Suitable Probability Distribution Research Articles

Assessing model selection techniques for distributions use in hydrological extremes in the presence of trimming and subsampling

A suitable probability distribution is required to quantify and estimate hydraulic structure design for risk evaluation and management. The inability of model selection criteria to differentiate, in some cases, among candidate distributions used in the analysis of hydrological extremes is often criticised. This study verifies, with the aid of model selection techniques, the potential utility of trimming and subsampling in distinguishing between candidate distributions, which might not be feasible using the traditional goodness of fit method alone, when samples available are small. The performance of the proposed method is evaluated through its application to real and simulated yearly peak rainfall datasets. The proposed approach is then compared with several standard model selection techniques. Results show that the model selection techniques with the aid of subsampling are effective in identifying the true parent distribution for the untrimmed samples given a two-parameter distribution; contrarily, they are inefficient where a distribution with a three-parameter is the parent distribution. However, as trimming is introduced, all model selection methods recognise the true parent distribution for a three-parameter distribution. Overall, utilising trimming and subsampling with the aid of model selection methods yields promising outcomes in the analysis of hydrological extreme frequencies. Drawing from the results of numerical simulation and examination of observed data, the use of trimming and subsampling can be a viable tool in differentiating among candidate distributions used in the investigation of hydrological extreme frequencies.

Probability of maximum values of stress concentration factors in tubular DKT-joints reinforced with FRP under axial loads

In the paper, the most suitable probability distributions for SCFmax in DKT-Joints stiffened with FRP under axial loading conditions are studied. To this end, 1296 models were generated and then analyzed in the ANSYS to provide five reliable samples. Created FE models were validated using previous laboratory and FE data. Afterward, to predict the appropriate probability distribution models, 12 different kinds of distributions were utilized to check the fit distribution curve to the samples. Next, the accuracy and reliability of the distribution curves were assessed with the help of goodness-of-fit tests, such as the Kolmogorov-Smirnov (KS) and the Chi-squared (CS). The results of the analysis reveal that the Birnbaum-Saunders distribution is the best option. Furthermore, based on the probability graphs, it was concluded that the developed distribution is suitable enough for the given data sets.

A suitable probability distribution for the prediction of rainfall in northern regions of Kashmir Himalaya

A suitable probability distribution for the prediction of rainfall in northern regions of Kashmir Himalaya

Using supervised machine learning for regional hydrological hazard estimation in metropolitan France

Study regionThis study is carried out for 1929 gauged catchments in France, ranging from 1 to 10,000 km², where quality hydrometric observations are available for flood frequency analysis. Study focusThe regional estimation of hydrological hazards is studied for flood risk management and prevention in hydrology. For gauged catchments, flow quantiles can be estimated from observations using statistical approaches based on suitable probability distributions or simulation approaches based on rainfall-runoff transformation models. For ungauged catchments, the lack of hydrological observations means that we have to extrapolate our knowledge of hazards from gauged catchments to ungauged catchments, using regionalization methods. It is therefore necessary to combine regionalization methods with the implemented hazard estimation approach. In this paper, two popular machine learning methods, Random Forest and Neural Networks, are tested and compared as regionalization methods. A classical regionalization method using multiple linear regression is also applied as a benchmark to evaluate the performance of all configurations. All these regionalization methods are applied to a simulation-based approach (the SHYREG method) and to a statistical-based approach using generalized extreme value distribution (GEV). New hydrological insights•Regionalization approaches based on multiple linear regression have limitations to explain parameters with environmental descriptors in Regional Flood Frequency Analysis (RFFA) domain.•Regionalizing RFFA parameters using Random Forest allows more explanatory variables to be considered through non-linear relationships, resulting in better parameter estimation.•Machine learning techniques can better handle environmental descriptors for regionalization, this providing a notable performance improvement, especially for the statistical approach.•The tested simulation-based approach is less sensitive to the choice of spatial interpolation method than the studied statistical approach.

Probability distributions of geotechnical properties for heterogeneous lignite mine spoils

ABSTRACT During surface coal mining, large amounts of overburden waste materials, known as spoils, are excavated and dumped, creating massive heaps. The sustainable management of these areas is a global priority, but a comprehensive geotechnical characterization has not yet been achieved due to limited research on the spoils’ geotechnical properties. This study conducts an advanced statistical analysis of lignite mine spoil materials from a large heap in Greece to determine suitable probability distributions. Five distributions - normal, lognormal, Weibull (2 and 3 parameters), and Gamma - were evaluated, considering the high variability of spoil material properties. The lognormal and Weibull distributions were found to best fit most geotechnical parameters, except for cohesion, which showed an uncommon distribution. In some cases, the normal distribution can be used for a simplified analysis. The findings can guide the analysis of spoil materials, as advanced numerical and stochastic methods are essential for robust geotechnical design.

Unveiling Torrential Flood Dynamics: A Comprehensive Study of Spatio-Temporal Patterns in the Šumadija Region, Serbia

This paper presents a comprehensive analysis of flood frequency and a spatio-temporal characterization of historical torrential floods in the Šumadija region using water discharge datasets and documented events. A chronology of 344 recorded torrential flood events, spanning from 1929 to 2020, illustrates the region’s vulnerability, with a death toll exceeding 43. The study defines the intra-annual primary and secondary peaks of torrential flood occurrences and explains their spatial distribution. Furthermore, the identification of suitable probability distribution functions underscores the necessity of tailored approaches for effective flood risk management in this diverse geographical environment. The study employed Flood Frequency Analysis (FFA) and goodness-of-fit tests, including the Kolmogorov–Smirnov (K-S) and Cramér–von Mises (CvM) tests, to assess the frequency and magnitude of flood events and evaluate diverse distribution functions. The main results include the identification of suitable probability distribution functions for each river within the region, emphasizing the need for tailored approaches in flood risk management. Additionally, discharge values for various return periods offer crucial insights for informed decision-making in flood risk management and infrastructure planning.

FORECASTING OF FLOOD EVENTS FOR WATER RESOURCES MANAGEMENT IN GUCHA-MIGORI RIVER BASIN, KENYA

Floods are devastating natural disasters frequently occurring in many river basins such as Gucha-Migori River Basin, Kenya. However, non-structural countermeasures such as hydrologic modelling, flood proofing, and continuous forecasting have not been fully explored and implemented to reduce risk, damage, and vulnerabilities of flood events. The major challenge in flood forecasting is the selection of the relevant probability distribution. The main objective of the study was to forecast flood events in Gucha-Migori River Basin for integrated water resource management. Daily hydrological datasets between 1969 and 2015 were obtained from Water Resources Authority. After the ranking of independent flood events had been achieved, return periods and flood frequencies were computed. The relationship between flood magnitudes and their respective frequencies was through modelling using probability distribution functions (PDFs). The evaluated PDFs include Normal Distribution, Log-Normal Distribution, Gumbel Distribution, and Log-Person Type III Distribution. Coefficient of Determination (R2), Goodness of Fit test, and Best-Fit Distribution Curve guided selection of the suitable distribution model. The suitable probability distribution model out of the tested four was then applied to predict P-percent annual exceedance probabilities corresponding to specific flood magnitudes. From the results, out of the 132 flood events that occurred in the period 1969 to 2015, only 42 were identified as independent flood events. Based on the frequency analysis, the highest and lowest, and the average and standard deviation of the recorded independent flood events were 423.90, 240.8, 284.80 m3/s, and 45.54 respectively. It was revealed that the probability of a flood event of a magnitude equal to or exceeding the lowest (240.8 m3/s) and the highest (423.90 m3/s) occurring for a particular year were 0.98 and 0.02 respectively. Gumbel distribution curve was selected to be the best fit for the Gucha-Migori River Basin. According to the derived Gumbel's formula, the estimated magnitude for the return period 1.05, 1.11, 1.25, 2, 5, 10, 25, 50, 100, and 200 years were 221.38, 231.92, 245.77, 278.01, 321.39, 350.11, 386.39, 413.31, 440.03, and 466.65 m3/s. This research provides useful information for enhanced determination of probability occurrence of flood events and thus proper water resources planning and management strategies in the Gucha-Migori river basin.

Distribution of Floods Frequency of Manafwa River, Uganda

The objective of this study was to analyse Manafwa River flood frequency in Eastern Uganda. Analysis of Manafwa River maximum annual flows from 1949-2015 was undertaken using Log Pearson 3 distribution in comparison with Gumbel, Normal and Log Normal distributions to determine frequency of occurrence and magnitude of extreme floods. Statistical analysis including goodness of fit tests of chi-square, Kolmogorov-Smirnov and Anderson-Darling tests were used to generate the most suitable probability distribution model. The results show quantile magnitudes lowest for Log Normal distribution at 43.59 m3/s and highest for Log Pearson 3 distribution at 51.67 m3/s. The 5-year quantile estimates are highest for Normal and Log Pearson at 70.37 m3/s and 63.99 m3/s respectively. The 10-year quantile estimates are highest for Log Normal and lowest for Log Pearson 3 distributions at 87.57 m3/s and 75.13 m3/s respectively. The 100-year quantile estimates are lowest for Normal and highest for Log Normal distributions at 108.57 m3/s and 154.66 m3/s respectively. The 200-year quantile estimates are lowest for Normal and highest for Log Normal distributions respectively at 114.980 m3/s and 177.16 m3/s respectively. Log Pearson 3 distribution emerged as best fit for data. From the statistical analysis, LP 3 probability distribution presents the most accurate regression coefficient at 0.8486 and the most suitable distribution of goodness of best fit using A-D, K-S and Chi square tests followed by the Gumbel distribution. The tests yield 0.15666, 0.04855 and 0.88502 for A-D, K-S and Chi square tests respectively for the LP 3 distribution. There is an increasing upward trend of the discharges at Manafwa River floodplains at higher probabilities of exceedance across all the probability distributions due to varrying climatic changes and rapid landuse changes in the Manafwa catchment. Manafwa river floodplains have the capacity to accommodate and boost crop production and productivity. Any nutrients lost to leaching could be gained from subsequent fallowing and sustainable soil fertility management including; proper drainage, crop rotation, adding organic manure, cover cropping and among others

Determination of Suitable Probability Distribution of Rainfall in Pakistan Considering Multiplicity

Extreme rainfall events are occasional, and understanding their intensity and frequency is important for long-term planning and for public safety. The current study aims to investigate the stability of extreme precipitation events in different regions of Pakistan, their independence and the uniformity of their occurrence. A rainfall frequency analysis (RFA) was conducted based on information from 8 weather stations namely, the distribution and moments L of the annual maximum precipitation in Pakistan were investigated; Using goodness-of-fit criteria, the study determined the possible distribution of precipitation. Relatively Absolute Error (RAE) results are based on the most appropriate GUM distribution, revealed that the Sialkot, Multan, Faisalabad and DI Khan stations produce very low errors (0.266, 0.847, 0.075, 0.856, 1.671, 2.522, 3.659, 4.524). It was found that the most suitable distribution is LN distribution for Peshawar, Sibbi and Karachi Stations. The GEV distribution also performs well with small errors for various return periods (0.266, 0.847, 0.075, 0.856, 1.671, 2.522, 3.659, 4.524), corresponding to the return periods 2, 5, 25, 50, 100, 200, 500 and 1000 years. In contrast, P3 has the advantage of a return period of 10 years for all stations, as they all produce similar results for this particular return period. The current research provides valuable insights into estimating extreme rainfall at stations where rainfall data are available

Identification of the most suitable probability distributions for ultimate strength of FRP-strengthened X-shaped tubular joints under axial loads

This paper focuses on identifying the most appropriate probability distributions for characterizing the ultimate strength of FRP-strengthened X-shaped tubular joints under axial loads. The study begins by verifying the accuracy of finite element (FE) models through a comparison with experimental results. Subsequently, 218 FE nonlinear static analyses are conducted to generate two reliable datasets for strength ratios. To determine the most suitable probability distribution models, 16 different distributions are fitted to the data, and their accuracy is evaluated using Kolmogorov-Smirnov and Chi-squared goodness-of-fit tests. The findings reveal that the Generalized Extreme Value distribution provides the best fit for describing the ultimate strength under compressive axial load, while the Burr distribution is identified as the most suitable model for tensile axial load. The analysis of probability differences revealed that the proposed probability distribution models for compressive and tensile loads exhibit minimal disparities, with a slight difference of 5.24% and 4.80%, respectively. The identified probability distribution models enhance the reliability and confidence in predicting the behavior and performance of these joints, contributing valuable insights to the field of structural engineering.

Propagation of material uncertainty in modal parameters and its influence in damage quantification of shear buildings

Structural health monitoring (SHM) plays a crucial role in post-disaster mitigation in investigating the serviceability of an existing structure. The physical properties, namely mass density, cross-sectional area, Poisson’s ratio, modulous of elasticity, etc., vary spatially throughout a real system. These uncertainties propagate in the modal parameters involved in SHM techniques. In this study, the elastic modulous has been considered the primary source of uncertainty. A detailed literature survey has been performed to identify the suitable probability distribution recommended for defining material uncertainty. Although the Gaussian distribution cannot replicate realistic material behaviour, its explicit implementation makes it prevalent. However, based on the principle of maximum entropy, the gamma distribution is the most appropriate recommendation among all non-Gaussian continuous distributions ranging [0, ∞). The Karhunen–Loève (KL) expansion has been adopted to discretize the stochastic field to determine the spectral mass and stiffness of a multi-damaged axially vibrating prismatic cantilever bar-type structure. Closed-form orthogonal pairs of eigensolutions of the KL expansion considering (Adhikari and Friswell, 2010) exponential covariance function have been derived. The expressions for spectral mass and stiffness for a multi-damaged cantilever bar considering material uncertainty have been developed. The uncertainty propagation in the modal properties has been estimated. The complete procedure has been followed to identify uncertainty propagation in damage severity of a common class of civil structures like shear buildings. The influence of material uncertainty in determining the damage severity of a structure with numerable degrees of freedom using a non-destructive approach has been explored. Extensive simulations have been performed to evaluate the effectiveness of the present study in a multi-damaged bar as well as shear building. Statistical parameters, namely mean and standard deviation of natural frequencies, modal displacement and damage severity have been determined. The uncertainty propagation has been reflected in the mean value for varying input stochasticity. However, the standard deviation remained consistent for undamaged as well as damaged structure irrespective of material uncertainty type. Considering gamma distribution in material stochasticity, the spread of uncertainty propagation has been more than the Gaussian distribution. However, experimental studies need to be performed further to investigate the stochastic behaviour in modal parameters of a real structure.

Copula-Based Severity–Duration–Frequency (SDF) Analysis of Streamflow Drought in the Source Area of the Yellow River, China

In classical severity–duration–frequency (SDF) analysis, the dependence between drought characteristics is not effectively considered. The present study aims to propose the SDF relationships of streamflow drought in the source area of the Yellow River (SAYR) using a copula-based approach. Comparison of multiple time-varying threshold levels and the integration and elimination of drought events were considered. Selection of marginal probability distribution and copula-based joint probability distribution was properly conducted with multiple means. Copula-based joint and conditional probabilities were computed. The findings support carrying out integration and elimination processing on the preliminarily identified streamflow droughts through a run analysis with a time-varying threshold level of the 80% quantile of daily streamflow. The Gaussian copula was selected as the optimal model for constructing bivariate joint probability distribution, with generalized extreme value and log-normal as the suitable marginal probability distributions of streamflow drought duration and severity. The proposed copula-based SDF relationships of streamflow drought events can provide more critical information in addition to univariate frequency analysis, benefitting from the joint and conditional probabilities. The multivariate probabilistic analyses can effectively consider the connection and interaction between drought characteristics, while conditional probability distribution allows analyzing the impact of one drought characteristic on another. The results also indicate a relatively high risk of streamflow drought with short duration and low severity in the region, requiring effective drought-mitigation strategies and measures.

Analysis of occupational radiation dose data and determination of suitable probability distribution.

The first study on fitting dose data for workers was performed by Gale( 1) in 1965 where log-normal and normal distributions were used. Since then, various models of dose distribution have been proposed. The log-normal distribution and its different forms have been widely used for fitting the dose data. Most of the studies included one or two distributions under consideration. In this study five distributions are considered for fitting and four distributions are selected based on observation of Cullen-Frey graph. The Akaike's Information criteria (AIC) and Bayesian Information criteria (BIC) are applied to find the suitable distribution to fit the occupational dose data. The maximum likelihood method was used for calculation of AIC, BIC values and parameter estimation. A computer code is written in R-language and environment for statistical computing and graphics for analysis of occupational dose data of three institutions.

Pooling probability distributions and partial information decomposition.

Notwithstanding various attempts to construct a partial information decomposition (PID) for multiple variables by defining synergistic, redundant, and unique information, there is no consensus on how one ought to precisely define either of these quantities. One aim here is to illustrate how that ambiguity-or, more positively, freedom of choice-may arise. Using the basic idea that information equals the average reduction in uncertainty when going from an initial to a final probability distribution, synergistic information will likewise be defined as a difference between two entropies. One term is uncontroversial and characterizes "the whole" information that source variables carry jointly about a target variable T. The other term then is meant to characterize the information carried by the "sum of its parts." Here we interpret that concept as needing a suitable probability distribution aggregated ("pooled") from multiple marginal distributions (the parts). Ambiguity arises in the definition of the optimum way to pool two (or more) probability distributions. Independent of the exact definition of optimum pooling, the concept of pooling leads to a lattice that differs from the often-used redundancy-based lattice. One can associate not just a number (an average entropy) with each node of the lattice, but (pooled) probability distributions. As an example, one simple and reasonable approach to pooling is presented, which naturally gives rise to the overlap between different probability distributions as being a crucial quantity that characterizes both synergistic and unique information.

A randomized method for the identification of switched NARX systems

The identification of switched systems is a complex optimization problem that involves both continuous (parametrizations of the local models, a.k.a. modes) and discrete variables (model structures, switching signal). In particular, the combinatorial complexity associated with the estimation of the switching signal grows exponentially with the number of samples, which makes data segmentation (i.e. estimating the number and location of mode switchings, and the mode sequence) a challenging problem. In this work, we extend a previously developed randomized approach for the identification of switched systems to encompass the estimation of the switching locations. The method operates by extracting samples from a probability distribution of switched models, and gathering information from the associated model performances to update the distribution, until convergence to a limit distribution associated to a specific model. A suitable probability distribution is employed to represent the likelihood of a mode switching at a certain time, and the update process is designed to correct the switching locations and remove redundant switchings. The proposed algorithm has been compared to existing state-of-the-art methods and has been tested on various benchmark examples, to demonstrate its effectiveness.

Optimal Allocation of Hybrid Renewable Energy System in Distribution Network using Arithmetic Optimization Algorithm

The main focus of this article is the optimal allocation of wind turbines, solar PV and storage in 33 bus radial distribution system. The arithmetic optimization algorithm (AOA) was employed in this work for the optimal sizing of wind turbine and solar PV units with batteries in 33 bus distribution networks. The 1 year period is divided into multiple time segments, and each time segment is evaluated independently. For each time slot solar irradiance and wind speed are generated using suitable probability distribution function. Sensitivity analysis was done to find potential buses that might be placed to reduce computation time and search space. The Backward-Forward sweep technique was used to conduct load flow analysis. For the sake of stability, a reasonable penetration level is selected. The overall energy loss is minimized by the AOA optimization method under equality and inequality constraints. The suggested technique was tested on 33 buses, and it was observed that correct sizing and placement of DG units results in a significant decrease in losses with improved voltage profile.

Buckling capacity model for timber screws loaded in compression: Experimental, analytical and FE investigations

This paper investigates the buckling of screws loaded in compression inserted into timber members. Screws are often used as a reinforcement in timber structures. However, under compression forces, they are prone to axial buckling. The current model for the screw buckling, enclosed in the EC5 proposal, is based on the general framework of EC3 for the instability of compressed steel members. The main shortcomings of the current formulation for the buckling of screws are the following. (1) The analytical expression for calculating the theoretical buckling load does not follow the observed modes. (2) Due to the need for dedicated studies, the value of the imperfection coefficient is arbitrarily chosen. This paper fills the above gaps. Firstly, a simple analytical expression for predicting the buckling of screws is proposed and validated against experimental and finite element (FE) findings. Furthermore, the formulation adopts a more accurate expression for lateral deformation based on experimental observation. Secondly, a FE model calibrated on experimental tests is used to estimate the defect coefficients of the instability curves as a function of the amplitude of the geometric defects of the screw, expressed as a fraction of its length. Finally, a Markov chain Monte Carlo analysis is carried out to simulate the capacity of screws with different sizes, assuming the uncertainty of all input parameters sampled from suitable probability distributions. The results are used to validate the proposed deterministic capacity model and estimate the uncertainty factors of the design equation.

Sparse Polynomial Approximations for Affine Parametric Saddle Point Problems

In this work we study convergence properties of sparse polynomial approximations for a class of affine parametric saddle point problems, including the Stokes equations for viscous incompressible flow, mixed formulation of diffusion equations for groundwater flow, time-harmonic Maxwell equations for electromagnetics, etc. Due to the lack of knowledge or intrinsic randomness, the (viscosity, diffusivity, permeability, permittivity, etc.) coefficients of such problems are uncertain and can often be represented or approximated by high- or countably infinite-dimensional random parameters equipped with suitable probability distributions, and the coefficients affinely depend on a series of either globally or locally supported basis functions, e.g., Karhunen–Loève expansion, piecewise polynomials, or adaptive wavelet approximations. We consider sparse polynomial approximations of the parametric solutions, in particular sparse Taylor approximations, and study their convergence properties for these parametric problems. Under suitable sparsity assumptions on the parametrization of the random coefficients, we show the algebraic convergence rates O(N−r) for the sparse polynomial approximations of the parametric solutions based on the results for affine parametric elliptic PDEs (Cohen, A. et al.: Anal. Appl. 9, 11–47, 2011), (Bachmayr, M., et al.: ESAIM Math. Model. Numer. Anal. 51, 321–339, 2017), (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015), (Chkifa, A., et al.: J. Math. Pures Appl. 103, 400–428, 2015), (Chkifa, A., et al.: ESAIM Math. Model. Numer. Anal. 47, 253–280, 2013), (Cohen, A., Migliorati, G.: Contemp. Comput. Math., 233–282, 2018), with the rate r depending only on a sparsity parameter in the parametrization, not on the number of active parameter dimensions or the number of polynomial terms N. We note that parametric saddle point problems were considered in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2) with the anticipation that the same results on the approximation of the solution map obtained for elliptic PDEs can be extended to more general saddle point problems. In this paper, we consider a general formulation of saddle point problems, different from that presented in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2), and obtain convergence rates for the two variables, e.g., velocity and pressure in Stokes equations, which are different for the case of locally supported basis functions.

Variability in the elastic and time-delayed properties of structural hemp fibre composites

Various papers in the open source literature have shown that the tensile properties of plant fibres are very scattered. To date, this difference has been viewed as an obstacle to the use of biobased composites in structural applications. However, this question of dispersion has yet not been extensively addressed at the composite scale. Therefore, this paper proposes a method to quantify the distribution of values of the monotonic and time-delayed properties of a woven hemp fabric-reinforced GreenPoxy composite. The variability in these properties is significantly lower than at the fibre scale and on the same order of magnitude as glass fibre composites. This paper also provides the suitable probability distribution laws for these properties, which are important inputs to ensure the optimized and reliable design of plant fibre composite structures.

Maximum Entropy Distributions with Applications to Graph Simulation

The problem of simulating graphs (networks) subject to constraints has been studied extensively across several areas. Applications of this problem include modeling inter-bank financial networks, predator-prey ecological graphs, contingency tables, and even studying larger networks such as the Internet. In “Maximum Entropy Distributions with Applications to Graph Simulation,” P. Glasserman and E. Lelo de Larrea study the more general problem of sampling uniformly from product sets under linear constraints, which includes simulating bipartite, directed, and undirected graphs with given degree sequences. For this purpose, they consider two suitable probability distributions: one that maximizes the entropy of the system, and another that maximizes the minimum probability of hitting the desired target set. Although apparently different, the authors provide conditions under which both distributions coincide. In addition, they propose a simple sequential algorithm to sample medium-sized graphs with fixed degrees.