Probability Density Research Articles

Machine learning predicts pedestrian wind flow from urban morphology and prevailing wind direction

Abstract Pedestrian-level wind plays a critical role in shaping the urban microclimate and is significantly influenced by urban form and geometry. The most common method for determining spatial wind speed patterns in cities relies on numerical computational fluid dynamics (CFD) simulations, which resolve Navier-Stokes equations around buildings. While effective, these simulations are computationally intensive and require specialised expertise, limiting their broader applicability. To address these limitations, this study proposes a more cost-effective alternative while achieving 90% performance in capturing the mean and maintaining spatial wind patterns captured by CFD. We developed a machine learning (ML) approach with U-net architecture to predict time mean wind speed patterns from prevailing wind directions and three-dimensional urban morphology, which are increasingly available for global cities. The model is trained and tested using a comprehensive dataset of 512 numerical simulations of urban neighbourhoods, representing diverse morphological configurations in cities worldwide. We find that the ML algorithm accurately predicts complex wind patterns, achieving a normalised mean absolute error of less than 10%, which is comparable to wind anemometer measurement in a low wind speed environment. In predicting wind statistics, the ML model also surpasses that of regression models based solely on statistical representations of urban morphology. The R2 values measuring grid-level agreement between ML and CFD range from 0.94-0.99 and 0.65-0.95 for the idealised and whole datasets, respectively. However, we find that grid-based R2 is not an effective metric for evaluating the 2D model performance due to localised biases arising from faster wind speed grid regions, which is revealed by the wind probability density function. These findings demonstrate that complex pedestrian wind patterns can be effectively predicted using an image-based ML approach, offering the potential to emulate physics-based LES models, which are computationally expensive, thereby significantly reducing computing costs.

Generative learning of continuous data by tensor networks

Beyond their origin in modeling many-body quantum systems, tensor networks have emerged as a promising class of models for solving machine learning problems, notably in unsupervised generative learning. While possessing many desirable features arising from their quantum-inspired nature, tensor network generative models have previously been largely restricted to binary or categorical data, limiting their utility in real-world modeling problems. We overcome this by introducing a new family of tensor network generative models for continuous data, which are capable of learning from distributions containing continuous random variables. We develop our method in the setting of matrix product states, first deriving a universal expressivity theorem proving the ability of this model family to approximate any reasonably smooth probability density function with arbitrary precision. We then benchmark the performance of this model on several synthetic and real-world datasets, finding that the model learns and generalizes well on distributions of continuous and discrete variables. We develop methods for modeling different data domains, and introduce a trainable compression layer which is found to increase model performance given limited memory or computational resources. Overall, our methods give important theoretical and empirical evidence of the efficacy of quantum-inspired methods for the rapidly growing field of generative learning.

Analytical model for pulse pileup spectra and count statistics in photon counting detectors with seminonparalyzable behavior.

Photon counting detectors (PCDs) with energy discriminating capabilities enable quantitative imaging of materials. However, the accuracy of estimates may be substantially degraded due to pulse pileup effects (PPEs) at high count rates. Accurate description of the output spectrum and count rate behavior of a PCD subject to pulse pileup is crucial to the development of photon counting computed tomography (PCCT). This study presents a fully analytical model to accurately predict the pulse pileup spectrum and count statistics (mean and covariance of energy-binned counts) for a non-paralyzable detector with nonzero pulse length and, therefore, seminonparalyzable behavior, that is, retriggering of dead time by pulses incident during the previous dead time. We recursively computed the probability density function (PDF) of pulse pileup spectra at different pulse pileup orders. To do this, we considered the following factors: the unipolar pulse shape, the incident pulse spectrum, the distribution of time intervals between incident pulses, and the trigger threshold. We then derived the count rate and spectrum-dependent expression of total count statistics (mean and variance of total counts) based on renewal theory. We simulated a non-paralyzable PCD using Monte Carlo simulation to separately validate the spectrum and count statistics model outputs. Finally, we investigated the model accuracy in predicting material decomposition noise using the Cramér-Rao lower bound (CRLB) and a multibin system model. A comparison between predictions of the proposed model and Monte Carlo simulation is presented. The results show excellent agreement between the proposed model prediction of pulse pileup spectrum and count statistics and Monte Carlo simulation for relative count rates (the average number of counts detected during one dead time, ) of up to . The coefficient of variation (CV) values between the spectra from model prediction and Monte Carlo simulation are less than , and the coefficient of determination values, , between the count statistics from model prediction and Monte Carlo simulation are greater than . The proposed model also accurately predicts material decomposition noise for a non-paralyzable PCD for relative count rates of up to , with relative error (RE) less than . We developed a fully analytical model of the pulse pileup spectrum and count statistics for a non-paralyzable detector model with nonzero pulse length. The model predictions agree with the Monte Carlo simulation outputs. This model could be used to correct and compensate for pulse pileup when imaging with PCDs.

SYNTHESIS OF LINGUISTIC MODELS FOR REPLICATING THE IMPACT OF LIGHTNING ON THE SHIELDING FAILURE RATE OF AN OVERHEAD POWER TRANSMISSION LINE

Background. Traditional methods for calculating lightning protection systems are continuously improving based on new knowledge about lightning and its characteristics. Growing demands for protection efficiency, especially for power transmission lines in the Smart Grid concept, create a need for new approaches. Fuzzy logic serves as one way to address these challenges, as it allows modeling complex and uncertain processes, which are typical for lightning protection systems. Objective. The main goal is to develop a linguistic model for predicting the failure frequency of shielding based on lightning current parameters, the width of the failure zone, and the current probability density function. Simulation helps calculate failure frequencies that could lead to insulation flashovers and equipment damage. Methods. An electrogeometric model was used to determine the width of the shielding failure zone. The synthesis of linguistic models for replicating the effect of lightning on the shielding failure rate was performed using a mathematical apparatus of fuzzy logic. The shielding failure rate is considered, which determines the probability that lightning will strike the phase conductor, bypassing the overhead shield wire. It is assumed that the minimum current for calculations is 3.0 kA, which is the threshold below which no initial lightning strikes have been observed on real power transmission lines. The probability of such a lightning current is calculated to be only 0.23%. For modeling the lightning current probability density function, formulas similar to the results of measurements on research towers were used. Results. A data about the width of the lightning shielding failure zone on a typical 330 kV power transmission line tower is obtained, demonstrating how this zone changes with different prospective lightning current values. Conclusions. Simulating the failure frequency of shielding using fuzzy logic and the Mamdani algorithm allows for more accurate failure predictions, taking into account variable conditions. Structural diagrams and transfer functions demonstrate how different parameters, such as current, current density, and failure zone width, affect the failure frequency. All these parameters and their interrelationships are described using linguistic variables and fuzzy rules, enabling more effective predictions for lightning protection systems. This approach provides more accurate and flexible modeling, meeting modern requirements for the safety of overhead power transmission lines, especially within the Smart Grid concept.

An Improved Set-Valued Observer and Probability Density Function-Based Self-Organizing Neural Networks for Early Fault Diagnosis in Wind Energy Conversion Systems

An Improved Set-Valued Observer and Probability Density Function-Based Self-Organizing Neural Networks for Early Fault Diagnosis in Wind Energy Conversion Systems

Neural signatures of temporal anticipation in human cortex represent event probability density

Temporal prediction is a fundamental function of neural systems. Recent results show that humans anticipate future events by calculating probability density functions, rather than hazard rates. However, direct neural evidence for this hypothesized mechanism is lacking. We recorded neural activity using magnetoencephalography as participants anticipated auditory and visual events distributed in time. We show that temporal anticipation, measured as reaction times, approximates the event probability density function, but not hazard rate. Temporal anticipation manifests as spatiotemporally patterned activity in three anatomically and functionally distinct parieto-temporal and sensorimotor cortical areas. Each of these areas revealed a marked neural signature of anticipation: Prior to sensory cues, activity in a specific frequency range of neural oscillations, spanning alpha and beta ranges, encodes the event probability density function. These neural signals predicted reaction times to imminent sensory cues. These results demonstrate that supra-modal representations of probability density across cortex underlie the anticipation of future events.

Seismic reliability analysis of reinforced slope considering soil parameter dependence structure

Research on the seismic performance of slopes is of great significance for ensuring the long-term stability of engineering projects, protecting human life and property, and reducing economic losses. Previous studies have shown that when studying slope stability, it is essential to consider the dependence structure among soil parameters. Therefore, in this study on the seismic reliability of reinforced slopes, the Copula function was introduced to represent the dependence structure between soil cohesion and the internal friction angle. In combination with the stochastic ground motion model of engineering physics, soil samples and ground motion samples were generated based on the Generalized F-discrepancy (GF-discrepancy) point selection method, and their assigned probabilities were obtained simultaneously. Subsequently, the dynamic response of the slope under the combined stochastic effects was simulated, and the Generalized Probability Density Evolution Method (GPDEM) was used to quantitatively evaluate the reliability of the reinforced slope structure under different failure thresholds. The results indicate: (1) The folded geogrid reinforced slope demonstrates superior seismic performance; (2) The Frank Copula function can better characterize the dependence structure between the soil parameters selected in this study, providing a sufficient number of effective samples for stochastic research; (3) Based on GPDEM, the structural reliability curves under different thresholds can be obtained rapidly. The research findings have important theoretical implications for the seismic reliability analysis and structural design of reinforced slopes.

A New Exponentiated Siya Distribution and Its Biomedical Application

This article introduces a new exponentiated distribution called the Siya Distribution by incorporating a new parameter into the existing two-parameter Gamma distribution. It is a versatile three-parameter model designed to capture various data behaviors encountered in biological and environmental studies. Siya distribution accounts for data that exhibit variable degrees of skewness and kurtosis, making it suitable for complex datasets such as medical measurements, reliability analysis, and survival times. We derive fundamental properties including the probability density function (PDF), moments, the cumulative distribution function (CDF), and moment generating function (MGF), along with the hazard function to allow comprehensive analytical exploration of the distribution’s behavior. Parameter estimation is conducted using Maximum Likelihood Estimation (MLE), providing robust estimators for real-world applications. Model performance is evaluated using two real datasets against three alternative existing parameter distributions using the Akaike Information Criterion (AIC), Corrected AIC (AICc), and Bayesian Information Criterion (BIC), demonstrating that the Siya Distribution consistently achieves a superior fit, especially with highly skewed data. Empirical applications to biological and medical data illustrate the model’s adaptability and potential to improve data representation in fields requiring precise distribution modelling. The Exponentiated Siya distribution thus offers a significant tool for advanced statistical analyses in applied sciences, supporting more accurate and nuanced interpretation of complex data trends.

Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution

In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.

Influence of Child-Level Factors and Lexical Characteristics on Vocabulary Knowledge of Children With Cochlear Implants and Hearing Aids.

Recent studies indicate children who are deaf and hard of hearing who use cochlear implants or hearing aids know fewer spoken words than their peers with typical hearing, and often those vocabularies differ in composition. To date, however, the interaction of a child's auditory profile with the lexical characteristics of words he or she knows has been minimally explored. The purpose of the present study is to evaluate how audiological history, phonological memory, and overall vocabulary knowledge interact with growth in types of spoken words known by children who are deaf and hard of hearing compared to children with typical hearing. Children with cochlear implants (n = 36) and hearing aids (n = 39) were compared to children with typical hearing (n = 47) at ages 4 and 6. Children participated in measures of phonological memory and vocabulary knowledge, inclusive of an experimental measure with words of varying phonotactic probability and neighborhood density. Results indicate that children with hearing aids and with cochlear implants tend to know fewer words across all lexical conditions than children with typical hearing. For children with cochlear implants, overall vocabulary knowledge was the best predictor of a mis-matched probability and density condition, whereas it was the best predictor of matched condition for children with hearing aids. Children with cochlear implants and children with hearing aids, then, appear to have different underlying skills that interact with the lexical characteristics of words to support vocabulary growth.

Characterizing the bulk angular distribution of metal flakes in pigmented coating systems leveraging RTI

A key factor in the appearance of metal flake-pigmented coatings (MF-PCs) is the orientation of the metal flakes, which affects the coating's luster (at large distances) and visual texture (up close). This study introduces a dome-based reflectance transformation imaging (RTI) system to estimate metal flakes’ orientation using intensity multi-light image collections (MLICs) and an algebraic reflectance map inversion algorithm. The flake's angles and 2D/3D probability density functions (PDFs) were derived from this, and a new orientation index (OI) for MF-PCs. The impact of metal flakes’ orientation on appearance was assessed using the lightness flop index (LFI), sparkle (SG), and graininess grades (GG). Results show that the LFI and GG increase as the OI decreases, while SG at 15° rises with higher orientation but decreases at 45° and 75°. Additionally, a non-linear negative exponential relationship between the OI and LFI was observed, highlighting orientation as a significant dimension in predicting MF-PC’s appearance.

Projected Malaria Transmission Risk Under Climate Intervention in South Asia

Abstract This study focuses on the impact of climate intervention under the ARISE-SAI-1.5 scenario of stratospheric aerosol injection (SAI) on projected malaria distribution in South Asia, relative to climate change under the SSP2-4.5 scenario, during the period 2045 to 2069. A dynamic malaria model is employed to assess the impacts of SAI and climate change on malaria redistribution. In addition to the entomological inoculation rate (EIR), the length of the transmission season (LTS) and malaria cases are considered as quantitative indicators of malaria transmission. The quantification of the projected malaria distribution employing several statistical techniques, including the probability density function technique, enables the assessment of malaria variability and risk across all seven highly climate-vulnerable countries of South Asia (Afghanistan, India, Iran, Bangladesh, Bhutan, Nepal, and Pakistan). Due to the lower temperatures achievable under ARISE-SAI-1.5 scenario relative to SSP2-4.5 scenario, the frequency of EIR occurrence shifts toward lower intensity values. This decrease in EIR is more pronounced in populous India and Bangladesh than in the other five South Asian countries during 2045-2069. The projected magnitude of LTS and the frequency of malaria case occurrences also diminish under ARISE-SAI-1.5 in South Asia.

Unraveling solar irradiance dynamics in arid atmospheres: A multi-decadal wavelet coherence and probability density functions analysis with implications for air quality, climate, and renewable energy

Unraveling solar irradiance dynamics in arid atmospheres: A multi-decadal wavelet coherence and probability density functions analysis with implications for air quality, climate, and renewable energy

Monte Carlo Neural PDE Solver for Learning PDEs Via Probabilistic Representation.

In scenarios with limited available data, training the function-to-function neural PDE solver in an unsupervised manner is essential. However, the efficiency and accuracy of existing methods are constrained by the properties of numerical algorithms, such as finite difference and pseudo-spectral methods, integrated during the training stage. These methods necessitate careful spatiotemporal discretization to achieve reasonable accuracy, leading to significant computational challenges and inaccurate simulations, particularly in cases with substantial spatiotemporal variations. To address these limitations, we propose the Monte Carlo Neural PDE Solver (MCNP Solver) for training unsupervised neural solvers via the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Compared to other unsupervised methods, MCNP Solver naturally inherits the advantages of the Monte Carlo method, which is robust against spatiotemporal variations and can tolerate coarse step size. In simulating the trajectories of particles, we employ Heun's method for the convection process and calculate the expectation via the probability density function of neighbouring grid points during the diffusion process. These techniques enhance accuracy and circumvent the computational issues associated with Monte Carlo sampling. Our numerical experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency compared to other unsupervised baselines.

Linear quadrature method of moments for solving the transported probability density function model for turbulent combustion

Large eddy simulation (LES) coupled with the transported probability density function (PDF) model has become a promising strategy for modelling turbulent combustion. For solving the high-dimensional PDF transport equation numerically, deterministic quadrature-based moment methods have arisen as an alternative choice besides stochastic particle and stochastic field methods. In this work, we propose a new quadrature-based moment method called linear Quadrature Method Of Moments (LQMOM) for the univariate PDF case. This method assumes that the underlying PDF distribution is smooth and it uses a set of predefined quadrature nodes and weights in the sample space to establish a linear system of equations for the PDF values at the quadrature nodes from the moments. The LQMOM is more efficient than the original QMOM which needs to solve a nonlinear system of equations for the moment inversion. However, it may generate negative PDF values for non-smooth PDF distributions. To remedy this flaw, we construct the LQMOM-QMOM hybrid algorithm by using detectors to detect single- and two-peak distributions. Furthermore, we extend the LQMOM to the multivariate PDF case by utilizing the main idea of the conditional QMOM (CQMOM) to obtain the linear CQMOM (LCQMOM), and then reduce it to the simplified version (LCQMOM-S) to gain efficiency. The LQMOM-QMOM hybrid algorithm for univariate PDFs is tested in a typical example and the LCQMOM-S-CQMOM hybrid algorithm for multivariate PDFs is checked in compressible turbulent reactive 2D shear layer flow and 3D air/H2 jet. It is shown that the hybrid algorithms can reduce 20–35% simulation time and produce similar results compared with the respective QMOM and CQMOM.

Workload of Queueing Systems with Autocorrelated Service Times

The queuing model with autocorrelated service times is studied with respect to workload, i.e., the time needed to serve all the customers in the queue. Specifically, new formulas for the probability density of workload, its tail, the average value, and entropy are derived and illustrated using numerical examples. Both time-dependent and steady-state cases are covered. It is also demonstrated that the average workload may reach surprisingly large values, exceeding several times the product of the average queue size and the average service time.

A Study on Exponential Poisson Lindley Distribution, Properties, Simulations and its Application in Engineering

This study extends the continuous Poisson Lindley distribution with the aid of an existing link function, the Exp-G family. It studied the probability density function (pdf) and the cumulative distribution function (cdf) of the new Extended Poisson Lindley distribution (ExPLinD) and presents appropriate plots. Statistical Properties of the ExPLinD were studied and presented. Some of these properties include its moments, moment generating function (mgf), characteristics function and quantile function. Reliability studies was carried out and relevant charts showing the plot of the survival function and hazard functions were presented. The ExPLinD has an increasing hazard shape among others displayed in the hazard shape plots, this makes the distribution useful in reliability studies. Maximum Likelihood method was employed to estimate the parameters of ExPLinD. Simulation studies of the ExPLinD was done using different parameter values and different sample sizes. Simulation studies of ExPLinD shows that the average estimates of the parameters of ExPLinD tends to the true parameters as sample size increases, this is in order with first order asymptotic theory. The ExPLinD was fitted to an Engineering data and compared to some closely related distributions. It provided a better fit compared to the related distribution which makes it an important distribution in Statistical modeling.

A New Bivariate Family of Distributions Based on the Clayton Archimedean Copula and Dagum Distribution

This study introduces a novel bivariate distribution combining the Clayton Archimedean copula and the Dagum distribution, addressing challenges in modeling complex dependencies, skewness, heavy tails, and multimodal distributions. The proposed NBCDagE distribution leverages the Clayton copula’s ability to capture asymmetric dependencies and the Dagum distribution’s flexibility to model diverse data behaviors, making it suitable for reliability, finance, and survival analysis applications. Key statistical properties of the NBCDagE distribution, including the probability density function (PDF), cumulative distribution function (CDF), product and joint moments, and Shannon entropy, were derived and analyzed. The model demonstrates sensitivity to parameter changes, with higher parameter values leading to sharper PDFs and lighter tails, while lower values result in flatter PDFs and heavier tails. Joint moments and entropy analyses revealed the distribution’s ability to adapt to varying data complexities, showcasing its robustness in capturing dependence structures and marginal characteristics. Visual representations, including contour plots and density curves, illustrate the flexibility of the NBCDagE model in handling a wide range of dependence patterns and data structures. The distribution’s performance was further validated through theoretical derivations and numerical examples, highlighting its adaptability and precision in multivariate data modeling. In conclusion, the NBCDagE distribution provides a robust framework for analyzing bivariate data with intricate dependency structures. Its flexibility and statistical rigor make it a valuable tool for diverse applications, paving the way for future research in higher-dimensional extensions and practical implementations.

Stationary Distribution and Probability Density for a Stochastic Crime Model

ABSTRACTIn this paper, we investigate the dynamic behavior of a criminal model (predator–prey model), in which the predators are the police, and the prey is gang members, and both predators and prey have infectious diseases, infected prey being more susceptible to predation, and infected predators hunting at a reduced rate. We get sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that both the police and gang members will be persistent and coexistent in the long term. What is more, we give the specific expression of the probability density function of the stochastic model around the unique endemic quasi‐equilibrium by solving the Fokker‐Planck equation. Finally, some numerical simulations are carried out to illustrate the theoretical results.

Enhancing Lymph Node Metastasis Risk Prediction in Early Gastric Cancer Through the Integration of Endoscopic Images and Real-World Data in a Multimodal AI Model.

Objectives: The accurate prediction of lymph node metastasis (LNM) and lymphovascular invasion (LVI) is crucial for determining treatment strategies for early gastric cancer (EGC). This study aimed to develop and validate a deep learning-based clinical decision support system (CDSS) to predict LNM including LVI in EGC using real-world data. Methods: A deep learning-based CDSS was developed by integrating endoscopic images, demographic data, biopsy pathology, and CT findings from the data of 2927 patients with EGC across five institutions. We compared a transformer-based model to an image-only (basic convolutional neural network (CNN)) model and a multimodal classification (CNN with random forest) model. Internal testing was conducted on 449 patients from the five institutions, and external validation was performed on 766 patients from two other institutions. Model performance was assessed using the area under the receiver operating characteristic curve (AUC), probability density function, and clinical utility curve. Results: In the training, internal, and external validation cohorts, LNM/LVI was observed in 379 (12.95%), 49 (10.91%), 15 (9.09%), and 41 (6.82%) patients, respectively. The transformer-based model achieved an AUC of 0.9083, sensitivity of 85.71%, and specificity of 90.75%, outperforming the CNN (AUC 0.5937) and CNN with random forest (AUC 0.7548). High sensitivity and specificity were maintained in internal and external validations. The transformer model distinguished 91.8% of patients with LNM in the internal validation dataset, and 94.0% and 89.1% in the two different external datasets. Conclusions: We propose a deep learning-based CDSS for predicting LNM/LVI in EGC by integrating real-world data, potentially guiding treatment strategies in clinical settings.