Bivariate Normal Probability Research Articles

A limit formula and a series expansion for the bivariate Normal tail probability

A limit formula and a series expansion for the bivariate Normal tail probability

A statistical model of routine process monitoring variates and conformance assessment for radiation processing

Modeling a routine radiation process monitoring variate consists of identifying a statistical model that accurately forecasts process variate solution space and provides an inference structure by which processed product conformance to dose range specifications can be assessed with a high degree of surety. This paper shall identify and describe the structural characteristics of the process variates associated with the various forms of routine radiation process monitoring and demonstrate the Bivariate Normal Joint Probability distribution model applicability to routine radiation processing end to end. The routine radiation process end to end consists of:a.Performance qualification dose mapping (PQ)b.Capability/reliability assessmentc.Process target optimization (PTO)d.Statistical process control (SPC) of the routine processe.Product conformance assessment of processed products to dose range specifications.The significance of this model is it is a probabilistic risk/confidence assessment of the process output variate. This treats the sampled variate output as expressing the aggregate variance of all expressing predictor variables (common cause), accurately treats output variates as random variates (solution space) and avoids the errors of the current routine radiation process monitoring industry model.

Estimation of order restricted location/scale parameters of a general bivariate distribution under general loss function: some unified results

We consider component-wise equivariant estimation of order restricted location/scale parameters of a general bivariate distribution under quite general conditions on underlying distributions and the loss function. This paper unifies various results in the literature dealing with sufficient conditions for finding improvements over arbitrary location/scale equivariant estimators. The usefulness of these results is illustrated through various examples. A simulation study is considered to compare risk performances of various estimators under bivariate normal and independent gamma probability models. A real-life data analysis is also performed to demonstrate applicability of the derived results.

Bivariate bias correction of the regional climate model ensemble over the Adriatic region

AbstractThe main goal of this analysis is to provide an assessment of the fine grid regional climate model (RCM) ensemble over the broader Adriatic region, which is the domain with orographically complex terrain and a developed coastline. The results of this study are a beneficial complement to earlier work on using modified bivariate bias correction methods. This analysis was based on an ensemble of 12 combinations of three RCMs and four global climate models (GCMs) from 1971 to 2004, divided into calibration and validation parts. A comparison of one univariate and three versions of bivariate bias correction methods of summer and winter monthly temperature and precipitation was performed. To examine the effect of various methods, we used different marginal distributions and distributions of dependence (copula). Bivariate bias correction was conducted with parametric and empirical marginal distributions. In the quantile mapping (univariate) and the parametric bivariate method, gamma (precipitation) and normal (temperature) probability distributions were used. We documented the bias and the influence of bias correction methods on precipitation and temperature, including their climate change signals in historical data. The considered methods maintained the spatial distribution of trends from the native ensemble. If the relationship between variables was not distinct and the correlation coefficient was low, bivariate bias correction methods tended to weaken the correlation coefficient from the model. Two experiments on how the bias correction methods influence statistical measures of the considered variables and their relationships showed that the bivariate method with empirical distributions (eeG) performed slightly better than other methods. Although the eeG method performed better than the other methods, none of the considered methods was able to fully preserve the correlation coefficient between variables detected from observations. Considering two copulas (Gaussian and Student's t), we conclude that the effect of different copula types on method performance is small.

The evaluation of bivariate normal probabilities for failure of parallel systems

The evaluation of bivariate normal probabilities for failure of parallel systems

A Sample Covariance-Based Approach For Spatial Binary Data

The field of landscape genetics enables the study of infectious disease dynamics by connecting the landscape features with evolutionary changes. Quantifying genetic correlation across space is helpful in providing insight into the rate of spread of an infectious disease. We investigate two genetic patterns in spatially referenced single-nucleotide polymorphisms (SNPs): isolation by distance and isolation by resistance. We model the data using a Generalized Linear Mixed effect Model (GLMM) with spatially referenced random effects and provide a novel approach for estimating parameters in spatial GLMMs. In this approach, we use the links between binary probit models and bivariate normal probabilities to directly compute the model-based covariance function for spatial binary data. Parameter estimation is based on minimizing sum of squared distance between the elements of sample covariance and model-based covariance matrices. We analyze data including Brucella Abortus SNPs from spatially referenced hosts in the Greater Yellowstone Ecosystem.

Wandering corrections from PIV measurements of tornado-like vortices

Wandering of tornado-like vortices consists in random oscillations of the vortex core from its time-averaged position, which complicates efforts to characterize vortex characteristics. A procedure is then necessary to retrieve characteristics of tornado-like vortices not affected by wandering smoothing effects. This study explores two procedures to correct wandering effects on Particle Image Velocimetry data obtained from a down-scaled model of the WindEEE Dome simulator. The first procedure re-centers the velocity data as a function of the instantaneous location of the vortex center. The second procedure treats the time-averaged vortex velocity field as the convolution of a bi-variate normal probability density function, which represents the distribution of vortex center locations over horizontal planes orthogonal to the vortex axis and the instantaneous tornado velocity field. Depending on swirl ratio and vortex height, wandering amplitude was generally from 5% to 9% of the updraft radius. The re-centering procedure was found to be more accurate than the deconvolution procedure. When applied to the turbulence statistics of the velocity field, the correction revealed that a higher level of turbulence at the center of low swirl tornadoes is a result of wandering. Also, the corrected shear stresses revealed a spiral pattern for cases of higher swirl. Wandering effects increases with reducing swirl ratio. For this experiment, swirl ratio is reduced from 0.96 down to 0.22 and errors on the vortex core radius as high as 50% and reduction of the maximum tangential velocity of 13% were observed.

Wandering of a Wing-Tip Vortex: Rapid Scanning and Correction of Fixed-Point Measurements

Oscillations of a wing-tip vortex over planes orthogonal to the freestream direction are typically ascribed to a specific fluid dynamic phenomenon, which is referred to as vortex wandering or meandering. Vortex wandering affects noticeably fixed-point measurements, producing measured vortices with larger size and weaker intensity than the actual ones, and thus methods for correction of wandering smoothing effects are needed. For this survey, the tip vortex generated from a tapered NACA 0012 half-wing is measured through the rapid scanning technique, which consists of quickly traversing a five-hole pressure probe through the vortex core to obtain velocity signals practically not affected by wandering. Rapid scanning allows evaluating probability distributions of the vortex center positions, which are found to be characterized by bivariate normal probability density functions. Estimate of wandering smoothing effects on fixed-point measurements is carried out by comparing rapid scanning data not affected by wandering with fixed-point three-sensor hot-film measurements. The vortex strength seems to be the main vortex parameter controlling wandering; wandering amplitude is reduced by increasing wing angle of attack or freestream velocity or by reducing streamwise distance from the wing. However, a sufficiently strong or concentrated vortex can be weakly affected by wandering.

A probabilistic approach to quantifying soil physical properties via time-integrated energy and mass input

Abstract. Soils form as the result of a complex suite of biogeochemical and physical processes; however, effective modeling of soil property change and variability is still limited and does not yield widely applicable results. We suggest that predicting a distribution of probable values based upon the soil-forming state factors is more effective and applicable than predicting discrete values. Here we present a probabilistic approach for quantifying soil property variability through integrating energy and mass inputs over time. We analyzed changes in the distributions of soil texture and solum thickness as a function of increasing time and pedogenic energy (effective energy and mass transfer, EEMT) using soil chronosequence data compiled from the literature. Bivariate normal probability distributions of soil properties were parameterized using the chronosequence data; from the bivariate distributions, conditional univariate distributions based on the age and flux of matter and energy into the soil were calculated and probable ranges of each soil property determined. We tested the ability of this approach to predict the soil properties of the original soil chronosequence database and soil properties in complex terrain at several Critical Zone Observatories in the US. The presented probabilistic framework has the potential to greatly inform our understanding of soil evolution over geologic timescales. Considering soils probabilistically captures soil variability across multiple scales and explicitly quantifies uncertainty in soil property change with time.

Abstract P5-11-10: Evaluation of the relationship between psychological distress and risk for breast cancer

Abstract Introduction: Women at increased risk for breast cancer (due to family history or having a pathogenic germline mutation in a cancer causing gene) have greater distress levels, which can impact quality of life and screening behavior. The goal of our study was to evaluate distress related to different risk factors for breast cancer and examine the stability of this distress over time. Methods: Women at increased risk for breast cancer who enrolled in the High-Risk Breast Program (HRBP) at the University of Vermont Medical Center were eligible for participation. Women with strong family history, a genetic mutation or atypia on biopsy (benign breast disease, BBD) who completed the Impact of Event Scale (IES) at least once were identified. The IES is a self-reported measure designed to measure subjective distress within the past seven days in relation to a stressor. The scale is based on a two-factor intrusion-avoidance model. Linear regression was used to examine change in distress over time and to compare distress levels between risk groups. Both risk measures and distress scores were examined using bivariate plots and normal probability plots. Pearson correlation coefficients and least squares regression coefficients were obtained along with 95% confidence intervals. In addition to treating the risk estimates as continuous measures, these estimates were grouped into low or average risk (<=2%) over five years, moderate risk (2%-4%) over five years, and high risk (>=5%) over five years. Using the ordinal classification into three risk groups, analysis of variance was conducted to detect group differences in average psychological distress using a 5% significance level. Results: The cohort comprised 344 women at increased risk for breast cancer due to a strong family history (80%), a genetic mutation (9%) or benign breast disease (16%). Mean IES score was 16.8 (CI 15.5, 18.2), and indicates high distress levels in this cohort. IES scores decreased over time (p<0.001 and p=0.003 after 4 and 8 years, respectively). Scores did not differ between the overall group and either women with a strong family history (p=0.06) or women with BBD (p=0.8). Women with genetic mutations had higher IES scores than those without (p=0.023). IES scores were positively associated with the number of risk factors an individual had (1, 2, or 3 risk factors: p=0.003). Conclusion: We demonstrated that distress is similar among different risk categories and that distress decreases over time; patterns not previously recognized. Women with benign breast disease have IES scores similar to women with strong family histories of breast cancer. Women with multiple risk factors are also noted to have higher scores. Women with genetic mutations appear to have the highest distress levels. These findings further our understanding of cancer related distress. Targeting women with highest distress levels may improve both quality of life and screening adherence. Citation Format: Vennard K, Crocker A, Ogheneruona A, Cuke M, Wood M. Evaluation of the relationship between psychological distress and risk for breast cancer [abstract]. In: Proceedings of the 2016 San Antonio Breast Cancer Symposium; 2016 Dec 6-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2017;77(4 Suppl):Abstract nr P5-11-10.

Bivariate categorical data analysis using normal linear conditional multinomial probability model.

Bivariate multinomial data such as the left and right eyes retinopathy status data are analyzed either by using a joint bivariate probability model or by exploiting certain odds ratio-based association models. However, the joint bivariate probability model yields marginal probabilities, which are complicated functions of marginal and association parameters for both variables, and the odds ratio-based association model treats the odds ratios involved in the joint probabilities as 'working' parameters, which are consequently estimated through certain arbitrary 'working' regression models. Also, this later odds ratio-based model does not provide any easy interpretations of the correlations between two categorical variables. On the basis of pre-specified marginal probabilities, in this paper, we develop a bivariate normal type linear conditional multinomial probability model to understand the correlations between two categorical variables. The parameters involved in the model are consistently estimated using the optimal likelihood and generalized quasi-likelihood approaches. The proposed model and the inferences are illustrated through an intensive simulation study as well as an analysis of the well-known Wisconsin Diabetic Retinopathy status data.

Bivariate conditioning approximations for multivariate normal probabilities

New formulas are derived for multivariate normal probabilities defined for hyper-rectangular probability regions. The formulas use conditioning with a sequence of bivariate normal probabilities. Th...

Parameter Dependence in Stochastic Modeling—Multivariate Distributions

We start with analyzing stochastic dependence in a classic bivariate normal density framework. We focus on the way the conditional density of one of the random variables depends on realizations of the other. In the bivariate normal case this dependence takes the form of a parameter (here the “expected value”) of one probability density depending continuously (here linearly) on realizations of the other random variable. The point is, that such a pattern does not need to be restricted to that classical case of the bivariate normal. We show that this paradigm can be generalized and viewed in ways that allows one to extend it far beyond the bivariate or multivariate normal probability distributions class.

Probabilistic Formulation for Emergency Facility Location in a Divided Area

Capacitated minimax facility location-allocation (LA) is a type of facility problem which is of prime importance for emergency situation, since it directly affects the service response time to customers. Capacitated minimax LA problem is concerned with locating some new facilities and allocating their capacity to customers when the maximum travelled distance from customers to facilities is minimized. This study involves a fixed line barrier in a region with some border crossings along it which divides the area into two subregions. Although several studies have recently been done on this problem in which the customer locations are known with certainty, much less attention has been devoted to developing a comprehensive mathematical model for the probabilistic extension of customer locations when there are some restrictions in the region. In order to fill this gap, a Mixed Integer Nonlinear Programming (MINLP) model is proposed for facility LA when customer locations are randomly distributed according to a bivariate normal probability distribution. Finally, the BARON solver in the GAMS software is used to solve the model, and a numerical example is provided to demonstrate its efficiency.

Morphometric Sex Determination of North American Golden Eagles

We made up to 12 measurements of 79 captured Golden Eagles (Aquila chrysaetos canadensis )t o evaluate best metrics for noninvasive, morphometric determination of sex. Sex of 43 male and 36 female Golden Eagles was confirmed post-release: 60 by DNA analysis and 19 by position during copulation. Eagles in adult plumage made up 57% of eagles of confirmed sex. All male and female morphometric means differed (P , 0.01) but most (n 5 10) metric ranges overlapped .10% between sexes. There was no overlap between sexes for the hallux claw (HAL) and head length (HEAD) metrics, regardless of age class. All male HAL and HEAD measurements were #51.7 mm and 119.5 mm, respectively. All female HAL and HEAD metrics were $51.6 mm and 119.8 mm, respectively. Multiple regression analysis indicated HAL and HEAD metrics were best of 12 morphometrics as indicators of sex (P , 0.04). Factorial ANOVAs showed no effect of age class and age class-by-sex interaction on dependent variables HAL and HEAD (P. 0.05). Sex assignments by plotting HAL and HEAD metrics of known-sex eagles relative to bivariate normal probability distribution (BNPD) percentile curves were 100% correct. Discriminate score (DS) derived from discrim- inate function analysis (DFA) incorporating HAL and HEAD metrics classified our sample eagles with 100% accuracy. Confirmatory analyses were 100% accurate. We also evaluated Bortolotti's (1984, Journal of Field Ornithology 55:54-66) methods of sex assignment using eagles we captured: culmen length (CL) and HAL correctly identified 89% of our known-sex eagles. Six of seven (86%) incorrect designations using his age- class dependent models were males classified as females, likely due to the variable effects of overgrown CLs or shrinkage in the museum specimens Bortolotti used. We propose using empirical data-driven BNPD plots first and then DS models if needed to assign sex to Golden Eagles .3.5 mo old in the field. Magnitude of the species' sexual dimorphism may mask clinal differences in intrasex HAL and HEAD metrics through- out the species' latitudinal range and these metrics may be accurate indicators of sex, regardless of age or region of origin in western North America.

Ensemble Model Output Statistics for Wind Vectors

Abstract A bivariate ensemble model output statistics (EMOS) technique for the postprocessing of ensemble forecasts of two-dimensional wind vectors is proposed, where the postprocessed probabilistic forecast takes the form of a bivariate normal probability density function. The postprocessed means and variances of the wind vector components are linearly bias-corrected versions of the ensemble means and ensemble variances, respectively, and the conditional correlation between the wind components is represented by a trigonometric function of the ensemble mean wind direction. In a case study on 48-h forecasts of wind vectors over the North American Pacific Northwest with the University of Washington Mesoscale Ensemble, the bivariate EMOS density forecasts were calibrated and sharp, and showed considerable improvement over the raw ensemble and reference forecasts, including ensemble copula coupling.

Minimizing supply airdrop collateral damage risk

Airdropped supplies are an essential part of military missions and humanitarian disaster support. This paper presents a methodology to determine the optimal location for making supply airdrops in order to minimize collateral damage from the drop while maintaining a high likelihood of successful recovery. Areas in the landing zone that are to be avoided are assigned weights and collateral damage risk is calculated as the weighted integral of the bivariate normal distribution over the undesirable areas in the scene. Optimum location of a drop consists of determining the angle of aircraft approach and (x, y) target. A series of non-linear optimization algorithms is presented along with their relative success in finding the optimal location in the airdrop problem. Specifically, two algorithms which leverage off of the bivariate normal probability distribution nature of airdrops are demonstrated to outperform other techniques in terms of speed and accuracy. The effects of changing airdrop parameters on expected collateral damage are discussed to lend advice to airdrop planners.

Correction of wandering smoothing effects on static measurements of a wing-tip vortex

Wandering is a typical feature of wing-tip vortices and it consists in random fluctuations of the vortex core. Consequently, vortices measured by static measuring techniques appear to be more diffuse than in reality, so that a correction method is needed. In the present paper statistical simulations of the wandering of a Lamb-Oseen vortex are first performed by representing the vortex core locations through bi-variate normal probability density functions. It is found that wandering amplitudes smaller than 60% of the core radius are well predicted by using the ratio between the RMS value of the mean cross-velocity and its slope measured at the mean vortex center. Furthermore, the principal axes of wandering can be accurately evaluated from the opposite of the cross-correlation coefficient between the spanwise and the normal velocities measured at the mean vortex center. The correction of the wandering smoothing effects is then carried out through four different algorithms that perform the deconvolution of the mean velocity field with the probability density function that represents the wandering. The corrections performed are very accurate for the simulations with wandering amplitudes smaller than 60% of the core radius, whereas errors become larger with increasing wandering amplitudes. Subsequently, the whole procedure to evaluate wandering and to correct the mean velocity field is applied to static measurements, carried out with a fast-response five-hole pressure probe, of a tip vortex generated from a NACA 0012 half-wing model. It is found that the wandering is predominantly in the upward-outboard to downward-inboard direction. Furthermore, the wandering amplitude grows with increasing streamwise distance from the wing, whereas it decreases with increasing angle of attack and free-stream velocity.

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.

Speed and accuracy comparison of bivariate normal distribution approximations for option pricing

Pricing compound and min–max options requires approximation of the bivariate normal probability. We compare the performance of five analytical approximation methods for bivariate normal probabilities used in the computation of compound and min–max options against an externally tested benchmark of Simpson numerical integration. Each of the methods is very accurate with all probability errors less than 10–6 and the average probability error less than 10–7. The maximum error in an option price calculation is US$0.01, the average error is less than 2.0 × 10–4, and an error of as large as US$0.01 is rare. The Divgi method is the most accurate method for compound options, and the Owen method is the most accurate for min–max options. The Drezner–Wesolowsky method performs well in terms of accuracy and best in terms of speed. No single method emerges as the best overall, though the more widely cited Drezner method is consistently the least accurate, as well as the second slowest method.