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There exist many bivariate parametric copulas to model bivariate data with different dependence features. We propose a new bivariate parametric copula family that cannot only handle various dependence patterns that appear in the existing parametric bivariate copula families, but also provides a more enriched dependence structure. The proposed copula construction exploits finite mixtures of bivariate normal distributions. The mixing operation, the distinct correlation and mean parameters at each mixture component introduce quite a flexible dependence. The new parametric copula is theoretically investigated, compared with a set of classical bivariate parametric copulas and illustrated on two empirical examples from astrophysics and agriculture where some of the variables have peculiar and asymmetric dependence, respectively.

For estimating bivariate survival function under random censor- ship, it is commonly believed that the Dabrowska estimator is among the best ones while the Volterra estimator is far from being computational ef- ficiency. As we will see, the Volterra estimator is a natural extension of the Kaplan-Meier estimator to bivariate data setting. We believe that the computational 'inefficiency' of the Volterra estimator is largely due to the formidable computational complexity of the traditional recursion method. In this paper, we show by numerical study as well as theoretical analysis that the Volterra estimator, once computed by dynamic programming technique, is more computationally efficient than the Dabrowska estimator. Therefore, the Volterra estimator with dynamic programming would be quite recom- mendable in applications owing to its significant computational advantages.

A new, exact and closed-form expression is presented for the Beaulieu-Xie (BX) phase-envelope joint distribution. In addition, bivariate statistics are also determined for the BX distribution: the bivariate probability density function (PDF) of the envelope, the bivariate PDF of the instantaneous signal to noise ratio (SNR) and the generalized joint moment of the envelope, with and without correlation between the in-phase and quadrature components. Theoretical curves are plotted for different parameter values that characterize the fading.

The primary objective is to propose and verify an ensemble approach based on fuzzy system and bivariate statistics for landslide susceptibility assessment (LSA) at Azarshahr Chay Basin (Iran). In this regard, various integrations of fuzzy membership value (FMV), frequency ratio (FR), and information value (IV) with index of entropy (IOE) were investigated. Aerial photograph interpretations and substantial field checking were used to identify the landslide locations. Out of 75 identified landslides, 52 (≈70%) locations were utilized for the training of the models, whereas the remaining 23 (≈30%) cases were employed for the validation of the models. Fourteen landslide conditioning factors including altitude, slope aspect, slope degree, lithology, distance to fault, curvature, land use, distance to river, topographic position index (TPI), topographic wetness index (TWI), stream power index (SPI), normalized difference vegetation index (NDVI), distance to road, and rainfall were prepared and utilized during the analysis. The $$\mathrm{FMV}\_\mathrm{IOE}$$ , $$\mathrm{FR}\_\mathrm{IOE}$$ , and $$\mathrm{IV}\_\mathrm{IOE}$$ models were designed utilizing the dataset for training. Finally, to validate as well as to compare the model’s predictive abilities, the statistical measures of receiver operating characteristic (ROC), including sensitivity, accuracy, and specificity, were employed. The accuracy of 92.7, 92.5, and 91.8% of the models such as $$\mathrm{FMV}\_\mathrm{IOE}$$ , $$\mathrm{FR}\_\mathrm{IOE}$$ , and $$\mathrm{IV}\_\mathrm{IOE}$$ ensembles, respectively, was by the area under the receiver operating characteristic (AUROC) values developed from the ROC curve. For the validation dataset, the $$\mathrm{FMV}\_\mathrm{IOE}$$ model had the maximum sensitivity, accuracy, and specificity values of 95.7, 91.3, and 87.0%, respectively. Thus, the ensemble of FMV_IOE was introduced as a promising and premier approach that could be used for LSA in the study area. Also, IOE results indicated that altitude, lithology, and slope degree were main drivers of landslide occurrence. The results of the present research can be employed as a platform for appropriate basined management practices in order to plan the highly susceptible zones to landslide and hence minimize the expected losses.

This paper extends the analysis of the bivariate Seemingly Unrelated (SUR) Tobit by modeling its nonlinear dependence structure through copula and assuming non-normal marginal error distributions. For model estimation, the use of copula methods enables the use of the (classical) Inference Function for Margins (IFM) method by Joe and Xu (1996), which is more computationally attractive (feasible) than the full maximum likelihood approach. However, our simulation study shows that the IFM method provides a biased estimate of the copula parameter in the presence of censored observations in both margins. In order to obtain an unbiased estimate of the copula association parameter, we propose/develop a modified version of the IFM method, which we refer to as Inference Function for Augmented Margins (IFAM). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome, we propose the use of resampling procedures (bootstrap methods) to obtain confidence intervals for the copula-based SUR Tobit model parameters. The satisfactory results from the simulation and empirical studies indicate the adequate performance of our proposed model and methods. We illustrate our procedure using bivariate data on consumption of salad dressings and lettuce by U.S. individuals.

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 paper we include dependency structures for electricity price forecasting and forecasting evaluation. We work with off-peak and peak time series from the German-Austrian day-ahead price, hence we analyze bivariate data. We first estimate the mean of the two time series, and then in a second step we estimate the residuals. The mean equation is estimated by OLS and elastic net and the residuals are estimated by maximum likelihood. Our contribution is to include a bivariate jump component on a mean reverting jump diffusion model in the residuals. The models' forecasts are evaluated using four different criteria, including the energy score to measure whether the correlation structure between the time series is properly included or not. In the results it is observed that the models with bivariate jumps provide better results with the energy score, which means that it is important to consider this structure in order to properly forecast correlated time series., 30 pages, 11 figures, 3 tables and Accepted in International Journal of Forecasting

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.

This article proposes a space‐efficient approximation to empirical tail dependence coefficients of an indefinite bivariate stream of data. The approximation, which has stream‐length invari...

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.

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.

Copula functions have been extensively used in applied statistics, becoming a good alternative for modeling the dependence of multivariate data. Each copula function has a different dependence structure. An important issue in these applications is the choice of an appropriate copula function model for each one; thus common classical or Bayesian discrimination methods might not be appropriate for determining the best copula. Considering only the special case of bivariate data, we propose a procedure obtained from a recently introduced dependence measure for selecting an appropriate copula for the statistical data analyses.

Outcrop analogs for aquifers allow measurements of permeability with relatively high-resolution and mapping sedimentary unit types. The spatial bivariate structure of permeability is defined by aquifer architecture. The architecture is often organized into a hierarchy of unit types, and associated permeability modes, across different spatial scales. The composite covariance (or semivariogram) is a linear summation of the auto- and cross-covariances (or semivariograms) of unit types defined at smaller scales, weighted by the related proportions and transition probabilities. It is well-known that an appreciable fraction of the composite variance arises from differences in mean permeability across unit types defined at smaller scales. Previous work has shown that the transition probabilities usually define the spatial bivariate correlation structure. The composite spatial bivariate statistics for permeability (covariance or semivariogram) will not be representative unless data locations allow proper definition of the transition probabilities of the units. Quantification of the stratal architecture can be used to better interpret the transition probabilities and thereby improve a model for the sample covariance and semivariogram. In this paper, we use an inverse modeling algorithm to fit the components of the hierarchical model, written as nested functions, in developing a hierarchical spatial correlation models. Specifically, the least-squares criterion along with prior information and other weighted constraints are used as the objective function for the inverse problem, which is solved by the Gauss–Newton–Levenberg–Marquardt method. The estimated covariance and transition probability models provides accurate representation of the spatial correlation structure of permeability for field-measured data from in Espanola Basin, New Mexico.

In this paper, we study the concomitants of dual generalized order statistics (and consequently generalized order statistics) when the parameters γ1,⋯,γn are assumed to be pairwise different from Huang–Kotz Farlie–Gumble–Morgenstern bivariate distribution. Some useful recurrence relations between single and product moments of concomitants are obtained. Moreover, Shannon’s entropy and the Fisher information number measures are derived. Finally, these measures are extensively studied for some well-known distributions such as exponential, Pareto and power distributions. The main motivation of the study of the concomitants of generalized order statistics (as an important practical kind to order the bivariate data) under this general framework is to enable researchers in different fields of statistics to use some of the important models contained in these generalized order statistics only under this general framework. These extended models are frequently used in the reliability theory, such as the progressive type-II censored order statistics.

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

A novel, general two-sample hypothesis testing procedure is established for testing the equality of tail copulas associated with bivariate data. More precisely, using an ingenious transformation of a natural two-sample tail copula process, a test process is constructed, which is shown to converge in distribution to a standard Wiener process. Hence, from this test process a myriad of asymptotically distribution-free two-sample tests can be obtained. The good finite-sample behavior of our procedure is demonstrated through Monte Carlo simulations. Using the new testing procedure, no evidence of a difference in the respective tail copulas is found for pairs of negative daily log-returns of equity indices during and after the global financial crisis.

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.

In this article we describe a new Hermite series based sequential estimator for the Spearman rank correlation coefficient and provide algorithms applicable in both the stationary and non-stationary settings. To treat the non-stationary setting, we introduce a novel, exponentially weighted estimator for the Spearman rank correlation, which allows the local nonparametric correlation of a bivariate data stream to be tracked. To the best of our knowledge this is the first algorithm to be proposed for estimating a time varying Spearman rank correlation that does not rely on a moving window approach. We explore the practical effectiveness of the Hermite series based estimators through real data and simulation studies demonstrating good practical performance. The simulation studies in particular reveal competitive performance compared to an existing algorithm. The potential applications of this work are manifold. The Hermite series based Spearman rank correlation estimator can be applied to fast and robust online calculation of correlation which may vary over time. Possible machine learning applications include, amongst others, fast feature selection and hierarchical clustering on massive data sets., Revised article incorporating comments by the reviewers, associate editor and editor of the Journal of Multivariate Analysis (accepted manuscript). Changes include a modified title and a significantly shortened article

This paper proposes a new class of covariance functions for bivariate random fields on spheres, having the same properties as the bivariate Matern model proposed in Euclidean spaces. The new class depends on the geodesic distance on a sphere; it allows for indexing differentiability (in the mean square sense) and fractal dimensions of the components of any bivariate Gaussian random field having such covariance structure. We find parameter conditions ensuring positive definiteness. We discuss other possible models and illustrate our findings through a simulation study, where we explore the performance of maximum likelihood estimation method for the parameters of the new covariance function. A data illustration then follows, through a bivariate data set of temperatures and precipitations, observed over a large portion of the Earth, provided by the National Oceanic and Atmospheric Administration Earth System Research Laboratory.

In this work, we investigate the asymptotic behavior of the extremes of a multivariate data by using the Reduced Ordering Principle (R-ordering). When, the sup-norm is used, we reveal the interrelation between the R-ordering principle and Marginal Ordering Principle (M-ordering). The asymptotic behavior of the maximum sup-norms corresponding to the bivariate data is completely determined. Finally, an application to real data illustrates and corroborates the theoretical results.

The present study attempts to measure wetland habitat vulnerability (WHV) in the Indian part of mature Ganges delta. Predictive algorithms belonging to bivariate statistics and machine learning (ML) algorithms were applied for fulfilling the data mining and generating the models. Results show that 60% of the wetland areas are covered by moderate to very high WHV, out of which > 300 km2 belong to very high WHV followed by a high vulnerability in almost 150 km2. This areal coverage increases by 10–15% from phase II to phase III. On the other hand, a relatively safe situation is confined to < 200 km2. The receiver operating characteristic curve, root-mean-square error, and correlation coefficient are used to assess the accuracy of these models and categorization of habitat vulnerability. Ensemble modeling is done using the individual models having a greater accuracy level in order to increase accuracy. A field-based model of the same is prepared by gathering information directly from the field which also exhibits similar results with the algorithm-based models. Analysis of residuals in standard regression strongly supports the relevance of the selected parameters and multi-parametric models.

Recently, a procedure of impounding a given marginal distribution, with the auxiliary density function generated from the assumed form of the distribution for the concomitant of an extreme order statistic of a sample, has been developed to create a bivariate distribution. In this paper we consider a marginal Rayleigh distribution and impound it with the pdf of an extended form of Lindley distribution to generate a new bivariate distribution. Some important properties of this new bivariate model together with some characterization theorems have been also derived. Most commonly used reliability functions associated with the generated bivariate distribution as well have been obtained. The well-known maximum likelihood (ML) method of estimation is applied for the estimation of the parameters of the generated bivariate distribution. A simulation study is also carried out to illustrate the closeness of ML estimates with the true value of the parameters. A real-life bivariate data set is used to demonstrate the application of the results of this paper in generating the appropriate bivariate distributional model in an innovative manner.

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.

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...

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.

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...

Financial time series are commonly modeled with methods assuming data normality. However, the real distribution can be nontrivial, also not having an explicitly formulated probability density function. In this work we introduce novel parameter estimation and high-powered distribution testing methods which do not rely on closed form densities, but use the characteristic functions for comparison. The approach applied to a pair of stock index returns demonstrates that such a bivariate vector can be a sample coming from a bivariate sub-Gaussian distribution. The methods presented here can be applied to any nontrivially distributed financial data, among others.

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 research article, the authors have implemented the Exploratory Data Analysis (EDA) techniques to examine the deviation of the monthly median noon and midnight values of the critical frequency of F 2 layer ionosphere (i.e. f 0 F 2 ) at the Wakkanai station (45.39°N, 141.68°E), Japan, during sunspot cycle 21 st (1976-1986) and 23 rd (1996-2008). Primarily, univariate analysis has been done, which shows the variations in f 0 F 2 at different local times, seasons and in the range of sunspot numbers (SSN), in which winter and semi-annual anomalies are detected in the months of December and March respectively. Secondly, the regression analysis is being used as a bivariate data analysis. The results proved a significantly nonlinear relationship exists between f 0 F 2 and SSN. In both solar cycles, saturation effects are seen in the month of March during the noontime period and immensely in June during the midnight time. The behavior of the ionosphere has been studied for different latitudes, seasonal effects and sunspot dynamic conditions, in which this paper plays an essential role in it.

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.

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.