Bivariate Normal Distribution Research Articles

Ensemble probability distribution of annual runoff for the past 70 years in two main watersheds of China

ABSTRACT Water resources in China, especially in the major basins of the Songhua and Yangtze Rivers, are characterized by uneven distribution both temporally and spatially, leading to notable challenges in per capita water availability. In this study, we investigate the ensemble probability distribution of annual runoff over the past 70 years in two of China's major watersheds: the Songhua River and the Yangtze River. By dividing each basin into several regions from upstream to downstream, based on annual mean discharge as a proxy for annual runoff, we observed a significant correlation between the annual runoff of the control sections and that of the upstream and downstream regions in these large watersheds. Consequently, this study establishes the probability distribution of annual runoff for each region within the basins, anchored on the design annual runoff of the control sections, using the joint bivariate logarithmic normal distribution of two interrelated random variables. Furthermore, we developed a relationship between the conditional probability distribution and correlation diagram data, and a regression equation correlating regional annual runoff with the watershed control section's runoff. This investigation into the historical patterns of runoff over the past 70 years provides a comprehensive understanding of the dynamics in these critical watersheds.

Bivariate cubic normal distribution for non-Gaussian problems

Probabilistic models play critical role in various engineering fields. Numerous critical issues exist in probabilistic modeling, especially for non-Gaussian correlated random variables. However, conventional parameter-based bivariate distribution models are generally developed for specific types of random variables, limiting their flexibility and applicability. A flexible bivariate distribution model is proposed, in which the joint cumulative distribution function is derived with the probability equivalently expressed as the summation of three basic probabilities corresponding to simple functions. These three basic probabilities are calculated with the aid of univariate cubic normal distribution, and thus the proposed model is named as bivariate cubic normal (BCN) distribution. The proposed BCN distribution has been applied in modeling several common bivariate distributions and actual engineering datasets. Results show that the BCN model accurately models both typical theoretical distributions and the JCDF of practical datasets, offering a significant improvement over existing models. Furthermore, the proposed BCN distribution has been utilized in seismic reliability assessment and the calculation of the mean recurrence interval and hazard curve of hurricane wind speed and storm size. Results demonstrate that the BCN distribution excels in modeling and matching capabilities, proving its accuracy and effectiveness in civil engineering applications.

Fusing linguistic and acoustic information for automated forensic speaker comparison

Verifying the speaker of a speech fragment can be crucial in attributing a crime to a suspect. The question can be addressed given disputed and reference speech material, adopting the recommended and scientifically accepted likelihood ratio framework for reporting evidential strength in court. In forensic practice, usually, auditory and acoustic analyses are performed to carry out such a verification task considering a diversity of features, such as language competence, pronunciation, or other linguistic features. Automated speaker comparison systems can also be used alongside those manual analyses. State-of-the-art automatic speaker comparison systems are based on deep neural networks that take acoustic features as input. Additional information, though, may be obtained from linguistic analysis. In this paper, we aim to answer if, when and how modern acoustic-based systems can be complemented by an authorship technique based on frequent words, within the likelihood ratio framework. We consider three different approaches to derive a combined likelihood ratio: using a support vector machine algorithm, fitting bivariate normal distributions, and passing the score of the acoustic system as additional input to the frequent-word analysis. We apply our method to the forensically relevant dataset FRIDA and the FISHER corpus, and we explore under which conditions fusion is valuable. We evaluate our results in terms of log likelihood ratio cost (Cllr) and equal error rate (EER). We show that fusion can be beneficial, especially in the case of intercepted phone calls with noise in the background.

Lancaster correlation: A new dependence measure linked to maximum correlation

AbstractWe suggest novel correlation coefficients which equal the maximum correlation for a class of bivariate Lancaster distributions while being only slightly smaller than maximum correlation for a variety of further bivariate distributions. In contrast to maximum correlation, however, our correlation coefficients allow for rank and moment‐based estimators which are simple to compute and have tractable asymptotic distributions. Confidence intervals resulting from these asymptotic approximations and the covariance bootstrap show good finite‐sample coverage. In a simulation, the power of asymptotic as well as permutation tests for independence based on our correlation measures compares favorably with competing methods based on distance correlation or rank coefficients for functional dependence, among others. Moreover, for the bivariate normal distribution, our correlation coefficients equal the absolute value of the Pearson correlation, an attractive feature for practitioners which is not shared by various competitors. We illustrate the practical usefulness of our methods in applications to two real data sets.

Improved memory-type ratio estimator for population mean in stratified random sampling under linear and non-linear cost functions

This paper offers an improved memory-type ratio estimator in stratified random sampling under linear and non-linear cost functions. The issue is given as all integer non-linear programming problems (AINLPPs). The sampling properties mainly the bias and the mean squared error of the introduced estimator are derived up to the first order of approximation. The optimum value of the characterizing scalar is obtained by the Lagrange method of maxima–minima. The least value of the MSE of the suggested estimator is also obtained for this optimum value of the charactering constant. The suggested estimator is compared both theoretically and empirically with the competing estimators. Under this setup, the optimum allocation with mean square error of the suggested estimator is attained, and the estimator is compared to other comparable estimators. The AINLPP is solved using the genetic programming approach, which is applied to both actual and simulated data sets from a bivariate normal distribution.

InfoSTGCAN: An Information-Maximizing Spatial-Temporal Graph Convolutional Attention Network for Heterogeneous Human Trajectory Prediction

Predicting the future trajectories of multiple interacting pedestrians within a scene has increasingly gained importance in various fields, e.g., autonomous driving, human–robot interaction, and so on. The complexity of this problem is heightened due to the social dynamics among different pedestrians and their heterogeneous implicit preferences. In this paper, we present Information Maximizing Spatial-Temporal Graph Convolutional Attention Network (InfoSTGCAN), which takes into account both pedestrian interactions and heterogeneous behavior choice modeling. To effectively capture the complex interactions among pedestrians, we integrate spatial-temporal graph convolution and spatial-temporal graph attention. For grasping the heterogeneity in pedestrians’ behavior choices, our model goes a step further by learning to predict an individual-level latent code for each pedestrian. Each latent code represents a distinct pattern of movement choice. Finally, based on the observed historical trajectory and the learned latent code, the proposed method is trained to cover the ground-truth future trajectory of this pedestrian with a bi-variate Gaussian distribution. We evaluate the proposed method through a comprehensive list of experiments and demonstrate that our method outperforms all baseline methods on the commonly used metrics, Average Displacement Error and Final Displacement Error. Notably, visualizations of the generated trajectories reveal our method’s capacity to handle different scenarios.

LOSS MODEL OF CLIMATE INSURANCE BASED ON EFFECT OF GROWING DEGREE DAYS INDEX

Climate change is a threat to agriculture, especially food crops such as rice. Climate index insurance is an alternative to cover the risk of agricultural losses due to crop failure due to climate change factors. The observed climate index is the effect of growing degree days which measures the impact of temperature on plant growth and development. The data used in this study is daily temperature data at Climatology Station Class 1 Darmaga, Bogor and Meteorological Station Class 3 Citeko, West Java, during the gadu (rice that is planted in the Gadu/Dry season) planting period. In determining the amount of loss, the average daily temperature on growing degree days is calculated using a combination of a time series model and a deterministic model. The deterministic model describes the trend and seasonality of the time series at each station. The parameters contained in the model will be estimated using least-square. To see the dependence of temperature at different stations using a normal bivariate distribution. The result show that the amount of loss based on the index of growing degree days per unit rupiah per degree Celsius (℃) for Meteorological Station Class 3 Citeko only occurs for certain percentages, namely 80%, 90%, and 95%, while for Climatology Station Class 1 Darmaga Bogor it can occur for each percentage. This indicates that the amount of losses obtained will depend on determining the strike level by using the mean and standard deviation of the growing degree days index distribution. Furthermore, these findings suggest that Climatology Station Class 1 Darmaga Bogor have higher risk of crop failure due to climate change than Meteorological Station Class 3 Citeko.

Value-at-Risk Effectiveness: A High-Frequency Data Approach with Semi-Heavy Tails

In the broader landscape of cryptocurrency risk management, this study delves into the nuanced estimation of Value-at-Risk (VaR) for a uniformly weighted portfolio of cryptocurrencies, employing the bivariate Normal Inverse Gaussian distribution renowned for its semi-heavy tails. Utilizing high-frequency data spanning between 1 January 2017 and 25 October 2022, with a primary focus on Bitcoin and Ethereum, our research seeks to accentuate the resilience of VaR methodology as a paramount risk assessment tool. The essence of our investigation lies in advancing the comprehension of VaR accuracy by quantitatively comparing the observed returns of both cryptocurrencies with their corresponding estimated values, with a central theme being the endorsement of the Normal Inverse Gaussian distribution as a potent model for risk measurement, particularly in the domain of high-frequency data. To bolster the statistical reliability of our results, we adopt a forward test methodology, showcasing not only a contribution to the evolution of risk assessment techniques in Finance but also underscoring the practicality of sophisticated distributional models in econometrics. Our findings not only contribute to the refinement of risk assessment methods but also highlight the applicability of such models in precisely modeling and forecasting financial risk within the dynamic realm of cryptocurrencies, epitomized by the case study of Bitcoin and Ethereum.

Calibration estimator of population mean in stratified extreme ranked set sampling with simulation study

Calibration estimation sets the original weights to include the known population characteristics of auxiliary variables using constraints. In this article, we have proposed a new calibration estimator of the population mean in stratified extreme ranked set sampling design, which is more efficient and costeffective design against other sampling designs in the literature. A detailed simulation study is carried out to observe the performance of proposed estimators. We have used the information of auxiliary variable to avoid ranking errors in our simulations. We have created samples from a bi-variate normal distribution with different values of ?xy. While one of these variables is taken as the variable of interest, the other is accepted as an auxiliary variable and used in ranking the sample units in each set. As a result of the simulation study using both synthetic and real data sets, we have found that our proposed estimators are more efficient than Sinha et al. [19] calibration estimator and classical stratified estimator.

Quantification in shooting precision for preferred and non-preferred foot in college soccer players using the 95% equal confidence ellipse.

Shooting precision is a fundamental characteristic in soccer, yet the probabilistic structure and magnitude of precision in soccer shooting remain quantitatively unexplored. This study aimed to quantify shooting precision using measures derived from the bivariate normal distribution for both preferred and non-preferred feet. Sixteen right-footed collegiate soccer players participated by performing instep kicks aiming at targets which are placed close to the left and right top corners of the soccer goal. We used bivariate normal distribution modeled the ball positions, revealing an ellipsoidal distribution, and the area of the 95% confidence ellipses served as an index of precision. Repeated measures ANOVAs revealed a significant main effect of the kicking foot. For shots aimed at the same side as the kicking foot, the area of the 95% confidence ellipse was 6.17 ± 1.93 m2 (mean ± SD) for the preferred foot and 10.22 ± 3.53 m2 for the non-preferred foot. Similar results were observed for shots aimed at the opposite side of the kicking foot. These quantitative findings hold promise for advancing soccer research and enhancing practical applications in soccer skill assessment.

Using Pearson’s System of Curves to Approximate the Distributions of the Difference between Two Correlated Estimates of Signal-to-Noise Ratios: The Cases of Bivariate Normal and Bivariate Lognormal Distributions

Using Pearson’s System of Curves to Approximate the Distributions of the Difference between Two Correlated Estimates of Signal-to-Noise Ratios: The Cases of Bivariate Normal and Bivariate Lognormal Distributions

A simple approach to probabilistic CPTu-based geotechnical stratigraphic profiling

Geotechnical profile delineation is an important aspect of the geotechnical engineering design process. The Cone Penetration Test (CPTu) is widely used for this purpose, as it is simple to classify soils based on their different responses to the cone. However, CPTu-based delineations typically involve engineering interpretation, as they are affected by the classification scheme applied, by inherent soil variability and by measurements error. A semi-automated CPTu data treatment tool is here presented aiming to simplify the soil delineation task, adding consistency and transparency to the involved engineering interpretation process and facilitating the transference of information between the different parties involved in design. The tool fits bivariate normal distributions to interpreted CPTu data according to some user specified criteria that quantify estimated classification and measurement uncertainties. Classification makes use of conventional class boundaries –here taken from Robertson, 1990 chart-, which are applied with user-specified refinement. Thin layers are consolidated into adjacent ones to further simplify the resulting profile. The operation of the proposed algorithm is illustrated by interpreting CPTu records from a deltaic sedimentary environment. The outputs of this simple method may be used as input for more advanced probabilistic methodologies, helping thus to bridge the gap between qualitative practice and quantitative research methods.

Iteratively Reweighted Least Squares Method for Estimating Polyserial and Polychoric Correlation Coefficients

An iteratively reweighted least squares (IRLS) method is proposed for the estimation of polyserial and polychoric correlation coefficients in this article. It calculates the slopes in a series of weighted linear regression models fitting on conditional expected values. For polyserial correlation, conditional expectations of the latent predictor is derived from the observed ordinal categorical variable, and the regression coefficient is obtained with the weighted least squares method. In estimating polychoric correlation coefficient, conditional expectations of the response variable and the predictor are updated in turns. Standard errors of the estimators are obtained using the delta method based on data summaries instead of the whole data. Conditional univariate normal distribution is exploited and a single integral is numerically evaluated in the proposed algorithm, comparing to the double integral computed numerically based on the bivariate normal distribution in the traditional maximum likelihood (ML) approaches. This renders the new algorithm very fast in the estimation of both polyserial and polychoric correlation coefficients. Thorough simulation studies are conducted to compare the performances of the proposed method with the classical ML methods. Real data analyses illustrate the advantage of the new method in computing speed. Supplementary materials for this article are available online.

Product of bi-dimensional VAR(1) model components. An application to the cost of electricity load prediction errors

Abstract The multi-dimensional vector autoregressive (VAR) time series is often used to model the impulse-response functions of macroeconomics variables. However, in some economical applications, the variable of main interest is the product of time series describing market variables, like e.g. the cost, being the product of price and volume. In this paper, we analyze the product of the bi-dimensional VAR(1) model components. For the introduced time series, we derive general formulas for the autocovariance function and study its properties for different cases of cross-dependence between the VAR(1) model components. The theoretical results are then illustrated in the simulation study for two types of bivariate distributions of the residual series, namely the Gaussian and Student’s t. The obtained results are applied for the electricity market case study, in which we show that the additional cost of balancing load prediction errors prior to delivery can be well described by time series being the product of the VAR(1) model components with the bivariate normal inverse Gaussian distribution.

Generalized Cramér’s coefficient via f-divergence for contingency tables

AbstractVarious measures in two-way contingency table analysis have been proposed to express the strength of association between row and column variables in contingency tables. Tomizawa et al. (2004) proposed more general measures, including Cramér’s coefficient, using the power-divergence. In this paper, we propose measures using the f-divergence that has a wider class than the power-divergence. Unlike statistical hypothesis tests, these measures provide quantification of the association structure in contingency tables. The contribution of our study is proving that a measure applying a function that satisfies the condition of the f-divergence has desirable properties for measuring the strength of association in contingency tables. With this contribution, we can easily construct a new measure using a divergence that has essential properties for the analyst. For example, we conducted numerical experiments with a measure applying the $$\theta$$ θ -divergence. Furthermore, we can give further interpretation of the association between the row and column variables in the contingency table, which could not be obtained with the conventional one. We also show a relationship between our proposed measures and the correlation coefficient in a bivariate normal distribution of latent variables in the contingency tables.

Using spatial point process models, clustering and space partitioning to reconfigure fire stations layout

AbstractFire stations (FS) are typically non-uniformly distributed across space, and their service area is, in general, defined based on administrative boundaries. Since the location of FS may considerably influence the readiness and the effectiveness of the provided services, national and regional governments need research-based information to adequately plan where to establish firefighting facilities. In this study, we propose a method to reconfigure the fire stations layout using spatial point process models, clustering and space partitioning. First, modelling fire intensity variation across space through a point process model enables to replicate the process independently by simulation. Subsequently, for each simulation, the k-means algorithm is used to define a siting location, minimizing the total within distance between the fire occurrences and the new position. This method allows to obtain a set of locations from which the respective distribution is inferred. Assuming a bivariate normal spatial distribution, we further define confidence siting regions. Ultimately, new FS service areas are defined by Voronoi tessellation. To exemplify the application of the method, we apply it to reconfigure the fire station layout at Aveiro, Portugal.

Thurstonian model for the four-interval oddity task

The four-interval oddity task is an extension of the Triangle method, which is the three-interval oddity task and is one of the most broadly used conventional sensory discrimination methods. It is a special case of the m-interval oddity task or the m-alternative odd-man-out task where m = 4. It is also a special situation of the unspecified ‘M + N’ with M = 3 and N = 1. Some new forms of the analytical psychometric function for the four-interval oddity task based on a normal distribution function, a bivariate normal distribution function, and a non-central F-distribution function are provided originally in this paper. The performance of the four-interval oddity task in both difference testing and similarity/equivalence testing is explored and compared with some unspecified forced-choice sensory discrimination methods including the Triangle. Tables and R codes are presented and provided for estimations of the probability of correct response, Pc, Thurstonian discriminal distance δ or d´, and the B value for estimating the variance of d´ for the four-interval oddity task.

A first-order Stein characterization for absolutely continuous bivariate distributions

A random variable X has a standard normal distribution if and only if E [ f ′ ( X ) ] = E [ X f ( X ) ] for any continuous and piecewise continuously differentiable function f such that the expectations exist. This first-order characterizing equation, called the Stein identity, has been extended to other univariate distributions. For the multivariate normal distribution, a number of Stein identities have already been developed, all of them second order equations. In this study, we developed a new Stein characterization for the bivariate normal distribution. Unlike many existing multivariate versions in the literature, ours is a system of first-order equations which has the univariate Stein identity as a special case. We also constructed a generalized Stein characterization for other absolutely continuous bivariate distributions. Finally, we illustrated how this Stein characterization looks like for some known absolutely continuous bivariate distributions.

Bayesian semiparametric joint model of multivariate longitudinal and survival data with dependent censoring.

We consider a novel class of semiparametric joint models for multivariate longitudinal and survival data with dependent censoring. In these models, unknown-fashion cumulative baseline hazard functions are fitted by a novel class of penalized-splines (P-splines) with linear constraints. The dependence between the failure time of interest and censoring time is accommodated by a normal transformation model, where both nonparametric marginal survival function and censoring function are transformed to standard normal random variables with bivariate normal joint distribution. Based on a hybrid algorithm together with the Metropolis-Hastings algorithm within the Gibbs sampler, we propose a feasible Bayesian method to simultaneously estimate unknown parameters of interest, and to fit baseline survival and censoring functions. Intensive simulation studies are conducted to assess the performance of the proposed method. The use of the proposed method is also illustrated in the analysis of a data set from the International Breast Cancer Study Group.

A Truncated Mixture Transition Model for Interval-Valued Time Series

Abstract We propose a model for interval-valued time series that specifies the conditional joint distribution of the upper and lower bounds as a mixture of truncated bivariate normal distributions. It preserves the interval natural order and provides great flexibility on capturing potential conditional heteroscedasticity and non-Gaussian features. The standard expectation maximization (EM) algorithm applied to truncated mixtures does not provide a closed-form solution in the M step. A new EM algorithm solves this problem. The model applied to the interval-valued IBM daily stock returns exhibits superior performance over competing models in-sample and out-of-sample evaluation. A trading strategy showcases the usefulness of our approach.