examples. In addition, we discuss how Bridge estimators naturally arise from the problem of f-divergence estimation.

Log-normalization constant estimation using the ensemble Kalman–Bucy filter with application to high-dimensional models

AbstractIn this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman–Bucy filter estimates based upon several nonlinear Kalman–Bucy diffusions. Using new conditional bias results for the mean of the aforementioned methods, we analyze the empirical log-scale normalization constants in terms of their $\mathbb{L}_n$ -errors and $\mathbb{L}_n$ -conditional bias. Depending on the type of nonlinear Kalman–Bucy diffusion, we show that these are bounded above by terms such as $\mathsf{C}(n)\left[t^{1/2}/N^{1/2} + t/N\right]$ or $\mathsf{C}(n)/N^{1/2}$ ( $\mathbb{L}_n$ -errors) and $\mathsf{C}(n)\left[t+t^{1/2}\right]/N$ or $\mathsf{C}(n)/N$ ( $\mathbb{L}_n$ -conditional bias), where t is the time horizon, N is the ensemble size, and $\mathsf{C}(n)$ is a constant that depends only on n, not on N or t. Finally, we use these results for online static parameter estimation for the above filtering models and implement the methodology for both linear and nonlinear models.

The Attraction Indian Buffet Distribution

We propose the attraction Indian buffet distribution (AIBD), a distribution for binary feature matrices influenced by pairwise similarity information. Binary feature matrices are used in Bayesian models to uncover latent variables (i.e., features) that explain observed data. The Indian buffet process (IBP) is a popular exchangeable prior distribution for latent feature matrices. In the presence of additional information, however, the exchangeability assumption is not reasonable or desirable. The AIBD can incorporate pairwise similarity information, yet it preserves many properties of the IBP, including the distribution of the total number of features. Thus, much of the interpretation and intuition that one has for the IBP directly carries over to the AIBD. A temperature parameter controls the degree to which the similarity information affects feature-sharing between observations. Unlike other nonexchangeable distributions for feature allocations, the probability mass function of the AIBD has a tractable normalizing constant, making posterior inference on hyperparameters straight-forward using standard MCMC methods. A novel posterior sampling algorithm is proposed for the IBP and the AIBD. We demonstrate the feasibility of the AIBD as a prior distribution in feature allocation models and compare the performance of competing methods in simulations and an application.

Centrality-Dependent Chemical Potentials of Light Hadrons and Quarks Based on pT Spectrum and Particle Yield Ratio in Au-Au Collisions at RHIC Energies

We analyze the pT spectra of π±, K±, p, and p¯ produced in different centralities’ Au-Au collisions at different collision energies from 7.7 to 62.4 GeV using a two-component Erlang distribution in the framework of a multi-source thermal model. The fitting results are consistent with the experimental data, and the yield ratios of negative to positive particles are obtained from the normalization constants. Based on the yield ratios, the chemical potentials of light hadrons (π, K, and p) and quarks (u, d, and s) are extracted. This study shows that only the yield ratios of p decrease with the increase in centrality. The logarithms of these yield ratios in the same centrality show obvious linear dependence on 1/sNN. The extracted chemical potentials (the absolute magnitude for π) of light hadrons and quarks decrease with the increase in energy. The curves of chemical potential vs. energy for all centralities derived from the linear fits of the logarithms of the yield ratio as a function of energy have their maximum (the absolute magnitude for π) at the same energy of 3.526 GeV, which is possibly the critical energy of phase transition from a liquid-like hadron state to a gas-like quark state in the collision system.

Integrable local and non-local vector Non-linear Schrödinger Equation with balanced loss and gain

The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-linearity are considered in presence of a linear term characterized, in general, by a non-hermitian matrix which under certain condition incorporates balanced loss and gain and a linear coupling between the complex fields of the governing non-linear equations. It is shown that the systems posses a Lax pair and an infinite number of conserved quantities and hence integrable. Apart from the particular form of the local and non-local reductions, the systems are integrable when the matrix representing the linear term is pseudo hermitian with respect to the hermitian matrix comprising the generic cubic non-linearity. The inverse scattering transformation method is employed to find exact soliton solutions for both the local and non-local cases. Further, it is shown that the presence of the linear term restricts the possible form of the norming constants and hence the polarization vector.

A Bayesian modified Ising model for identifying spatially variable genes from spatial transcriptomics data

A recent technology breakthrough in spatial molecular profiling (SMP) has enabled the comprehensive molecular characterizations of single cells while preserving spatial information. It provides new opportunities to delineate how cells from different origins form tissues with distinctive structures and functions. One immediate question in SMP data analysis is to identify genes whose expressions exhibit spatially correlated patterns, called spatially variable (SV) genes. Most current methods to identify SV genes are built upon the geostatistical model with Gaussian process to capture the spatial patterns. However, the Gaussian process models rely on ad hoc kernels that could limit the models' ability to identify complex spatial patterns. In order to overcome this challenge and capture more types of spatial patterns, we introduce a Bayesian approach to identify SV genes via a modified Ising model. The key idea is to use the energy interaction parameter of the Ising model to characterize spatial expression patterns. We use auxiliary variable Markov chain Monte Carlo algorithms to sample from the posterior distribution with an intractable normalizing constant in the model. Simulation studies using both simulated and synthetic data showed that the energy-based modeling approach led to higher accuracy in detecting SV genes than those kernel-based methods. When applied to two real spatial transcriptomics (ST) datasets, the proposed method discovered novel spatial patterns that shed light on the biological mechanisms. In summary, the proposed method presents a new perspective for analyzing ST data.

Microscopic description of the H(α,γ)2Li6 radiative capture reaction

The $^{2}\mathrm{H}(\ensuremath{\alpha},\ensuremath{\gamma})^{6}\mathrm{Li}$ radiative capture responsible for the $^{6}\mathrm{Li}$ production during the big-bang nucleosynthesis is comprehensively studied within a microscopic approach. The approach implements microscopically clustering aspects of nuclear structure and dynamics in an oscillator-basis representation. The total astrophysical $S$ factor of the reaction is calculated. All allowed partial electric quadrupole and magnetic dipole transitions between the $^{4}\mathrm{He}+^{2}\mathrm{H}$ continuum and the $^{6}\mathrm{Li}$ ground state are considered in the standard long-wavelength limit. Isospin-forbidden electric dipole transitions are taken into account in two ways. The first method is based on the expression for the electric dipole operator at the leading order of the long-wavelength approximation with the usage of the exact-mass prescription. In the second method, this operator is written at the first order beyond the leading-order approximation. Contributions of the transitions are compared to each other. The $^{4}\mathrm{He}+^{2}\mathrm{H}$ nuclear phase shifts for the initial channels of the considered reaction are computed. Important properties of the $^{6}\mathrm{Li}$ nucleus, such as the breakup threshold, the asymptotic normalization constants, and the electric quadrupole moment are also described. Deformation effects and their manifestations in $^{6}\mathrm{Li}$ are discussed. The obtained results are shown to be in good agreement with a large set of experimental data.

On a Parametrization of Partial-Sums Discrete Probability Distributions

For every discrete probability distribution, there is one and only one partial summation which leaves the distribution unchanged. This invariance property is reconsidered for distributions with one parameter. We show that if we change the parameter value in the function which defines the summation, two families of distributions can be observed. The first of them consists of distributions which are resistant to the change. For these distributions, the change of the parameter is reversed by the normalization constant, and the distributions remain unchanged. The other contains distributions sensitive to the change. Partial summations with the changed parameter value applied to sensitive distributions result in new distributions with two parameters. A necessary and sufficient condition for a distribution to be resistant to the parameter change is presented.

Resonant states 3+ and 2− of the Boron isotope 8B

The Jost functions, constructed by fitting available partial cross-sections for the elastic p 7Be scattering with [Formula: see text], are analytically continued to complex energies, where the resonances are located as their zeros. In addition to the resonance energies and widths, the residues of the [Formula: see text]-matrix at the corresponding poles, as well as the Asymptotic Normalization Constants (ANC) are determined. The fitting is done using the semi-analytic representation of the Jost function with proper analytic structure, defined on the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction.

Normalization of ZZ instanton amplitudes in minimal string theory

We use insights from string field theory to analyze and cure the divergences in the cylinder diagram in minimal string theory with both boundaries lying on a ZZ brane. We focus on theories with worldsheet matter consisting of the (2, p) minimal model plus Liouville theory, with total central charge 26, together with the usual bc-ghosts. The string field theory procedure gives a finite, purely imaginary normalization constant for non-perturbative effects in minimal string theory, or doubly non-perturbative effects in JT gravity. We find precise agreement with the prediction from the dual double-scaled one-matrix integral. We also make a few remarks about the extension of this result to the more general (p′, p) minimal string.

Effect of Charge Transfer and Non-Covalent Interactions of the Synthesized NLO Compound p-Nitrophenol Sodium-Bisulfate Using DFT Method

In the present work a new semiorganic single crystal p-nitrophenol sodium-bisulfate (PNSB) was synthesized and fully characterized by means of single crystal XRD, powder XRD, FT-IR, FT-Raman and UV-visible spectroscopic techniques. The optimized geometrical parameters were obtained at the DFT/B3LYP level of theory and correlate them with similar structures. Normal coordinate analysis, force constants, and potential energy distributions have been used to support the complete vibrational assignments for all vibrational modes. The computed IR and Raman frequencies correlate well with the experiments, as indicated by the correlation factor. The optimized molecular structure reveals the presence of C-H…O hydrogen bonding interaction. The stability of the molecule arising from hyper conjugative interaction and charge delocalization has been analyzed using natural bonding orbital analysis. NBO analysis proved the presence of intermolecular O-H…Na22 hydrogen bonds caused by the interaction of the lone pair of oxygen with the anti-bonding orbital. HOMO, LUMO and ESP analyses were done using DFT, whereas, the absorption band in the UV–vis spectrum was predicted and compared with the experimental data. ESP is used to identify the nucleophilic region, which is located around the sodium atom, and the electrophilic region, which is primarily located at the oxygen atoms of the PNSB molecule. The frontier orbital gap denotes the charge transfer interaction within the compound required for optical activity. Electron Localization Function, Localized Orbital Locator and Reduced Density Gradient analysis were discussed. In order to study the nonlinear optical activity of PNSB, the dipole moment, polarizability and fihyperpolarizabilities were computed. It is found that the calculated first order hyperpolarizability of PNSB is 51.75 times greater than that of urea.

Deep Investigation Into the Electrodeposition Model of Mg Battery Anode.

Analysis of nucleation/growth dynamics is important to understand the molecular mechanism on the electrode surface. The electrocrystallization mechanism of Mg anode in aqueous electrolyte was comprehensively investigated which can help us understand the surface discharge mechanism of Mg anode and provide a new theoretical idea for the development of high performance magnesium ion battery. The influence of applied potential signals on normal growth constant and active site numbers was studied using i-t transient curves. The dimensionless processed transient curves confirmed that the initial nucleation/growth process of Mg electrode in aqueous solution followed the diffusion-controlled three-dimensional instantaneous nucleation model.

Manakov system with parity symmetry on nonzero background and associated boundary value problems

We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-defined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark–bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x) and the ‘mixed parity’ cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions.

Model Based Screening Embedded Bayesian Variable Selection for Ultra-high Dimensional Settings

We develop a Bayesian variable selection method, called SVEN, based on a hierarchical Gaussian linear model with priors placed on the regression coefficients as well as on the model space. Sparsity is achieved by using degenerate spike priors on inactive variables, whereas Gaussian slab priors are placed on the coefficients for the important predictors making the posterior probability of a model available in explicit form (up to a normalizing constant). Embedding a unique model based screening and using fast Cholesky updates, SVEN produces a highly scalable computational framework to explore gigantic model spaces, rapidly identify the regions of high posterior probabilities and make fast inference and prediction. A temperature schedule is used to further mitigate multimodal posterior distributions. The temperature value is guided by our model selection consistency results which hold even when the norm of mean effects solely due to the unimportant variables diverges. An appealing byproduct of SVEN is the construction of novel model weight adjusted prediction intervals. The performance of SVEN is demonstrated through a number of simulation experiments and a real data example from a genome wide association study with over half a million markers. Supplementary materials for this article are available online.

Growth and Mortality as Causes of Variation in Metabolic Scaling Among Taxa and Taxonomic Levels.

Metabolic rate (MR) usually changes (scales) out of proportion to body mass (BM) as MR=aBMb, where a is a normalisation constant and b is the scaling exponent that reflects how steep this change is. This scaling relationship is fundamental to biology, but over a century of research has provided little consensus on the value of b, and why it appears to vary among taxa and taxonomic levels. By analysing published data on fish and taking an individual-based approach to metabolic scaling, I show that variation in growth of fish under naturally restricted food availability can explain variation in within-individual (ontogenetic) b for standard (maintenance) metabolic rate (SMR) of brown trout (Salmo trutta), with the fastest growers having the steepest metabolic scaling (b≈1). Moreover, I show that within-individual b can vary much more widely than previously assumed from work on different individuals or different species, from -1 to 1 for SMR among individual brown trout. The negative scaling of SMR for some individuals was caused by reductions in metabolic rate in a food limited environment, likely to maintain positive growth. This resulted in a mean within-individual b for SMR that was significantly lower than the across-individual ("static") b, a difference that also existed for another species, cunner (Tautogolabrus adspersus). Interestingly, the wide variation in ontogenetic b for SMR among individual brown trout did not exist for maximum (active) metabolic rate (MMR) of the same fish, showing that these two key metabolic traits (SMR and MMR) can scale independently of one another. I also show that across-species ("evolutionary") b for SMR of 134 fishes is significantly steeper (b approaching 1) than the mean ontogenetic b for the brown trout and cunner. Based on these interesting findings, I hypothesise that evolutionary and static metabolic scaling can be systematically different from ontogenetic scaling, and that the steeper evolutionary than ontogenetic scaling for fishes arises as a by-product of natural selection for fast-growing individuals with steep metabolic scaling (b≈1) early in life, where size-selective mortality is high for fishes. I support this by showing that b for SMR tends to increase with natural mortality rates of fish larvae within taxa.

Scale evolution of T-odd gluon TMDs at small x

We study the energy or TMD evolution of the three leading twist dipole type T-odd gluon TMDs inside a transversely polarized nucleon, all of which at small $x$ dynamically originate from the spin dependent odderon. Their energy dependence presents a unique opportunity to study the polarization dependent TMD evolution in the small-$x$ region, where the distributions are identical up to a normalization constant at tree level. We further propose to study their evolution via azimuthal asymmetries in virtual photon-jet production in polarized proton-proton collisions at RHIC. We present model predictions for the asymmetries as functions of the large jet or photon transverse momentum and $Q^2$ which set the hard scales in this process.

Index spaces and standard indices in metric modelling

We analyze the basic structure of certain metric models, which are constituted by an index I acting on a metric space (D; d) representing a relevant property of the elements of D. We call such a structure (D; d; I) an index space and define on it normalization and consistency constants that measure to what extent I is compatible with the metric d. The “best” indices are those with such constants equal to 1 (standard indices), and we show an approximation method for other indices using them. With the help of Lipschitz extensions, we show how to apply these tools: a new model for the triage process in the emergency department of a hospital is presented.

Extremal properties of the beta-normal distribution

Asymptotic behaviors of the extremes of the beta-normal distribution are derived. The higher-order asymptotic expansions of the probability density and cumulative distribution functions for the maximum are given under an optimal normalizing constants. In particular, the associated rates of convergence are explicitly calculated.

Varying Relationship Between Vascular Plant Leaf Area and Leaf Biomass Along an Elevational Gradient on the Eastern Qinghai-Tibet Plateau.

The altitudinal gradient is one of the driving factors leading to leaf trait variation. It is crucial to understand the response and adaptation strategies of plants to explore the variation of leaf traits and their scaling relationship along the altitudinal gradient. We measured six main leaf traits of 257 woody species at 26 altitudes ranging from 1,050 to 3,500 m within the eastern Qinghai-Tibet Plateau and analyzed the scaling relationships among leaf fresh weight, leaf dry weight, and leaf area. The results showed that leaf dry weight increased significantly with elevation, while leaf fresh weight and leaf area showed a unimodal change. Leaf dry weight and fresh weight showed an allometric relationship, and leaf fresh weight increased faster than leaf dry weight. The scaling exponent of leaf area and leaf fresh weight (or dry weight) was significantly greater than 1, indicating that there have increasing returns for pooled data. For α and normalization constants (β), only β of leaf area vs. leaf fresh weight (or dry weight) had significantly increased with altitude. All three paired traits had positive linear relationships between α and β. Our findings suggest that plants adapt to altitudinal gradient by changing leaf area and biomass investment and coordinating scaling relationships among traits. But leaf traits variation had a minor effect on scaling exponent.