Approximate Normal Distribution Research Articles

On the logarithm of the Riemann zeta-function near the nontrivial zeros

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log|\zeta(\rho+z)|)$ and $(\arg\zeta(\rho+z)).$ Here $\rho=\frac12+i\gamma$ runs over the nontrivial zeros of the zeta-function, $0<\gamma \leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\ll 1/\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $\rho$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\log (|\zeta^\prime(\rho)|/\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\frac12\log\log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.

On the suitability of using the diagonal Gaussian approximation in the calculation of the magnitude-based likelihood function.

Statistical likelihood maximization is currently one of the main tools in computational procedures in biological crystallography. In these procedures, the likelihood function is calculated, as a rule, within the framework of a diagonal Gaussian approximation (DGA) of the joint probability distribution of the real and imaginary parts of a set of structure factors. This approximation assumes pairwise uncorrelated values of various structure-factor components. In this paper, exact formulas are derived for pairwise correlations of structure factors, and conditions under which these correlations can be considered to be negligible are discussed. It is shown that in the case where the probability distribution of the atomic coordinates is related to the region of the molecule or its domains, the correlation of the structure factors of reflections s and w is determined mostly by the magnitudes of the Fourier transform of the probability distribution calculated at the points 2s, 2w, s - w and s + w. However, in the case where the probability distribution describes small corrections to the coordinates of the existing preliminary atomic model, the correlation is determined by the values of the structure factors of the preliminary model that correspond to the 2s, 2w, s - w and s + w reflections rather than by the Fourier transform of the probability distribution. Test cases demonstrate that the practice of using the DGA for calculation of the likelihood when based on sets containing neighbouring reflections may be unjustified in some crystallographic applications, especially in single-particle studies.

Infrared image denoising based on the variance-stabilizing transform and the dual-domain filter

In the past decades, image denoising has been widely studied as a basic technology for image processing. However, most denoising methods are designed for Gaussian noise, while few researches focus on the suppression of mixed Poisson-Gaussian noise, which usually found in infrared images. In order to remove the mixed noise in infrared images, this paper proposes a denoising method based on the variance-stabilizing transform (VST) and the dual-domain filter (DDF). We transform the mixed noise data into an approximate Gaussian distribution with uniform variance through the VST and then adopt the improved DDF to denoise the transformed data. Finally, we apply the closed approximation inverse of the VST to the denoised data for the final denoising estimate. The denoising results of infrared images with different intensity all effectively suppress the mixed noise and retain abundant details. The quality and quantity comparisons with nine other methods reveal that our method can achieve a superior performance.

Establishment of the reference intervals of four biochemical parameters for liver function among adults in Changchun, China based on the indirect method

Objective To establish the reference intervals (RIs) of the four biochemical parameters for liver function [aspartate aminotransferase (AST), alanine aminotransferase (ALT), gamma-glutamyl transpeptidase (GGT), and alkaline phosphatase (ALP)] in adults by using the indirect method based on the data of subjects undergoing physical examination in laboratory information system (LIS). Methods With the help of the LIS, related data were collected from the healthy adults who underwent physical examination in Physical Examination Center of The First Hospital of Jilin University from October 2019 to October 2020. The Kolmogorov-Smirnov test was used to check the normal distribution of the original data; data with skewed distribution were transformed into data with approximate normal distribution using the BOX-COX method; the Turkey method was used to remove outliers; the Mann-Whitney U test or the Z-test was used for comparison between groups; the non-parametric method was used to calculate RIs. Results A total of 27 218 subjects were included after the removal of outliers. All four parameters showed age and sex differences. The RIs of AST were 14-39 U/L for male individuals aged 20-79 years, 12-32 U/L for female individuals aged 20-49 years, and 14-39 U/L for female individuals aged 50-79 years; the RIs of ALT were 10-71 U/L for male individuals aged 20-54 years, 10-49 U/L for male individuals aged 55-79 years, 7-43 U/L for female individuals aged 20-49 years, and 9-49 U/L for female individuals aged 50-79 years; the RIs of GGT were 11-70 U/L for male individuals aged 20-64 years, 10-64 U/L for male individuals aged 65-79 years, 6-45 U/L for female individuals aged 20-49 years, and 7-54 U/L for female individuals aged 50-79 years; the RIs of ALP were 38-96 U/L for male individuals aged 20-79 years, 33-89 U/L for female individuals aged 20-49 years, and 40-106 U/L for female individuals aged 50-79 years. The RIs of AST, ALT, and GGT established in this study were similar to those in the industry standards of China (relative deviation Conclusion The RIs of the four biochemical parameters for liver function established by the indirect method are relatively consistent with those established by the direct method, which holds promise for application in clinical laboratory.

MIxBN: library for learning Bayesian networks from mixed data

This paper describes a new library for learning Bayesian networks from data containing discrete and continuous variables (mixed data). In addition to the classical learning methods on discretized data, this library proposes its algorithm that allows structural learning and parameters learning from mixed data without discretization since data discretization leads to information loss. This algorithm based on mixed MI score function for structural learning, and also linear regression and Gaussian distribution approximation for parameters learning. The library also offers two algorithms for enumerating graph structures - the greedy Hill-Climbing algorithm and the evolutionary algorithm. Thus the key capabilities of the proposed library are as follows: (1) structural and parameters learning of a Bayesian network on discretized data, (2) structural and parameters learning of a Bayesian network on mixed data using the MI mixed score function and Gaussian approximation, (3) launching learning algorithms on one of two algorithms for enumerating graph structures - Hill-Climbing and the evolutionary algorithm. Since the need for mixed data representation comes from practical necessity, the advantages of our implementations are evaluated in the context of solving approximation and gap recovery problems on synthetic data and real datasets.

Transport of oil droplets from a jet in crossflow: Dispersion coefficients and Vortex trapping

Understanding the trajectory of oil droplets in crossflow jets is important to estimate the pathways of hydrocarbons and to plan countermeasures. We report experimental results of an oil jet with release velocity around 1.5 m/s in a crossflow of 0.3 m/s. The hydrodynamics of the jet obtained with the Large Eddy Simulation (LES) were used to predict the migration of the oil droplets. Two Lagrangian techniques were explored, one with the inertia of the droplet is considered and the other that treats the droplets as massless particles with rising velocities corresponding to their size. We did not note a large difference between the two approaches. The droplets showed stronger segregation in the vertical direction, which renders the usage of a Gaussian distribution approximation in the vertical inapplicable. The dispersion coefficient at each direction was computed for different-sized droplets. The eddy diffusivity computed based on Boussinesq gradient approximation using the LES data was compared with the dispersion coefficients obtained based on Lagrangian tracking. We also found that droplets 500 μm and larger escape the vortex while smaller ones get trapped within the vortex. A similar outcome was observed using a vortex trapping function based on inward–outward force balancing at the elevation of the vortex core. The counter-rotating vortex pair (CVP) altered the distribution of droplets of 1 mm and smaller significantly, and bimodal concentration distributions with peaks near the CVP vortex cores and minimum concentration near the center plane were obtained in the lateral–horizontal​ direction. Therefore, measurements of the oil droplet size distribution (DSD) in the center plane of crossflow jets could underestimate the number of small droplets in the whole plume.

Reliability modelling for multi-component systems subject to stochastic deterioration and generalized cumulative shock damages

For a complex system consisting of multiple components, it is often unrealistic that one type of environmental shocks affects all the components at the same time. Correspondingly, random shocks are categorized into several distinct sets according to their functions, attributes or sizes. This study develops generalized reliability models for multi-component systems, where each component is subject to two dependent competing failure processes, i.e., a soft failure process caused jointly by internal performance degradation and an incremental damage due to effective external shock sets, and a hard failure process caused by the same random shocks. A damage improvement coefficient and a damage aggravation coefficient are respectively introduced to extend the standard cumulative shock damage model into two more generalized shock cases. Analytical representations of system reliability for a series–parallel system and a parallel–series system are derived based on a gamma to normal distribution approximation approach. To quantitatively compare the effects of these two damage coefficients, a block replacement policy is further adopted by searching for the optimal replacement intervals with a Nelder–Mead downhill simplex method. Finally, an illustrative example of micro-electro-mechanical systems (MEMS) consisting of four silicon micro-mechanical resonators is provided to examine the effects of self-healing ability in the materials of polymer binder on system reliability and replacement period.

Quantitative Detection of Minor Defects in Metal Materials Based on Variation Coefficient of CT Image

In order to perform precise quantitative detection of internal minor defects in additive manufacturing of metal materials based on industrial CT, a quantitative detection method for minor defects in metal materials based on gray-scale variation coefficient of minor defects in CT image is presented. Firstly, based on the analysis of gray-scale features of minor defect regions in CT image, a minor defect image response model is constructed by combining the point spread function (PSF) and noise approximate Gaussian distribution, and the correspondence between the minor defects of different sizes and the local variation coefficient of CT image is derived, and a quantitative detection method based on the coefficient of variation of minor defect region in CT image is established for minor defects of AM metal materials. Industrial CT experiments were then conducted by processing a series of minor defect reference test blocks of different sizes with TC4 titanium. Experiment results show that the defect size and the coefficient of variation reflect a good linear relationship of monotonous increase, with correlation more than 0.996, and the quantitative results of minor defects based on the coefficient of variation method accord well with the measurement results of the microscope, with relative error controlled within 4%. Obviously, this method characterized by simple measurement process and high practical value has improved the quantitative precision of minor defects in industrial CT.

EDA on the asymptotic normality of the standardized sequential stopping times, Part-II: Distribution-free models

In sequential analysis, an experimenter gathers information regarding an unknown functional (parameter) by observing random samples in successive steps. We discuss a number of distribution-free scenarios under a variety of loss functions. The number of observations gathered upon termination is a positive integer-valued random variable, customarily denoted by N. Often, a standardized version of N would follow an approximate normal distribution in the asymptotic sense. We provide exploratory data analysis (EDA) with the help of a number of interesting illustrations. We do so via large-scale simulation studies to demonstrate broad applicability of the purely sequential methodologies along with the appropriateness of asymptotic normality of the standardized stopping variables as a practical and useful guideline.

Generalised discretisation of continuous‐time distributions

In this study, the definition of discretisation that was proposed recently for continuous-time distributions is made applicable not only to ordinary functions but to a variety of distributions including weak derivatives such that they could be viewed from a unified perspective under useful theorems. While it is not absolutely necessary to introduce distributions for discrete-time signals having finite values, it turns out that it is insightful to introduce discrete-time equivalents in appreciating their richness, which culminates into continuous-time distributions as the sampling-interval approaches zero. For instance, a discretisation of a derivative of a distribution can be found as a discrete derivative of a discretisation of a distribution. This is much easier than the traditional approach, where an ordinary function must first be found to approximate the derivative of a distribution. Simulations show that, by changing a single parameter of the proposed model, different types of signals that are similar to traditional ones developed separately by approximating distributions by ordinary functions, such as Dirichlet’ kernel, Gaussian distribution and sinc approximation, can be obtained.

Residual Learning With Multifactor Extreme Learning Machines for Waveheight Prediction

We explore the use of extreme learning machines (ELMs) for ocean waveheight prediction. Ocean waveheight variations are affected by multiple factors, such as air pressure, wind speed, and wind direction. To investigate the effects of these factors on waveheight prediction from a machine learning perspective, we develop a multifactor ELM (MFELM), which generalizes the vector inputs of existing ELMs into those of matrix inputs. The generalization enables the MFELM to admit the “multiple factor and multiple time” observed data (e.g., the air pressure, wind speed, and wind direction time sequences) for comprehensive waveheight prediction. Furthermore, we define the concept of numerical residuals in terms of the difference between observed data and waveheight numerical predictions, which are generated from equations of ocean dynamics. In contrast to the commonly used training schemes based on observed data, we propose to exploit numerical residuals for training the MFELM. The motivation for the training strategy arises from the observation that the numerical residuals follow an approximate Gaussian distribution and an ELM (or its generalization MFELM) can generate Gaussian random input layer weights and biases that have common distribution with the numerical residuals. This enhances the representational power of the ELM related methodologies in terms of effectively capturing numerical residual variations. The trained MFELM predicts future numerical residuals that are used for correcting numerical predictions, and finally result in more accurate predictions. Experimental results validate the effectiveness of our proposed method.

EEG Self-Adjusting Data Analysis Based on Optimized Sampling for Robot Control

Research on electroencephalography (EEG) signals and their data analysis have drawn much attention in recent years. Data mining techniques have been extensively applied as efficient solutions for non-invasive brain–computer interface (BCI) research. Previous research has indicated that human brains produce recognizable EEG signals associated with specific activities. This paper proposes an optimized data sampling model to identify the status of the human brain and further discover brain activity patterns. The sampling methods used in the proposed model include the segmented EEG graph using piecewise linear approximation (SEGPA) method, which incorporates optimized data sampling methods; and the EEG-based weighted network for EEG data analysis, which can be used for machinery control. The data sampling and segmentation techniques combine normal distribution approximation (NDA), Poisson distribution approximation (PDA), and related sampling methods. This research also proposes an efficient method for recognizing human thinking and brain signals with entropy-based frequent patterns (FPs). The obtained recognition system provides a foundation that could to be useful in machinery or robot control. The experimental results indicate that the NDA–PDA segments with less than 10% of the original data size can achieve 98% accuracy, as compared with original data sets. The FP method identifies more than 12 common patterns for EEG data analysis based on the optimized sampling methods.

Technology and threshold: An empirical study of road passenger transport emissions

This paper analyzes the influencing factors of road transport carbon emissions from the perspective of “human-vehicle-environment”. The investigation data of Chongqing Transportation Group passenger transport emissions is used to carry out stepwise regression analysis. Influencing factors are selected to determine the main factors affecting road transport emissions; meanwhile, the road passenger transport emissions model is established. The influence degree of the main factors and the stability of the model are studied, and the calculation effect of the actual emission test model is compared. The results show that: a. the main influencing factors are the driving mileage of the secondary trunk road, the service life of the vehicle, the load quality and the oil products, among which the driving age, the driving mileage of the main trunk road and the highway are not the main influencing factors. b. The regression model is tested for multicollinearity, equal variance and residual distribution. It is found that the regression model does not have multicollinearity, which meets the hypothesis of equal variance and residual approximate normal distribution. It indicates that the model was stable. c. The simulation results show that the calculated value and the actual value almost coincide, and the fitting effect is optimal.

Adaptation of robust Z’ factor for assay quality assessment in microelectrode array based screening using adult dorsal root ganglion neurons

BackgroundCell-based assays comprising primary sensory neurons cultured in vitro are an emerging tool for the screening and identification of potential analgesic compounds and chronic pain treatments. High-content screening (HCS) platforms for drug screening are characterized by a measure of assay quality indicator, such as the Z’-factor, which considers the signal dynamic range and data variation using control compounds only. Although widely accepted as a quality metric in high throughput screening (HTS), standard Z’-factor are not well-suited to indicate the quality of complex cell-based assays. New MethodThe present study describes a method to assess assay quality in the context of extracellular recordings from dorsal root ganglion (DRG) sensory neurons cultured on multi-well microelectrode arrays. Data transformations are applied to electrophysiological parameters, such as electrode and well spike rates, for valid normality assumptions and suitability for use as a sample signal. Importantly, using transformed well-wide metrics, a robust version of the Z’-factor was applied, based on the median and median absolute deviation, to indicate assay quality and assess hit identification of putative pharmacological compounds. ResultsApplication of appropriately scaled data and robust statistics ensured insensitivity to data variation and approximation of normal distribution. The use median and median absolute deviation of log transformed well spike rates in computing the Z’-factor revealed a value of 0.61, which is accepted as an “excellent assay.” Known antagonists of nociceptor-specific voltage-gated sodium ion channels were identified as true hits in the present assay format under both spontaneous and thermally stimulated conditions. Comparison with Existing MethodsThe present approach demonstrated a large signal dynamic range and reduced sensitivity to data variation compared to standard Z’-factor used widely in HTS. ConclusionOverall, the present study provides a statistical basis for the implementation of a HCS platform utilizing adult DRG neurons on microelectrode arrays.

A GIS-based 3D slope stability analysis method based on the assumed normal stress on the slip surface

The study proposes a geographic information systems (GIS)-based slope stability analysis method assuming a normal stress distribution acting on the slip surface. Compared with traditional methods, the three-dimensional (3D) safety factor acquired through this method will more closely approximate the actual value. First, a 3D slope stability analysis model is developed using grid column units, and the spatial expression of calculation parameters based on the grid column is given by the spatial analysis capability of GIS. Then, four equilibrium equations are derived under the limit equilibrium condition. The normal stress distribution acting on the slip surface is analyzed to construct a reasonable normal stress distribution approximation function. The 3D safety factor is obtained through the approximation function and the Mohr-Coulomb strength criterion. Moreover, we develop a GIS-based extension module which combines the grid-based data with the 3D slope stability analysis model. The accuracy and feasibility of the module are verified by three typical cases.

Describing the Pearson R distribution of aggregate data.

Ecological studies and epidemiology need to use group averaged data to make inferences about individual patterns. However, using correlations based on averages to estimate correlations of individual scores is subject to an "ecological fallacy". The purpose of this article is to create distributions of Pearson R correlation values computed from grouped averaged or aggregate data using Monte Carlo simulations and random sampling. We show that, as the group size increases, the distributions can be approximated by a generalized hypergeometric distribution. The expectation of the constructed distribution slightly underestimates the individual Pearson R value, but the difference becomes smaller as the number of groups increases. The approximate normal distribution resulting from Fisher's transformation can be used to build confidence intervals to approximate the Pearson R value based on individual scores from the Pearson R value based on the aggregated scores.

Enhancement of Spectral Efficiency and Power Budget in WDN-PON Employing LDPC-Coded Probabilistic Shaping PAM8

A $5\times60$ Gb/s low-density parity-check (LDPC) coded WDM-PON using 8-level pulse amplitude modulation (PAM8) signals with probabilistic shaping (PS) over 20-km standard single-mode fiber (SSMF) with direct detection is proposed and experimentally demonstrated. Each optical carrier at the optical line terminal (OLT) carries a 20-Gbaud probabilistically-shaped LDPC-coded PS-PAM8 signal from a conventional uniform PAM16 signal following a designed approximate Gaussian distribution (AGD). A LDPC-based bit-interleaved coded modulation scheme with iterative decoding (BICM-ID) is utilized to recover the inherently overlapped symbols due to the proposed PS mapping. Combining decision-directed least-mean-squares (DD-LMS) and constant modulus algorithm (CMA), a blind feedback pre-equalizer is adopted to reduce the frequency loss of the signal owing to imperfect electro-optical components in experiment. As for experimental verification and application, 1.48-dB superior receiver power sensitivity and 1.7-dB launch fiber power optimization are gained respectively by using LDPC-coded PS-PAM8 signals in WDM-PON, which can bring higher spectral efficiency and power loss budget for the optical distribution networks.

Gaussian approximation of the distribution of strongly repelling particles on the unit circle

В статье рассматривается модель $n$ упорядоченных сильно отталкивающихся частиц $\{e^{i \theta_j}\}_{j=0}^{n-1}$ с плотностью $$ p(\theta_0,…, \theta_{n-1})=Z_n^{-1} \exp \{-\frac{\beta}2\sum_{j \neq k} \sin^{-2} (\frac{\theta_j-\theta_k}2)\},\qquad \beta&gt;0. $$ Пусть $\theta_j=2\pi j/n+x_j/n^2+\mathrm{const}$ таково, что $\sum_{j=0}^{n-1}x_j=0$. Определим $\zeta_n(2\pi j/n)=x_j/\sqrt{n}$ и продолжим $\zeta_n$ кусочно линейным образом на $[0, 2 \pi]$. Доказывается функциональная сходимость $\zeta_n(t)$ к $\zeta(t)=\sqrt{2/\beta} \operatorname{Re} (\sum_{k=1}^{\infty} (1/k) e^{ikt} Z_k)$, где $Z_k$ - независимые одинаково распределенные комплексные стандартные гауссовские случайные величины.

A Quadratic Polynomial Receiving Scheme for Sinusoidal Signals Enhanced by Stochastic Resonance Under Color Noise

In this paper, a receiving scheme for intermediate frequency (IF) signals enhanced by stochastic resonance (SR) is proposed. The proposed scheme mitigates the reception failure of these signals, which can occur in radio and communication systems under extremely low signal-to-noise ratio (SNR). The SR mechanism for enhancing sinusoidal signals is analyzed. An analytic solution with time parameters of the Fokker-Planck Equation (FPE) is obtained by introducing the decision time from the non-autonomous FPE into an autonomous one. A quadratic polynomial receiving structure for sinusoidal signals enhanced by SR is proposed by comparing the characteristics of energy detection and matched filter detection. And the polynomial coefficients of the quadratic system are obtained by maximizing the deflection. Based on the idea of “the average of $N$ samples” and the assumption of Gaussian distribution approximation under the law of large numbers, a quadratic polynomial receiving scheme for sinusoidal signals enhanced by SR is proposed. The conclusions are as below: 1) when the noise intensity is constant, the smaller the correlation time, the bigger the local SNR around the IF frequency due to the better performance of the low-pass filter; 2) The error bit ratio of the quadratic polynomial receiver is less than $1\times 10^{-2}$ when $N =20$ and the SNR is above −14 dB, which can be applied to the military emergency communication under extremely low SNR. Experiment verifies the theory.

Estimation and Inference in Mixed Effect Regression Models Using Shape Constraints, with Application to Tree Height Estimation

SummaryEstimation of tree height given diameter is an important part of the forest inventory analysis of the US Forest Service. Existing methods use parametric models to estimate the curve. We propose a semiparametric model in which log (height) is a smooth, increasing and concave function of diameter, with a random-plot component and fixed effect covariates. Large sample properties and inference methods that work well in practice are derived. Proposed inference methods use approximate normal distributions for the fixed effects and a likelihood ratio test for the significance of the random effect. A closed form approximate prediction method is provided and overall it outperformed competitors for both a simulation and a real data application. The methods are implemented by the cgamm routine in the R package cgam and can be used for a wide range of mixed model applications.