Search Results (169)

The objective of this study is to analyze spatial entropy flows that reveal the directional dynamics of climate change—patterns that remain obscured in traditional statistical analyses. This approach enables the identification of pathways for “climate information transport”, highlights associations with atmospheric circulation types, and allows for the localization of both sources and “informational voids”—regions where entropy is dissipated. The analytical framework is grounded in a quantitative assessment of long-term climate variability across Europe over the period 1901–2010, utilizing Shannon entropy as a measure of atmospheric system uncertainty and variability. The underlying assumption is that the variability of temperature and precipitation reflects the inherently dynamic character of climate as a nonlinear system prone to fluctuations. The study focuses on calculating entropy estimated within a 70-year moving window for each calendar month, using bivariate distributions of temperature and precipitation modeled with copula functions. Marginal distributions were selected based on the Akaike Information Criterion (AIC). To improve the accuracy of the estimation, a block bootstrap resampling technique was applied, along with numerical integration to compute the Shannon entropy values at each of the 4165 grid points with a spatial resolution of 0.5° × 0.5°. The results indicate that entropy and its derivative are complementary indicators of atmospheric system instability—entropy proving effective in long-term diagnostics, while its derivative provides insight into the short-term forecasting of abrupt changes. A lag analysis and Spearman rank correlation between entropy values and their potential supported the investigation of how circulation variability influences the occurrence of extreme precipitation events. Particularly noteworthy is the temporal derivative of entropy, which revealed strong nonlinear relationships between local dynamic conditions and climatic extremes. A spatial analysis of the information entropy field was also conducted, revealing distinct structures with varying degrees of climatic complexity on a continental scale. This field appears to be clearly structured, reflecting not only the directional patterns of change but also the potential sources of meteorological fluctuations. A field-theory-based spatial classification allows for the identification of transitional regions—areas with heightened susceptibility to shifts in local dynamics—as well as entropy source and sink regions. The study is embedded within the Fokker–Planck formalism, wherein the change in the stochastic distribution characterizes the rate of entropy production. In this context, regions of positive divergence are interpreted as active generators of variability, while sink regions function as stabilizing zones that dampen fluctuations. Full article

In semi-arid regions, flood events are often characterized by rapid runoff and high hydrological variability, posing significant challenges for infrastructure safety and flood risk assessment. Traditional flood frequency analysis methods, typically based on univariate models using annual flood peak, may fail to capture the full complexity of such events. This study investigates the limitations of the univariate approach through the analysis of the 2004 flood event in the Jaguaribe River basin (Brazil), which caused the Castanhão Reservoir to receive a discharge of more than 5 hm³ and fill from 4.5% to over 70% of its capacity in just 55 days. Although the peak discharge in 2004 was not an exceptional record, the combination of high flood volume and short duration revealed a much rarer event than suggested by peak flow alone. To improve compound flood risk assessment, a bivariate frequency analysis based on copula functions was applied to jointly model flood peak and average flood intensity. The latter is a variable newly proposed in this study to better capture the short-duration but high-volume flood until peak that can strongly influence dam safety. Specifically, for the 2004 event, the univariate return period of flood peak was only 35 years, whereas the joint return period incorporating both peak flow and average flood intensity reached 995 years—underscoring a potential underestimation of flood hazard when relying solely on peak flow metrics. Our bivariate return periods and the average flood intensity metric provide actionable information for climate adaptation, supporting adaptive rule curves and risk screening during initial impoundment and high-inflow events in semi-arid reservoirs. Collectively, the proposed methodology offers a more robust framework for assessing extreme floods in intermittent river systems and offers practical insights for dam safety planning in climatically variable regions such as the Brazilian Semi-Arid. Full article

We develop and evaluate a copula-based multistate model for illness–death processes with dependent transition times. The framework couples Cox proportional hazards models for the marginal transition intensities with Archimedean copulas to capture dependence, and it is estimated via the Inference Functions for Margins (IFM) approach under right censoring. A Monte Carlo study shows that assuming independence between transitions can severely underestimate joint survival, yielding coverage as low as $40\%$ under strong dependence, compared with $92\%$ to $97\%$ when copulas are used. We apply the method to a large Colombian cohort of COVID-19 patients (2021 to 2022) that includes sociodemographic, clinical, and vaccination data. The Gumbel copula best captures the strong positive dependence between hospitalization and death, producing more accurate joint survival estimates than independence-based models. Model diagnostics, including proportional hazards tests, Kaplan-Meier comparisons, hazard rate functions, and TTT plots, support the adequacy of the Cox margins. We also discuss limitations and avenues for extension, such as parametric or cure-fraction margins, nested or vine copulas, and full-likelihood estimation. Overall, the results underscore the methodological and applied value of integrating copulas into multistate models, offering a robust framework for analyzing dependent event times in epidemiology and biomedicine. Full article

The dynamic environment of coastal regions subjects infrastructure to multiple interacting natural hazards, with the simultaneous occurrence of windstorms and earthquakes posing a particularly critical challenge. Unlike inland hazards, these coastal threats frequently exhibit irregular statistical behavior and terrain-induced anomalies. This study proposes a novel probabilistic framework to assess compound hazard effects, advancing beyond traditional single-hazard analyses. By integrating maximum entropy theory with bivariate Copula models, a unified return period analysis is developed to capture the joint probability structure of seismic and wind events. The model is calibrated using long-term observational data collected from a representative coastal zone since 2000. For the PGA marginal distribution, our sixth-moment maximum-entropy model achieved an R² of 0.90, compared with 0.57 for a conventional GEV fit—reflecting a 58% increase in explained variance. Analysis shows the progressive evolution of damage from slight damaged through moderate damaged and severe damaged to collapse for an 18-story reinforced concrete frame structure, and shows that the combined effect of seismic and wind loads results in risk probabilities of aforementioned damage state of approximately 2 × 10⁻³, 6 × 10⁻⁴, 2 × 10⁻⁴, and 3 × 10⁻⁵, respectively, under a 0.4 g ground motion and a concurrent wind speed of 15 m/s. Furthermore, when both the uncertainty of loss ratios and structural parameters are incorporated, the standard deviation of the economic loss ratio reaches up to 0.015 in the transition region (PGA 0.2–0.4 g), highlighting considerable variability in economic loss assessment, whereas the mean economic loss ratio rapidly saturates above 0.8 with increasing PGA. These findings demonstrate that uncertainty in economic loss is most pronounced within the transition region, while remaining much lower outside this zone. Overall, this study provides a robust framework and quantitative basis for comprehensive risk assessment and resilient design of coastal infrastructure under compound wind and seismic hazards. Full article

This paper introduces a new unit-Burr distortion function constructed via a transformation of the Burr random variable. The distortion can be applied to existing base copulas to create new copula families. The relationships of tail dependence coefficients and tail orders between the base bivariate copula and the unit-Burr distorted copula are derived. The unit-Burr distortion-induced family of copulas includes well-known copula classes, such as the BB1, BB2, and BB4 copulas, as special cases. The unit-Burr distortion of existing bivariate copulas may result in a family of copulas with both lower and upper tail coefficients ranging from 0 to 1. An empirical application to the rates of return for Microsoft and Google stocks is presented. Full article

Many applications in health research involve the analysis of multivariate distributions of random variables. In this paper, we review the basic theory of copulas to illustrate their advantages in deriving a joint distribution from given marginal distributions, with a specific focus on bivariate cases. Particular attention is given to the Archimedean family of copulas, which includes widely used functions such as Clayton and Gumbel–Hougaard, characterized by a single association parameter and a relatively simple structure. This work differs from previous reviews by providing a focused overview of applied studies in biomedical research that have employed Archimedean copulas, due to their flexibility in modeling a wide range of dependence structures. Their ease of use and ability to accommodate rotated forms make them suitable for various biomedical applications, including those involving survival data. We briefly present the most commonly used methods for estimation and model selection of copula’s functions, with the purpose of introducing these tools within the broader framework. Several recent examples in the health literature, and an original example of a pediatric study, demonstrate the applicability of Archimedean copulas and suggest that this approach, although still not widely adopted, can be useful in many biomedical research settings. Full article

This paper addresses the limitations of existing bivariate generalized exponential (GE) distributions for modeling lifetime data, which often exhibit rigid dependence structures or non-GE marginals. To overcome these limitations, we introduce four new bivariate GE distributions based on the Farlie–Gumbel–Morgenstern, Gumbel–Barnett, Clayton, and Frank copulas, which allow for more flexible modeling of various dependence structures. We employ a Bayesian framework with Markov Chain Monte Carlo (MCMC) methods for parameter estimation. A simulation study is conducted to evaluate the performance of the proposed models, which are then applied to a real-world dataset of electrical treeing failures. The results from the data application demonstrate that the copula-based models, particularly the one derived from the Frank copula, provide a superior fit compared to existing bivariate GE models. This work provides a flexible and robust framework for modeling dependent lifetime data. Full article

Ensuring the security of agricultural systems is essential for achieving national food security and sustainable development. Given that agricultural systems are inherently complex and composed of coupled subsystems—such as water, land, and energy—a comprehensive and multidimensional assessment of system security is necessary. This study focuses on Northeast China, a major food-producing region, and introduces the concept of agricultural system coupling security, defined as the integrated performance of an agricultural system in terms of resource adequacy, internal coordination, and adaptive resilience under external stress. To operationalize this concept, a coupling security evaluation framework is constructed based on three key dimensions: reliability (Rel), coordination (Cor), and resilience (Res). An Agricultural System Coupling Security Index (AS-CSI) is developed using the entropy weight method, the Criteria Importance Through Intercriteria Correlation (CRITIC) method, and the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method, while obstacle factor diagnosis is employed to identify key constraints. Furthermore, bivariate and trivariate Copula models are used to estimate joint risk probabilities. The results show that from 2001 to 2022, the AS-CSI in Northeast China increased from 0.38 to 0.62, indicating a transition from insecurity to relative security. Among the provinces, Jilin exhibited the highest CSI due to balanced performance across all Rel-Cor-Res dimensions, while Liaoning experienced lower Rel, hindering its overall security level. Five indicators, including area under soil erosion control, reservoir storage capacity per capita, pesticide application amount, rural electricity consumption per capita, and proportion of agricultural water use, were identified as critical threats to regional agricultural system security. Copula-based risk analysis revealed that the probability of Rel–Cor reaching the relatively secure threshold (0.8) was the highest at 0.7643, and the probabilities for Rel–Res and Cor–Res to reach the same threshold were lower, at 0.7164 and 0.7318, respectively. The probability of Rel–Cor-Res reaching the relatively secure threshold (0.8) exceeds 0.54, with Jilin exhibiting the highest probability at 0.5538. This study provides valuable insights for transitioning from static assessments to dynamic risk identification and offers a scientific basis for enhancing regional sustainability and economic resilience in agricultural systems. Full article

We establish a correspondence between Lorenz curves and Pickands dependence functions, thereby reframing the construction of any bivariate extreme-value copula as an inequality problem. We discuss the conditions under which a Lorenz curve generates a closed-form Pickands model, considerably expanding the small set of tractable parametrizations currently available. Furthermore, the Pickands measure-generating function M can be written explicitly in terms of the quantile function underlying the Lorenz curve, providing a constructive route to model specification. Finally, classical inequality indices like the Gini coincide with scale-free, rotation-invariant indices of global upper-tail dependence, thereby complementing local coefficients such as the upper tail dependence index ${\lambda }_U$. Full article

The classical exponential model, despite its flexibility, fails to describe data with non-constant failure or between-event dependency. To overcome this limitation, two new bivariate lifetime distributions are introduced in this paper. The Farlie–Gumbel–Morgenstern (FGM)-based and Ali–Mikhail–Haq (AMH)-based modified Fréchet–exponential (MFE) models, by embedding the flexible MEF margin in the FGM and AMH copulas. The resulting distributions accommodate a wide range of positive or negative dependence while retaining analytical traceability. Closed-form expressions for the joint and marginal density, survival, hazard, and reliability functions are derived, together with product moments and moment-generating functions. Unknown parameters are estimated through the maximum likelihood estimation (MLE) and inference functions for margins (IFM) methods, with asymptotic confidence intervals provided for these parameters. An extensive Monte Carlo simulation quantifies the bias, mean squared error, and interval coverage, indicating that IFM retains efficiency while reducing computational complexity for moderate sample sizes. The models are validated using two real datasets, from the medical sector regarding the infection recurrence times of 30 kidney patients undergoing peritoneal dialysis, and from the economic sector regarding the growth of the gross domestic product (GDP). Overall, the proposed copula-linked MFE distributions provide a powerful and economical framework for survival analysis, reliability, and economic studies. Full article

The aim of this study is to quantitatively analyze the long-term climate variability in Poland during the period 1901–2010, using Shannon entropy as a measure of uncertainty and complexity within the atmospheric system. The analysis is based on the premise that variations in temperature and precipitation reflect the dynamic nature of the climate, understood as a nonlinear system sensitive to fluctuations. This study focuses on monthly distributions of temperature and precipitation, modeled using the bivariate Clayton copula function. A normal marginal distribution was adopted for temperature and a gamma distribution for precipitation, both validated using the Anderson–Darling test. To improve estimation accuracy, a bootstrap resampling technique and numerical integration were applied to calculate Shannon entropy at each of the 396 grid points, with a spatial resolution of 0.25° × 0.25°. The results indicate a significant increase in Shannon entropy during the summer months, particularly in July (+0.203 bits) and January (+0.221 bits), compared to the baseline period (1901–1971), suggesting a growing unpredictability of the climate. The most pronounced trend changes were identified in the years 1985–1996 (as indicated by the Pettitt test), while seasonal trends were confirmed using the Mann–Kendall test. A spatial analysis of entropy at the levels of administrative regions and catchments revealed notable regional disparities—entropy peaked in January in the West Pomeranian Voivodeship (4.919 bits) and reached its minimum in April in Greater Poland (3.753 bits). Additionally, this study examined the relationship between Shannon entropy and global climatic indicators, including the Land–Ocean Temperature Index (NASA GISTEMP) and the ENSO index (NINO3.4). Statistically significant positive correlations were observed between entropy and global temperature anomalies during both winter ($\rho$ = 0.826) and summer ($\rho$ = 0.650), indicating potential linkages between local climate variability and global warming trends. To explore the direction of this relationship, a Granger causality test was conducted, which did not reveal statistically significant causality between NINO3.4 and Shannon entropy ($p$ > 0.05 for all lags tested), suggesting that the observed relationships are likely co-varying rather than causal in the Granger sense. Further phase–space analysis (with a delay of $\tau$ = 3 months) allowed for the identification of attractors characteristic of chaotic systems. The entropy trajectories revealed transitions from equilibrium states (average entropy: 4.124–4.138 bits) to highly unstable states (up to 4.768 bits), confirming an increase in the complexity of the climate system. Shannon entropy thus proves to be a valuable tool for monitoring local climatic instability and may contribute to improved risk modeling of droughts and floods in the context of climate change in Poland. Full article

Metals in deep-sea polymetallic nodules are indispensable for battery production and play a crucial role in facilitating the socio-economic green transition. A three-dimensional laser scanning model of nodules in the northwest Pacific Ocean has yielded an amount of data on volume, shape, and particle size. To deeply mine the correlation between the characteristics of the nodules, a joint probability density function (JPDF) based on copula theory is used. A univariate probability density function (PDF) linked to the particle size, burred depth, shape factor, and mass of the ores is established. The trend of nodule density with particle size is analyzed. Then, bivariate joint distribution using the copula method is constructed for mass and particle size. Furthermore, trivariate joint distribution using the copula method for nodule mass, particle size, and shape factor is derived. The results of this paper provide data to support the resource assessment of polymetallic nodules and optimize the design of mining systems. Full article

As an extension of the (univariate) Birnbaum–Saunders distribution, the Type-II generalized crack (GCR2) distribution, built on an appropriate base density, provides a sufficient level of flexibility to fit various distributional shapes, including heavy-tailed ones. In this paper, we develop a bivariate extension of the Type-II generalized crack distribution and study its dependency structure. For practical applications, three specific distributions, GCR2-Generalized Gaussian, GCR2-Student’s t, and GCR2-Logistic, are considered for marginals. The expectation-maximization algorithm is implemented to estimate the parameters in the bivariate GCR2 models. The model fitting results on a catastrophic loss dataset show that the bivariate GCR2 distribution based on the generalized Gaussian density fits the data significantly better than other alternative models, such as the bivariate lognormal distribution and some Archimedean copula models with lognormal or Pareto marginals. Full article

This paper considers comparing properties and characterizations of the bivariate functions under Archimedean copula. It is shown that some results of the usual stochastic order for the bivariate functions in the independent case are generalized to the Archimedean copula-linked dependent case, and we also derive some characterizations of different bivariate functions composed by Archimedean copula-linked dependent random variables. These results generalize some existing results in the literature and bring conclusions closer to reality. Two applications in scheduling problems are also provided to illustrate the main results. Full article

In this research, we present a new distribution, which is the bivariate alpha power Burr-XII distribution, based on the alpha power Burr-XII distribution. We thoroughly examine the key features of our newly developed bivariate model. We introduce a new class of bivariate models, which are built with the copula function. The statistical properties of the proposed distribution, such as conditional distributions, conditional expectations, marginal distributions, moment-generating functions, and product moments were studied. This was accomplished with two datasets of real data that came from two distinct devices. We employed Bayesian, maximum likelihood estimation, and least squares estimation strategies to obtain estimated points and intervals. Additionally, we generated bootstrap confidence intervals and conducted numerical analyses using the Markov chain Monte Carlo method. Lastly, we compared this novel bivariate distribution’s performance to earlier bivariate models, to determine how well it fit the real data. Full article