Empirical Probability Distribution Research Articles

Effects of hydrological variability on the sustainable use of water in a regional economy. An application to Tuscany

Existing input-output (IO) models have mainly focused on water demand. Some studies have incorporated water supply (availability), but do not take into account its natural variability, an essential element when performing a water stress analysis. The present study integrates the hydrological variability of water availability into a hydroeconomic IO model, considering its exogenous effects on water supply and its exogenous effects on water demand. Two endogenous effects are considered: i) changes in blue water requirements in the agricultural industry due to variations in precipitation and evapotranspiration, and ii) changes in grey water requirements in all discharging industries due to variations in runoff and groundwater recharge. By means of a T-years hydrological series and Monte Carlo simulations, the model allows estimating T values of the Extended Water Exploitation Index (EWEI), obtaining its empirical probability distribution and confronting it with scarcity thresholds. Additionally, the model includes a methodology to incorporate intra-annual variation, obtaining the critical month EWEI and defining a more transparent and endogenous scarcity threshold. Empirically tested for the Italian region of Tuscany considering a multivariate hydrological model for the generation of a 100-year hydrological series, our results allow a more in-depth analysis of water scarcity in the region.

What is the actual composition of specific land cover? An evaluation of the accuracy at a national scale – Remote sensing in comparison to topographic land cover

Satellite imagery allows us to capture and collect land cover information for increasingly large areas. This allows us to represent current land cover on maps in a simple and standardized way; however, any land cover determined in this way is subject to some algorithmic uncertainty. This paper aims, for the first time, to indicate the magnitude of this uncertainty through the empirical probability distribution of a given land cover at a given location. By analyzing 3 data sources, i.e. the Corine Land Cover map, the POLSA land cover map and the classic map - the BDOT10k database of topographic objects. Empirical distributions of the occurrence of land cover class data in areas with a given land use on a topographic map were determined. The work was carried out on a large scale, i.e. on the maximum possible sample for Poland, i.e. on the area of the whole country. This makes it possible to introduce and quantify uncertainties. Spatial analyses were carried out using satellite-based methods to determine land cover or using a topographic map. This work and its results will be useful to all users who want to assess the occurrence of a phenomenon in a given area, taking into account the uncertainty of the land cover, and thus obtain more accurate and reliable results. It also provides, for the first time, a methodology for verifying such map correspondences, which can be replicated in work by other researchers, using the confusion matrix and as evaluation metrics the true positive rate (TPR) and weighted accuracy have been adopted. The paper proposes a link between land cover classes in all databases. It was shown that the TPR for BDOT10k was higher than 50% only with CLC Level 1 (72.0%) and POLSA Land Cover (61%), while the TPR for RS classes for each remote sensing data was always higher than 60% with BDOT10k. The class with the highest remote sensing classes was related to water, especially marine (92.0% for POLSA and 85.3% for CLC level 3), arable land (98% for POLSA, lowest for CLC level 3 (80%), and forests (coniferous POLSA – 89%, CLC level 1 and 2–85%), while low values were obtained for wetlands, peatbogs. The authors do not state which land cover approach is better, as each may have multiple uses, but the values presented in this work must raise awareness of uncertainties in land cover and critical implementation in decision-making processes for multiple areas of human activity. The study provides ready-to-use values of the probability of a given land cover class being present on a topographic map, given that remote sensing has classified it as such. These functions can also be used in reverse, to determine the probability of a given land cover class being present in remote sensing, given that a specific class has been identified on a topographic map. The results of the consistency assessment, with the composition structure, can be used by a wide range of users, including public administration, land managers, land architects, public services, academia and individuals.

K-Means NANI: An Improved Clustering Algorithm for Molecular Dynamics Simulations.

One of the key challenges of k-means clustering is the seed selection or the initial centroid estimation since the clustering result depends heavily on this choice. Alternatives such as k-means++ have mitigated this limitation by estimating the centroids using an empirical probability distribution. However, with high-dimensional and complex data sets such as those obtained from molecular simulation, k-means++ fails to partition the data in an optimal manner. Furthermore, stochastic elements in all flavors of k-means++ will lead to a lack of reproducibility. K-means N-Ary Natural Initiation (NANI) is presented as an alternative to tackle this challenge by using efficient n-ary comparisons to both identify high-density regions in the data and select a diverse set of initial conformations. Centroids generated from NANI are not only representative of the data and different from one another, helping k-means to partition the data accurately, but also deterministic, providing consistent cluster populations across replicates. From peptide and protein folding molecular simulations, NANI was able to create compact and well-separated clusters as well as accurately find the metastable states that agree with the literature. NANI can cluster diverse data sets and be used as a standalone tool or as part of our MDANCE clustering package.

Observed and future shifts in climate zone of Borneo based on CMIP6 models

Climate change has significantly altered the characteristics of climate zones, posing considerable challenges to ecosystems and biodiversity, particularly in Borneo, known for its high species density per unit area. This study aimed to classify the region into homogeneous climate groups based on long-term average behavior. The most effective parameters from the high-resolution daily gridded Princeton climate datasets spanning 65 years (1950–2014) were utilized, including rainfall, relative humidity (RH), temperatures (Tavg, Tmin, Tmax, and diurnal temperature range (DTR)), along with elevation data at 0.25° resolution. The FCM clustering method outperformed K-Mean and two Ward's hierarchical methods (WardD and WardD2) in classifying Borneo's climate zones based on multi-criteria assessment, exhibiting the lowest average distance (2.172–2.180) and the highest compromise programming index (CPI)-based correlation ranking among cluster averages across all climate parameters. Borneo's climate zones were categorized into four: ‘Wet and cold’ (WC) and ‘Wet’ (W) representing wetter zones, and ‘Wet and hot’ (WH) and ‘Dry and hot’ (DH) representing hotter zones, each with clearly defined boundaries. For future projection, EC-Earth3-Veg ranked first for all climate parameters across 961 grid points, emerging as the top-performing model. The linear scaling (LS) bias-corrected EC-Earth3-Veg model, as shown in the Taylor diagram, closely replicated the observed datasets, facilitating future climate zone reclassification. Improved performance across parameters was evident based on MAE (35.8–94.6%), MSE (57.0–99.5%), NRMSE (42.7–92.1%), PBIAS (100–108%), MD (23.0–85.3%), KGE (21.1–78.1%), and VE (5.1–9.1%), with closer replication of empirical probability distribution function (PDF) curves during the validation period. In the future, Borneo's climate zones will shift notably, with WC elongating southward along the mountainous spine, W forming an enclave over the north-central mountains, WH shifting northward and shrinking inland, and DH expanding northward along the western coast. Under SSP5-8.5, WC is expected to expand by 39% and 11% for the mid- and far-future periods, respectively, while W is set to shrink by 46%. WH is projected to expand by 2% and 8% for the mid- and far-future periods, respectively. Conversely, DH is expected to expand by 43% for the far-future period but shrink by 42% for the mid-future period. This study fills a gap by redefining Borneo's climate zones based on an increased number of effective parameters and projecting future shifts, utilizing advanced clustering methods (FCM) under CMIP6 scenarios. Importantly, it contributes by ranking GCMs using RIMs and CPI across multiple climate parameters, addressing a previous gap in GCM assessment. The study's findings can facilitate cross-border collaboration by providing a shared understanding of climate dynamics and informing joint environmental management and disaster response efforts.

The added value of simulated near-surface wind speed over the Alps from a km-scale multimodel ensemble

The advancement of computational resources has allowed researchers to run convection-permitting regional climate model (CPRCM) simulations. A pioneering effort promoting a multimodel ensemble of such simulations is the CORDEX Flagship Pilot Studies (FPS) on “Convective Phenomena over Europe and the Mediterranean” over an extended Alps region. In this study, the Distribution Added Value metric is used to determine the improvement of the representation of all available FPS hindcast simulations for the daily mean near-surface wind speed. The analysis is performed on normalized empirical probability distributions and considers station observation data as the reference. The use of a normalized metric allows for spatial comparison among the different regions (coast and inland), altitudes and seasons. This approach permits a direct assessment of the added value between the CPRCM simulations against their global driving reanalysis (ERA-Interim) and respective coarser resolution regional model counterparts. In general, the results show that CPRCMs add value to their global driving reanalysis or forcing regional model, due to better-resolved topography or through better representation of ocean-land contrasts. However, the nature and magnitude of the improvement in the wind speed representation vary depending on the model, the season, the altitude, or the region. Among seasons, the improvement is usually larger in summer than winter. CPRCMs generally display gains at low and medium-range altitudes. In addition, despite some shortcomings in comparison to ERA-Interim, which can be attributed to the assimilation of wind observations on the coast, the CPRCMs outperform the coarser regional climate models, both along the coast and inland.

Bathymetric Modelling of High Mountain Tropical Lakes of Southern Ecuador

Very little is known on high mountain tropical lakes of South America. Thus, the main motivation of this research was obtaining base bathymetric data of 119 tropical lakes of the Cajas National Park (CNP), Ecuador, that could be used in future geomorphological studies. Eleven interpolation methods were applied with the intention of selecting the best one for processing the scattered observations that were collected with a low-cost fishing echo-sounder. A split-sample (SS) test was used and repeated several times considering different proportions of available observations, selected randomly, for training of the interpolation methods and accuracy evaluation of the respective products. This accuracy was assessed through the use of empirical exceedance probability distributions of the mean absolute error (MAE). A single best interpolation method could not be identified. Instead, the study suggested six better-performing methods, including the complex methods Kriging (ordinary), minimum curvature (spline), multiquadric, and TIN with linear interpolation but also the much simpler methods natural neighbour and nearest neighbour. A sensitivity analysis (SA), considering several data error magnitudes, confirmed this. This advocated that sophisticated interpolation methods do not always produce the best products as geomorphological characteristics of the study site(s) together with observation data characteristics are likely to play important roles in their performance. As such, this type of assessment should be carried out in any terrestrial mapping of bathymetry that is based on the interpolation of scattered observations. Upon the analysis of the relative hypsometric curves of the 119 study lakes, they were classified into three average form categories: convex, concave, and mixed. The separated accuracy analysis of these three groups of lakes did not help in identifying a single best method. Finally, the interpolated bathymetries of 114 of the study lakes were incorporated into the best DEM of the study site by equalising their elevation reference systems. It is believed that the resulting enhanced DEM could be a very useful tool for a more appropriate management of these very beautiful but fragile high mountain tropical lakes.

Data-driven approach enabling post-operation evaluation of air conditioning performance regarding thermal conditions attained indoors

Data-driven approaches become more and more attractive in building performance analysis including performance assessment of heat, ventilation, and air conditioning (HVAC) systems. Their popularity is associated with the increasing availability of large amounts of high-quality building-related measurement data. The study is focused on a data-driven approach that enables post-operation evaluation of AC performance based on indoor air monitoring. It allows for the identification of the classifying solution which recognizes the monitoring data that is representative of thermal conditions during AC operation. The major novelty consists in proposing the classification performance criterion, which is F1w-Score (the weighted F1-Score), and the procedure of weights selection. The procedure utilizes Hellinger distance (H). It is applied to measure the similarity between the binary empirical probability distribution of T and RH (EPDtr), associated with AC operation, and the prediction of this distribution (EPDes), made by the classifier. The approach was tested on three years of data from continuous indoor air monitoring in the IT engineers' room equipped with an AC unit. The best weights of F1w-Score were wPPV = 0.85 and wTPR = 0.15. The best classifying solution was identified using F1w-Score. It involved: Logistic Regression as the classifier and feature vector being Fourier spectrum of 16 min long fragments of RH and T time series. The performance of the best classifying solution was high as indicated by F1w-Score = 0.89 ± 0.05. The corresponding H(EPDtr, EPDes) = 0.17 ± 0.04 pointed at the capability to identify the measurement data that characterizes indoor conditions during AC operation well. This work contributes to the development of data-driven approaches serving diagnostics of HVAC systems. It draws attention to the problem of defining the classification performance criterion which takes into account the ultimate, domain-specific use of the classification results.

Sparse species interactions reproduce abundance correlation patterns in microbial communities

During the last decades, macroecology has identified broad-scale patterns of abundances and diversity of microbial communities and put forward some potential explanations for them. However, these advances are not paralleled by a full understanding of the dynamical processes behind them. In particular, abundance fluctuations of different species are found to be correlated, both across time and across communities in metagenomic samples. Reproducing such correlations through appropriate population models remains an open challenge. The present paper tackles this problem and points to sparse species interactions as a necessary mechanism to account for them. Specifically, we discuss several possibilities to include interactions in population models and recognize Lotka-Volterra constants as a successful ansatz. For this, we design a Bayesian inference algorithm to extract sets of interaction constants able to reproduce empirical probability distributions of pairwise correlations for diverse biomes. Importantly, the inferred models still reproduce well-known single-species macroecological patterns concerning abundance fluctuations across both species and communities. Endorsed by the agreement with the empirically observed phenomenology, our analyses provide insights into the properties of the networks of microbial interactions, revealing that sparsity is a crucial feature.

Geomagnetically induced potential on pipelines: A probabilistic approach

The paper presents a probabilistic approach for the determination of the potential induced along a pipeline route during a geomagnetic storm; particular emphasis is given to intense events such as 100-year and 10-year storms. By assuming that the induced geoelectric field orientation at the ground level and at a given latitude can be treated as a random variable with normal probability distribution, by means of a Monte Carlo algorithm it is possible to build up the empirical probability distribution of the induced potential along the pipeline. The main result is that, in case of an extreme event such as a 100-year storm, even at middle latitudes, potentials of several tens of Volts have a not negligible probability of being induced, at least one time during the pipeline life, along the plant route with possible consequences of damages to apparatuses and equipment connected to the pipeline itself.

A Practical Approach of Probabilistic Seismic Hazard Analysis for Vector IMs Regarding Mainshock with Potentially Largest Aftershock

ABSTRACT Seismic hazard and risk may be underestimated if aftershocks are ignored within the traditional probabilistic seismic hazard analysis (PSHA). In this study, a practical approach is proposed to calculate the mean annual rate of the joint exceedance of different ground motion vector intensity measures (IMs) for mainshocks and their potentially largest aftershock. The proposed approach is built based on an indirect vector-valued PSHA framework, which was originally used to compute the joint hazard of multiple IMs for mainshocks. Within the proposed approach, the overall hazard caused by mainshock-aftershock scenarios is calculated by including two key components, namely: 1) the occurrence rate of a given hazard level of mainshock as calculated by the standard PSHA; and 2) the occurrence probability of a given hazard level of aftershock conditioned on the mainshock, which is calculated by establishing the joint distribution of the IMs for both mainshocks and aftershocks using the Copula theory. Based on 662 groups of mainshock-aftershock ground motions from 13 earthquake events in the NGA-West2 database, the empirical joint probability distribution model between the peak ground accelerations of mainshocks and aftershocks (i.e. PGA MS and PGA AS) is developed. The Clayton Copula function is calibrated as the optimal choice and the marginal distributions of PGA MS or PGA AS could be properly modelled using the generalized extreme value distributions. The proposed approach is implemented to construct the joint hazard surface regarding both PGA MS and PGA AS for a hypothetical site. The results indicate that using different types of copula functions and marginal distributions of PGA MS and PGA AS has a noticeable impact on the final joint hazard level. Moreover, a coarse assumption that PGA MS and PGA AS follow a joint Gaussian distribution in the logarithm scale would lead to considerable bias in the mainshock-aftershock joint hazard especially at low mean annual rates of exceedance.

Quantification of Equivalent Strut Modeling Uncertainty and Its Effects on the Seismic Performance of Masonry Infilled Reinforced Concrete Frames

ABSTRACT Quantifying aleatory and epistemic uncertainty in nonlinear structural response simulation is key to robust performance-based seismic assessments. This paper focuses on the modeling parameters that are used to describe the nonlinear force-deformation response of the equivalent infill struts used in models of reinforced concrete frames with infills. The variability in several parameters is characterized by developing empirical and theoretical multivariate probability distributions based on the deduced-to-predicted ratios derived using data from 113 physical experiments. The effect of the uncertainty in the infill strut modeling parameters on maximum story drift ratios and associated limit state fragility functions is investigated for a 3-story reinforced concrete frame building with infills. Multiple stripe analysis is performed using hazard-consistent ground motions. Relative to when only record-to-record variability is considered, modeling parameter uncertainty has a non-negligible effect on the dispersion of the maximum story drift ratios, and affects both the median and dispersion of the limit state fragilities.

Construction of an Intelligent Evaluation Model of Yield Risk Based on Empirical Probability Distribution

Construction of an Intelligent Evaluation Model of Yield Risk Based on Empirical Probability Distribution

An active-learning reliability method based on support vector regression and cross validation

To reduce the high computational cost of the probability density evolution method (PDEM) for structural reliability analysis, an active-learning reliability method that combines the support vector regression (SVR) and the PDEM is proposed, which is abbreviated as the ASVR-PDEM. First, the empirical probability distribution of the SVR is proposed based on the leave-one-out cross-validation (LOOCV) strategy. On this basis, two different learning functions are proposed according to the concept of the PDEM-oriented expected improvement function, which accommodate to any type of metamodels in essence. Then, the pseudo-interpolation replacement is devised to avoid the prediction error of the SVR at the representative points in the region of interest and, thus, maintains the precision of failure probability. Moreover, the influences of four key ingredients on the proposed method are investigated, namely the type of metamodels, the form of learning functions, the deployment of the pseudo-interpolation replacement and the training scheme in the LOOCV strategy. Three numerical examples are investigated to show the feasibility of the ASVR-PDEM. Comparisons are made against several conventional reliability methods. The results highlight the superiority of the ASVR-PDEM in terms of computational accuracy, efficiency and computational time, especially in the case of the time-consuming dynamic reliability problems.

On the Statistics of Elsasser Increments in Solar Wind and Magnetohydrodynamic Turbulence

In this Letter we investigate the dependency with scale of the empirical probability distribution functions (PDF) of Elsasser increments using large sets of WIND data (collected between 1995 and 2017) near 1 au. The empirical PDF are compared to the ones obtained from high-resolution numerical simulations of steadily driven, homogeneous reduced MHD turbulence on a 20483 rectangular mesh. A large statistical sample of Alfvénic increments is obtained by using conditional analysis based on the solar wind average properties. The PDF tails obtained from observations and numerical simulations are found to have exponential behavior in the inertial range, with an exponential decrement that satisfies power laws of the form α l ∝ l −μ , where l is the scale size, with μ between 0.17 and 0.25 for observations and 0.43 for simulations. PDF tails were extrapolated assuming their exponential behavior extends to arbitrarily large increments in order to determine structure function scaling laws at very high orders. Our results point to potentially universal scaling laws governing the PDF of Elsasser increments and to an alternative approach to investigate high-order statistics in solar wind observations.

Value-at-Risk Estimation Using an Interpolated Distribution of Financial Returns Series

Abstract This paper develops a model for estimating Value-at-Risk (VaR) from the historical return series. The proposed method uses spline interpolation to represent the empirical probability distribution of the return series. The approach developed in this paper is easy to implement using available programming platforms, and it can be generalized to other applications that involve estimating empirical distribution. In order to check the validity of the model, we use established back-testing methods and show that the model is robust to the changes in sample size and significance levels used to estimate VaR. We test the model against some similar distribution-based models using historical data from S&P500 index. We show that Value-at-Risk estimation based on the proposed method can outperform common historical, parametric, and kernel-based methods. As a result, the method can be useful in the context of validation of market risk models. JEL classification numbers: C52, C63, G17, G32. Keywords: Value-at-Risk, Non-parametric estimation, Empirical distribution, Spline Interpolation.

Sizing capacities of renewable generation, transmission, and energy storage for low-carbon power systems: A distributionally robust optimization approach

To decrease carbon dioxide emission, a high penetration level of renewable energy will be witnessed over the world in the future. By then, energy storage will play an important role in power balancing and peak shaving. This paper considers the capacity sizing problem during the transition to a low-carbon power system: the retirement plan of conventional fossil-fuel generators and the growth of demands are given. The renewable generation capacities at given sites are to be determined in coordination with the upgrade of transmission lines and installation of energy storage units. In order to capture the inaccuracy of empirical probability distributions for uncertain renewable output and load profiles, a novel distributionally robust bi-objective sizing method using Wasserstein-metric-based ambiguity sets is proposed. The total investment cost and expected carbon dioxide emission subject to operating conditions and a load shedding risk constraint are minimized. The distributionally robust shortfall risk of load shedding and the worst-case expectation of carbon dioxide emission are reformulated into computable forms based on calculating the Lipschitz constants. The final problem comes down to solving mixed-integer linear programming problems. The numerical results demonstrate the effectiveness of the proposed method and the necessity of using distributionally robust optimization.

Imprecise subset simulation

The objective of this work is to quantify the uncertainty in probability of failure estimates resulting from incomplete knowledge of the probability distributions for the input random variables. We propose a framework that couples the widely used Subset simulation (SuS) with Bayesian/information theoretic multi-model inference. The process starts with data used to infer probability distributions for the model inputs. Often such data sets are small. Multi-model inference is used to assess uncertainty associated with the model-form and parameters of these random variables in the form of model probabilities and the associated joint parameter probability densities. A sampling procedure is used to construct a set of equally probable candidate probability distributions and an optimal importance sampling distribution is determined analytically from this set. Subset simulation is then performed using this optimal sampling density and the resulting conditional probabilities are re-weighted using importance sampling. The result of this process are empirical probability distributions of failure probabilities that provide direct estimates of the uncertainty in failure probability estimates that result from inference on small data sets. The method is demonstrated to be both computationally efficient – requiring only a single subset simulation and nominal cost of sample re-weighting – and to provide reasonable estimates of the uncertainty in failure probabilities.

An empirical mean-field model of symmetry-breaking in a turbulent wake.

Improved turbulence modeling remains a major open problem in mathematical physics. Turbulence is notoriously challenging, in part due to its multiscale nature and the fact that large-scale coherent structures cannot be disentangled from small-scale fluctuations. This closure problem is emblematic of a greater challenge in complex systems, where coarse-graining and statistical mechanics descriptions break down. This work demonstrates an alternative data-driven modeling approach to learn nonlinear models of the coherent structures, approximating turbulent fluctuations as state-dependent stochastic forcing. We demonstrate this approach on a high–Reynolds number turbulent wake experiment, showing that our model reproduces empirical power spectra and probability distributions. The model is interpretable, providing insights into the physical mechanisms underlying the symmetry-breaking behavior in the wake. This work suggests a path toward low-dimensional models of globally unstable turbulent flows from experimental measurements, with broad implications for other multiscale systems.

Probabilistic Evaluation of Hydraulic Fracture Performance Using Ensemble Machine Learning

Evaluation of hydraulic fracture (HF) performances is critical to develop unconventional resources such as tight oil and gas. We present a probabilistic evaluation approach that integrates ensemble machine learning with Monte Carlo simulation. In the method, we employ the ensemble learning to develop a predictive model between well productivity and its influential factors including both geological properties and HF treatment parameters. Next, coupling the built prediction model with Monte Carlo simulation, an empirical cumulative probability distribution of the well productivity is generated. The well HF performance is assessed by estimating its cumulative probability value. The proposed method is applied to evaluate the HF performances in a developed block of the eastern Sulige region. The study shows that 19% of the wells were fractured with good quality and 55% of the wells were fractured with average quality, while the rest were stimulated with poor quality. The evaluations provide a guideline for optimization of HF designs of wells that have not been hydraulically stimulated in the region.

Coupled modeling of the seasonal transmission loss in the Beaufort Sea

An eight-element drifting vertical line array was deployed overwinter 2019–20 in the Beaufort Sea to record transmissions from two moored 35 Hz sources deployed as part of the Coordinated Arctic Acoustic Thermometry Experiment (CAATEX), and seven 925 Hz sources deployed by the Arctic Mobile Observing Systems (AMOS) experiment. Transmissions were received every day on the array at ranges of 10 to 850 km. A probabilistic rough-ice canopy was integrated to a Parabolic Equation code to predict the propagation loss. Sound speed profiles and percent ice cover along the propagation paths were obtained at daily resolution from data-assimilative Global Ice Ocean Prediction System (GIOPS, 20 km spatial resolution). Sea ice roughness and keel properties are empirically determined from the GIOPS ice thickness. The ice roughness is generated using a random pulse-train model with a characteristic wavenumber spectrum slope determined by ice age. Empirical probability distributions of ice keel depth, slope, and spatial densities are used to randomly add keels. Spatial averaging is used to account for signal bandwidth and uncertainty in receiver positions. The environment files and modeling are performed in piece-wise steps to manage memory footprints over long ranges. Modeled propagation losses compare well with the data and reproduce the seasonal variability observed in the measured results.