Bivariate Function Research Articles

Source apportionment of PM2.5 using DN-PMF in three megacities in South Korea

AbstractPM2.5 pollution is problematic in megacities on the western coast in South Korea (Seoul, Incheon, and Gwangju). As these megacities are located downwind of China, their air quality is easily affected by local and long-range transport sources. PM2.5 samples collected in Seoul (n = 222), Incheon (n = 221), and Gwangju (n = 224) from September 2020 to March 2022, were chemically characterized. Dispersion normalized positive matrix factorization was applied to these PM2.5 speciated data to provide source apportionments. Nine common sources (including secondary nitrate, secondary sulfate, biomass burning, mobile, and waste incinerator) were identified at all sites. The conditional bivariate probability function helped to identify each site’s local sources. Joint potential source contribution function analysis identified northeast China and Inner Mongolia as potential source areas of long-range transport pollutants affecting all sites. Forced lifestyle changes due to the pandemic such as limited gatherings while increased recreational activities may have caused different patterns on the biomass burning source. The constraints on old vehicles during the policy implementation periods likely reduced the mobile source contributions in cities that adopted the policy. Secondary nitrate accounted for 40% of the PM2.5 mass at all sites, implying a significant impact from NOX sources. While the current policy focuses primarily on controlling primary emission sources, it should include secondary sources as well which may include precursor emissions control. Healthier air quality would be achieved if the policy effects are not limited to local, but also to foreign sources in regions upwind of Korea by intergovernmental collaboration.

A Low-Rank Matrix Approach to Compute Polynomial Approximations of Smooth Two-Dimensional Functions

Polynomial approximation of smooth functions is becoming increasingly important in fields like numerical analysis and scientific computing. These approximations are vital in models that rely on spectral methods. To reduce the memory costs for large dimensional problems, various methods to provide data-sparse representations have been proposed, including methods based on singular value decomposition, adaptive cross approximation, and matrices with hierarchical low-rank structures, to mention a few. This work presents implementation details on the polynomial approximation of univariate smooth functions through the Polynomial1 class, and of bivariate smooth functions by low-rank matrix representation via the Polynomial2 class. These approaches are explained within Tau Toolbox, a mathematical software library for solving integro-differential problems by the spectral Tau method.

Source identification of ultrafine particulate matter and ammonia emissions in the metropolitan area

Previous studies have estimated the sources of particulate matter in the atmosphere. Among these, studying the sources of secondary aerosols harmful to human health is important. However, there is a lack of research on ammonia (NH3), a precursor to secondary aerosol formation. This study uses positive matrix factorization (PMF) model and conditional bivariate probability function (CBPF) model to estimate the sources of particulate matter and ammonia. The results showed that about 40% of the PM2.5 mass at both sites was attributable to secondary aerosol. To estimate the emission sources of ammonia that contribute to the generation of secondary aerosols, CBPF was utilized to model and compare the emission characteristics of categorized pollution sources and ammonia, and it was found that SMA had similar emission trends to industry, road dust, oil combustion, and biomass combustion, while GRA had similar emission trends to oil combustion and vehicle (diesel). Considering the results from these two regions, ammonia in the metropolitan area is more likely to be emitted from daily activities than from long distances. The study results demonstrate the major role of secondary aerosols on ambient PM2.5 concentrations and can help develop effective management strategies and policies for air pollution mitigation.

Sharp upper and lower estimates for the approximation of bivariate functions by sums of univariate functions

Sharp upper and lower estimates for the approximation of bivariate functions by sums of univariate functions

Attribution of aerosol particle number size distributions to main sources using an 11-year urban dataset

Abstract. Source apportionment was performed using size-segregated atmospheric particle number concentrations (PNCs) in 27 size channels over a diameter range of 6–1000 nm augmented by air pollutants with a time resolution of 1 h in Budapest for 11 full years. The input dataset was treated for the effect of the local meteorology using dispersion correction. Both the uncorrected dataset and corrected dataset were evaluated using positive matrix factorization for separate seasons. Six source types including nucleation, two road vehicle emission sources separated into a semi-volatile fraction and a solid-core fraction, a diffuse urban source, a secondary inorganic aerosol (SIA) source, and an ozone-associated secondary aerosol source were identified, characterized and quantified. The dispersion correction did not considerably change the profiles, diel variations or patterns of the sources, while it substantially modified the relative shares of the nucleation source in all seasons. The mean relative contributions of the traffic emissions (60 %) indicate that on-road motor vehicles were the leading source of particle numbers. The nucleation was responsible for 24 % of the PNC annually as a lower estimate. It exhibited a compound character consisting of photochemically induced nucleation and traffic-related nucleation. Its contributions were the highest in spring and the lowest in winter. The shares of the urban diffuse and SIA source types were the largest in autumn and winter and in spring and summer, respectively, but they were typically ⪅ 10 %. The O3-associated secondary aerosol made up the smallest (⪅ 3 %) contributions. The conditional bivariate probability function analysis showed considerable spatial variations in the source origin. The combination of the size-segregated particle number concentrations, wide overall range of the size channels, considerably long dataset, dispersion correction and modelling over separate seasons led jointly to a unique adaptation of the source apportionment and yielded novel and valuable insights into the urban aerosol sources and processes both for Budapest and in general.

Day–night variation, size distribution, source identification, and health risk of polycyclic aromatic hydrocarbons in particulate matter in Ulsan, South Korea

Particulate matter (PM) in 11 different sizes was collected at a semi-rural site in Ulsan, a multi-industrial city in South Korea, over one year from 2019 to 2020 to investigate the temporal patterns in size distribution and potential sources of polycyclic aromatic hydrocarbons (PAHs). The mean concentration of Σ21 PAHs was higher in winter (0.29 and 0.45 ng/m3 for coarse and fine PM, respectively) than in summer (0.06 and 0.07 ng/m3). Although there was no statistically significant difference between day and night for the concentration of individual PAHs in both coarse and fine PM, the concentrations were generally higher during the day than at night. During the day, Σ21 PAHs had two distinct peaks (0.056–0.10 μm and 3.20–5.60 μm) in winter, but they tended to partition to larger particles (1.00–1.80 μm and >18.0 μm) in summer. This result suggests that the volatilization of PAHs in smaller particles was enhanced by high-temperature conditions in summer, followed by their condensation onto larger particles. At night, Σ21 PAH peaks were mainly around the size of 0.7 μm (droplet mode), suggesting the importance of heterogeneous-aqueous reactions. The results of the conditional bivariate probability function and concentration-weighted trajectory indicate that the high concentrations of PAHs in summer and winter were more affected by local pollution and long-range atmospheric transport, respectively. The results of the health risk assessment demonstrate that PAHs in fine PM contributed significantly to cancer risk compared to those in coarse PM. In conclusion, investigating the size distribution of PAHs in particles is necessary to understand their environmental behaviors and evaluate potential health risks.

Investigating the depositional environments using particle-size analysis of Lower Cretaceous sandstone reservoirs, Biyadh Formation, Saudi Arabia

This study employed particle-size analysis on twenty friable sandstone beds from two outcrops of the Biyadh Formation, which acts as a water aquifer, petroleum reservoir, and CO2 capturing reservoir. Hydrodynamic processes and depositional environments were identified using bivariate plots, Visher diagrams, Passega diagrams, and linear discriminant functions (LDF). The results reveal that the beds of Outcrop A mainly consist of medium- to well-sorted, medium-grained sandstone that is near-symmetrical and mesokurtic. However, Outcrop B comprises medium-sorted, medium-grained sandstone beds that are near-symmetrical and leptokurtic. All the beds in Outcrop A and B are unimodal. Furthermore, Visher and Passega diagrams indicate that the particles of Outcrop A were mostly transported by rolling, saltation, and suspension. In contrast, the particles of Outcrop B were transported by rolling, saltation, suspension, and graded suspension. The LDF technique shows that the depositional environments were beach, aeolian, fluvial, and shallow marine.

Means-based class of modified Weerakoon-Fernando method for determining the zero of nonlinear equation

This paper introduces a generalized class of modified Weerakoon and Fernando iterative method (WFM) for determining the zero of equations that are nonlinear. The idea applied in the modification, involves the composition of a perturbed Newton method with a corrector iterative function that is based on parametric bi-variate rational function of degree one and a power means generating function. The convergence analysis on the developed class of methods, shows that it is of convergence order four with efficiency index of 1.5874. Some existing modified WFM were established to be concrete members of the developed class of methods. Numerical results obtained when the new methods were implemented are provided to compare their effectiveness and efficiency.

Entropic velocity distribution for sediment-laden flows with a modified dip estimation model at data-scarce river sites

ABSTRACT This study proposes an entropic velocity-distribution and dip-position (y max) model for sediment-laden (SL) flows. Unlike previous studies, the proposed entropic distribution incorporates the effect of suspended sediment (SS) and y max, to improve velocity predictions near bed and water surface in SL flows. A new dampened mixing length for SL flows is proposed in terms of SS concentration. Moreover, the effect of channel bed roughness (k s), slope (S), SS, width (B) and flow depth (H) on y max is studied. Further, a comprehensive model for y max is proposed as a bivariate function of B/H and k s/H for natural channels. A comparative analysis is presented with the existing models, which shows the proposed equations provide higher accuracy with fewer parameters and a better representation of physical processes in SL flows. Subsequently, the proposed equations are applied at three gauged sites (Mahanadi River, India), showing their suitability to obtain velocity at data-scarce river sites.

Association of Combined Per- and Polyfluoroalkyl Substances and Metals with Chronic Kidney Disease.

Background: Exposure to environmental pollutants such as metals and Per- and Polyfluoroalkyl Substances (PFAS) has become common and increasingly associated with a decrease in the estimated Glomerular Filtration Rate (eGFR), which is a marker often used to measure chronic kidney disease (CKD). However, there are limited studies involving the use of both eGFR and the urine albumin creatinine ratio (uACR), which are more comprehensive markers to determine the presence of CKD and the complexity of pollutant exposures and response interactions, especially for combined metals and PFAS, which has not been comprehensively elucidated. Objective: This study aims to assess the individual and combined effects of perfluorooctanoic acid (PFOA), perfluorooctanesulfonic acid (PFOS), Cadmium (Cd), Mercury (Hg), and Lead (Pb) exposure on CKD using data from the National Health and Nutritional Examination Survey (NHANES) 2017-2018. Methods: We employed the use of bivariate logistic regression and Bayesian Kernel Machine Regression (BKMR) in our analysis of the data. Results: Logistic regression results revealed a positive association between PFOA and CKD. Our BKMR analysis revealed a non-linear and bi-phasic relationship between the metal exposures and CKD. In our univariate exposure-response function plot, Cd and Hg exhibited a U and N-shaped interaction, which indicated a non-linear and non-additive relationship with both low and high exposures associated with CKD. In addition, the bivariate exposure-response function between two exposures in a mixture revealed that Cd had a U-shaped relationship with CKD at different quantiles of Pb, Hg, PFOA, and PFOS, indicating that both low and high levels of Cd is associated with CKD, implying a non-linear and complex biological interaction. Hg's interaction plot demonstrated a N-shaped association across all quantiles of Cd, with the 75th quantile of Pb and the 50th and 75th quantiles of PFOA and PFOS. Furthermore, the PIP results underscored Cd's consistent association with CKD (PIP = 1.000) followed by Hg's (PIP = 0.9984), then PFOA and PFOS with a closely related PIP of 0.7880 and 0.7604, respectively, and finally Pb (PIP = 0.6940), contributing the least among the five environmental pollutants on CKD, though significant. Conclusions: Our findings revealed that exposure to environmental pollutants, particularly Hg and Cd, are associated with CKD. These findings highlight the need for public health interventions and strategies to mitigate the cumulative effect of PFAS and metal exposure and elucidate the significance of utilizing advanced statistical methods and tools to understand the impact of environmental pollutants on human health. Further research is needed to understand the mechanistic pathways of PFAS and metal-induced kidney injury and CKD, and longitudinal studies are required to ascertain the long-term impact of these environmental exposures.

Fragility assessment of sea-crossing cable-stayed bridge subjected to multi-hazard action via TKC and R-vine copula

Seismic fragility analysis methods commonly used for assessing structural components and systems have certain limitations. This paper introduces a new semi-parametric approach for structural component fragility analysis, the TKC method. This method is based on log-transformed kernel density estimation (log-TKDE) and copula functions, which establish the joint probability distribution (JPD) of a structural response value (d) and a disaster-inducing factor strength parameter (α) using log-TKDE and match the marginal cumulative distributions (MCDs) of d and a via bivariate copula functions. A novel method for evaluating structural system fragility is also proposed based on the R-vine copula function. This approach accounts for the intricate correlations among different components, providing a precise estimation of the multidimensional JPD. The reliability and accuracy of the proposed methods are demonstrated through a seismic fragility analysis of a sea-crossing cable-stayed bridge under earthquake load. Furthermore, a multi-hazard fragility analysis of bridge components and systems is conducted considering hazards such as earthquake, wind, and wave loads. A comparison is made with the seismic fragility of bridges subjected to earthquake loads alone. The results show that the median and 95% confidence of the damage probability of structural components calculated by the TKC method and Monte Carlo simulation (MCS) closely match, which validates the proposed method. The TKC method is also more accurate and computationally efficient than the MCS method with the same Bootstrap sample size and confidence level (95%). The proposed structural system fragility assessment method based on the R-vine copula accounts for complex correlations between components and yields more accurate results than the first-order boundary method. Load spectrum characteristics, structural component characteristics, and other factors cause the multi-hazard fragility of bridge components and systems to differ substantially from their seismic fragility. The difference rate of system damage probability can reach 28.82%. In evaluating the anti-disaster performance of sea-crossing bridges, it is necessary to incorporate the marine environmental load and consider the dynamic response coupling effect of the bridge excited by seafloor ground motions with extreme wind and wave loads. This approach can guide the optimization of bridge designs and facilitate multi-hazard fragility analysis in accordance with real damage probabilities.

On Correlation Aversion and Insurance Demand

This is a study of decision problems under two-dimensional risk. We use an existing index of absolute correlation aversion to conveniently classify bivariate preferences, with respect to attitudes toward this risk. This classification seems to be more important than whether decision makers are correlation-averse or correlation-seeking for the study of insurance demand when a loss has a multidimensional impact. On this note, we also re-examine Mossin’s theorem under bivariate preferences, where full insurance is preferred with a fair premium, while less than full coverage is preferred with a proportional premium loading. Furthermore, based on the comparative statics of this two-dimensional insurance model for changes in correlation aversion, we derive testable implications about the classification of bivariate utility functions. For the particular case when the two-dimensional risk can be interpreted as risk on income and health, we identify the form of separable utility functions depending on health status and income that is consistent with household disability insurance decisions.

On bivariate compound exponentiated survival function of the beta distribution: estimation and prediction

A bivariate distribution is a probability distribution that describes the joint behavior of two random variables. It provides information about the simultaneous variation of two variables, allowing us to analyze their relationship and dependencies. This article discussed the bivariate compound exponentiated survival function of the beta distribution. the joint cumulative distribution function and the joint probability density function were found in closed forms. Several characteristics of this distribution have been discussed. The maximum likelihood (ML) estimators of the parameters and two sample ML predictions of the future observations are derived. The Bayes estimators (BEs) of the parameters based on the squared error loss function and two sample Bayesian predictions of the future observations are presented. The performance of the proposed bivariate distributions is examined using a simulation study. Finally, two data sets are considered in the framework of the proposed distributions to demonstrate their flexibility for real-life applications.

Stability analysis of discrete-time delayed systems via matrix-injection-based transformation method for bivariate quadratic functions

In this paper, the stability analysis of discrete-time systems with a time-varying delay is studied. First, a selectable matrix-injection-based transformation lemma is proposed to guarantee the negative (or positive) definiteness of bivariate quadratic function. This lemma allows a selection of conditions that combine different delay sets, which is more relaxed than that in most previous studies. Then, an improved Lyapunov–Krasovskii functional (LKF) with delay-squared-product summation is given, which introduces extra cross-terms about delay-variation square for stability analysis. Considering the role of injection matrices in the proposed lemma, together with two types of allowable delay sets, four stability criteria are established. Finally, the less conservatism of the proposed criteria is verified by two numerical examples, which demonstrate the superiority of the presented methods.

Cumulative Histograms under Uncertainty: An Application to Dose–Volume Histograms in Radiotherapy Treatment Planning

In radiotherapy treatment planning, the absorbed doses are subject to executional and preparational errors, which propagate to plan quality metrics. Accurately quantifying these uncertainties is imperative for improved treatment outcomes. One approach, analytical probabilistic modeling (APM), presents a highly computationally efficient method. This study evaluates the empirical distribution of dose–volume histogram points (a typical plan metric) derived from Monte Carlo sampling to quantify the accuracy of modeling uncertainties under different distribution assumptions, including Gaussian, log-normal, four-parameter beta, gamma, and Gumbel distributions. Since APM necessitates the bivariate cumulative distribution functions, this investigation also delves into approximations using a Gaussian or an Ali–Mikhail–Haq Copula. The evaluations are performed in a one-dimensional simulated geometry and on patient data for a lung case. Our findings suggest that employing a beta distribution offers improved modeling accuracy compared to a normal distribution. Moreover, the multivariate Gaussian model outperforms the Copula models in patient data. This investigation highlights the significance of appropriate statistical distribution selection in advancing the accuracy of uncertainty modeling in radiotherapy treatment planning, extending an understanding of the analytical probabilistic modeling capacities in this crucial medical domain.

On the analysis of fractional calculus operators with bivariate Mittag Leffler function in the kernel

Bivariate Mittag-Leffler (ML) functions are a substantial generalization of the univariate ML functions, which are widely recognized for their significance in fractional calculus. In the present paper, our initial focus is to investigate the fractional calculus properties of the integral and derivative operators with kernels including the Bivariate ML functions. Further, certain fractional Cauchy-type problems including these operators are considered. Also the numerical approximations of the Caputo type derivative operator are investigated. The theoretical results are justified by applications on examples. Furthermore, the theory of applying the same operators with respect to arbitrary monotonic functions is analyzed in this research.

Generalized bivariate mixture model of directional wind speed in mixed wind climates

Probabilistic modeling of the directional wind speed has been an important topic in the fields of wind energy assessment as well as structural wind-resistant design. The mixture model is required for the probabilistic modeling on the directional wind speed as the wind climates often show the mixture nature. For accuracy and robustness concerns in engineering applications, a generalized bivariate mixture (GBM) model of the directional wind speed is developed by using the copula-based bivariate distribution function. In this model, the marginal distributions and connection function for each sub-model can be flexibly defined. In parametrizing the model, the data-driven parametric estimation including a data preprocess and an automatic unsupervised fitting process is presented. The performance of GBM model is verified using two data sets and compared with the offset elliptical normal mixture (OENM) model. The robustness of GBM model is found for its good performance in fitting to the targets with or without smoothing process, and in tackling directional wind speed with the extremely complex probability characteristic.

On approximation of bivariate functions by Abel–Poisson and conjugate Abel–Poisson means

On approximation of bivariate functions by Abel–Poisson and conjugate Abel–Poisson means

Multivariate compactly supported C∞ functions by subdivision

This paper discusses the generation of multivariate C∞ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports [0,1+ϵ] in comparison to the support [0,2] of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of C∞ compactly supported wavelets of small support in any dimension.

Impacts of Nudged Sea Surface Temperature on Tropical Precipitation, Moisture, and Vertical Velocity in an Earth System Model

Abstract Understanding the effect of the nudged sea surface temperature (SST) on precipitation simulation is important in Earth system models. In this study, we analyzed the role of nudged SST and air–sea interaction in precipitation simulation by using the Chinese Academy of Sciences Earth System Model version 2 (CAS-ESM2). CAS-ESM2 fully coupled runs with and without nudged tropical SST to observations, and the atmosphere-only runs (AMIP) are comprehensively evaluated against observations and reanalysis over the tropical ocean from 1982 to 2015. The precipitation is affected by the nudged SST and air–sea interaction by means of changes in both thermodynamic and dynamic processes. Simulation with the nudged SST dramatically improves the climatological precipitation fields over the tropics and four regional areas. As the simulated SSTs are nudged toward the observations, CAS-ESM2 well reproduces the mean annual values in column water vapor (CWV), vertical structure of specific humidity, and large-scale circulation. The nudged SST run also captures a wide range of the complex interactions between SST, precipitation, moisture, and large-scale circulations, as measured by the monthly relationships and bivariate probability distribution function. The biases in heavy precipitation associated with the CWV induced under moist conditions are decreased by applying the nudging technique. In particular, the nudging run performs better in reproducing the SST–precipitation covariability than the control and AMIP runs. However, CAS-ESM2 with nudged tropical SST simulates the degradation of strong ascending motions, which is similar to the AMIP runs and reflects the deviation of the atmospheric model. Our results confirm that the fully coupled run using the nudged SST is a method for improving the precipitation simulation across the tropical oceans. Significance Statement Understanding the sources of precipitation errors in Earth system models is challenging because of the complex error compensation and air–sea interaction. When the simulated SST is nudged toward observations, few studies have focused on the changes in both thermodynamic and dynamic processes via atmospheric large-scale circulation, moisture, and changes in the SST–precipitation relationship at the same time. Here, we carry out two CMIP6 historical experiments (a fully coupled control run and an SST-nudged run) and the atmosphere-only runs by using CAS-ESM2 from 1982 to 2015. Then, we validate and analyze the climatology of SST, precipitation, moisture, large-scale circulation, and their complex interactions over the tropical oceans and different regional areas. The fully coupled nudged-SST run outperforms the fully free coupled run in reproducing the climatology and complex interactions between SST, precipitation, moisture, and large-scale circulations across the tropics. Moreover, the biases in the SST–precipitation covariability are significantly alleviated in the nudging experiments. We confirmed that the SST nudging method is a potential way to improve precipitation simulation in CAS-ESM2.