Multivariate Distribution Model Research Articles

Constructing multivariate distribution of rainfall characteristics: A Bayesian vine algorithm

Constructing a multivariate distribution model for rainfall characteristics is an important task for risk-based hydrologic design. However, most stochastic rainfall models are limited to bivariate frequentist distribution and cannot be easily generalized to multivariate (≥3) Bayesian settings. Towards this problem, this study presents a Bayesian vine algorithm. The presented algorithm translates the high-dimensional Bayesian update task into multiple lower-dimensional update tasks using the vine approach. The algorithm also computes the evidence, allowing Bayesian model selection of marginals and copulas in the joint distribution. Two numerical examples, a synthetic rainfall data from Canada, and a real rainfall data from China are utilized to demonstrate the algorithm’s capability. The results suggest that the algorithm can satisfactorily identify and characterize the underlying marginal and dependence for the rainfall characteristics. Since the high-dimensional Bayesian update task is translated into lower-dimensional Bayesian updates, the algorithm is scalable, stable, and computationally efficient in higher dimensions. A 20-year continuous hourly rainfall record can be characterized for its depth, duration, and intensity in around 60 s without any convergence or tuning issues. Overall, this study provides a powerful Bayesian tool for constructing stochastic rainfall models, which in turn can be used in multiple civil engineering disciplines concerned with rainfall interaction. This study does not consider temporal/serial dependence but only the cross-dependence among the rainfall characteristics.

Using Copula functions to predict climatic change impacts on floods in river source regions

Flood frequency in river source regions is significantly affected by rainfall and snowmelt as part of climatic changes. A traditional univariate flood frequency analysis cannot reflect the complexity of floods, and when used in isolation, it can only underestimate flood risk. For effective flood prevention and mitigation, it is essential to consider the combined effects of precipitation and snowmelt. Copula functions can effectively quantify the joint distribution relationship between floods and their associated variables without restrictions on their distribution characteristics. This study uses copula functions to consider a multivariate probability distribution model of flood peak flow (Q) with cumulative snowmelt (CSm) and cumulative precipitation (CPr) for the Hutubi River basin located in northern Xinjiang, China. The joint frequencies of rainfall and snowmelt floods are predicted using copula models based on the Coupled Model Intercomparison Project Phase 6 data. The results show that Q has a significant positive correlation with 24-d CSm (r = 0.559, p = 0.002) and 23-d CPr (r = 0.965, p < 0.05). Flood frequency will increase in the future, and mid- (2050–2074) and long-term (2075–2099) floods will be more severe than those in the near-term (2025–2049). The probability of flood occurrence is higher under the SSP2-4.5 and SSP1-2.6 scenarios than under SSP5-8.5. Precipitation during the historical period (1990–2014) led to extreme floods, and increasing future precipitation trends are found to be insignificant. Snowmelt increases with rising temperatures and occurs earlier than estimated, leading to an earlier flood period in the basin and more frequent snowmelt floods. The Q under the joint return period is larger than that during the same univariate return period. This difference indicates that neglecting the interaction between precipitation and snowmelt for floods leads to an underestimation of the flood risk (with underestimations ranging from 0.3% to 22%). The underestimations decrease with an increase in the return period. The joint risks of rainfall or snowmelt according to various flood periods should be considered for rivers with multi-source runoff recharge in flood control design. This study reveals the joint impact of precipitation and snowmelt on extreme floods under climate change in river source regions. This study also provides a scientific basis for regional flood prevention and mitigation strategies, as well as for the rational allocation of water resources.

Crystal structure, cation distribution and electrical properties in NixAlyMn3−x−yO4 (0

NixAlyMn3−x−yO4 (0 < x < 0.6; 0 < y < 0.3) series NTC chip thermistors with homogeneous chemical compositions and high-pure spinel phases were fabricated via a tape casting-lamination-sintering process. The multi-variate cation distribution model was established as (Mn1-x+v2+Nix-v2+)Mn2-2v-y3+Mnv4+Niv2+Aly3+O4 (0 < v < 0.5). The conduction mechanism was proposed based on a nearest neighbor hopping (NNH) model. They were verified coherently by cation-oxyanion chemical compositions (XRF/EDS), valence states (XPS), phase structures and Rietveld refinements (XRD), IR vibrations and electrical analyses. The obtained NTC chips were well suitable for commercial applications with a high reliability and with regulable ρ25 (4080 ± 135 ∼ 15,760 ± 526 Ω cm) and B25/50 (4016 ± 20 ∼ 4251 ± 13 K).

Structured stochastic models based on multi-source heterogeneous data for predicting internal electricity load of non-residential buildings

Electricity load patterns in buildings are increasingly diverse and inherently present building-by-building and temporal variations. This study develops structured stochastic models to capture individual and temporal variations in non-residential building groups (e.g., education, office, culture, and retail buildings). Hourly electricity load data of non-residential buildings, categorized by building use, are sourced from two open datasets (Ministry of Land, Infrastructure and Transport and Korea Electric Power Corporation). First, heterogeneous public datasets are processed and utilized to provide both detailed building information and high-resolution electricity load data. Second, a structured set of four stochastic models are formulated using linear regression and multivariate normal distribution models. Model 1 utilizes the beta distribution to derive daily average electricity load per unit area for each building use. Model 2 utilizes the burr distribution to determine the daily variation of electricity load from the daily average. Models 3 and 4 utilize a multivariate normal distribution to represent the hourly average electricity load profile and its time variation, respectively. The coefficient of variation of the root mean square error (CVRMSE) for the structured probabilistic models ranges from 5.2 % to 14.8 % regarding five key load characterization variables. CVRMSEs for near-peak load, near-base load, rise time, high-load duration, and fall time are 8.5–13.9, 9.5–14.7, 7.3–14.8, 6.4–13.9, and 5.2–11.4 %, respectively. The validation results demonstrate that the proposed probabilistic models capture actual variations in measurements. Structured statistical models improve prediction performance by distinctly reflecting individual and temporal variations in electricity loads.

Data-driven predictions of shield attitudes using Bayesian machine learning

Shield attitude prediction is essential in shield tunnelling to prevent excessive attitude deviations and is highly correlated with shield operational parameters. Furthermore, geological and geotechnical uncertainty are the primary sources of attitude deviation risks; however, these uncertainties were neglected in previous studies. In this paper, a novel data-driven Bayesian machine learning method for site-specific shield attitude predictions is proposed by simultaneously considering the spatial variability of geotechnical parameters and the generic cross-correlations between shield attitude parameters and operational parameters. The Gaussian process model is first used to effectively capture the spatial variability of geotechnical parameters using field borehole data. The multivariate probability distribution model is then constructed to characterise the generic cross-correlation between four attitude parameters, four geotechnical parameters and eight shield operational parameters. In the Bayesian theory framework, a hybrid strategy is proposed to fuse the generic data from other shield tunnelling projects with the ongoing site-specific data, ensuring that the shield attitude prediction is dominated by site-specific data when those data are abundant or by generic data when the site-specific data are sparse. Finally, the Hamiltonian Monte Carlo sampling method is employed to generate the posterior distributions of shield attitudes conditioned on geotechnical parameters and shield operational parameters. The proposed method has the benefit of rigorously quantifying the uncertainties of shield attitude predictions. The proposed approach is examined on a real-world tunnelling project in Nanjing City, indicating that the proposed method outperforms existing techniques in terms of both accuracy and robustness. Parametric analyses are conducted to explore the impacts of key factors on the performance of the proposed approach.

A comprehensive comparison of copula models and multivariate normal distribution for geo-material parametric data

Multivariate distribution models for geo-material parametric data play a crucial role in reliability-based design. Currently, two major models are commonly used: the conventional multivariate normal distribution model and the copula model (including subtypes like the elliptical copula model and Vine copula model). This study aims to assess the performance of different models including three specific copula models (Gaussian copula, t copula, and D-Vine copula models) and the conventional normal distribution model. To this end, this study employs an incomplete database comprising 693 data samples related to lacustrine soils near Dianchi Lake in Kunming city, China. Results indicate that copula models generally exhibit lower model uncertainty compared to the conventional multivariate normal distribution model. Using Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), the elliptical copula ranks first, followed by Gaussian copula, with D-Vine copula ranking third. For a balanced evaluation of performance and complexity, the conventional multivariate normal distribution model is favored. This study provides valuable insights for selecting appropriate models for geo-material parametric data, aiding reliability-based design in geotechnical engineering. Additionally, it is demonstrated that sampling with replacement in the Bootstrapping technique has only a slight effect on the estimation of correlation matrices for the incomplete database.

From point forecasts to multivariate probabilistic forecasts: The Schaake shuffle for day-ahead electricity price forecasting

Modeling price risks is crucial for economic decision making in energy markets. Besides the risk associated with a single price, the dependence structure of multiple prices is often relevant. We therefore propose a generic and easy-to-implement method for generating multivariate probabilistic forecasts based on univariate point forecasts of day-ahead electricity prices. While each univariate point forecast refers to one of the day’s 24 hours, the multivariate forecast distribution models dependencies across hours. The proposed method is based on simple copula techniques and an optional time series component. We illustrate the method for five benchmark data sets. Furthermore, we demonstrate an example for constructing realistic prediction intervals for the weighted sum of consecutive electricity prices as needed for pricing individual load profiles.

Efficient Bayesian method for characterizing multiple soil parameters using parametric bootstrap

This study proposes an efficient Bayesian method using parametric bootstrap in place of the inefficient conventional Bayesian method for characterizing the multivariate PDF of multiple soil parameters. The improvement of the efficiency is achieved by replacing the complex multi-dimensional integral in the conventional Bayesian method with a simple arithmetic average in the proposed method. The good efficiency of the proposed method makes it possible to characterize the multivariate PDF of multiple soil parameters using their limited multivariate site-specific test data and prior knowledge on the site. Moreover, the proposed method has no limitation on the multivariate distribution model and prior distribution of multiple soil parameters. It does not require that the multivariate distribution model is a multivariate normal distribution and the prior distribution is conjugate to the multivariate normal distribution. Two practical examples using real multivariate site-specific test data of multiple soil parameters and one numerical example with known multivariate distribution are presented to illustrate and demonstrate the proposed method. The results indicate that the proposed method performs well in characterizing the multivariate PDF of multiple soil parameters. The role of informative prior knowledge is that it can produce more accurate results than non-informative prior knowledge and thus reduce the demand for the multivariate site-specific test data of multiple soil parameters.

Power laws for accurate determination of landslide volume based on high-resolution LiDAR data

Landslide volume is critical in landslide risk management due to its close association with the landslide mobility and damage to elements at risk. Limited by accessibility, site conditions and availability of equipment, directly measuring the landslide volume on site is challenging. Instead, many empirical area-volume power-law models have been developed to estimate the landslide volume. Yet the validity of these power-law models for small-scale shallow landslides is in question considering the substantial uncertainty of field-estimated landslide volume data for developing these empirical power laws. In this study, two high-resolution LiDAR-derived digital terrain models taken in 2010 and 2020 are leveraged to interpret the volumes of 1326 shallow landslides that occurred in Hong Kong during the same period. The high precision of the digital terrain models ensures the accurate interpretation of landslides and the identification of landslide source and transport areas. On the basis of the interpreted landslide characteristics, new power-law models and a multivariate distribution model are proposed to accurately estimate the volumes of small-scale shallow landslides based on easy-to-obtain landslide source features such as source area, length, width or maximum depth. The multivariate distribution model rigorously expresses the conditional probability distribution of the landslide volume and quantifies the uncertainty of volume estimation. The models therefore enhance the capability of landslide risk assessment and management.

Prediction of soil thermal conductivity based on multivariate probability distribution models

Soil thermal conductivity (λ) is an important parameter for determining the thermal properties of rock and soil materials. In this study, multivariate probability distribution (MPD) models were established based on the factors influencing λ. The performance of the MPD models was evaluated by testing parameters and comparing them with the traditional empirical relationship model of λ to verify the effectiveness of the MPD models. According to the research results, MPD models can accurately predict λ. With the increase in influencing factors considered by the MPD models, the prediction accuracy significantly improved, the correlation coefficient (R2) increased from 0.7125 to 0.9248, the E(ε) value was reduced to 1.0208, and the COV(ε) value was reduced to 0.2336. Among the established MPD models, the performance of the λ-{w, ρd, n, Sr, c, sa, qc} model was the best, and the prediction accuracy of the MPD models was better than that of the traditional empirical relationship model. The results of this study suggest that different types of MPD model should be chosen to estimate the thermal conductivities of different types of soil with significant differences in engineering properties and complex sedimentary environments.

Life history traits modulate the influence of environmental stressors on biodiversity: The case of fireflies, climate and artificial light at night

AbstractAimArtificial light at night (ALAN) is an unprecedented stressor recently introduced in the abiotic milieu of natural landscapes. As such, understanding how ALAN and other natural stressors act in concert to shape the spatial distribution of biodiversity is a core goal in conservation ecology. Here, we aim at understanding how ALAN and climate interact with life history traits and courtship signalling systems to dictate the composition of firefly communities in a global biodiversity hotspot.LocationAn extensive elevational gradient in the Atlantic rain forest (Brazil) currently known as the hottest hotspot of fireflies on Earth.MethodsWe used multivariate species distribution models to understand how species traits and courtship signalling systems interact with climate and ALAN to determine species abundance within firefly communities. We also investigated how species‐specific responses to climate and ALAN scale up to determine compositional changes in firefly communities along the elevational gradient.ResultsWe found that climate shapes communities by filtering species according to their body size and trophic position. ALAN dictates the dominant courtship signalling system within communities by affecting the abundance of species that use bioluminescence or a combination of bioluminescence and pheromones in courtship. We also found that associations between beta‐diversity and ALAN were non‐stationary, being higher in regions under low levels of light pollution. This suggests that even incipient increases in ALAN within protected areas can yield fast changes in the composition of firefly communities.Main ConclusionsFirefly responses to climate and ALAN are modulated by traits associated with different facets of their life histories. Given the alarming changes in both stressors predicted for the foreseeable future, our findings indicate that firefly communities are vulnerable to compositional changes even within protected areas.

Multivariate distribution models of soil electrical resistivity

Soil electrical resistivity is an important parameter in geotechnical engineering. In this paper, a multivariate distribution model is used to predict soil electrical resistivity. The Box-Cox method was used to transform the soil parameters, which were then analysed by a correlation matrix. Multivariate distribution models with different input parameters were established, verified and analysed, and compared with conventional empirical models. The results show that the predictive accuracy of the multivariate distribution models was improved significantly by increasing the number of parameters. The correlation coefficient (R2) of the calibration data increased from 0.05 to 0.88, and that of the validation data increased from 0.06 to 0.83. The model with the best predictive accuracy was the RE−{G, F, Sr} model (R2 = 0.87, E(ε) = 1.22, COV(ε) = 0.98). The predictive accuracy of the multivariate distribution model was obviously higher than that of the conventional empirical model. The multivariate distribution model is an effective and simple way to predict soil electrical resistivity. The proposed model exhibits good performance using limited electrical resistivity data from frozen soils.

Predicted distribution of plains spotted skunk in Arkansas and Missouri

AbstractThe plains spotted skunk (Spilogale interrupta) is of conservation concern because of widespread population declines and is being considered for listing under the United States Endangered Species Act. Although the taxon is relatively rare and difficult to study, recent research and reports provided some information about spotted skunks in Arkansas and Missouri, USA. Using this information, we employed full and simplified multivariate species distribution models to evaluate cover type associations and identify regions of high predicted presence for plains spotted skunks. The simple model contained percent forest and percent development within 5 km, land cover category, and distance to water. Percent forest within 5 km was the most important variable based on permutation importance in both models, indicating that plains spotted skunk habitat may persist in contiguous forest at the landscape scale. Regions predicted to have high presence occurred in northern, western, and southern Arkansas and southern Missouri, totaling &gt;300,000 ha. The resulting plains spotted skunk distribution map can be used for research and management efforts in areas of high probability of occurrence, and future statewide survey efforts may validate our results.

Constructing Quasi-Site-Specific Multivariate Probability Distribution Using Hierarchical Bayesian Model

In geotechnical engineering, it is challenging to construct a site-specific multivariate probability distribution model for soil/rock properties because the site-specific data are usually sparse and incomplete. In contrast, there are abundant generic soil/rock data in the literature for the construction of a generic multivariate probability distribution model, but this model is typically biased and/or imprecise for a specific site. A hybridization method has been proposed to combine these two sources of soil/rock data (site-specific data and a generic database) to produce a quasi-site-specific model, but this method is essentially heuristic. In the current paper, a more rational method that exploits the geologic origin of soil/rock data is proposed. There is a tendency for data to be more similar within a single site and less similar between sites. This is called site uniqueness in geotechnical engineering practice, but no data-driven methods exist to quantify this data feature currently. The hierarchical Bayesian model (HBM) is a natural model to exploit this group information. The grouping criterion can be site localization, soil/rock types, or others. This paper only studies the group criterion based on site localization. This means that a generic database is now viewed as a collection of data groups labeled by qualitative site labels. This site label does not contain any quantitative information such as GPS location, it merely demarcates each group as distinct. The novel contribution is the development of an efficient HBM with closed-form conditional probabilities based on suitably chosen conjugate priors that can handle multivariate, uncertain and unique, sparse, incomplete, and potentially corrupted (MUSIC) data containing site labels. Numerical comparisons between the hybridization method (which cannot incorporate group information) and HBM show that even the simple qualitative knowledge that data belong to a geographically constrained site can improve the estimation of soil/rock properties. The GPS location of each site is not needed.

Quasi-site-specific multivariate probability distribution model for sparse, incomplete, and three-dimensional spatially varying soil data

ABSTRACT In a previous work, the first two authors proposed a data-driven method that can construct a site-specific multivariate probability density function model for soil properties using sparse, incomplete, and spatially variable site investigation data. The spatial variability was limited to the depth direction (horizontal variability was not considered). This data-driven method is referred to as GPR-MUSIC-X. In the current paper, two improvements with respect to GPR-MUSIC-X are made. First, the one-dimensional spatial variability considered by GPR-MUSIC-X is extended to three-dimensional spatial variability (denoted by GPR-MUSIC-3X). Second, a hierarchical Bayesian model (HBM) is adopted to learn the cross-correlation (correlation among different soil parameters) behaviour of generic sites in a soil database accounting for site differences (or uniqueness), and the learning outcome is incorporated into GPR-MUSIC-3X. The resulting model is a quasi-site-specific model (denoted by HBM-MUSIC-3X) because it not only is based on site-specific data but also is informed by the soil database in a manner sensitive to site uniqueness. A case history is used to illustrate the effectiveness of the proposed HBM-MUSIC-3X.

Study on wind-induced fatigue of heliostat based on artificial neural network

Heliostat is a reflection device in tower solar power station. It is generally arranged in a flat and open area, with an independent column structure. Since the angle needs to be adjusted continuously during use, and the mirror panel suffers a large wind pressure, its support structure is prone to wind-induced fatigue. In this paper, a multivariate joint distribution model was first established based on the law of movement of the heliostat with the sun and the distribution of wind speed and direction. Through a combination of wind tunnel test and finite element analysis, the wind-induced fatigue of typical working conditions was analyzed, and then artificial neural network was used to predict the fatigue life of unknown working conditions. The neural network was improved and tested to make it more consistent with the actual situation, so that it can replace a large amount of work of finite element analysis. Finally, the distribution law of wind-induced fatigue life of heliostats under different factors was summarized, and statistics were performed on the working conditions with high probability fatigue, so as to provide a basis for practical engineering application.

Assessing and improving model fitness in MOCABA data assimilation

A mathematical and computational framework is introduced to assess and improve the fitness of multivariate distribution models used in combination with the Bayesian data assimilation framework MOCABA. This is achieved by expanding the distribution model using invertible vairable transformations. Such model expansions enable us to generalize the basic MOCABA framework in order to make it applicable also to observables whose uncertainty distributions significantly deviate from normal distributions. Comparing inferences made on the basis of a normal distribution model with inferences made on the basis of a generalized model enables us to assess the fitness of the normal distribution model. Moreover, in cases where the normal distribution model performs poorly, we may switch to a generalized model with a better performance and assess its fitness by comparing its inferences to inferences from other generalized models. The presented methodology can be used for non-perturbative Bayesian updating of basic input parameters (e.g. nuclear data) or, more generally, of integral functions of these parameters (e.g. the power distribution in a nuclear reactor). In this paper, we focus on generalized distribution models based on variable transformations related to the Johnson SU distribution. To give an illustration of the generalized MOCABA method and to demonstrate its practical benefit, we apply it to the criticality safety analysis of a spent fuel pool for PWR fuel assemblies using data from a large number of different criticality safety benchmark experiments.

An Adaptive Cuckoo Search-Based Optimization Model for Addressing Cyber-Physical Security Problems

One of the key challenges in cyber-physical systems (CPS) is the dynamic fitting of data sources under multivariate or mixture distribution models to determine abnormalities. Equations of the models have been statistically characterized as nonlinear and non-Gaussian ones, where data have high variations between normal and suspicious data distributions. To address nonlinear equations of these distributions, a cuckoo search algorithm is employed. In this paper, the cuckoo search algorithm is effectively improved with a novel strategy, known as a convergence speed strategy, to accelerate the convergence speed in the direction of the optimal solution for achieving better outcomes in a small number of iterations when solving systems of nonlinear equations. The proposed algorithm is named an improved cuckoo search algorithm (ICSA), which accelerates the convergence speed by improving the fitness values of function evaluations compared to the existing algorithms. To assess the efficacy of ICSA, 34 common nonlinear equations that fit the nature of cybersecurity models are adopted to show if ICSA can reach better outcomes with high convergence speed or not. ICSA has been compared with several well-known, well-established optimization algorithms, such as the slime mould optimizer, salp swarm, cuckoo search, marine predators, bat, and flower pollination algorithms. Experimental outcomes have revealed that ICSA is superior to the other in terms of the convergence speed and final accuracy, and this makes a promising alternative to the existing algorithm.

Learning-Based Probabilistic Power Flow Calculation Considering the Correlation Among Multiple Wind Farms

Wind power generation provides a new route for the sustainable development of energy. However, with the large scale integration of wind farms, the volatile wind energy and the correlation among various wind speeds bring a considerable quantity of uncertainty for the power system operation. In this paper, a probabilistic power flow (PPF) analysis model for the power system incorporating the correlation among various wind speeds is proposed. As distinct from existing studies, in this paper, we introduce the relevance vector machine (RVM) into correlation modeling of wind speeds via historical learning samples to construct bivariate joint distribution. Compared with the conventional parameter estimation methods, the proposed method has higher flexibility and computational efficiency. On this basis, the regular vine copula approach is adopted further to build the multivariate joint distribution model of wind speeds. To calculate the PPF of power system with the integration of wind power, we employ the three-point estimation method (3PEM) while the Rosenblatt Transformation technique is proposed to transform the input variables into independent variables. The effectiveness of the proposed calculation framework is examined through simulation studies, and the obtained results illustrate the advantages of the proposed method.

Evaluating multispecies survey designs using a joint species distribution model

Survey designs should be efficient as marine survey programs are usually expensive and time-consuming; however, surveys have rarely been evaluated for multiple species. In the present study, we evaluated multispecies fisheries surveys with respect to three influential factors, i.e., sampling methods, estimation methods and sample size. A joint species distribution model (JSDM) developed in north Yellow Sea, China was used as the operating model to simulate the spatial distribution of multiple species simultaneously. We examined the precision of multispecies abundance estimation using diverse sampling methods [random sampling (RDS), systematic sampling (SYS), stratified random sampling (SRS), generalized random-tessellation stratified sampling (GRT) and spatial coverage sampling (SPC)], estimation methods [arithmetic mean (Arm), universal kriging (Ukr), multivariate distribution model (Mvd), and boral model (Brm)], and a range of sample sizes (from 30 to 300). The results showed significant differences in estimation among sampling methods, where GRT and SYS yielded less relative absolute bias (RAB) over all and RDS showed the least precision. Regarding estimation methods, Mvd and Arm showed the best performances and Brm yielded the least precision. Significant interactions existed between sampling and estimation methods. Arm worked best with GRT, likewise Mvd with SYS and Ukr with SPC. SPC and Mvd showed the best performances for a small sample size (N = 30), and all sampling and estimation methods provided similar results for a large sample size (N = 300). Generally, doubling sample size resulted in a decrease of RAB by 0.097 on average, a rate depending on species, sampling and estimation methods. This study contributed to an integrative framework for evaluating designs of multispecies fisheries surveys.