Search Results (15)

Generalized Laplace and discrete generalized Laplace models are described and some of their properties are reviewed. For the analysis of directional data, it is natural to consider wrapped versions of these models. A well-known result for the circular and, more recently, for multivariate circular distributions that mixing and wrapping commute, allows us to readily determine the nature of these wrapped models. Some bivariate extensions of these models are discussed, together with some consideration of the feasibility of wrapping such models. Multivariate versions of the models can be envisioned. Full article

This study introduces two bivariate extensions of the recently proposed weighted discretized Fréchet–Weibull distribution, termed as bivariate weighted discretized Fréchet–Weibull (BWDFW) distributions. These models are specifically designed for analyzing two-dimensional discrete datasets and are developed using two distinct structural approaches: the minimum operator (BWDFW-I) and the maximum operator (BWDFW-II). A rigorous mathematical formulation is presented, encompassing the joint cumulative distribution function, joint probability mass function, and joint (reversed) hazard rate function. The dependence structure of the models is investigated, demonstrating their capability to capture positive quadrant dependence. Additionally, key statistical measures, including covariance, Pearson’s correlation coefficient, Spearman’s rho, and Kendall’s tau, are derived using the joint probability-generating function. For robust statistical inferences, the parameters of the proposed models are estimated via the maximum likelihood estimation method, with extensive simulation studies conducted to assess the efficiency and accuracy of the estimators. The practical applicability of the BWDFW distributions is demonstrated through their implementation in two real-world datasets: one from the aviation sector and the other from the security and safety domain. Comparative analyses against four existing discrete bivariate Weibull extensions reveal the superior performance of the BWDFW models, with BWDFW-I (minimum operator based) exhibiting greater flexibility and predictive accuracy than BWDFW-II (maximum operator based). These findings underscore the potential of the BWDFW models as effective tools for modeling and analyzing bivariate discrete data in diverse applied contexts. Full article

In the peer-to-peer (P2P) lending market, current studies focus on two categories of approaches to evaluate the loans, thus providing investment suggestions to the investors: credit scoring (i.e., predicting the credit risk) and profit scoring (i.e., predicting the profitability). However, relying on a single scoring approach may bias the loan evaluation conclusion. In this paper, we propose a bivariate model based on the integration of two scoring approaches. We first formulate the loan evaluation task as a multi-target problem, in which loan_status (i.e., default or not default) is used as the discrete outcome for the credit risk measure while the annualized rate of return (ARR) is used as the continuous outcome for the profitability measure. Then to solve the multi-target problem, we design a novel loss function based on the assumption that the discrete outcome follows a Bernoulli distribution, and the continuous outcome is normally distributed conditional on the discrete output. The effectiveness of the proposed model is examined using the real-world P2P data from the Lending Club. Results indicate that our approach outperforms the sole scoring methods by identifying loans with higher profit and lower default risk. Therefore, the proposed method can serve as an alternative for loan evaluation. Full article

This paper introduces a novel four-parameter discrete bivariate distribution, termed the bivariate discretized Fréchet–Weibull distribution (BDFWD), with marginals derived from the discretized Fréchet–Weibull distribution. Several statistical and reliability properties are thoroughly examined, including the joint cumulative distribution function, joint probability mass function, joint survival function, bivariate hazard rate function, and bivariate reversed hazard rate function, all presented in straightforward forms. Additionally, properties such as moments and their related concepts, the stress–strength model, total positivity of order 2, positive quadrant dependence, and the median are examined. The BDFWD is capable of modeling asymmetric dispersion data across various forms of hazard rate shapes and kurtosis. Following the introduction of the mathematical and statistical frameworks of the BDFWD, the maximum likelihood estimation approach is employed to estimate the model parameters. A simulation study is also conducted to investigate the behavior of the generated estimators. To demonstrate the capability and flexibility of the BDFWD, three distinct datasets are analyzed from various fields, including football score records, recurrence times to infection for kidney dialysis patients, and student marks from two internal examination statistical papers. The study confirms that the BDFWD outperforms competitive distributions in terms of efficiency across various discrete data applications. Full article

Information theory explains how systems encode and transmit information. This article examines the neuronal system, which processes information via neurons that react to stimuli and transmit electrical signals. Specifically, we focus on transfer entropy to measure the flow of information between sequences and explore its use in determining effective neuronal connectivity. We analyze the causal relationships between two discrete time series, $\mathit{X}:=\left\{X_t:t\in \mathbb{Z}\right\}$ and $\mathit{Y}:=\left\{Y_t:t\in \mathbb{Z}\right\}$, which take values in binary alphabets. When the bivariate process $(\mathit{X},\mathit{Y})$ is a jointly stationary ergodic variable-length Markov chain with memory no larger than k, we demonstrate that the null hypothesis of the test—no causal influence—requires a zero transfer entropy rate. The plug-in estimator for this function is identified with the test statistic of the log-likelihood ratios. Since under the null hypothesis, this estimator follows an asymptotic chi-squared distribution, it facilitates the calculation of p-values when applied to empirical data. The efficacy of the hypothesis test is illustrated with data simulated from a neuronal network model, characterized by stochastic neurons with variable-length memory. The test results identify biologically relevant information, validating the underlying theory and highlighting the applicability of the method in understanding effective connectivity between neurons. Full article

The spatial accessibility of urban parks is an important indicator of the livability level of cities. In this paper, we propose a comprehensive multimodal two-step floating catchment area (CM2SFCA) method which integrates supply capacity, the selection probability of individuals, and variable catchment sizes into the traditional multimodel 2SFCA method. This method is used to measure park accessibility in Wuhan, China. The results show that the spatial distribution of park accessibility under the proposed method is variant. High accessibility areas are clustered near the Third Ring Road with strong supply capacity parks, and low accessibility areas are distributed in the western and southern regions. Compared with the single-model accessibility (bicycling, driving, and public transit) method, we found that the multimodal spatial accessibility, combining the characteristics of three single transportations, can provide a more realistic evaluation. We also explore the spatial relationship between park accessibility and population density by bivariate local Moran’s I statistic and find that the Low Ai-High Pi area is located in the center of the study area, and the Low Ai-Low Pi area is located at the edge of the study area, with a relatively discrete distribution of parks and weak supply capacity. These findings may provide some insights for urban planners to formulate effective policies and strategies to ease the spatial inequity of urban parks. Full article

The notion that the occurrence of an event is surprising has been discussed in the literature without adequate details. By definition, a surprise index is an index by which how surprising an event is may be determined. Since its inception, this index has been evaluated for univariate discrete probability models, such as the binomial, negative binomial, and Poisson probability distributions. In this article, we derive and discuss using numerical studies, in addition to the above-mentioned probability models, surprise indices for several other univariate discrete probability models, such as the zero-truncated Poisson, geometric, Hermite, and Skellam distributions, by adopting a established strategy and using the Mathematica, version 12 software. In addition, we provide symbolical expressions for the surprise index for several univariate continuous probability models, which has not been previously discussed. For illustrative purposes, we present some possible real-life applications of this index and potential challenges to extending the notion of the surprise index to bivariate and higher dimensions, which might involve ubiquitous normalizing constants. Full article

In this article, a new family of bivariate discrete distributions is proposed based on the copula concept, in the so-called bivariate discrete odd generalized exponential-G family. Some distributional properties, including the joint probability mass function, joint survival function, joint failure rate function, median correlation coefficient, and conditional expectation, are derived. After proposing the general class, one special model of the new bivariate family is discussed in detail. The maximum likelihood approach is utilized to estimate the family parameters. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood estimators. Finally, the importance of the new bivariate family is explained by means of two distinctive real data sets in various fields. Full article

Discrete-valued time series modeling has witnessed numerous bivariate first-order integer-valued autoregressive process or BINAR(1) processes based on binomial thinning and different innovation distributions. These BINAR(1) processes are mainly focused on over-dispersion. This paper aims to propose new bivariate distributions and processes based on a recently proposed over-dispersed distribution: the Poisson 2S-Lindley distribution. The new bivariate distributions, denoted by the abbreviations BP2S-L(I) and BP2S-L(II), are then used as innovation distributions for the BINAR(1) process. Properties are investigated for both distributions as well as for the BINAR(1) processes. The distribution parameters are estimated using the maximum likelihood method, and the BINAR(1)BP2S-L(I) and BINAR(1)BP2S-L(II) process parameters are estimated using the conditional least squares and conditional maximum likelihood methods. Monte Carlo simulation experiments are conducted to study large and small sample performances and for the comparison of the estimation methods. The Pittsburgh crime series and candy sales datasets are then used to compare the new BINAR(1) processes to some other existing BINAR(1) processes in the literature. Full article

In this paper, different mathematical expressions are derived to compute the residual magnitude of voltage caused by faults along the line and on the bus. Symmetrical and unsymmetrical faults are taken into consideration, and the consequences of the various fault distributions are considered. A new way of assessing a sag is proposed that incorporates the method of fault position and mathematical expression based on sequence currents and voltages. The fault impedance is introduced to obtain a better result. A fast and efficient load flow analysis technique produces quick computational results. In addition, the sag analysis is performed using the bivariate joint discrete probability distribution method that gives a clear idea about the probability of occurrence of sag in a meshed network. The suggested approach is applied in the IEEE 39-bus system and with an existing real-time electrical power distribution system in India. Full article

In-depth studies have been conducted on the risk of exposure to air pollution in urban residents, but most of them are static studies based on the population of residential units. Ignoring the real environmental dynamics during daily activity and mobility of individual residents makes it difficult to accurately estimate the level of air pollution exposure among residents and determine populations at higher risk of exposure. This paper uses the example of the Wuhan metropolitan area, high-precision air pollution, and population spatio-temporal dynamic distribution data, and applies geographically weighted regression models, bivariate LISA analysis, and Gini coefficients. The risk of air pollution exposure in elderly, low-age, and working-age communities in Wuhan was measured and the health equity within vulnerable groups such as the elderly and children was studied. We found that ignoring the spatio-temporal behavioral activities of residents underestimated the actual exposure hazard of PM_2.5 to residents. The risk of air pollution exposure was higher for the elderly than for other age groups. Within the aging group, a few elderly people had a higher risk of pollution exposure. The high exposure risk communities of the elderly were mainly located in the central and sub-center areas of the city, with a continuous distribution characteristic. No significant difference was found in the exposure risk of children compared to the other populations, but a few children were particularly exposed to pollution. Children’s high-exposure communities were mainly located in suburban areas, with a discrete distribution. Compared with the traditional static PM_2.5 exposure assessment, the dynamic assessment method proposed in this paper considers the high mobility of the urban population and air pollution. Thus, it can accurately reveal the actual risk of air pollution and identify areas and populations at high risk of air pollution, which in turn provides a scientific basis for proposing planning policies to reduce urban PM_2.5 and improve urban spatial equity. Full article

We propose several numerical algorithms to compute the distribution of gross loss in a positively dependent catastrophe insurance portfolio. Hierarchical risk aggregation is performed using bivariate copula trees. Six common parametric copula families are studied. At every branching node, the distribution of a sum of risks is obtained by discrete copula convolution. This approach is compared to approximation by a weighted average of independent and comonotonic distributions. The weight is a measure of positive dependence through variance of the aggregate risk. During gross loss accumulation, the marginals are distorted by application of insurance financial terms, and the value of the mixing weight is impacted. To accelerate computations, we capture this effect using the ratio of standard deviations of pre-term and post-term risks, followed by covariance scaling. We test the performance of our algorithms using three examples of complex insurance portfolios subject to hurricane and earthquake catastrophes. Full article

Medical studies often involve a comparison between two outcomes, each collected from a sample. The probability associated with, and confidence in the result of the study is of most importance, since one may argue that having been wrong with a percent could be what killed a patient. Sampling is usually done from a finite and discrete population and it follows a Bernoulli trial, leading to a contingency of two binomially distributed samples (better known as $2\times 2$ contingency table). Current guidelines recommend reporting relative measures of association (such as the relative risk and odds ratio) in conjunction with absolute measures of association (which include risk difference or excess risk). Because the distribution is discrete, the evaluation of the exact confidence interval for either of those measures of association is a mathematical challenge. Some alternate scenarios were analyzed (continuous vs. discrete; hypergeometric vs. binomial), and in the main case—bivariate binomial experiment—a strategy for providing exact p-values and confidence intervals is proposed. Algorithms implementing the strategy are given. Full article

Through an urban tunnel-driving experiment, this paper studies the changing trend of drivers’ visual characteristics in tunnels. A Tobii Pro Glasses 2 wearable eye tracker was used to measure pupil diameter, scanning time, and fixation point distribution of the driver during driving. A two-step clustering algorithm and the data-fitting method were used to analyze the experimental data. The results show that the univariate clustering analysis of the pupil diameter change rate of drivers has poor discrimination because the pupil diameter change rate of drivers in the process of “dark adaptation” is larger, while the pupil diameter change rate of drivers in the process of “bright adaptation” is relatively smooth. The univariate and bivariate clustering results of drivers’ pupil diameters were all placed into three categories, with reasonable distribution and suitable differentiation. The clustering results accurately corresponded to different locations of the tunnel. The clustering method proposed in this paper can identify similar behaviors of drivers at different locations in the transition section at the tunnel entrance, the inner section, and the outer area of the tunnel. Through data-fitting of drivers’ visual characteristic parameters in different tunnels, it was found that a short tunnel, with a length of less than 1 km, has little influence on visual characteristics when the maximum pupil diameter is small, and the percentage of saccades is relatively low. An urban tunnel with a length between 1 and 2 km has a significant influence on visual characteristics. In this range, with the increase in tunnel length, the maximum pupil diameter increases significantly, and the percentage of saccades increases rapidly. When the tunnel length exceeds 2 km, the maximum pupil diameter does not continue to increase. The longer the urban tunnel, the more discrete the distribution of drivers’ gaze points. The research results should provide a scientific basis for the design of urban tunnel traffic safety facilities and traffic organization. Full article

Red foxes are a well-established species of urban ecosystems in the UK and worldwide. Understanding the spatial ecology of foxes in urban landscapes is important for enhancement of urban biodiversity and effective disease management. The Resource Dispersion Hypothesis (RDH) holds that territory (home range) size is linked to distribution and richness of habitat patches such that aggregation of rich resources should be negatively associated with range size. Here, we tested the RDH on a sample of 20 red foxes (Vulpes vulpes) in the city of Brighton and Hove. We focused on residential garden areas, as foxes were associated with these in previous studies. We equipped 12 male and 8 female foxes with GPS collars recording at 15 min intervals during discrete seasons over four years. We regressed fox core area size against garden size, number of garden patches, and edge density within and between patches as extracted from GIS in a series of bivariate linear mixed models. We found that foxes used smaller core areas where gardens were large and well-connected and larger core areas where numerous, smaller gardens were fragmented by internal barriers (e.g., fences, walls) or bisected by other habitats such as managed grassland or built-up areas. Our findings confirm the RDH and help to inform future urban planning for wildlife. Full article