Joint Probability Distribution Research Articles

Field theory of active chiral hard disks: A first-principles approach to steric interactions

Abstract A first-principles approach for active chiral hard disks is presented, that 
explicitly accounts for steric interactions on the two-body level. We derive an 
effective one-body equation for the joint probability distribution of positions 
and angles of the particles. By projecting onto the angular modes, we write a 
hierarchy for the lowest hydrodynamic modes, i.e. particle density, polarisation, 
and nematic tensor. Introducing dimensionless variables in the equations, we
highlight the assumptions, which - though inherent - are often included implicit in 
typical closure schemes of the hierarchy. By considering 
different regimes of the Péclet number, the well-known models in active 
matter can be obtained through our consideration. Explicitly, we derive 
an effective diffusive description and by going to higher 
orders in the closure scheme, we show that this first-principles approach 
results in the recently introduced Active Model B +, a natural extension of the 
Model B for active processes. Remarkably, here we find that chirality can 
change the sign of the phenomenological activity parameters.

Causal Structure Learning with Conditional and Unique Information Groups-Decomposition Inequalities

The causal structure of a system imposes constraints on the joint probability distribution of variables that can be generated by the system. Archetypal constraints consist of conditional independencies between variables. However, particularly in the presence of hidden variables, many causal structures are compatible with the same set of independencies inferred from the marginal distributions of observed variables. Additional constraints allow further testing for the compatibility of data with specific causal structures. An existing family of causally informative inequalities compares the information about a set of target variables contained in a collection of variables, with a sum of the information contained in different groups defined as subsets of that collection. While procedures to identify the form of these groups-decomposition inequalities have been previously derived, we substantially enlarge the applicability of the framework. We derive groups-decomposition inequalities subject to weaker independence conditions, with weaker requirements in the configuration of the groups, and additionally allowing for conditioning sets. Furthermore, we show how constraints with higher inferential power may be derived with collections that include hidden variables, and then converted into testable constraints using data processing inequalities. For this purpose, we apply the standard data processing inequality of conditional mutual information and derive an analogous property for a measure of conditional unique information recently introduced to separate redundant, synergistic, and unique contributions to the information that a set of variables has about a target.

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.

Strongly correlated multielectron bunches from interaction with quantum light.

Strongly correlated electron systems are a cornerstone of modern physics, being responsible for groundbreaking phenomena from superconducting magnets to quantum computing. In most cases, correlations in electrons arise exclusively because of Coulomb interactions. In this work, we reveal that free electrons interacting simultaneously with a light field can become highly correlated via mechanisms beyond Coulomb interactions. In the case of two electrons, the resulting Pearson correlation coefficient for the joint probability distribution of the output electron energies is enhanced by more than 13 orders of magnitude compared to that of electrons interacting with the light field in succession (one after another). These highly correlated electrons are the result of momentum and energy exchange between the participating electrons via the external quantum light field. Our findings pave the way to the creation and control of highly correlated free electrons for applications including quantum information and ultrafast imaging.

An application-time-window based credibility assessment method for equipment digital twins in manufacturing

To effectively assess the credibility of manufacturing equipment digital twins (DT), a new method fully considering the dynamic evolution process is urgently needed. In this paper, the influence of the evolution process on the applications is scrutinized, the connotation of equipment DT credibility assessment is expanded, and an assessment method to evaluate the general evolution process of equipment DTs in manufacturing is proposed. The method obtains the application-time-window (ATW) by fitting a joint probability distribution based on observation of actual application time points. Then the method uses the probability of application occurrence as a weight to quantitatively assess the model performance throughout the evolution process. The effectiveness of the method was validated by ablation experiments in manufacturing simulation scenes of workpieces transferring with robotic arms. Compared with existing assessment methods, the error of quantified DT credibility value is reduced by at least 65% and the assessment result can be obtained once the evolution is completed.

On the Bivariate Extension of the Extended Standard U-quadratic Distribution

This paper derives the bivariate version of the extended standard U-quadratic (eSU) distribution using the method of compounding or pseudo family of distributions. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape with the following properties: $(i)$ $X$ and $Y$ have bathtub shapes; $(ii)$ $X$ has constant and $Y$ has bathtub shapes; and $(iii)$ $X$ has inverted bathtub and $Y$ has bathtub shapes. Moreover, some properties of the derived distribution such as the marginal distribution, conditional distributions, conditional moments, product and ratio moments are derived. Further, the maximum likelihood estimation is performed to estimate the parameters of the derived distribution. Also, a simulation study is carried out to evaluate the behavior of the estimates of the parameters. Moreover, we derive a new bivariate Kumaraswamy distribution and use it to simulate bivariate data with $X$ and $Y$ having bathtub shapes. Furthermore, A new bivariate version of the Cubic Transmuted Uniform (CTU) distribution is also derived. Finally, the proposed Bivariate eSU distribution is applied to simulated data and compared with the said Bivarite Cubic Transmuted Uniform distribution. The result shows that the proposed Bivariate eSU distribution provides a better fit for the said simulated dataset as compared with the Bivariate CTU distribution.

Reliability-Based Design Optimization for a Vertical-Type Breakwater with an Emphasis on Sliding, Overturn, and Collapse Failure

To promote the application of reliability-based design within the Korean coastal engineering community, the author conducted reliability analyses and optimized the design of a vertical-type breakwater, considering multiple limit states in the seas off of Pusan and Gunsan – two representative ports in Korea. In this process, rather than relying on design waves of a specific return period, the author intentionally avoided such constraints. Instead, the author characterized the uncertainties associated with wave force, lift force, and overturning moment – key factors significantly influencing the integrity of a vertical-type breakwater. This characterization was achieved by employing a probabilistic model derived from the frequency analysis results of long-term in-situ wave data. The limit state of the vertical-type breakwater encompassed sliding, overturning, and collapse failure, with the close interrelation between wave force, lift force, and moment described using the Nataf joint probability distribution. Simulation results indicate, as expected, that considering only sliding failure underestimates the failure probability. Furthermore, it was shown that the failure probability of vertical-type breakwaters cannot be consistently secured using design waves with a specific return period. In contrast, breakwaters optimally designed to meet the reliability index requirement of <i>β</i>-3.5 to 4 consistently achieve a consistent failure probability across all sea areas.

Stochastic dynamics of mechanical systems with impacts via the Step Matrix multiplication based Path Integration method

AbstractIn this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics.

Fragility Analysis of a Transmission Tower-Line System Subjected to Wind and Ice Loads Considering Fatigue Damage

Disasters such as ice and wind can pose a serious threat to the normal operation of power transmission lines. Wind-induced fatigue damage can further reduce the load-carrying capacity of transmission towers, increasing their fragility to ice and wind disasters. To assess the comprehensive capability of transmission towers to resist extreme ice and wind disasters, this paper proposes a failure probability evaluation framework for transmission towers considering wind-induced fatigue damage under the coupled effect of ice and wind. Taking a certain transmission line in Hunan as an example, the failure probability caused by ice and wind disasters is calculated for different years of service. First, based on the historical meteorological data in three cities in Hunan, a joint probability model of wind speed and wind direction considering their correlation is established using the copula function. Then, based on this probability model, the wind-induced fatigue damage of transmission towers is calculated using the Miner linear damage theory and the S-N curve. Subsequently, the fragility of the tower under the coupled load of ice and wind is calculated for different years of service. Finally, by combining the structural fragility function with the joint probability distribution model of ice thickness and wind speed, the collapse probability of the transmission tower under the action of ice and wind disasters is calculated. The results indicate that the influence of wind-induced fatigue damage cannot be ignored when transmission towers encounter ice and wind disasters. With increasing service time, the ability of transmission towers to resist ice and wind disasters gradually decreases, and the failure probability also increases under extreme ice and wind conditions.

Rapid simulation of two-dimensional spectra with correlated anisotropic dimensions.

A new algorithm has been developed to simulate two-dimensional (2D) spectra with correlated anisotropic frequencies faster and more accurately than previous methods. The technique uses finite-element numerical integration on the sphere and an interpolation scheme based on the Alderman-Solum-Grant algorithm. This method is particularly useful for numerical calculations of joint probability distribution functions involving quantities with a parametric orientation dependence. The technique's efficiency also allows for practical least-squares fitting of experimental 2D solid-state nuclear magnetic resonance (NMR) datasets. The simulation method is illustrated for select 2D NMR methods, and a least-squares analysis is demonstrated in the extraction of paramagnetic shift and quadrupolar coupling tensors and their relative orientation from the experimental shifting-d echo 2H NMR spectrum of a NiCl2 · 2D2O salt.

Deconvolution and Analysis of the 1H NMR Spectra of Crude Reaction Mixtures.

Nuclear magnetic resonance (NMR) spectroscopy is an important analytical technique in synthetic organic chemistry, but its integration into high-throughput experimentation workflows has been limited by the necessity of manually analyzing the NMR spectra of new chemical entities. Current efforts to automate the analysis of NMR spectra rely on comparisons to databases of reported spectra for known compounds and, therefore, are incompatible with the exploration of new chemical space. By reframing the NMR spectrum of a reaction mixture as a joint probability distribution, we have used Hamiltonian Monte Carlo Markov Chain and density functional theory to fit the predicted NMR spectra to those of crude reaction mixtures. This approach enables the deconvolution and analysis of the spectra of mixtures of compounds without relying on reported spectra. The utility of our approach to analyze crude reaction mixtures is demonstrated with the experimental spectra of reactions that generate a mixture of isomers, such as Wittig olefination and C-H functionalization reactions. The correct identification of compounds in a reaction mixture and their relative concentrations is achieved with a mean absolute error as low as 1%.

Multifrequency Very Long Baseline Interferometry Imaging of the Subparsec-scale Jet in the Sombrero Galaxy (M104)

We report multifrequency and multiepoch very long baseline interferometry studies of the subparsec jet in the Sombrero galaxy (M104, NGC 4594). Using Very Long Baseline Array data at 12, 22, 44, and 88 GHz, we study the kinematics of the jet and the properties of the compact core. The subparsec jet is clearly detected at 12 and 22 GHz, and the inner jet base is resolved down to ∼70 Schwarzschild radii (R s) at 44 GHz. The proper motions of the jet are measured with apparent subrelativistic speeds of 0.20 ± 0.08c and 0.05 ± 0.02c for the approaching and the receding jet, respectively. Based on the apparent speed and jet-to-counterjet brightness ratio, we estimate the jet viewing angle to be larger than ∼37°, and the intrinsic speed to be between ∼0.10c and 0.40c. Their joint probability distribution suggests the most probable values of the viewing angle and intrinsic speed to be 66°−6°+4° and 0.19 ± 0.04c, respectively. We also find that the measured brightness temperatures of the core at 12, 22, and 44 GHz are close to the equipartition brightness temperature, indicating that the energy density of the radiating particles is comparable to the energy density of the magnetic field in the subparsec jet region. Interestingly, the measured core size at 88 GHz (∼25 ± 5 R s) deviates from the expected frequency dependence seen at lower frequencies. This may indicate a different origin for the millimeter emission, which can be explained by an advection-dominated accretion flow (ADAF) model. This model further predicts that at 230 and 340 GHz, the ADAF may dominate the radio emission over the jet.

Distilling dynamical knowledge from stochastic reaction networks

Stochastic reaction networks are widely used in the modeling of stochastic systems across diverse domains such as biology, chemistry, physics, and ecology. However, the comprehension of the dynamic behaviors inherent in stochastic reaction networks is a formidable undertaking, primarily due to the exponential growth in the number of possible states or trajectories as the state space dimension increases. In this study, we introduce a knowledge distillation method based on reinforcement learning principles, aimed at compressing the dynamical knowledge encoded in stochastic reaction networks into a singular neural network construct. The trained neural network possesses the capability to accurately predict the state conditional joint probability distribution that corresponds to the given query contexts, when prompted with rate parameters, initial conditions, and time values. This obviates the need to track the dynamical process, enabling the direct estimation of normalized state and trajectory probabilities, without necessitating the integration over the complete state space. By applying our method to representative examples, we have observed a high degree of accuracy in both multimodal and high-dimensional systems. Additionally, the trained neural network can serve as a foundational model for developing efficient algorithms for parameter inference and trajectory ensemble generation. These results collectively underscore the efficacy of our approach as a universal means of distilling knowledge from stochastic reaction networks. Importantly, our methodology also spotlights the potential utility in harnessing a singular, pretrained, large-scale model to encapsulate the solution space underpinning a wide spectrum of stochastic dynamical systems.

On joint distribution of bids in some k-th price auction model

In the era of increasingly widespread use of electronic communication tools, constant growth of e-commerce turnover is hardly surprising. One of its mechanisms are online auctions whose turnover, on a global scale, is enormous. This paper proposes a stochastic model of the value of subsequent actual bids appearing in the course of a dynamic internet k-th price auction. On this basis, it was possible to determine unconditional joint probability distributions, as well as univariate marginal distributions of successive bids appearing in the course of the auction type under consideration.

Fragility assessment of sea-crossing cable-stayed bridge subjected to multi-hazard action via TKC and R-vine copula

Seismic fragility analysis methods commonly used for assessing structural components and systems have certain limitations. This paper introduces a new semi-parametric approach for structural component fragility analysis, the TKC method. This method is based on log-transformed kernel density estimation (log-TKDE) and copula functions, which establish the joint probability distribution (JPD) of a structural response value (d) and a disaster-inducing factor strength parameter (α) using log-TKDE and match the marginal cumulative distributions (MCDs) of d and a via bivariate copula functions. A novel method for evaluating structural system fragility is also proposed based on the R-vine copula function. This approach accounts for the intricate correlations among different components, providing a precise estimation of the multidimensional JPD. The reliability and accuracy of the proposed methods are demonstrated through a seismic fragility analysis of a sea-crossing cable-stayed bridge under earthquake load. Furthermore, a multi-hazard fragility analysis of bridge components and systems is conducted considering hazards such as earthquake, wind, and wave loads. A comparison is made with the seismic fragility of bridges subjected to earthquake loads alone. The results show that the median and 95% confidence of the damage probability of structural components calculated by the TKC method and Monte Carlo simulation (MCS) closely match, which validates the proposed method. The TKC method is also more accurate and computationally efficient than the MCS method with the same Bootstrap sample size and confidence level (95%). The proposed structural system fragility assessment method based on the R-vine copula accounts for complex correlations between components and yields more accurate results than the first-order boundary method. Load spectrum characteristics, structural component characteristics, and other factors cause the multi-hazard fragility of bridge components and systems to differ substantially from their seismic fragility. The difference rate of system damage probability can reach 28.82%. In evaluating the anti-disaster performance of sea-crossing bridges, it is necessary to incorporate the marine environmental load and consider the dynamic response coupling effect of the bridge excited by seafloor ground motions with extreme wind and wave loads. This approach can guide the optimization of bridge designs and facilitate multi-hazard fragility analysis in accordance with real damage probabilities.

Abstract 7373: Generative Bayesian networks for augmentation of molecular data from commercial genetics panels

Abstract Background: Computational models in medical research which use molecular features to predict patient sensitivities or outcomes are traditionally limited to producing a scalar output for a given disease phenotype, e.g. progression-free survival or drug response. Even for relatively intelligible models, it is generally difficult to place confidence limits on the prediction or develop an intuition for how a prediction varies with respect to “what-if” changes in those features or how sensitive a prediction is to changes in input variables. Moreover, while essential for understanding and modeling tumor behavior, molecular features from patient tumors in real-world settings are typically limited to profiles of hundreds of genes, while the biologically relevant inputs to the patient’s condition can be much larger. Approach: To generate data required to enhance the interpretation of predictive models and extend the utility of real-world patient oncology datasets, a distribution of conditions consistent with the observed features and their incidence across cancer patient cohorts is required. We hypothesized we could synthesize patient samples drawn from the joint probability distribution of a broader universe of features when a subset of them is held fixed at the observed values. We model this joint distribution by learning a Bayesian network over a broad feature set for which some training data is available and generate feature profiles by applying Gibbs sampling to the learned network. Results: We assessed the potential clinical utility of using these generated feature profiles in predicting drug response and enhancing biological interpretation of different model outputs. We found generative somatic mutations were useful for predicting cancer patient outcomes, including drug response predictions in breast cancer tumors from AACR Project GENIE. Synthesized patient feature profiles enhanced biological interpretability of nominal panel data (100-200 genes), providing a means to assign probabilities to outcomes and uncovering previously described mechanisms of drug response in real-world patient data. Conclusion: We introduce an approach that leverages Bayesian networks to synthesize richly annotated patient feature profiles from limited molecular data routinely collected in real-world settings. This approach addresses challenges associated with limited molecular data, biological interpretability and evaluation of predictive models and enhances the utility of real-world datasets for cancer research. Citation Format: Dillon H. Tracy, Jeff Sherman, Maayan Baron. Generative Bayesian networks for augmentation of molecular data from commercial genetics panels [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 7373.

Serviceability analysis of sea-crossing bridges under correlated wind and wave loads

With coupled wind and wave loads suffered by sea-crossing bridges, it is challenging to include uncertainties for assessing bridge serviceability. In this study, an efficient probabilistic peak response framework with surrogate models is developed for the serviceability analysis of sea-crossing bridges under coupled wind and wave loads. The joint probability distribution functions (JPDF) of the mean wind speed, significant wave height, and peak wave period are first derived based on long-term field measurement data and using a C-vine copula model. The JPDF serves as an input to the bridge to calculate the peak response of the bridge through finite-element analyses. To improve computational efficiency and make the problem manageable, the surrogate models are then generated by both the support vector machine and response surface method to predict the bridge peak responses. The exceeding probabilities for the serviceability assessment are finally obtained by the Monte-Carlo method using the pre-defined threshold values of bridge serviceability. A real long-span sea-crossing bridge is used as a case study to demonstrate the feasibility and effectiveness of the proposed framework. The results elucidate the effects of correlated wind and wave loads on bridge serviceability. Ignoring the correlation of wind and wave parameters will significantly overestimate the exceeding probability of bridge serviceability.

A fast uncertainty quantification methodology and sampling technique for joint probability distribution of the Arrhenius rate expression: a case study applied to H2/CO kinetic mechanism

This work proposes a fast, novel, mathematically robust, and elegant unconstrained Method of Uncertainty Quantification (MUQ) for the temperature-dependent Arrhenius rate constant using Cholesky Decomposition (CD) of the covariance matrix of the Arrhenius parameters. The Arrhenius parameters of a reaction are treated as normally distributed correlated random variables. The MUQ method lends itself to an approach for Sampling of Arrhenius Curves (SAC), which automatically ensures that the generated samples are consistent with the distribution of Arrhenius parameters. The Method of Uncertainty Quantification and Sampling of Arrhenius Curves (MUQ-SAC) is used to train a quadratic polynomial response surface (PRS). Three important classes of Arrhenius curves possible within the SAC method are identified. From the total set of targets, a small subset of targets is first used to study the combinations of different classes of Arrhenius curves and their proportions within the design matrix, which is used to train PRS. Based on this understanding, response surfaces are generated for 64 ignition delay targets (Tig), which comprises of a wide range of pressure (P: 1--32.7 atm), temperature (T: 916-DIFadd-2869 K), and equivalence ratio (ϕ: 0.5--6.11), as well as for 74 laminar flame speed targets (Fls), spanning P: 1--25 atm, T: 285--600 K, and ϕ: 0.6--5. The H 2 /CO sub-mechanism of FFCM1.0 is chosen for the case study in which 22 reactions (66 Arrhenius parameters) are considered active parameters. The quality of the generated PRS is measured using the relative Maximum Residual Error (MRE). The quality of PRS is also checked against the search iterations encountered during mechanism optimisation. Accurate PRS could be generated over the search iterations, validating that the SAC method is able to span the entire uncertainty domain while training the PRS. The implementation of the MUQ-SAC method is made available at https://github.com/KrunalPanchal1995/MUQ-SAC.git.

Investigating the Main Features of the Correlation Between Electron Density and Temperature in the Topside Ionosphere Through Swarm Satellites Data

AbstractElectron density (Ne) and electron temperature (Te) observations collected by Langmuir Probes on board the European Space Agency (ESA) Swarm B satellite are used to characterize their correlation in the topside ionosphere at an altitude of about 500 km. Spearman correlation coefficient values (RSpearman) are calculated on joint probability distributions between Ne and Te for selected conditions. The large data set of Swarm B observations at 2‐Hz rate, covering the years 2014–2022, allowed investigating the correlation properties of the topside ionospheric plasma on a global scale, for different diurnal and seasonal conditions, with both a coverage and a detail never reached before. Results are given as maps of RSpearman as a function of the Quasi‐Dipole (QD) magnetic latitude and magnetic local time (MLT) coordinates. The characterization of the correlation at high latitudes, along with the description of the diurnal trend at all latitudes, are the new findings of this study. The main correlation features point out a negative correlation at the morning overshoot, during daytime at mid latitudes, and during nighttime at the ionospheric trough and subauroral latitudes. Conversely, a positive correlation dominates the nighttime hours at mid and low latitudes and, to a minor extent, the low latitudes from 09 MLT onwards. A seasonal dependence of the correlation is noticeable only at very high latitudes where the general pattern of the negative correlation does not hold around ±75° QD latitude in the summer season. Results from Swarm B have been statistically compared and discussed with observations from the Arecibo, Jicamarca, and Millstone Hill incoherent scatter radars.

A Joint Probability Distribution for Multivariate Wind-Wave Conditions and Discussions on Uncertainties

Abstract This article presents a joint statistical model, which is needed in probabilistic design and structural risk assessment, that has been fitted to data of wind and wave conditions for an offshore location off South Brittany. The data are from a numerical model and contain hourly values for several wind and wave variables over a period of 32 years. The joint distribution presented in this article considers the variables wind direction, mean wind speed, significant wave height, wave direction, and peak period. A conditional model for turbulence given wind speed is introduced to yield an additional variable for the joint model. The joint model is constructed as a product of marginal and conditional models for the various variables. Additionally, the fitted models will be used to construct environmental contours for some of the variables. For significant wave height, various models are used to obtain different extreme value estimates, illustrating the uncertainties involved in extrapolating statistical models beyond the support of the data, and a discussion on the use of nonparametric copulas for the joint distribution is presented. Moreover, bootstrap has been performed to estimate the uncertainty in estimated model parameters from sampling variability. The effect of changing which variable to model as the marginal in a conditional model is illustrated by switching from wind speed to significant wave height. Such joint distribution models are important inputs for design of offshore structures, and in particular for offshore wind turbines, and the influence of the joint model in design is illustrated by a simple case study. This article is an extension of the conference paper by Vanem et al. (2023, “A Joint Probability Distribution Model for Multivariate Wind and Wave Conditions,” 42nd International Conference on Ocean, Offshore and Arctic Engineering).