Standard Normal Distribution Research Articles

Gaussian approximation and moderate deviations of Poisson shot noises with application to compound generalized Hawkes processes

Abstract In this article, we give explicit bounds on the Wasserstein and Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard normal distribution, using the results of Last et al. (Prob. Theory Relat. Fields165, 2016). Relying on the theory developed by Saulis and Statulevicius in Limit Theorems for Large Deviations (Kluwer, 1991) and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities, and normal approximation bounds with Cramér correction terms for the same variables. The aforementioned results are then applied to Poisson shot noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results of Hillairet et al. (ALEA19, 2022) and Khabou et al. (J. Theoret. Prob.37, 2024) regarding the normal approximation and those of Zhu (Statist. Prob. Lett.83, 2013) for moderate deviations.

Generalised drought index: a novel multi-scale daily approach for drought assessment

Abstract. Drought is a complex climatic phenomenon characterised by water scarcity and is recognised as the most widespread and insidious natural hazard, posing significant challenges to ecosystems and human society. In this study, we propose a new daily based index for characterising droughts, which involves standardising precipitation and/or precipitation minus potential evapotranspiration (PET) data. The new index proposed here, the generalised drought index (GDI), is computed for the entire period available from the Iberian Gridded Dataset (1971 to 2015). Comparative assessments are conducted against the daily Standardised Precipitation Index (SPI), the Standardised Precipitation Evapotranspiration Index (SPEI), and a simple Z-Score standardisation of climatic variables. Seven different accumulation periods are considered (7, 15, 30, 90, 180, 360, and 720 d) with three drought levels: moderate, severe, and extreme. The evaluation focuses mainly on the direct comparison amongst indices in terms of their ability to conform to the standard normal distribution, added value assessment using the distribution added value (DAV), and a simple bias difference for drought characteristics. Results reveal that the GDI, together with the SPI and SPEI, follows the standard normal distribution. In contrast, the Z-Score index depends on the original distribution of the data. The daily time step of all indices allows the characterisation of flash droughts, with the GDI demonstrating added value when compared to the SPI and SPEI for the shorter and longer accumulations, with a positive DAV up to 35 %. Compared to the Z-Score, the GDI shows expected greater gains, particularly at lower accumulation periods, with the DAV reaching 100 %. Furthermore, the spatial extent of drought for the 2004–2005 event is assessed. All three indices generally provide similar representations, except for the Z-Score, which exhibits limitations in capturing extreme drought events at lower accumulation periods. Overall, the findings suggest that the new index offers improved performance and comparatively adds value to similar indices with a daily time step.

Parameter-adaptive variational autoencoder for linear/nonlinear blind source separation

AbstractBlind source separation (BSS) serves as an important technique in the field of structural health monitoring (SHM), particularly for solving modal decomposition tasks. This study proposes a novel approach to both linear and nonlinear BSS problems in the Variational Autoencoder (VAE) framework, where the encoding and decoding processes of VAE are interpreted as procedures for inferring sources from observations and remixing these sources, respectively. In this way, the distribution of latent variables inferred by VAE is equivalent to the distribution of sources. We make improvements to the vanilla VAE to augment its effectiveness for BSS. First, we substitute standard normal distributions with trainable Gaussian processes (GP) as priors for latent variables and implement an exponential function as the activation function for adaptive parameters in the GP kernel functions. While the form of the priors is set as GP, the parameters of their kernel functions are not fixed but automatically converge to suitable values during the model training process. Additionally, a hyperparameter is introduced to balance the terms in the loss function. The proposed method is referred to as parameter-adaptive VAE (PAVAE). Then, upon different assumptions of the variances of sources, the proposed PAVAE is branched into two types: homoscedastic PAVAE (Ho-PAVAE) and heteroscedastic PAVAE (He-PAVAE). Through numerical and laboratory experiments, we demonstrate the effectiveness of the proposed method in solving BSS problems and their potential to underpin future research in SHM.

An Adaptive Kernel-Based Structural Change Test for Copulas

This paper proposes a structural change test for copula models based on the kernel smoothing method. The proposed approach enables adaptable estimation of the dynamic marginal distributions, either parametrically or semi-parametrically. The test statistic is formulated via the weighted quadratic distance between the local smoothing copula and the empirical copula function, utilizing pseudo-observations of marginal distributions. The test statistic is pivotal with an asymptotic standard Normal distribution, irrespective of the marginal distributions, parameters, and estimations, and is consistent against a wide range of smoothly transitioning structural changes as well as abrupt structural breaks for copula models. Monte Carlo simulations show that the test performs well in finite samples and outperforms existing tests in the case of periodic changes.

Crystal Structure Prediction Using Generative Adversarial Network with Data-Driven Latent Space Fusion Strategy.

Crystal structure prediction (CSP) is an important field of material design. Herein, we propose a novel generative adversarial network model, guided by a data-driven approach and incorporating the real physical structure of crystals, to address the complexity of high-dimensional data and improve prediction accuracy in materials science. The model, termed GAN-DDLSF, introduces a novel sampling method called data-driven latent space fusion (DDLSF), which aims to optimize the latent space of generative adversarial networks (GANs) by combining the statistical properties of real data with a standard Gaussian distribution, effectively mitigating the "mode collapse" problem prevalent in GANs. Our approach introduces a more refined generation mechanism specifically for binary crystal structures such as gallium nitride (GaN). By optimizing for the specific crystallographic features of GaN while maintaining structural rationality, we achieve higher precision and efficiency in predicting and designing structures for this particular material system. The model generates 9321 GaN binary crystal structures, with 16.59% reaching a stable state and 24.21% found to be metastable. These results can significantly enhance the accuracy of crystal structure predictions and provide valuable insights into the potential of the GAN-DDLSF approach for the discovery and design of binary, ternary, and multinary materials, offering new perspectives and methods for materials science research and applications.

Constrained Bayesian Method for Testing Equi-Correlation Coefficient

The problem of testing the equi-correlation coefficient of a standard symmetric multivariate normal distribution is considered. Constrained Bayesian and classical Bayes methods, using the maximum likelihood estimation and Stein’s approach, are examined. For the investigation of the obtained theoretical results and choosing the best among them, different practical examples are analyzed. The simulation results showed that the constrained Bayesian method (CBM) using Stein’s approach has the advantage of making decisions with higher reliability for testing hypotheses concerning the equi-correlation coefficient than the Bayes method. Also, the use of this approach with the probability distribution of linear combinations of chi-square random variables gives better results compared to that of using the integrated probability distributions in terms of providing both the necessary precisions as well as convenience of implementation in practice. Recommendations towards the use of the proposed methods for solving practical problems are given.

Ultrasonic characterization of defects in polycrystalline materials based on TFM image reconstruction using denoising diffusion probabilistic models

This paper presents a refined ultrasonic total focusing method (TFM) for accurately characterizing key defect parameters including size, orientation, and location in complex materials. The proposed approach aims to accurately reconstruct the defect shape in images compromised by grain interference. First, a TFM image database was generated with varying defect parameters and grain noise levels by adopting a custom forward modeling process based on superposition of defect and grain scattering data. Second, a new architecture called TFM-DDPM was then proposed which incorporates TFM and the conditional denoising diffusion probabilistic model (DDPM). Third, pixel-level uncertainty can be quantified through repeated sampling of noise which follows a standard normal distribution. An acceptance criterion for the reconstruction was further established by analyzing the quantified uncertainty. In simulation, the median Dice coefficient between the de-noised TFMs of 30° cracks and their corresponding ground truth images increased from 0.5585 to 0.9137 using the proposed approach, and the median IoU metric rose from 0.3875 to 0.8408. By applying the 6-dB drop approach to the reconstructed images, the root mean squared errors (RMSEs) of size, angle and location were decreased by 93.59%, 86.90% and 86.25%, respectively. It is also demonstrated that our method can reject 69.19% of incorrect characterization results caused by high levels of noise and/or images with steeply inclined (e.g., 45°) cracks, while accepting 97.29% accurate results. Experimental results obtained on nickel-based alloy K403 specimens also exhibited significant improvements. For 2–3 mm cracks, the average sizing error reduced from 0.46 mm to 0.13 mm, and the error in crack angle decreased from 28.63° to 1.50°. Comparable performance enhancements were observed for 4–5 mm cracks using the proposed approach.

Explicit Formulas of Average Run Length for Triple Moving Average Control Chart to Monitor Changes in Mean Parameter of Normal and Non-normal Process

In this research, a formula is presented for calculating the average run length of a triple moving average (TMA) control chart, which is used to measure control chart performance and determine control chart width. To apply appropriately to data in various processes, a TMA control chart is used to detect small changes in a process by finding the average value of the data over a specified period of time (w). However, determining measure control chart performance and control chart width is difficult because the formula for calculation is not created. The performance of the control charts is measured using the standard normal distribution probability properties and compared with the moving average (MA) control charts and the double moving average (DMA) control charts. The analysis results show that TMA is very effective at detecting small changes, when w is set to have a small value. The study determines that the data within the process are normally and non-normally distributed. Detecting changes on real data by TMA charts has the best detection performance compared to MA and DMA.

Structured orthogonal random features based on DCT for kernel approximation

Random Fourier features are a popular technique used to improve nonlinear kernel methods in large-scale problems. Recent studies have shown that replacing random Gaussian matrices of random feature maps with appropriately scaled random orthogonal matrices, such as SORF, can significantly improve kernel approximation performance. However, the SORF method theoretically requires zero-padding for datasets whose input feature dimension does not meet d=2p for some p∈N+, resulting in increased memory and computation time. To address this limitation, we propose a new structured orthogonal random features method based on discrete cosine transform (SORF-DCT) to approximate Gaussian kernel functions. The SORF-DCT method does not require the input feature dimension to be d=2p. The key to SORF-DCT is that we combine the DCT matrix with diagonal ”sign-flipping” matrices and scale them appropriately using a diagonal matrix whose diagonal elements follow the chi distribution, resulting in a structured orthogonal matrix whose elements follow the standard Gaussian distribution instead of a random Gaussian matrix. SORF-DCT generates D-dimensional structured orthogonal random features in O(Dlogd) time and O(D) memory by employing fast discrete cosine transform, where d and D represent the input feature dimension and expansion dimension, respectively. We prove that SORF-DCT is an unbiased estimator of the Gaussian kernel function and analyze the concentration error bounds. Experimental results on eleven benchmark datasets show that SORF-DCT achieves lower kernel approximation errors, requires less time, and has comparable nonlinear learning capabilities.

INFERENCE IN MILDLY EXPLOSIVE AUTOREGRESSIONS UNDER UNCONDITIONAL HETEROSKEDASTICITY

Mildly explosive autoregressions have been extensively employed in recent theoretical and applied econometric work to model the phenomenon of asset market bubbles. An important issue in this context concerns the construction of confidence intervals for the autoregressive parameter that represents the degree of explosiveness. Existing studies rely on intervals that are justified only under conditional homoskedasticity/heteroskedasticity. This paper studies the problem of constructing asymptotically valid confidence intervals in a mildly explosive autoregression where the innovations are allowed to be unconditionally heteroskedastic. The assumed variance process is general and can accommodate both deterministic and stochastic volatility specifications commonly adopted in the literature. Within this framework, we show that the standard heteroskedasticity- and autocorrelation-consistent estimate of the long-run variance converges in distribution to a nonstandard random variable that depends on nuisance parameters. Notwithstanding this result, the corresponding t-statistic is shown to still possess a standard normal limit distribution. To improve the quality of inference in small samples, we propose a dependent wild bootstrap-t procedure and establish its asymptotic validity under relatively weak conditions. Monte Carlo simulations demonstrate that our recommended approach performs favorably in finite samples relative to existing methods across a wide range of volatility specifications. Applications to international stock price indices and U.S. house prices illustrate the relevance of the advocated method in practice.

Implications of data topology for deep generative models

Many deep generative models, such as variational autoencoders (VAEs) and generative adversarial networks (GANs), learn an immersion mapping from a standard normal distribution in a low-dimensional latent space into a higher-dimensional data space. As such, these mappings are only capable of producing simple data topologies, i.e., those equivalent to an immersion of Euclidean space. In this work, we demonstrate the limitations of such latent space generative models when trained on data distributions with non-trivial topologies. We do this by training these models on synthetic image datasets with known topologies (spheres, torii, etc.). We then show how this results in failures of both data generation as well as data interpolation. Next, we compare this behavior to two classes of deep generative models that in principle allow for more complex data topologies. First, we look at chart autoencoders (CAEs), which construct a smooth data manifold from multiple latent space chart mappings. Second, we explore score-based models, e.g., denoising diffusion probabilistic models, which estimate gradients of the data distribution without resorting to an explicit mapping to a latent space. Our results show that these models do demonstrate improved ability over latent space models in modeling data distributions with complex topologies, however, challenges still remain.

Plant biodiversity assessment of locally cultivated crops and household food security in Northern Iran

BackgroundLocally cultivated crops play an important role in the food security. The biodiversity of these crops can be important for the livelihood of households in current and future generations. This research aims to study the socio-economic, agronomic, and ecological aspects which contribute to the maintenance of crop diversification and food security in the study area.MethodBased on latitude and topography 10 villages were randomly selected, which was done in a study with 227 household farm managers. Food security was evaluated based on the species richness, and area under cultivation of food groups, and the probit logistic regression model was used for evaluation. Probit regression, also known as probit models, is used when the output or dependent variable of the model is bivariate. In inverse probit models, the standard normal distribution is modeled as a linear combination of predictor variables. In this situation, the application of normal regression methods is not applicable because the distribution occurred in two levels. In this study, it is assumed that the protection of biodiversity of local plants is related to food security, and for this reason, we have used this model.ResultsThe findings showed that socio-economic status of different households and farms in this coastal area is one of the key factors affecting the biodiversity of locally cultivated crops. Local cultivars, especially those of rice, are resistant to most environmental factors and contribute to family food security. Food security and rice abundance are significantly correlated in all of the research area’s communities.ConclusionsTo ensure the sustainability, and health of production, and to ensure food security, planting various crops in this study area is recommended. Future research is needed to focus on solutions and technologies rich in diversity tailored to the socio-economic and environmental factors of locally cultivated crops.

Central Limit Theorem and Law of Large Numbers Analogues for the Total Progeny in the Q-Processes

In this paper we are dealing with population size growth systems called Q-processes. The class of trajectories of these systems is a subset of the family of all possible trajectories of the ordinary Galton-Watson branching system, provided that they do not decay in the remote future. We observe the total progeny of a single founder-individual, generated by the reproduction law of the Q-process up to time n. By analogy with branching systems models, this variable is of great interest in studying the deep properties of the Q-process. Our main results are analogues of Central Limit Theorem and Law of Large Numbers for Sn, denoting the total progeny of a single founder-individual, generated by the reproduction law of the Q-process up to time n. We find that the total progeny as a random variable approximates the standard normal distribution function under a second moment assumption for the initial Galton-Watson system offspring law. We estimate the speed rate of this approximation. We also prove an analogue of the law of large numbers with an estimate of the approximation rate to the degenerate distribution.

Directed likelihood statistic to test the concentration parameter in von Mises regressions

We derive explicit expressions for the directed likelihood statistic and its modified version for testing several hypotheses on the concentration parameter in the von Mises regression model. We verify that the standard normal distribution gives a poor approximation to the true distribution of the usual directed likelihood statistic to test the concentration parameter, while its modified version leads to very accurate inference even for very small samples. An empirical application is considered for illustrative purposes.

Reliability measurements of the furniture frames with selected joint types

The reliability was evaluated for frame construction with mortise and tenon (MT), dowel, and non-glued and glued staple joints made of beech wood. The moment capacities of the T-shaped joints were determined under static vertical loads. The MT joint had an average moment capacity of 204 Nm, followed by dowel joints (154 Nm). In staple joints, joint strength increased after gluing. A three-seat sofa frame was defined and theoretically subjected to loads on arm rails, side rails, and back posts. Moment levels on the joint of the frame were obtained by using the stiffness method. To measure the reliability, these moment levels were assumed to be normally distributed with a coefficient of variation of 10%; accordingly, the normal distribution of the data sets was transformed into a normal standard distribution, and then, the reliability of each joint on the frame was obtained by using probabilistic approaches. MT joints were found to have the highest moment capacity and, correspondingly, the highest reliability (99.99%). Gluing the staple joints increased the strength, so their reliabilities were increased. In designing frame construction, the critical joints should be determined, and then, the joinery system with the higher reliability should be used.

Machine learning and traditional approaches in shear reliability of steel fiber reinforced concrete beams

In the field of structural engineering, the exploration of steel fibre reinforced concrete (SFRC) beams has recently intensified, particularly due to their improved tension and shear performance of structure. This study pioneers a novel reliability analysis of shear capacity predictions for SFRC beams, distinctively classifying the datasets into high-strength (HSFRC) and normal-strength (NSFRC) categories. A comprehensive database of 142 HSFRC and 265 NSFRC beams serves as the foundation for this analysis, which critically examines the standard Gaussian distribution in shear design models and proposes the Lognormal and Weibull distributions as more precise alternatives. Employing advanced First-order (FORM) and Second-order Reliability Methods (SORM), the study covers a broad spectrum of load conditions, including dead, live, snow, wind, and seismic loads, to evaluate various empirical, semi-empirical and machine learning proposed shear capacity prediction formulas. One of the key innovations of this research is the development of differentiated resistance coefficients for various risk levels in the reliability analysis, allowing future structural designers to tailor their designs according to specific risk profiles. This approach significantly enhances the balance between economic efficiency and structural safety based on the evolution of different target reliability indexes. The study reveals that existing design equations for SFRC beams generally lean towards conservatism. This study found that formulas derived from machine learning exhibited superior prediction ability compared to traditional theoretically derived regression formulas. When compared the proposed machine learning formulas, the Tarawneh's formula demonstrated better prediction ability than the Kara's formula when applied to larger datasets. However, high predictive power does not necessarily equate to high reliability. Machine learning formulas prioritise predictive accuracy, often at the expense of insufficient redundancy for ensuring safety. Overall, it introduces Kara's formula for NSFRC beams, which stands out with its superior predictive performance, offering an optimal balance of safety and cost-efficiency. Ashour's formula is also identified as a more effective and safer option for HSFRC beams. Complementing these findings, the extensive sensitivity analysis of the collected data not only confirms the robustness of the conclusions but also prepares for the integration of broader datasets in the future.

Способи побудови розподілу максимальної відстані між випадковими нормальними точками на площині

Many practical problems call for constructing the maximum interpoint distance distribution for random pints in a plane. In the literature, the case of a great number of points is considered, for which an asymptotic distribution is determined. This paper addresses the problem of constructing the maximum interpoint distance distribution for a small number of random points in a plane whose coordinates are independent random quantities that obey the standard normal distribution. The special case of three random points in a plane is considered as the basic one, for which three ways to construct the maximum interpoint distance distribution are studied. The first way is to construct the distribution function from geometrical considerations. To do this, the loci of three points are considered from the condition that the maximum distance between them shall not exceed a certain value. The position of the third point in the plane is determined relative to the two other points: the leftmost and the lowermost one. In this case, the construction of the distribution function involves the successive evaluation of several integrals using numerical methods. The obtained results are in good agreement with those of statistical simulation. The second way is based on studying the distance between pairs of random normal points in a plane. Taken separately, the distances between each pair of random normal points obey one-dimensional Rayleigh distributions, but in the aggregate they prove to be correlated because they are determined from the same pint coordinates. A joint distribution of the squared distances between three points is constructed using the three-dimensional Moran-Downton distribution. Using it, a distribution function of the squared maximum distance between three random normal points, which is identical with the maximum interpoint distance distribution, is obtained. It is found that for small values it underestimates the actual probability of the maximum distance not exceeding a certain value. For great distance values, the above probabilities coincide. The third way uses the Rice distribution (a generalization of the Rayleigh distribution) to approximate the unknown maximum interpoint distance distribution for three random normal points in a plane. The Rice distribution parameters found by the least-squares method are in good agreement with those obtained by statistical simulation. The results for three random normal points are generalized to a greater number of points (up to 30). It is shown that in this case the third way is most efficient.

Investigation on vibration characteristics induced by gas–liquid two-phase flows in a horizontal pipe

The intrinsic characteristics of multiphase flow inevitably result in vibrations within the pipe structure. The evolution of vibration characteristics is crucial for monitoring the stability of the multiphase flow pipelines. However, under high liquid velocity condition, the influence of flow pattern on the gas–liquid two-phase flow-induced vibration (GTFIV) of the pipe remains unclear. In this paper, the vibration signals in the horizontal pipe under different flow patterns at high liquid velocity are measured by a vibration test system. Synchronously, images inside the pipe are captured to illustrate the flow mechanism. The results show that the α stable distribution model holds higher fitting accuracy than the standard normal distribution for the GTFIV. The random behavior of small-scale bubbles determines the structural richness of the GTFIV. The multifractal characteristics of the GTFIV are strongly dependent on the flow pattern. The formation of the gas plug leads to significant unevenness in the fractal structure of the GTFIV. The multifractal strength of the GTFIV gradually increases as the flow pattern transforms from dispersed bubble flow to slug flow. Some multifractal spectrum parameters of the vibration signal can be applied for the identification of flow patterns in the pipe.

Establishment of a Comprehensive Evaluation System for Low Nitrogen and Screening of Nitrogen-Efficient Germplasm in Peanut

Screening for nitrogen (N)-efficient germplasm to achieve high yield and high N efficiency is an important strategy to enhance the sustainability of modern agriculture. In this study, 127 peanut (Arachis hypogaea L.) germplasm resources were comprehensively evaluated by seedling hydroponics and field. At the seedling stage, with the range of low-nitrogen screening concentrations gradually narrowed through a comprehensive membership function analysis, standard normal distribution test, and variance analysis, we found that 0.15 mM N for 24 days could be the optimal condition for evaluating the N efficiency of peanuts. Through principal component analysis and correlation analysis, dry matter weight, root/shoot ratio, N content, N accumulation, N-use efficiency, and N use index were considered to be the N efficiency parameters, and a regression mathematical model was established accordingly. In the field, peanut genotypes that differ in resistance to low-nitrogen stress were evaluated by a yield nitrogen efficiency index under normal nitrogen and no nitrogen applications to verify the results at the seedling stage. Based on the multiple phenotypic analysis, N-efficient and N-inefficient peanut genotypes among germplasm were screened, and a comprehensive evaluation system was established to provide the theoretical basis for peanut breeding and cultivation techniques.

Research on Binary Black Hole Systems by Analysis of Gravitational Waves

The mass ratio, effective spin, and chirp mass are three important properties to describe the binary systems of black holes. The gravitational wave sources can be a breakthrough in the investigation of black holes. The formation of black holes is a significant problem to solve as the electromagnetic waves (light) cannot escape from the black hole, and the gravitational waves can be emitted when the binary systems of black holes or neutron stars merge. Based on the detection of gravitational waves, the process of the formation of black holes can be traced and studied in the binary black holes and primordial black holes formed in the early universe. The mass ratio distribution is plotted to investigate the difference between the binary black hole and neutron star black hole and the mass ratio which makes it easier for the merger of the black holes to happen. The investigation of the effective spin helps classify the model of the formation of black holes as there are five models of the black hole. The statistical model can be tested with the chosen data, and the cumulative standard Gaussian distribution is used in this paper. The result states that the model is not suitable for the chosen data and another statistical model should be introduced to analyse the effective spin. These parameters are all important for determining the stage and formation process of the binary systems.