Bivariate Probability Function Research Articles

Screening of short-lived climate pollutants in a street canyon in a mid-sized city in Brazil

Black carbon (BC) and tropospheric ozone (O3) are two main short-lived climate pollutants also linked to health effects. They are ubiquitous in street canyons, since this environment is a hotspot for traffic-related pollutants due to their particular airflow characteristics, location within the cities and the high density of vehicles and population.We report on BC and O3 concentrations measured in a Brazilian city in November 2014. Measurements of BC at 880 nm wavelength were conducted in a street canyon on the north and south façades and at rooftop level (7 wavelengths, including 880 nm) whereas O3 was recorded only on the south façade. Concurrent meteorological data were gathered at a suburban and a rooftop sites.Clear diurnal patterns were found for BC related to traffic emissions and atmospheric mixing conditions. Ozone peaked in the afternoon in response to maximum photochemical production and at night most likely linked to vertical and/or horizontal transport. By using conditional bivariate probability functions, we identified on-road traffic as the main local source for BC during daytime, and at night an intermittent signal was associated with local waste and biomass burned on the city's outskirts. A complementary air backward trajectory analysis helped conclude that locally produced O3 was enhanced by regional transport from large cities and/or biomass smoke.Mitigation strategies for BC and O3 depletion should target the vehicle fleet, particularly diesel buses, reduction of biomass and waste burning at local level, and decrease of open biomass burning in large areas in Brazil and neighbor countries.

BTX in urban areas of eastern Spain: a focus on time variations and sources.

Seasonal and daily cycles of BTX were studied in a non-industrialized city (Alicante) and an urban area near an oil refinery plant (Castellón) in order to evaluate the influence of different sources on time variations. Lower levels were observed in summer than in winter at both locations due to higher dispersion conditions and photochemical removal of BTX during the summer season. Daily patterns showed seasonal differences and were controlled by traffic emissions and the evolution of the mixing layer height, with no influence of the petroleum refinery plant in the city of Castellón. The results of the conditional bivariate probability function suggest that the influence of this source on BTX concentrations was limited to point impacts. At both sites, benzene exhibited a different behavior from toluene and xylenes, most likely due to its significantly lower chemical reactivity.

Composition and source apportionment of PM1 at urban site Kanpur in India using PMF coupled with CBPF

This study addresses the three major questions: (1) what are the emission sources of PM1 which are affecting the study area; (2) where do these emission sources come from; and (3) is there any temporal variation in the emission sources. To address these issues, two advanced statistical methods are described in this paper. Identification of emission sources was performed by EPA PMF (v 5.0) and to understand the temporal variability, sampling was done for three winter seasons 2008–09, 2009–10 and 2011–12 within Kanpur city. To identify the possible source directions, Conditional Bivariate Probability function (CBPF) was used. The average PM1 concentration was higher in 2008–09 followed by 2011–12 and 2009–10 winter seasons. 2008–09 winter showed sources such as secondary sources mixed with power plant emission (42.8%), industrial emission (32.3%), coal combustion, brick kilns and vehicular emission (13.2%) and residual oil combustion and road dust (11.7%). The major contributors during winter season 2009–10 were secondary sources (33.1%), biomass burning (23.3%), heavy oil combustion (13%), vehicular emission mixed with crustal dust (11.3%), leather tanning industries (10.3%), industrial emission (4%), coal combustion and brick kilns (3.4%) and solid waste burning and incineration (1.5%) compared to secondary sources mixed with biomass burning (42.3%), industrial emission and crustal dust (35.1%) and vehicular emission and brick kilns (22.6%) during 2011–12 winter season. PMF model revealed that secondary sources were the main contributors for all the three winter seasons followed by biomass burning and power plant emission. The results of CBPF analysis agreed well with the locations of known local point sources., e.g. in the case of industrial emissions, the maximum probability was in the direction between NES direction where almost all the major industries are located in and around Kanpur while in the opposite direction the probability of biomass burning was high due to a rural area in NWS direction.

Carbonaceous PM2.5 and secondary organic aerosol across the Veneto region (NE Italy)

Organic and elemental carbon (OC–EC) were measured in 360 PM2.5 samples collected from April 2012 to February 2013 at six provinces in the Veneto region, to determine the factors affecting the carbonaceous aerosol variations. The 60 daily samples have been collected simultaneously in all sites during 10 consecutive days for 6months (April, June, August, October, December and February). OC ranged from 0.98 to 22.34μg/m3, while the mean value was 5.5μg/m3, contributing 79% of total carbon. EC concentrations fluctuated from 0.19 to 11.90μg/m3 with an annual mean value of 1.31μg/m3 (19% of the total carbon). The monthly OC concentration gradually increased from April to December. The EC did not vary in accordance with OC. However the highest values for both parameters were recorded in the cold period. The mean OC/EC ratio is 4.54, which is higher than the values observed in most of the other European cities. The secondary organic carbon (SOC) contributed for 69% of the total OC and this was confirmed by both the approaches OC/EC minimum ratio and regression. The results show that OC, EC and SOC exhibited higher concentration during winter months in all measurement sites, suggesting that the stable atmosphere and lower mixing play important role for the accumulation of air pollutant and hasten the condensation or adsorption of volatile organic compounds over the Veneto region. Significant meteorological factors controlling OC and EC were investigated by fitting linear models and using a robust procedure based on weighted likelihood, suggesting that low wind speed and temperature favour accumulation of emissions from local sources. Conditional probability function and conditional bivariate probability function plots indicate that both biomass burning and vehicular traffic are probably the main local sources for carbonaceous particulate matter emissions in two selected cities.

Thirteen years of air pollution hourly monitoring in a large city: Potential sources, trends, cycles and effects of car-free days

Thirteen air pollutant concentrations were measured hourly for 13years (2000–2013) at an urban background site of a large city in the eastern Po Valley (Italy) and results were chemometrically analysed. The pollutant list includes CO, NO, NO2, NOx, O3, SO2, benzene, toluene, ethylbenzene, o-, m- and p-xylenes and PM10, all known or suspected of having adverse effects on human health. The hourly data were statistically processed to detect the long-term trends in relation to the changes in the emission scenarios occurred in the last decade. The most probable emission sources and atmospheric photochemical processes were investigated by analyzing the seasonal, weekly, diurnal cycles of pollutants and the lagged correlations amongst pollutants. The role of micro-meteorological factors upon the air quality was assessed by analyzing the relationships with key weather parameters, while the location of the potential sources was studied by matching atmospheric circulation and pollution data through bivariate polar plots and conditional probability functions. In addition, a new statistical procedure is presented and tested to analyze the periods when common mitigation measures were adopted in the city (e.g., the total stop of traffic and car-free days) and to evaluate their real effect upon the air quality. By providing direct information on the levels and trends of key pollutants, this study finally enables some general considerations about air pollution in an important hotspot of Southern Europe, the eastern Po Valley, where the levels of some key pollutants are still far from meeting the EC limit and target values. It may help policy-makers to take successful mitigation measures.

On a Simple Construction of a Bivariate Probability Function With a Common Marginal

We introduce a family of bivariate discrete distributions whose members are generated by a decreasing mass function p, and with margins given by p. Several properties and examples are obtained, including a family of seemingly novel bivariate Poisson distributions.

Data Assimilation by Use of the Iterative Ensemble Smoother for 2D Facies Models

Summary The ensemble Kalman filter (EnKF) and related ensemble-based smoothers are well-suited to history match reservoir models that are multivariate Gaussian. Estimating categorical variables such as facies types is much more difficult with the EnKF, especially when the variables have complex transitional dependencies. In a previous study, the EnKF was used for updating third-order Markov-chain models in one dimension with an efficient post-processing step to ensure that the posterior samples are constrained by the prior. The efficiency of the post-processing step depended on the use of an optimization algorithm (Viterbi algorithm) that is not directly applicable in higher dimensions. In this paper, the post-processing step is carried out with a sequential noniterative optimization algorithm that readily extends to higher dimensions. An iterative ensemble-based data-assimilation method by use of Levenberg-Marquardt (LM) regularization—namely, LM ensemble randomized maximum likelihood (LM-EnRML)—is used to update reservoir properties to honor production data disregarding the categorical feature of the facies model. The ensemble of realizations updated with LM-EnRML is used to approximate the likelihood of the model variables given data, and bivariate transition probability functions are used to represent the joint probability of the prior facies model. At the post-processing step, the facies type at each gridblock is determined sequentially to maximize the posterior probability, given the approximate prior and likelihood. We demonstrate the approach by conditioning three binary-facies models with channel structure and a three-facies model of sand dunes to nonlinear observations. Our results show the updated facies models honor production data very well, and the transitions among facies are consistent with the prior model.

Conditional bivariate probability function for source identification

In this paper a new receptor modelling method is developed to identify and characterise emission sources. The method is an extension of the commonly used conditional probability function (CPF). The CPF approach is extended to the bivariate case to produce a conditional bivariate probability function (CBPF) plot using wind speed as a third variable plotted on the radial axis. The bivariate case provides more information on the type of sources being identified by providing important dispersion characteristic information. By considering intervals of concentration, considerably more source information can be revealed that is absent in the basic CPF or CBPF. We demonstrate the application of the approach by considering an area of high source complexity, where many new sources can be identified and characterised compared with currently used techniques. Dispersion model simulations are undertaken to verify the approach. The technique has been made available through the openair R package.

Validation of three new measure-correlate-predict models for the long-term prospection of the wind resource

The estimation of the long-term wind resource at a prospective site based on a relatively short on-site measurement campaign is an indispensable task in the development of a commercial wind farm. The typical industry approach is based on the measure-correlate-predict (MCP) method where a relational model between the site wind velocity data and the data obtained from a suitable reference site is built from concurrent records. In a subsequent step, a long-term prediction for the prospective site is obtained from a combination of the relational model and the historic reference data. In the present paper, a systematic study is presented where three new MCP models, together with two published reference models (a simple linear regression and the variance ratio method), have been evaluated based on concurrent synthetic wind speed time series for two sites, simulating the prospective and the reference site. The synthetic method has the advantage of generating time series with the desired statistical properties, including Weibull scale and shape factors, required to evaluate the five methods under all plausible conditions. In this work, first a systematic discussion of the statistical fundamentals behind MCP methods is provided and three new models, one based on a nonlinear regression and two (termed kernel methods) derived from the use of conditional probability density functions, are proposed. All models are evaluated by using five metrics under a wide range of values of the correlation coefficient, the Weibull scale, and the Weibull shape factor. Only one of all models, a kernel method based on bivariate Weibull probability functions, is capable of accurately predicting all performance metrics studied.

Health care utilization and self-assessed health: specification of bivariate models using copulas

The discernment of relevant factors driving health care utilization constitutes one important research topic in health economics. This issue is frequently addressed through specification of regression models for health care use (y—often measured by number of doctor visits) including, among other covariates, a measure of self-assessed health (sah). However, the exogeneity of sah within those models has been questioned, due to the possible presence of unobservables influencing both y and sah, and because individuals’ health assessments may depend on the quantity of medical care received. This article addresses the possible simultaneity of (sah,y) by adopting a full information approach, through specification of the bivariate probability function (p.f.) of these discrete variables, conditional on a set of exogenous covariates (x). The approach is implemented with copula functions, which afford separate consideration of each variable margin and their dependence structure. The specification of the joint p.f. of (sah,y) enables estimation of several quantities of potential economic interest, namely features of the conditional p.f. of y given sah and x. The adopted models are estimated through maximum likelihood, with cross-sectional data from the Portuguese National Health Survey of 1998–1999. Estimates of the margins’ parameters do not vary much among different copula models, while, in accordance with theoretical expectations, the dependence parameter is estimated to be negative across the various joint models.

Preisach Modeling of AFC Magnetic Recording Media

The ability of the Preisach formalism to model hysteresis processes in antiferromagnetically coupled (AFC) recording media is investigated in this paper. Two modeling approaches are proposed, each one using a weighed mixture of two normal bivariate probability function densities as the characteristic density and either classical or appropriately modified one-dimensional or two-dimensional hysteresis operators. The models are identified for three samples of different lower layer thickness, t/sub L/. In all cases, the resulting weighing factor w/sub L/ correlates with t/sub L/ as do some of the density parameters with material parameters. The calculated major loops compare well with the experimental data while the discrepancy of the calculated minor loops at fields lower than the coercive field indicates the limitations of the proposed models.

Hysteresis modeling of Gd films and AFC thin-film recording media

This paper presents simulations of hysteresis processes in thin film media using 1D and 2D Preisach models. In the 2D version, a vector operator and superposition of angularly distributed models are used. The characteristic density of the material being modeled is reconstructed via a curve-fitting least-squares procedure that determines the parameters of a bivariate normal probability function density or a weighed mixture of normal densities based on major loop data only. The models have been identified for several samples of Gd films annealed at various temperatures and AFC thin-film recording media consisting of a hard and a soft phase antiferromagnetically coupled. The major and minor hysteresis loops calculated for all samples are in good agreement with experimental data.

On Bivariate Cumulative Damage Models With Application

Bivariate cumulative damage models are proposed where the responses given the damages are independent random variables. The bivariate damage process can be either bivariate Poisson or bivariate gamma. A bivariate continuous cumulative damage model is investigated in which the responses given the damages have gamma distributions. In this case evaluation of the joint density function and bivariate tail probability function is facilitated by expanding the gamma distributions of the conditional responses by Laguerre polynomials. This approach also leads to evaluation of associated survival models. Moments and estimating equations are discussed. In addition, a bivariate discrete cumulative damage model is investigated in which the responses given the damages have a distribution chosen from a class that includes the negative binomial, the Neyman Type-A, the Polya-Aeppli, and the Lagrangian Poisson. Probabilities are obtained from recursive formulas which do not involve cancellation error as all quantities are non-negative. Moments and estimating equations are presented for these models also. The continuous and the discrete models are applied to describe the rise of systolic and diastolic blood pressure with age.

From snakes to stars: the statistics of collapsed objects -- II. Testing a generic scaling ansatz for hierarchical clustering

We develop a diagrammatic technique to represent the multipoint probability density function of mass fluctuations in terms of the statistical properties of individual collapsed objects, and relate this to other statistical descriptors such as cumulants, cumulant correlators and factorial moments. We use this approach to establish key scaling relations describing various measurable statistical quantities if clustering follows a simple general scaling ansatz, as expected in hierarchical models. We test these detailed predictions against high-resolution numerical simulations. We show that, when appropriate variables are used, the count probability distribution function (CPDF) shows clear scaling properties in the non-linear regime. We also show that analytic predictions made using the scaling model for the behaviour of the void probability function (VPF) also match the simulations very well. We generalize the results for the CPDF to the two-point (bivariate) count probability distribution function (2CPDF), and show that its behaviour in the simulations is also well described by the theoretical model, as is the bivariate void probability function (2VPF). We explore the behaviour of the bias associated with collapsed objects in the limit of large separations, finding that it depends only on the intrinsic scaling parameter associated with collapsed objects, and that the bias for two different objects can be expressed as a product of the individual biases of the objects. Having thus established the validity of the scaling ansatz in various different contexts, we use its consequences to develop a novel technique for correcting finite-volume effects in the estimation of multipoint statistical quantities from observational data.

A class of log-linear models with constrained marginal distributions

In the log-linear model for bivariate probability functions the conditional and joint probabilities have a simple form. This property make the log-linear parametrization useful when modeling these probabilities is the focus of the investigation. On the contrary, in the log-linear representation of bivariate probability functions, the marginal probabilities have a complex form. So the log-linear models are not useful when the marginal probabilities are of particular interest. In this paper the previous statements are discussed and a model obtained from the log-linear one by imposing suitable constraints on the marginal probabilities is introduced.

Discrete bivariate distributions with given marginals and correlation

Using convex linear combinations of the probability functions for the discrete Frechet boundary distributions and the probability function for independent random variables, we construct bivariate probability functions for dependent discrete random variables with arbitrary marginals and any correlation between the theoretical minimum and maximum. The technique is elementary and the results can be adapted readily for simulation.

A model function for ocean radar cross sections at 14.6 GHz

The relationship between the ocean's normalized radar cross section (NRCS) at 14.6 GHz and the surface wind vector is derived using the 3 months of Seasat microwave scatterometer (SASS) measurements. The derivation is based on the statistics of the SASS observations, and no in situ measurements are required, other than a mean global wind speed, which comes from climatology. The frequency distribution of the global wind vectors observed by SASS is assumed to be a bivariate normal probability function. A NRCS model function is found that maps the assumed wind vector statistics into the observed SASS NRCS statistics. This function is compared with a NRCS model coming from the Joint Air Sea Interaction Experiment (JASIN) and with aircraft scatterometer measurements. The results indicate that the statistically derived NRCS model is an improvement over the JASIN model, which was based on a limited number of in situ anemometer measurements.

Calculation of Univariate and Bivariate Normal Probability Functions

Mill's ratio is expressed as a convergent series in orthogonal polynomials. Truncation of the series provides an approximation for the complemented normal distribution function $Q(x)$, with its maximum error at a finite value of $x$. The analogous approximation for $xQ(x)$ is used to obtain a new method of calculating the bivariate normal probability function.

A Note on the Trinomial Distribution

In a first course on probability and distribution theory, when bivariate distributions are discussed, the trinomial distribution is presented as an example of a bivariate distribution with binomial marginals. Later, the student encounters Aitken and Gonin's (1935) bivariate binomial distribution arising from fourfold sampling with replacement (c.f. Kendall and Stuart, 1963, p. 141). It is unfortunate, however, that textbooks do not relate the trinomial distribution to Aitken and Gonin's bivariate binomial distribution. This may lead to the misunderstanding that the two distributions are bivariate binomial distributions which are different in structure. This misunderstanding grows further when students are introduced to the canonical expansion of Aitken and Gonin's bivariate binomial probability function, that is, as a series bilinear in Krawtchouk's polynomials. A student would think that the trinomial distribution, which is very simple in nature, cannot be related to something as complex as Krawtchouk's polynomials. The aim of the present note is to point out the fact that the two distributions have similar structures, and in fact the trinomial distribution is a special case of Aitken and Gonin's bivariate binomial distribution. Hence, the canonical structure of the trinomial distribution is indicated. Aitken and Gonin's (1935) bivariate binomial is based on a population in which each individual may be classified as either A or A and at the same time as either B or B. Each individual in a sample of size n is then classified in one of the four mutually exclusive classes AB, AB, AB, AB with probabilities pit, Pto, Pox, Poo, respectively. If X1 is the number of occurrences of A and X2 is the number of occurrences of B, then the marginal distributions of Xx and X2 are binomial (n, pt) and binomial (n, P2), where

On Calculating the Probability Distribution of Extreme Magnitudes of Switching Surge Overvoltages

A mathematical method for evaluating the probability of extreme magnitudes of switching surge overvoltages is proposed which includes correlation among the three phase circuit breaker pole closing times. The model utilizes published data for trivariate and bivariate normal probability functions. Examples are presented which illustrate critical aspects of the calculations, and suggestions are made regarding specific applications.