The unique demands for analyzing nonlinear time series produced by complex systems have generated a paradigm shift in statistical theory and application in much the same way as nonlinear dynamics have augmented the understanding of specific phenomena in the life and social sciences. Topics covered include: the statistical computation of the fractal dimensions, ergodicity, strategic use of nonlinear model libraries, identifying oscillators, state space analysis and entropy, time delays and the production of emergents, and quantum computing of fractal images. Substantive applications include US unemployment, political affiliation in the Netherlands, bipolar disorder, biomechanics, heart rate complexity, Bitcoin and other market prices, temperature anomalies and climate change, and economic growth.

Context. The relevance of the topic lies in the need to improve radio direction finders to increase accuracy, resistance to interference and adaptation to changing operating conditions. Modern scientific achievements require the development of methods for statistical synthesis and analysis of stochastic signal processing in multi-antenna systems, which will allow to take into account the uncertainty of real conditions. It is important to expand the capabilities of such systems for use in radar, radio navigation, communications and other industries. This will facilitate the creation of new approaches for direction finding of unknown signal sources in complex operating scenarios.Objective. The study is based on improving the measurements of the angular position accuracy of radio sources of stochasticsignals.Method. The approach is is based on the statistical theory of optimization of radio remote sensing and radar systems. Signal and noise models are constructed for stochastic signal sources, and the likelihood functional in the spectral domain is formulated, taking into account the structure of inverse correlation matrices. The Cramer-Rao inequality is used to determine the limiting errors of estimation of the angular position of the radio source.Results. For the first time the approach to statistical optimization of the structure of multi-antenna radio systems for directionfinding of stochastic radiation sources is theoretically justified, allowing to take into account the spatial orientation, antenna array geometry and radiation pattern. An optimal method of processing the observation equations for estimating the angular position of stochastic signal sources is constructed. A generalized structure of a single-antenna direction finder containing a matched filter, a decoherence filter and a digital calculator is proposed. It is proved that the use of decorelating processing allows to increase the estimation accuracy by increasing the number of independent signal samples. Analytical expressions for estimation and limiting errors, which take into account the spectrum width and directional pattern parameters, are obtained.Conclusions. This paper presents the latest theoretical foundations for the synthesis of radio direction finders of arbitrary configuration, which take into account the variety of radiation pattern shapes, spatial location and orientation of direction finders. The developed models of signals and noise using the maximum likelihood function criterion for the first time allow solving optimisation problems of synthesis with consideration the physical content consideration of correlation matrices. The obtained results are confirmed by solving the problem of measuring the radiation source angular position, which proves the proposed approaches effectiveness.

Quantum materials exhibit a rich dynamic of physical parameters, which, when combined, can lead to entirely different behaviors. These parameters constantly compete with each other, with the most influential parameters determining the state of the system. For example, in the case of metal–insulator transitions, electron–electron interactions compete with the kinetic energy of the electrons and disorder. Understanding these complex dynamics is crucial for both fundamental physics and the development of novel technological applications, particularly given the role of disorder in tuning critical temperatures, a property with significant potential benefit in the fabrication of new devices where temperature requirements are still the bottleneck. In this article, properties of the Mott metal–insulator transition within disordered electron systems are explored using the disordered Hubbard model, the simplest Hamiltonian for capturing the metal–insulator transition. The model solutions are obtained using the self-consistent statistical dynamical mean-field theory (statDMFT). statDMFT incorporates local electronic correlation effects while allowing for Anderson localization due to disorder.

Abraham Wald’s formalization of the sequential probability ratio test in the crucible of World War II is one of the more famous cases in the history of statistics of the interplay of statistical theory and real-world applications. Focusing entirely on the moments around its creation, however, obscures the way in which it was also a continuation of previous work he had done in the late 1930s, and in particular, Wald’s development of decision theory and his approach to using inverse probability. By situating the origins of the sequential probability ratio test in a broader history, we see not only how inverse probability initially made its way into sequential analysis but also the ongoing importance of the role of applications in motivating the development of statistical theory.

A comparative analysis of methods for evaluating an integral indicator of population health, based on the summation of weighted arithmetic group averages, has shown that their advantage lies in the use of weighting coefficients, which improve the accuracy of the integral assessment. However, their disadvantage is the quality of expert assessments of the weighting coefficients, which depend on the segment selection and the number of experts, which requires their further improvement. The goal of the work is to develop a comprehensive integral indicator to assess the state of population health for the purpose of prompt situation assessment and decision-making. Methods Used: Modeling of the integral indicator of population health, based on the theory of mathematical statistics, systems of linear algebraic equations, and regression-correlation analysis. Novelty: The developed model of the integral indicator of population health, in the form of a function of individual statistical health indicators over time, allows for comparison and analysis of population health in spatial and temporal aspects. Results: The use of the presented methodological approach for creating a research base and developing an algorithm for calculating the integral indicator using regression-correlation analysis, the range amplitude of statistical indicators, situation analysis methods, event theory, and averages, allows for the assessment of the medical-demographic state both at the current moment and based on trend assessments using linear trends in the forecast period.

The proliferation of data in modern industries has created a demand for statistical inference methods that are both predictive and scalable. This paper aims to close the gap between statistical inference theory and machine learning practice by utilizing Python’s rich functionality. This paper contributes and empirically demonstrates an end-to-end framework for a data scientist’s workflow, ranging from data cleaning and feature engineering to model construction and statistical validation. Through three real-world case studies in e-commerce, healthcare and finance, the paper empirically compares the relative merits of regularized regression, Bayesian classifiers, and ensemble methods. The findings reveal that Bayesian models offer superior uncertainty estimation in healthcare, where data is often scarce, whereas ensembles such as Gradient Boosting achieve state-of-the-art predictive accuracy in financial applications with big data. The paper emphasizes that statistical validation remains a mandatory step in building reliable machine learning systems. It also discusses practical challenges such as scalability, model interpretability, and data quality, and proposes mitigation solutions and future research directions. This research provides a practical guide to implementing statistical validation in data science workflows.

This paper compares the large-sample asymptotic convergence of Bayesian linear regression and ordinary least squares regression. It analyzes their fundamental differences from three aspects: theoretical foundations, convergence paths, and asymptotic performance, and points out the advantages of Bayesian methods in scenarios with moderate sample sizes and reliable prior information. Furthermore, a unified framework is constructed to reveal the relative efficiency of both methods under different model settings through mathematical derivation and simulation studies. Based on the Bernstein-von Mises theorem and asymptotic statistical theory, it is demonstrated that the two methods are asymptotically equivalent under regularity conditions. This provides a theoretical basis for method selection in practical applications.

A rich variety of amorphous solids are found throughout nature, science, and technology, including those formed via the vulcanization of long, flexible polymer molecules. A special class—those featuring a wide separation between the very long timescales on which constraining bonds release and the much shorter timescales on which unconstrained degrees of freedom relax—exhibit equilibrium states and are therefore amenable to equilibrium statistical mechanics. A review is given of the least detailed (and thus most general) approach to equilibrium amorphous solids: statistical field theory. The field at the center of this theory is motivated by the aim of characterizing the amorphous solid state. This field, and the theory that governs it, turn out to be rather unusual in essential ways. What the statistical field theory approach predicts—and can predict—is discussed, including the following: the emergence of the solid and its intrinsic heterogeneity; fluctuations and connections with percolation; symmetry breaking and elasticity; and correlations and the information they furnish. Emphasis is placed on the idea, particular to amorphous solids, that such solids are naturally characterized in terms of distributions that describe the spatial heterogeneity of the thermal motions of their constituents. This information is subtly encoded in the wave vector dependencies of the average field and its correlations. The review concludes with some reflections on the applicability—or otherwise—of the ideas and results it explores to a variety of amorphous solids and related systems.

A large and growing number of recent studies has embraced a statistical learning view of reading, revealing that readers utilize an array of regularities that are available in writing systems as they process printed words and texts. However, previous studies have focused on the impact of one regularity (or an otherwise small number of cues). Therefore, we currently have a limited understanding of (1) whether different regularities each carry unique explanatory power, beyond other (collinear) cues; (2) how do regularities at different levels of the input contribute to reading behavior; and (3) whether regularities vary in their contributions across processing stages. To answer these questions, we employ Random Forests analyses on a large-scale, eye-movement, passage-reading database from English first- and second-language readers, evaluating the relative importance of a large number of regularities on multiple eye-movement dependent variables. First, our findings demonstrate that, each regularity uniquely contributes to the model's performance. Second, we show that both text-level regularities (e.g., predictability) and word-level regularities (including print-speech and print-meaning regularities), contribute to continuous text reading. Third, we document varying contributions of some regularities over time, with later reading measures being more impacted by text-level regularities. These results support and extend statistical learning theories of reading, showing that readers are attuned to a range of regularities in their writing system, which jointly guide naturalistic reading behavior.

Hydrogen Cyanide Could Be Efficiently Produced by the Sequential Reaction of CH ( ² Π) Radicals with N ₂ Molecules in the Atmosphere of Titan: Investigations by Quantum-Chemical and Statistical Rate Theories

Quantum-like modeling (QLM)—quantum theory applications outside of physics—are intensively developed with applications in biology, cognition, psychology, and decision-making. For cognition, QLM should be distinguished from quantum reductionist models in the spirit of Hameroff and Penrose, as well as Umezawa and Vitiello. QLM is not only concerned with just quantum physical processes in the brain but also with QL information processing by macroscopic neuronal structures. Although QLM of cognition and decision-making has seen some success, it suffers from a knowledge gap that exists between oscillatory neuronal network functioning in the brain and QL behavioral patterns. Recently, steps toward closing this gap have been taken using the generalized probability theory and prequantum classical statistical field theory (PCSFT)—a random field model beyond the complex Hilbert space formalism. PCSFT is used to move from the classical “oscillatory cognition” of the neuronal networks to QLM for decision-making. In this study, we addressed the most difficult problem within this construction: QLM for entanglement generation by classical networks, that is, “mental entanglement.” We started with the observational approach to entanglement based on operator algebras describing “local observables” and bringing into being the tensor product structure in the space of QL states. Moreover, we applied the standard states entanglement approach: entanglement generation by spatially separated networks in the brain. Finally, we discussed possible future experiments on “mental entanglement” detection using the EEG/MEG technique.

Quantum statistical theory of dislocation mobility in discrete lattices

The green function theory was developed by the scientist George Green 1793-1841 and is a mathematical function. Where this theory was developed to improve and manage ordinary and partial differential equations with different dimensions and for a time-dependent and time-independent problem .The theory was also developed to serve physics and mechanics, largely in quantum field theory and thermodynamics, as well as in statistical field theory. Thus, during the understanding and analysis of the theory of green function, its role and importance in science were shown, and many applications were presented to understand it, such as i.e. Boundary and Initial Value problem, Wave Equation, Kirchhoff Diffusion Equation, Diffraction Theory, Helmholtz Equation and etc. So the green function have many important roles in many aspect of sciences, so we try to cover the theory from all of it's aspect.

Abstract Statistical regularities can be acquired from usage. To examine language speakers’ statistical metacognition about multiword expressions (MWEs), we collected ratings for frequency, dispersion, and directional association strength of English binomials from L1, advanced and intermediate L2 speakers. Mixed-effects modeling showed all speakers had limited speaker-to-corpus consistency but significant sensitivity to statistical regularities of language, supporting usage-based (Gries & Ellis, 2015) and statistical learning theories (Christiansen, 2019). Their statistical metacognition was also shaped by word-level cues, consistent with dual-route model (Carrol & Conklin, 2014). Despite similarities, frequency metacognition showed the strongest speaker-to-corpus consistency, while dispersion metacognition was the hardest to develop. Advanced L2 speakers showed the greatest speaker-to-corpus consistency and sensitivity, while lower-proficiency speakers relied more on word-level cues in metacognitive judgments, supporting the shallow-structure hypothesis (Clahsen & Felser, 2006). Overall, L1 and L2 speakers develop diverse statistical metacognition, with L2 speakers not necessarily inferior, suggesting that statistical metacognition is not solely shaped by usage-based experience.

Neural networks that include a combinatorial optimization layer can give surprisingly efficient heuristic policies for difficult combinatorial optimization problems. Three questions remain open: which architecture should be used, how should the parameters of the machine learning model be learned, and what performance guarantees can we expect from the resulting algorithms? Following the intuitions of geometric deep learning, we explain why equivariant layers should be used when designing such policies, and illustrate how to build such layers on routing, scheduling, and network design applications. We introduce a learning approach that enables to learn such policies when the training set contains only instances of the difficult optimization problem and not their optimal solutions, and show its numerical performance on our three applications. Finally, using tools from statistical learning theory, we prove a theorem showing the convergence speed of the estimator. As a corollary, we obtain that, if an approximation algorithm can be encoded by the neural network for some parametrization, then the learned policy will retain the approximation ratio guarantee. On our network design problem, our machine learning policy has the approximation ratio guarantee of the best approximation algorithm known and the numerical efficiency of the best heuristic.

Modern machine learning, fueled by large datasets and complex models, faces a critical tension. The statistical principles underpinning learning (generalization, efficiency, robustness) often clash with the computational realities of optimization, especially in a resource constrained environment or when data exhibits inherent geometric structure. This work addresses the theme "Statistics Meets Optimization" by employing an optimization framework explicitly designed to leverage statistical data properties, particularly group invariances/equivariances common in real world data (e.g., spatial rotations in satellite imagery, temporal shifts in sensor data), to achieve significant gains in sample efficiency and convergence speed. We theoretically derive generalization bounds linking the exploitation of data geometry to reduced sample complexity. Empirically, we demonstrate the efficacy of our method on a challenging real world case study, i.e., on predicting crop yield anomalies in Delta State, Nigeria, using limited, noisy, and spatially heterogeneous satellite and meteorological data. Our optimizer achieved a significant performance with 40% less data compared to adaptive baselines (Adam, RMSProp), highlighting the practical impact of statistically-informed optimization, especially for regions facing data scarcity. This work provides a concrete bridge between statistical theory (data structure, efficiency) and optimization practice (algorithm design, scalability), demonstrating that geometry-aware algorithms can democratize effective ML for resource-limited applications.

We introduce a method for describing eigenvalue distributions of correlation matrices from multidimensional time series. Using our newly developed matrix H theory, we improve the description of eigenvalue spectra for empirical correlation matrices in multivariate financial data by considering an informational cascade modeled as a hierarchical structure akin to the Kolmogorov statistical theory of turbulence. Our approach extends the Marchenko-Pastur distribution to account for distinct characteristic scales, capturing a larger fraction of data variance, and challenging the traditional view of noise-dressed financial markets. We conjecture that the effectiveness of our method stems from the increased complexity in financial markets, reflected by new characteristic scales and the growth of computational trading. These findings not only support the turbulent market hypothesis as a source of noise but also provide a practical framework for noise reduction in empirical correlation matrices, enhancing the inference of true market correlations between assets.

Background. The article analyzes the inefficiency of transport and logistics production in Russia, particularly in the road freight sector. It demonstrates that the current structure, driven by the interests of individual companies, fails to ensure long-term stability. The significant growth in freight turnover (126.5% since 2001) alongside minimal increases in shipping volumes (5.9%) indicates extensive development and inefficient resource utilization. Purpose. To identify the causes of inefficiency in Russia’s road freight system and propose restructuring measures based on statistical data and an analysis of the industry’s organizational framework. Materials and methods.The study utilizes data from Rosstat, the Russian Ministry of Transport, and analytical centers (2001–2023) on freight turnover, shipping volumes, and the transport sector’s share of GDP. A comparative analysis of indicator trends revealed a disparity between GDP growth (20-fold) and freight volume growth (10.4%). The structure of the road freight system was examined, including the distribution of trucks and transport work between sole proprietors (36.6%) and legal entities (63.4%). Methods included statistical analysis, systems theory, and expert assessments (e.g., an Ernst Young survey). Results. The study found that Russia’s road freight system is highly fragmented, dominated by sole proprietors (56% of vehicles), and suffers from low market transparency. A modified management model was proposed, emphasizing direct coordination between the macro level (state regulation) and micro level (enterprises) to enhance industry resilience.

Abstract Machine learning methods, in particular deep learning methods such as artificial neural networks (ANNs) with many layers, have become widespread and useful tools in nuclear physics. However, these ANNs are typically treated as "black boxes", with their architecture (width, depth, and weight/bias initialization) and the training algorithm and parameters chosen empirically by optimizing learning based on limited exploration. We test a non-empirical approach to understanding and optimizing nuclear physics ANNs by adapting a criticality analysis based on renormalization group flows in terms of the hyperparameters for weight/bias initialization, training rates, and the ratio of depth to width. This treatment utilizes the statistical properties of neural network initialization to find a generating functional for network outputs at any layer, allowing for a path integral formulation of the ANN outputs as a Euclidean statistical field theory. We use a prototypical example to test the applicability of this approach: a simple ANN for nuclear binding energies. We find that with training using a stochastic gradient descent optimizer, the predicted criticality behavior is realized, and optimal performance is found with critical tuning. However, the use of an adaptive learning algorithm leads to somewhat superior results without concern for tuning and thus obscures the analysis. Nevertheless, the criticality analysis offers a way to look within the black box of ANNs, which is a first step towards potential improvements in network performance beyond using adaptive optimizers.

This paper examines age discrimination in physics education in Nigeria, analysing how educational policy impacts participation, equity, and achievement. Guided by Statistical Discrimination Theory, it evaluates how rigid age limits in university admissions marginalise learners regardless of merit, experience, or academic potential. The study highlights that such policies exclude non-traditional learners, late bloomers, and those delayed by socio-economic challenges, thereby reducing diversity essential for innovation in physics education. Drawing from existing literature, it argues for inclusive reforms aligned with lifelong learning and educational justice. Recommendations include eliminating fixed age limits, implementing merit-based and flexible entry pathways, and promoting awareness of age diversity in educational planning. Removing structural age barriers is essential to fostering equity, broadening participation, and enhancing productivity in physics education.