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Tinh Tuc town, Cao Bang province, Vietnam is prone to landslides due to the complexity of its climatic, geological, and geomorphological conditions. In this study, in order to produce a landslide susceptibility map, the modified analytical hierarchy process and landslide susceptibility analysis methods were used together with the layers, including: landslide inventory, slope, weathering crust, water storage, geology, land use, and distance from the road. In the study area, 98% of landslides occurred in highly or completely weathered units. Geology, land use, and water storage data layers were found to be important factors that are closely related with the occurrence of landslides. Although the weight of the “distance from the road” factor has a low value, the weight of layer “

I used linear and nonlinear regression to re-examine the allometric relationship between length of the cephalic horn and width of the pronotum (a measure of body size) for males of 28 species of rhinoceros beetle (Coleoptera, Scarabaeidae, Dynastinae). An earlier investigation reported that variation in relative size of the horn conforms in most of the 28 species to a pattern of discontinuous, loglinear allometry, which implies, in turn, that two different power equations are needed to describe the full range in untransformed data. My analyses indicate, however, that allometric variation is continuous in all 28 species and that pattern in each of the original bivariate distributions is well described by a single mathematical function. Allometric variation on the original, arithmetic scale is linear for some species; either linear or slightly sigmoidal for other species; and decidedly sigmoidal in still other species. So-called major and minor morphs merely represent the extremes in absolute and relative size for the horn on animals drawn from the same, continuous distribution. The different findings reported in the earlier investigation resulted from insidious problems introduced by logarithmic transformation of the original bivariate data coupled with an overly complex statistical protocol for classifying individual males as either “major” or “minor.” Attempts at classification lead inevitably to oversimplification and misrepresentation of the pattern of allometric variation. Focus needs to return to describing and interpreting pattern in untransformed observations because the relationship between the original variables is the relationship of interest.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiment, twin or family data), the shared frailty models were suggested. These models are based on the assumption that frailty acts multiplicatively to hazard rate. In this paper, we assume that frailty acts additively to hazard rate. We introduce the positive stable shared frailty models with three different baseline distributions namely, the generalized log-logistic and the generalized Weibull distributions. We introduce the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We apply these models to a real-life bivariate survival data set of McGilchrist and Aisbett (Biometrics 47:461–466, 1991) related to the kidney infection data and a better model is suggested for the data.

In this work, we utilise a recently developed method of constructing a bivariate distributional model to generate a new bivariate distribution with Weibull and Burr type XII distributions as its marginals. Some general characteristics of this newly generated distribution together with some characterization results are further derived. The maximum likelihood method of estimation is applied for the estimation of the parameters of the constructed distribution. The closeness of the maximum likelihood estimators with the true values of the parameters have been illustrated through a simulation study. We have identified the generated model as a suitable model for a real life bivariate data set reported in the literature. We have also used another bivariate data set on geoelectrical variables X and Y in which X represents the aquifer thickness and Y represents the coefficient of anisotropy to illustrate the results described in this paper for bivariate distributional modelling.

In this work, we investigate the asymptotic behaviour of the intermediate and central order statistics of a bivariate data by using the Reduced Ordering Principle (R-ordering). When, the sup-norm i...

Methods to compare two samples of interval-valued data are discussed. In the interval-valued data, observations have the form of intervals and are often read as two-dimensional vector of lower and upper bounds. We consider four methods, two of which are the applications of existing methods for bivariate data, and the other two are based on the marginalization (univariate distributional representation) of the interval-valued data. We conduct a comprehensive numerical study and analysis of real data to understand the performance of four methods.

The relationship of bivariate data ordinarily measured using correlation coefficient. The most commonly used correlation coefficient is the Pearson correlation coefficient. This coefficient is well-known as the best coefficient for interval or ratio bivariate data with a linear relationship. Even though this coefficient is good under the mentioned condition, it also becomes very sensitive to a small departure from linearity. Usually, this is because of the existence of an outlier. For that reason, this paper provides new robust correlation coefficients which combine the elements of nonparametric technique from the Hodges Lehmann estimator and the parametric technique based on the Pearson correlation coefficient. This paper also introduces different scale estimators such as median and median absolute deviation (MADn) and denoted by rHL(med) and rHL(MADn) respectively. The performance of the proposed correlation coefficients is measured by the coefficient values and these values are also being compared to the Pearson correlation coefficient and several existing robust correlation coefficients. The results show that the Pearson correlation coefficient (r) with no doubt is very good under perfect data condition, but with only 10% outliers, it not only give poor correlation value but turns the direction of the relationship to negative. While the rHL(med) and rHL(MADn) offer the highest coefficient values and these values are robust to the existence of outliers by up to 30%. With very good performance under all data conditions yet simple in the calculation, the rHL(med) and rHL(MADn) is considered a good alternative to the r when need to deal with outliers. Keywords: correlation coefficient; Hodges Lehmann; median; median absolute deviation (MADn)

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the s...

Bivariate data consist of 2 random variables that are obtained from the same population. The relationship between 2 bivariate data can be measured by correlation coefficient. A correlation coefficient computed from the sample data is used to measure the strength and direction of a linear relationship between 2 variables. However, the classical correlation coefficient results are inadequate in the presence of outliers. Therefore, this study focuses on the performance of different correlation coefficient under contaminated bivariate data to determine the strength of their relationships. We compared the performance of 5 types of correlation, which are classical correlations such as Pearson correlation, Spearman correlation and Kendall’s Tau correlation with other robust correlations, such as median correlation and median absolute deviation correlation. Results show that when there is no contamination in data, all 5 correlation methods show a strong relationship between 2 random variables. However, under the condition of data contamination, median absolute deviation correlation denotes a strong relationship compared to other methods.

A three-parameter bivariate distribution is derived from the marginal and conditional Weibull distributions. Its joint properties are derived and method of estimation of its parameters discussed. It is generalised for the multivariate case when the assumptions of piecewise conditional independence of the variables is tenable. It is applied to a bivariate data set of wind speed and solar radiation, where the conditional dependence of the two variables involved is suspected. The result shows that the model is compatible with the data. It is proposed for application when there is need to allow for such conditional structure in a bivariate model.

Resumen Existen diferentes métodos que permiten establecer a escalas regionales la susceptibilidad a la ocurrencia de movimientos en masa. Entre los métodos más utilizados se encuentran los métodos estadísticos bivariado y multivariado, los cuales exigen un inventario de procesos de remoción en masa. En el presente estudio se evalúa y zonifica la susceptibilidad por movimientos en masa en el norte de los Andes colombianos, región conocida como valle de Aburrá, por dos métodos estadísticos, uno de ellos bivariado, denominado Peso de la Evidencia, y recomendado por el Servicio Geológico Colombiano para estudios de amenaza en suelos rurales; y un segundo método estadístico tipo multivariado, denominado Regresión Logística, de amplio uso a nivel mundial. Para ambos casos, la construcción del modelo de susceptibilidad se realizó soportado en el histograma de frecuencias, correlación de Pearson, Análisis Discriminante y Análisis de Componentes Principales. Para evaluar el desempeño, la capacidad de predicción y los criterios de zonificación en alto, medio y bajo de cada uno de los métodos utilizados se utilizó el análisis ROC. Para la regresión logística se obtuvo un área bajo la curva del 76.8 % para el desempeño y 77.5 % para la capacidad de predicción, mientras que para el Peso de la Evidencia se obtuvo un 77.8 % en el desempeño y 77.5 % en la predicción, señalando resultados satisfactorios que permiten la incorporación de dichos resultados en los estudios básicos necesarios para la ordenación del territorio. Abstract Evaluating and zoning mass movement landslide is a fundamental tool for land planning. There are different methods that help to establish in a regional scale landslide hazard. The most common methods are bivariate statistics and multivariate statistic, which need a landslide historical inventory. This study takes a place in the North of the Andes in a region of Colombia called Aburrá Valley, for evaluating and zoning landslide susceptibility by two methods, one of them is bivariate statistics called Weights of Evidence which is recommended by the Geological Service of Colombia for rural area, and the second one is a multivariate statistic method, called logistic regression, which is widely used worldwide. Both methods are supported in frequency histogram, Pearson correlation, Discriminant Analysis, and Principal Component Analysis. The accuracy of the landslide susceptibility maps produced from the two models is classified in high, medium and low by ROC analysis. The AUC plot estimation results showed that the susceptibility map using Logistic regression has a training accuracy of 76.5% and a prediction capacity of 77.5 %. The Weights of evidence method has the highest training accuracy of 77.8 % and a prediction of 77.5 %. This result allows to include the methods in territorial planning studies.

The definition of multivariate quantiles has gained considerable attention in previous years as a tool for understanding the structure of a multivariate data cloud. Due to the lack of a natural ordering for multivariate data, many approaches have either considered geometric generalisations of univariate quantiles or data depths that measure centrality of data points. Both approaches provide a centre-outward ordering of data points but do no longer possess a relation to the cumulative distribution function of the data generating process and corresponding tail probabilities. We propose a new notion of bivariate quantiles that is based on inverting the bivariate cumulative distribution function and therefore provides a directional measure of extremeness as defined by the contour lines of the cumulative distribution function which define the quantile curves of interest. To determine unique solutions, we transform the bivariate data to the unit square. This allows us to introduce directions along which the quantiles are unique. Choosing a suitable transformation also ensures that the resulting quantiles are equivariant under monotonically increasing transformations. We study the resulting notion of bivariate quantiles in detail, with respect to computation based on linear programming and theoretical properties including asymptotic behaviour and robustness. It turns out that our approach is especially useful for data situations that deviate from the elliptical shape typical for ‘normal-like’ bivariate distributions. Moreover, the bivariate quantiles inherit the robustness of univariate quantiles even in case of extreme outliers.

Hormone assay results below the assay detection limit (DL) can introduce bias into quantitative analysis. Although complex maximum likelihood estimation methods exist, they are not widely used, whereas simple substitution methods are often used ad hoc to replace the undetectable (UD) results with numeric values to facilitate data analysis with the full data set. However, the bias of substitution methods for steroid measurements is not reported. Using a large data set (n = 2896) of serum testosterone (T), DHT, estradiol (E2) concentrations from healthy men, we created modified data sets with increasing proportions of UD samples (≤40%) to which we applied five different substitution methods (deleting UD samples as missing and substituting UD sample with DL, DL/√2, DL/2, or 0) to calculate univariate descriptive statistics (mean, SD) or bivariate correlations. For all three steroids and for univariate as well as bivariate statistics, bias increased progressively with increasing proportion of UD samples. Bias was worst when UD samples were deleted or substituted with 0 and least when UD samples were substituted with DL/√2, whereas the other methods (DL or DL/2) displayed intermediate bias. Similar findings were replicated in randomly drawn small subsets of 25, 50, and 100. Hence, we propose that in steroid hormone data with ≤40% UD samples, substituting UD with DL/√2 is a simple, versatile, and reasonably accurate method to minimize left censoring bias, allowing for data analysis with the full data set.

In Romania, as in the rest of the world, the flood frequency has increased considerably. Prahova river basin is among the most exposed catchments of the country to flood risk. It also represents the area of the present study for which the identification of surfaces with high susceptibility to flood phenomena was attempted by applying 2 hybrid models (adaptive neuro-fuzzy inference system and fuzzy support vector machine hybrid) and 2 bivariate statistical models (certainty factor and statistical index). The computation of Flood Potential Index (FPI) was possible by considering a number of 10 flood conditioning factors together with a number of 158 flood pixels and 158 non-flood pixels. Generally, the high and very high flood potential appears on around 25% of the upper and middle basin of Prahova river. The validation of the results was made through the ROC Curve model. One of the novelties of this research is related to the application of Fuzzy Support Vector Machine ensemble for the first time in a study concerning the evaluation of the susceptibility to a certain natural hazard.

The complex relationship between water quality and river discharge has long been an active area of research. This research applies bivariate statistics to river nitrate concentration and discharge, and develops a procedure for selecting the best bivariate distribution for the chosen models. Nitrate concentration (NO3) data from five stations on the Lower Illinois River, USA, collected from 1972 to 2012 were used for demonstration of this procedure. To explore the statistical relationship between discharge and nitrate concentration, three copula statistical models were applied to determine optimal bivariate distributions. The generalized Pareto (GPA), generalized extreme value (GEV), and lognormal two- (LN2) and three-parameter (LN3) distributions were used as marginal distributions, and were combined with the Clayton, Frank, and Gumbel copula models. The probability plot correlation coefficient (PPCC) test and the S n statistic were used to select the most suitable copula models. As a result, the GPA distributions for streamflow and nitrate concentration in the Frank copula model were more accurate for representing the data from the two mainstem stations, Havana (D-31) and Valley City (D-32), and those from the Oakford station along the Sangamon River (E-25). However, the Clayton copula model using NL3 for representing streamflow and GPA for representing nitrate concentration was more accurate for stations along the Spoon (DJ-08) and La Moine Rivers (DG-01). This study enabled the development of a procedure for determining an optimal copula for application in the environmental sciences and also provided a case study for applying copula models using discharge and nutrient concentration as hydrologic and water quality variables, respectively.

The bivariate Weibull distribution is an important lifetime distribution in survival analysis. In this paper, Farlie–Gumbel–Morgenstern (FGM) copula and Weibull marginal distribution are used for creating bivariate distribution which is called FGM bivariate Weibull (FGMBW) distribution. FGMBW distribution is used for describing bivariate data that have weak correlation between variables in lifetime data. It is a good alternative to bivariate several lifetime distributions for modeling real-valued data in application. Some properties of the FGMBW distribution are obtained such as product moment, skewness, kurtosis, moment generation function, reliability function and hazard function. Three different estimation methods for parameters estimation are discussed for FGMBW distribution namely; maximum likelihood estimation, inference function for margins method and semi-parametric method. To evaluate the performance of the estimators, a Monte Carlo simulations study is conducted to compare the preferences between estimation methods. Also, a real data set is introduced, analyzed to investigate the model and useful results are obtained for illustrative purposes.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival time...

Landslide susceptibility assessment was conducted in Canada Hill, Sarawak, Malaysia through a combined bivariate statistics and expert consultation approach using geographical information system, which captures landslide-conditioning parameters specific to the study area; to ensure its usefulness in practice. Over the past four decades, many landslide parameters and increasingly sophisticated statistical methods have been used in landslide research. However, the findings have had very limited use in practice as the actual ground conditions are not well represented. The weakness is due to poor quality of data in landslide inventories and inadequate understanding of landslide-conditioning parameters. In this study, bivariate statistical method was used in conjunction with an iterative process of expert consultation. Thirteen original landslide-conditioning parameters were narrowed down to six, with the addition of a unique parameter, planar failure potential, which was selected based on expert input. The parameter captures planar failure landslides, which has the highest impact in the study area, causing loss of lives and property destruction. The inaugural landslide susceptibility map for the study area has five classes; very low, low, moderate, high and very high susceptibility. All major planar failures and most smaller circular failures fall within the very high susceptibility class, with a success rate of 75.8%. The approach used in this study has improved the quality of the landslide inventory and delineated key conditioning parameters. The resultant map captures local conditions, which is useful for landslide management.

Many clinical decisions are taken based on the results of continuous diagnostic tests. Usually, only the results of one single test is taken into consideration, the interpretation of which requires a reference range for the healthy population. However, the use of two different tests, can be necessary in the diagnosis of certain diseases. This obliges a bivariate reference region be available for their interpretation. It should also be remembered that reference regions may depend on patient variables (eg, age and sex) independent of the suspected disease. However, few proposals have been made regarding the statistical modeling of such reference regions, and those put forward have always assumed a Gaussian distribution, which can be rather restrictive. The present work describes a new statistical method that allows such reference regions to be estimated with no insistence on the results being normally distributed. The proposed method is based on a bivariate location-scale model that provides probabilistic regions covering a specific percentage of the bivariate data, dependent on certain covariates. The reference region is estimated nonparametrically and the nonlinear effects of continuous covariates via polynomial kernel smoothers in additive models. The bivariate model is estimated using a backfitting algorithm, and the optimal smoothing parameters of the kernel smoothers selected by cross-validation. The model performed satisfactorily in simulation studies under the assumption of non-Gaussian conditions. Finally, the proposed methodology was found to be useful in estimating a reference region for two continuous diagnostic tests for diabetes (fasting plasma glucose and glycated hemoglobin), taking into account the age of the patient Óscar Lado-Baleato is funded by a predoctoral grant (ED481A-2018) from the Galician Government (Plan I2C)-Xunta de Galicia. This research was also supported by grants from the Carlos III Health Institute, Spain (PI16/01404 and RD16/0017/0018), and by the project MTM2017-83513-R cofinanced by the Ministry of Economy and Competitiveness (SPAIN) and the European Regional Development Fund (FEDER). This work was also supported by grants from the Galician Government: RED INBIOEST (ED341D-R2016/032), Grupo de Referencia Competitiva (ED431C 2016-025), and Grupo de Potencial Crecimiento (IN607B 2018-1). Javier Roca-Pardiñas acknowledges financial support by the Grant MTM2017-89422-P (MINECO/AEI/FEDER, UE) SI

Chapter 6 deals with the descriptive statistics of bivariate data and regression analysis. It presents the concepts of covariance and correlation, and their implementation in Python. Then, it shows how to perform linear and nonlinear regression. Finally, it ends with an example of nonlinear regression in earth science: the application of the crystal-lattice-strain model to interpret experimental data in petrology.

Episodes of frequent flooding continue to increase, often causing serious damage and tools to identify areas affected by such disasters have become indispensable in today's society. Using the latest techniques can make very accurate flood predictions. In this study, we introduce four effective methods to evaluate the flood susceptibility of Poyang County, in China, by integrating two independent models of frequency ratio and index of entropy with multilayer perceptron and classification and regression tree models. The flood locations of the study area were identified through the flood inventory process, and 12 flood conditioning factors were used in the training and validation processes. According to the results of the linear support vector machine, elevation, slope angle, and soil have the highest predictive ability. The experimental results of the four hybrid models demonstrate that between 20% and 50% of the study area has high and very high flood susceptibility. The multilayer perceptron-probability density hybrid model is the most effective among the six comparative methods.

Damage assessment is a key element in structural health monitoring of various industrial applications to understand well and predict the response of the material. The big uncertainty in carbon fiber composite materials response is because of variability in the initiation and propagation of damage. Developing advanced tools to design with composite materials, methods for characterizing several damage modes during operation are required. While there is a significant amount of work on the analysis of acoustic emission (AE) from different composite materials and many loading cases, this research focuses on applying an unsupervised clustering method for separating AE data into several groups with distinct evolution. In this paper, we develop an adaptive sampling and unsupervised bivariate data clustering techniques to characterize the several damage initiations of a composite structure in different lay-ups. An adaptive sampling technique pre-processes the AE features and eliminates redundant AE data samples. The reduction of unnecessary AE data depends on the requirements of the proposed bivariate data clustering technique. The bivariate data clustering technique groups the AE data (dependent variable) with respect to the mechanical data (independent variable) to assess the damage of the composite structure. Tensile experiments on carbon fiber reinforced composite laminates (CFRP) in different orientations are carried out to collect mechanical and AE data and demonstrate the damage modes. Based on the mechanical stress-strain data, the results show the dominant damage regions in different lay-ups of specimens and the definition of the different states of damage. In addition, the states of the damage are observed using Scanning Electron Microscope (SEM) analysis. Based on the AE data, the results show that the strong linear correlation between AE and mechanical energy, and the classification of various modes of damage in all lay-ups of specimens forming clusters of AE energy with respect to the mechanical energy. Furthermore, the validation of the cluster-based characterization and improvement of the sensitivity of the damage modes classification are observed by the combined knowledge of AE and mechanical energy and time-frequency spectrum analysis.

Across the world, the flood magnitude is expected to increase as well as the damage caused by their occurrence. In this case, the prediction of areas which are highly susceptible to these phenomena becomes very important for the authorities. The present study is focused on the evaluation of flood potential within Trotuș river basin in Romania using six ensemble models created by the combination of Analytical Hierarchy Process (AHP), Certainty Factor (CF) and Weights of Evidence (WOE) on one hand, and Gradient Boosting Trees (GBT) and Multilayer Perceptron (MLP) on the other hand. A number of 12 flood predictors, 172 flood locations and 172 non-flood locations were used. A percentage of 70% of flood and non-flood locations were used as input in models. From the input data, 70% were used as training sample and 30% as validating sample. The highest accuracy was obtained by the MLP-CF model in terms of both training (0.899) and testing (0.889) samples. A percentage between 21.88% and 36.33% of study area is covered with high and very high flood potential. The results validation, performed through the ROC Curve method, highlights that the MLP-CF model provided the most accurate results.

Researchers often focus on bivariate normal correlation (r) to evaluate bivariate relationships. However, these techniques assume linearity and depend on parametric assumptions. We propose a new nonparametric statistical model that can be more intuitively understood than the conventional r: probability of bivariate superiority (PBS). Our development of Bp, the estimator of a PBS relationship, extends Dunlap’s (1994) common-language transformation of r (CLr) by providing a method to directly estimate PBS—the probability that when x is above (or below) the mean of all X, its paired y score will also be above (or below) the mean of all Y. Probability of superiority is an important form of bivariate relationship that until now could only be accurately estimated when data met the parametric assumptions for r. We specify the copula that forms the theoretical basis for PBS, provide an algorithm for estimating PBS from a sample, and describe the results of a Monte Carlo experiment that evaluated our algorithm across 448 data conditions. The PBS estimate, Bp, is robust to violations of parametric assumptions and offers a useful method for evaluating the significance of probability-of-superiority relationships in bivariate data. It is critical to note that Bp estimates a different form of bivariate relationship than does r. Our working examples show that a PBS effect can be significant in the absence of a significant correlation, and vice versa. In addition to utilizing the PBS model in future research, we suggest that this new statistical procedure be used to find theoretically important but previously overlooked effects from past studies.

A simple sorting approach for inducing any desired Pearson or Spearman correlation to independent bivariate data, whose marginals can be of any distributional type and nature is described and illus...

Cylindrical data are bivariate data from the combination of circular and linear variables. However, up to now no work has been done on the detection of outlier in cylindrical data. We intro...

In this study we introduce a new method of adding two shape parameters to any baseline bivariate distribution function (df) to get a more flexible family of bivariate df's. Through the additional parameters we can fully control the type of the resulting family. This method is applied to yield a new two-parameter extension of the bivariate standard normal distribution, denoted by BSSN. The statistical properties of the BSSN family are studied. Moreover, via a mixture of the BSSN family and the standard bivariate logistic df, we get a more capable family, denoted by FBSSN. Theoretically, each of the marginals of the FBSSN contains all the possible types of df's with respect to the signs of skewness and excess kurtosis. In addition, each possesses very wide range of the indices of skewness and kurtosis. Finally, we compare the families BSSN and FBSSN with some important competitors (i.e., some generalized families of bivariate df's) via real data examples. AMS 2010 Subject Classification: 62-07; 62E10; 62F99.

Most data in environmental sciences and geochemistry are compositional. Already the unit used to report the data (e.g., μg/l, mg/kg, wt%) implies that the analytical results for each element are not free to vary independently of the other measured variables. This is often neglected in statistical analysis, where a simple log-transformation of the single variables is insufficient to put the data into an acceptable geometry. This is also important for bivariate data analysis and for correlation analysis, for which the data need to be appropriately log-ratio transformed. A new approach based on the isometric log-ratio (ilr) transformation, leading to so-called symmetric coordinates, is presented here. Summarizing the correlations in a heat-map gives a powerful tool for bivariate data analysis. Here an application of the new method using a data set from a regional geochemical mapping project based on soil O and C horizon samples is demonstrated. Differences to 'classical' correlation analysis based on log-transformed data are highlighted. The fact that some expected strong positive correlations appear and remain unchanged even following a log-ratio transformation has probably led to the misconception that the special nature of compositional data can be ignored when working with trace elements. The example dataset is employed to demonstrate that using 'classical' correlation analysis and plotting XY diagrams, scatterplots, based on the original or simply log-transformed data can easily lead to severe misinterpretations of the relationships between elements.

Measures of direction dependence enable researchers to determine the directionality of linear effects in bivariate data. Existing fourth moment-based approaches assume that regression error...

Learning to work with bivariate data, a key goal of middle-grades statistics curricula, is aided by a sequence of lessons.

A purely data-based index for detecting bivariate association is proposed for preliminary data exploration when seeking to model a dependent variable, associated with a possibly large number of independent variables. No particular form of association between the dependent and independent variables is assumed. The proposed bivariate association index is the value p , which is the probability that a scatter plot created by an X -randomization will generate a smaller mean nearest neighbour distance. The rationale is that randomizing an existing X-Y association will result in a scatter plot which will usually have a greater mean nearest neighbour distance. The process is then repeated for all other independent variables to give a specific p for each one. A subset of potentially informative independent variables is then obtained by noting all those with low p values, but just how small p should be is left to the user.

Weighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this paper, we propose an alternative method to construct a new family of bivariate and multivariate weighted distributions. For illustrative purposes, some examples of the proposed method are presented. Several structural properties of the bivariate weighted distributions including marginal distributions together with distributions of the minimum and maximum, evaluation of the reliability parameter and verification of total positivity of order two are also presented. In addition, we provide some multivariate extensions of the proposed models. A real life data set is used to show the applicability of these bivariate weighted distributions.

Recently, Mahmoudi and Mahmoodian [7] introduced a new class of distributions which contains univariate normal–geometric distribution as a special case. This class of distributions are very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the normal–geometric distribution marginals. Different properties of this new bivariate distribution have been studied. This distribution has five unknown parameters. The EM algorithm is used to determine the maximum likelihood estimates of the parameters. We analyze one series of real data set for illustrative purposes.

Bivariate counts are collected in many sectors of research but the analysis of such data is often challenging because each series of counts may exhibit different levels and types of dispersion. This paper addresses this problem by proposing a flexible bivariate COM-Poisson model that may handle any combination of over-, equi- and under-dispersion at any levels. In this paper, the bivariate COM-Poisson is developed via Archimedean copulas. The Generalized Quasi-Likelihood (GQL) approach is used to estimate the unknown mean parameters in the copula-based bivariate COM-Poisson model while the dependence parameter is estimated using the copula likelihood. We further introduce a Monte Carlo experiment to generate bivariate COM-Poisson data under different dispersion levels. The performance of the GQL approach is assessed on the simulated data. The model is applied to analyze real-life epileptic seizures data.

Interval-censored failure time data arise in a number of fields and many authors have discussed various issues related to their analysis. However, most of the existing methods are for univariate data and there exists only limited research on bivariate data, especially on regression analysis of bivariate interval-censored data. We present a class of semiparametric transformation models for the problem and for inference, a sieve maximum likelihood approach is developed. The model provides a great flexibility, in particular including the commonly used proportional hazards model as a special case, and in the approach, Bernstein polynomials are employed. The strong consistency and asymptotic normality of the resulting estimators of regression parameters are established and furthermore, the estimators are shown to be asymptotically efficient. Extensive simulation studies are conducted and indicate that the proposed method works well for practical situations. Supplementary materials for this article are ...

In this study, we propose a new class of flexible bivariate distributions for discrete random variables. The proposed class of distribution is based on the notion of conditional failure rate for a discrete-type random variable. We derive general formulae for the joint distributions belonging to the proposed class that, unlike other discrete bivariate models already proposed in the literature such as the well-known and most popular Holgate's bivariate Poisson distribution, can model both positive and negative dependence. We discuss general statistical properties of the proposed class as well. Specific families of bivariate distributions can be generated from the general class proposed in this paper just by specifying the ‘baseline distributions’. Furthermore, specific discrete bivariate distributions belonging to the proposed class are applied to analyze three real data sets, and the results are compared with those obtained from conventional models. Copyright © 2017 John Wiley & Sons, Ltd.

In this chapter, statistical methodology for bivariate data and calibration of experimental systems is developed. Experimental systems include analytical instruments useful in geochemistry. From the freely available BiDASys software, we can apply both the conventional ordinary least-squares linear regression (OLR) and the new uncertainty weighted least-squares linear regression (UWLR) models. BiDASys was used for achieving and comparing the OLR and UWLR models for the calibration of a high-performance liquid chromatography equipment for the determination of rare-earth elements. Equations are provided for both regressions. The advantages of the UWLR model over the OLR are clearly documented. This is followed by linearity tests, which are useful for deciding whether a linear or a curvilinear fit is more appropriate. ANOVA for the evaluation of fitting is finally presented and exemplified from citations of literature on new precise and accurate critical values for the F and t tests.

Summary This research is motivated by discovering and underpinning genetic causes for the progression of a bilateral eye disease, age-related macular degeneration (AMD), of which the primary outcomes, progression times to late-AMD, are bivariate and interval-censored due to intermittent assessment times. We propose a novel class of copula-based semiparametric transformation models for bivariate data under general interval censoring, which includes the case 1 interval censoring (current status data) and case 2 interval censoring. Specifically, the joint likelihood is modeled through a two-parameter Archimedean copula, which can flexibly characterize the dependence between the two margins in both tails. The marginal distributions are modeled through semiparametric transformation models using sieves, with the proportional hazards or odds model being a special case. We develop a computationally efficient sieve maximum likelihood estimation procedure for the unknown parameters, together with a generalized score test for the regression parameter(s). For the proposed sieve estimators of finite-dimensional parameters, we establish their asymptotic normality and efficiency. Extensive simulations are conducted to evaluate the performance of the proposed method in finite samples. Finally, we apply our method to a genome-wide analysis of AMD progression using the Age-Related Eye Disease Study data, to successfully identify novel risk variants associated with the disease progression. We also produce predicted joint and conditional progression-free probabilities, for patients with different genetic characteristics.

Flash-flood is considered to be one of the most destructive natural hazards in the world, which is difficult to accurately model and predict. The objective of the present research is to propose new ensembles of bivariate statistics and artificial intelligences and to introduce a comprehensive methodology for predicting flood susceptibility. The Putna river catchment of Romania is selected as a case study. In this regard, a total of six ensemble models were proposed and verified: Multilayer Perceptron neural network-Frequency Ratio (MLP-FR), Multilayer Perceptron neural network -Weights of Evidence (MLP-WOE), Rotation Forest-Frequency Ratio (RF-FR), Rotation Forest-Weights of Evidence (RF-WOE), Classification and Regression Tree-Frequency Ratio (CART-FR), and Classification and Regression Tree-Weights of Evidence (CART-WOE). In a first step, a geospatial database was created for the study area. This database includes 132 flood locations and 14 conditioning factors (lithology, slope angle, plan curvature, hydrological soil group, topographic wetness index, landuse, convergence index, elevation, distance from river, profile curvature, rainfall, aspect, stream power index, and topographic position index). In the next step, the Information Gain Ratio was used to evaluate the predictive ability of these factors. Subsequently, the database was used to train and validate the six ensemble models. The Receiver operating characteristic (ROC) curve, area under the curve (AUC), and statistical measures were used to evaluate the performance of the models. The results show that the prediction capability of the proposed ensemble models varied from 86.8% (the RF-FR model) to 93.9% (the RF-WOE model). These values indicate a high prediction performance for all the models. Therefore, we can state that the proposed ensemble models are new reliable tools which can be used for flood susceptibility modelling.

We develop an empirical likelihood (EL) approach to test independence of two univariate random variables X and Y versus the alternative that X and Y are strictly positive quadrant dependent (PQD). Establishing this type of ordering between X and Y is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague, we create a distribution-free test statistic that integrates a localized EL ratio test statistic with respect to the empirical joint distribution of X and Y. When compared to well-known existing tests and distance-based tests we develop by using copula functions, simulation results show the EL testing procedure performs well in a variety of scenarios when X and Y are strictly PQD. We use three datasets for illustration and provide an online R resource practitioners can use to implement the methods in this article. Supplementary materials for this article are available online.

To properly set the guaranteed quality of service for different distribution system operators (DSOs), the regulator has to determine the influence of extreme events on the annual reliability indices and predict the reliability of these events. In this paper, the spatial correlation of lightning storms and annual reliability indices for different DSOs are determined. The correlation is calculated using the global annual reliability and lightning data aggregated from individual DSOs. Using Gaussian and t copula, the joint distribution of individual reliability indices (SAIFI and SAIDI) and lightning density for different DSOs and geographical regions has been determined. After the distribution fitting, it is possible to calculate the risk of not attaining the guaranteed quality of service for every individual DSO on condition that the lightning activity does not surpass the required level. The methodology is tested by using 7 year's period real operational data from 27 Serbian distribution utilities.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g.,...

A new bivariate beta distribution capable of providing better fits than all its competitors is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and conditional moments. The method of maximum likelihood is used to derive the associated estimation procedure. Applications to six bivariate data sets are illustrated.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g. matched pairs experiments, twin or family data), the shared frailty models were suggested. These models are based on the assumption that frailty act multiplicatively to hazard rate. In this paper we assume that frailty acts additively to hazard rate. We introduce the shared gamma frailty models with two different baseline distributions namely, the generalized log logistic and the generalized Weibull. We introduce the Bayesian estimation procedure using Markov Chain Monte Carlo technique to estimate the parameters involved in these models. We apply these models to a real life bivariate survival data set of McGilchrist and Aisbett (Biometrics 47:461–466, 1991) related to the kidney infection data and a better model is suggested for the data.

An expression for the stress-strength reliability R=P(X1