Continuous Probability Distribution Function Research Articles

Nonequilibrium Study of the J1−J2 Ising Model with Random J2 Couplings in the Square Lattice

We studied the phase transitions of the [Formula: see text] spin-1/2 Ising model in the square lattice by considering [Formula: see text] fixed and [Formula: see text] as random interactions following discrete and continuous probability distribution functions. The configuration of [Formula: see text] in the lattice evolves in time through a competing kinetics using the Metropolis algorithm leading to a steady state without reaching the free-energy minimum. So, different phase diagrams were obtained according to the distribution of [Formula: see text]. For fixed and random values of [Formula: see text], we noted that the amplitude of the correlation length is different from the Ising value at the critical point of the superantiferromagnetic–paramagnetic frontier.

Convex Stochastic Approaches for the Optimal Allocation of Distributed Energy Resources in AC Distribution Networks with Measurements Fitted to a Continuous Probability Distribution Function

This paper addresses the optimal stochastic allocation of distributed energy resources in distribution networks. Typically, uncertain problems are analyzed in multistage formulations, including case generation routines, resulting in computationally exhaustive programs. In this article, two probabilistic approaches are proposed–range probability optimization (RPO) and value probability optimization (VPO)–resulting in a single-stage, convex, stochastic optimal power flow problem. RPO maximizes probabilities within a range of uncertainty, whilst VPO optimizes the values of random variables and maximizes their probabilities. Random variables were modeled with hourly measurements fitted to the logistic distribution. These formulations were tested on two systems and compared against the deterministic case built from expected values. The results indicate that assuming deterministic conditions ends in highly underestimated losses. RPO showed that by including ±10% uncertainty, losses can be increased up to 40% with up to −72% photovoltaic capacity, depending on the system, whereas VPO resulted in up to 85% increases in power losses despite PV installations, with 20% greater probabilities on average. By implementing any of the proposed approaches, it was possible to obtain more probable upper envelopes in the objective, avoiding case generation stages and heuristic methods.

The Method of Probabilistic Nodes Combination in Biomechanics

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

The Method of Probabilistic Nodes Combination in Handwritten Character Reconstruction

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

A New Extension Form for Continuous Probability Distributions: Uniform-X Distributions

In this paper, generating extension forms for continuous probability distribution functions is studied. The considered transformer function is applied to three well-known probability distributions- Normal, Kumaraswamy, Weibull- and new extensions of these distributions are obtained. The related functions of the new extensions are defined, random samples are generated from the new extensions and the results are presented. Parameter estimation procedures of the extensions are studied, and likelihood equations are obtained. To demonstrate the modeling performance of the extensions, three different data sets are considered, separately. Each data set is modeled by both the corresponding probability distribution and its extensions. The new extensions give the best fit to the corresponding data over the well-known probability distributions.

Deep non-crossing probabilistic wind speed forecasting with multi-scale features

Clean and renewable wind energy has made an outstanding contribution to alleviating the energy crisis. However, the randomness and volatility of wind brings great risk to the integration of wind power to the grid. Therefore, it is essential to obtain reliable and efficient wind speed forecasts. Quantile-based machine learning techniques, which usually produce satisfied quantile-based prediction intervals (PIs) for wind energy, have received widespread attention. However, the obtained PIs are usually crossed and violate the monotonicity of different conditional quantiles. In addition, the completeness and quality of features directly affect the forecasting performance of the models. Therefore, mining effective and sufficient information from the limited input data helps to improve the forecasting performance. In this paper, a novel method is developed for probabilistic wind speed forecasting based on deep learning, non-crossing quantile loss, multi-scale feature (MSF) extraction, and kernel density estimation (KDE). In terms of feature extraction, sufficient MSFs with simple pattern will be extracted based on a multi-layer convolutional neural network. Attention-based long short-term memory is used to further extract and encode temporal information for features of each scale and reduce computational cost. The final feature is obtained by concatenating all the encoded feature vectors. Instead of directly outputting different conditional quantiles, this study obtains the positive difference of adjacent conditional quantiles. On this basis, a non-crossing quantile loss is designed to ensure the monotonicity of different conditional quantiles. To understand the forecasting uncertainty comprehensively, KDE is used to estimate the continuous probability distribution function for various PIs. The proposed method is verified on four wind speed datasets collected form South Dakota. The results demonstrate that the proposed method has an excellent ability of generating high-quality, high-precision, and non-crossing probabilistic wind speed forecasts.

A Novel Method of 2D Computation and Reconstruction

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

Black carbon aerosols from the coal seam of eastern India: A real-time analysis with statistical validation

Underground coal seam fire is one of the serious problems caused by coal mining activities. Hence, the present study is focused on investigating the potential emission of black carbon (BC) and its dispersion from the fire-affected area located in the Jharia coal field, Dhanbad district of the Jharkhand state of India. The real-time BC concentration measurements in the active fire area and the adjacent neighbouring urban area were studied and reported. Results showed an average concentration of BC was 12.54 $$\upmu $$ g m $$^{-3}$$ in the fumes/smokes emitting from the surface cracks and vents at the fire-affected zone. The continuous probability distribution functions, i.e., lognormal, gamma and Weibull, were used to fit the BC data individually, and among these lognormal fits, the best were for the medium-range values. Furthermore, the robustness of the obtained BC concentration was also explained by these models. All fitness was in good agreement, in the studied area, with the coefficient of regression ranging between 0.84 and 0.96. Both the experimental and predicted results clearly indicated the contribution of the BC that discharged from the coal seam fire as a source of particulate pollutant. Spatial modelling was also performed to validate the experimental consistency of BC concentration while the model was in good agreement with the underground coal fire zone as compared to the rest.

Simple and Low-Cost Procedure for Monthly and Yearly Streamflow Forecasts during the Current Hydrological Year

Accurately forecasting streamflow values is essential to achieve an efficient, integrated water resources management strategy and to provide consistent support to water decision-makers. We present a simple, low-cost, and robust approach for forecasting monthly and yearly streamflows during the current hydrological year, which is applicable to headwater catchments. The procedure innovatively combines the use of well-known regression analysis techniques, the two-parameter Gamma continuous cumulative probability distribution function and the Monte Carlo method. Several model performance statistics metrics (including the Coefficient of Determination R2; the Root-Mean-Square Error RMSE; the Mean Absolute Error MAE; the Index of Agreement IOA; the Mean Absolute Percent Error MAPE; the Coefficient of Nash-Sutcliffe Efficiency NSE; and the Inclusion Coefficient IC) were used and the results showed good levels of accuracy (improving as the number of observed months increases). The model forecast outputs are the mean monthly and yearly streamflows along with the 10th and 90th percentiles. The methodology has been successfully applied to two headwater reservoirs within the Guadalquivir River Basin in southern Spain, achieving an accuracy of 92% and 80% in March 2017. These risk-based predictions are of great value, especially before the intensive irrigation campaign starts in the middle of the hydrological year, when Water Authorities have to ensure that the right decision is made on how to best allocate the available water volume between the different water users and environmental needs.

Anti-diffusion in continuous opinion dynamics

Considerable effort using techniques developed in statistical physics has been aimed at numerical simulations of agent-based opinion models and analysis of their results. Such work has elucidated how various rules for interacting agents can give rise to steady state behaviors in the agent populations that vary between consensus and fragmentation. At the macroscopic population level, analysis has been limited due to the lack of an analytically tractable governing macro-equation for the continuous population state. We use the integro-differential equation that governs opinion dynamics for the continuous probability distribution function of agent opinions to develop a novel nonlinear partial differential equation for the evolution of opinion distributions. The highly nonlinear equation allows for the generation of a system of approximations. We consider three initial population distributions and determine their small-time behavior. Our analysis reveals how the generation of clusters results from the interplay of diffusion and anti-diffusion and how initial instabilities arise in different regions of the population distribution.

Statistical analysis of annual maximum daily rainfall for Nelspruit and its environs.

Nelspruit and its environs frequently experience extreme high annual maximum daily rainfall (AMDR) events resulting in flood hazards. These flood hazards have caused flood disasters that have resulted in loss of property and lives. The main objective of this study was to carry out statistical analysis of extreme high AMDR events that have caused flood hazards, which in turn have caused flood disasters in Nelspruit and its environs. Empirical continuous probability distribution functions (ECPDF) and theoretical continuous probability distribution functions (TCPDF) were applied to carry out the statistical analysis of the extreme high AMDR events. Annual maximum daily rainfall event of magnitude 100 mm was identified as a threshold. Events > 100 mm were considered as extreme high events resulting in flood disasters. The results of empirical frequency analysis showed that the return period of flood disasters was 10 years. The occurrence probability of flood disaster event at least once in 1, 2, 3, 4 and 5 years was 0.10, 0.19, 0.27, 0.34 and 0.41, respectively. Generalised logistic PDF was identified as the best-fit theoretical PDF for statistical analysis of the extreme high AMDR events in Nelspruit and its environs. The results of this study contributed to the understanding of frequency and magnitude of extreme high AMDR events that could lead to flood disasters. The results could be applied in developing flood disaster management strategies in Nelspruit and its environs.

Combined cumulants and Laplace transform method for probabilistic load flow analysis

The simultaneity of power systems development and uncertainty of system elements has promoted the importance of probabilistic load flow (PLF) in the operating and planning studies of the system. This clarifies that the use of the fast and accurate approaches for PLF computation is necessary. To achieve this objective, this study presents an analytical technique, based on the properties of Laplace transform (LT). The suggested methodology is applicable for every continuous probability distribution function as the input random variable. The proposed procedure is applied to the MATPOWER 9- and 118-bus test systems. To validate the combined cumulants and LT (CCLT) technique, the results are compared with the Monte Carlo simulation and the cumulants method combined with the maximum entropy (CCME) principle. The test results show that the proposed approach gives accurate results, with the lower computational burden comparing CCME. Furthermore, the method formulation and case study results demonstrate that the CCLT method is mathematically straightforward and computationally efficient.

Dynamic multi‐stage under frequency load shedding considering uncertainty of generation loss

Under frequency load shedding (UFLS) as a special protection scheme is executed to arrest declining frequency. The setting parameters of a multi-stage load shedding plan are determined using a discretised model of dynamic system frequency response considering uncertainty of generation loss. The proposed probabilistic UFLS plan is formulated as a mixed integer linear programming optimisation problem to minimise the expected amount of load curtailment. The uncertainty of the proposed discretised dynamic load shedding plan is modelled using the point estimate method, and the obtained results are compared with those of the Monte-Carlo simulation method (MCSM). The utilised point estimate technique converts the continuous probability distribution functions of the uncertain parameter (i.e. the generation loss) to discrete statistical moments. The proposed probabilistic UFLS is solved using the BONMIN solver. The performance of the proposed dynamic UFLS plan is investigated for the IEEE 39-bus test system.

Evaluation of statistical distributions to analyze the pollution of Cd and Pb in urban runoff.

Heavy metal pollution in urban runoff causes severe environmental damage. Identification of these pollutants and their statistical analysis is necessary to provide management guidelines. In this study, 45 continuous probability distribution functions were selected to fit the Cd and Pb data in the runoff events of an urban area during October 2014-May 2015. The sampling was conducted from the outlet of the city basin during seven precipitation events. For evaluation and ranking of the functions, we used the goodness of fit Kolmogorov-Smirnov and Anderson-Darling tests. The results of Cd analysis showed that Hyperbolic Secant, Wakeby and Log-Pearson 3 are suitable for frequency analysis of the event mean concentration (EMC), the instantaneous concentration series (ICS) and instantaneous concentration of each event (ICEE), respectively. In addition, the LP3, Wakeby and Generalized Extreme Value functions were chosen for the EMC, ICS and ICEE related to Pb contamination.

Three new diagnostic verification diagrams

ABSTRACTForecast evaluation is inherently multidimensional, so that graphical representations of verification datasets, which can simultaneously show multiple dimensions of forecast verification, are helpful for diagnosing the strengths and weakness of the forecasts. This paper presents three new diagnostic verification diagrams for settings that previously lacked them. The H‐F diagram is a graphical representation of the likelihood–base rate factorization, and the T‐M diagram is a graphical representation of the calibration–refinement factorization, of the joint distribution of forecasts and observations in the setting of non‐probabilistic forecasts for dichotomous events. The Conditional Boxplots diagram represents graphically the calibration–refinement factorization for probability forecasts of continuous outcomes, where each forecast is itself a continuous probability distribution function. The three new diagrams are described, and illustrated with example verification datasets.

Improving quality of sample entropy estimation for continuous distribution probability functions

Entropy is a one of the key parameters characterizing state of system in statistical physics. Although, the entropy is defined for systems described by discrete and continuous probability distribution function (PDF), in numerous applications the sample entropy is estimated by a histogram, which, in fact, denotes that the continuous PDF is represented by a set of probabilities. Such a procedure may lead to ambiguities and even misinterpretation of the results. Within this paper, two possible general algorithms based on continuous PDF estimation are discussed in the application to the Shannon and Tsallis entropies. It is shown that the proposed algorithms may improve entropy estimation, particularly in the case of small data sets.

On the gap and time interval between the first two maxima of long continuous time random walks

We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval T where at each time step the walker waits a random time τ, before performing a jump drawn from a symmetric continuous probability distribution function (PDF) , of Lévy index . Our study includes the case where the waiting time PDF has a power law tail, , with , such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if (and super-diffusive if ). We investigate the joint PDF of the gap g between the first two highest positions of the CTRW and the time t separating these two maxima. We show that this PDF reaches a stationary limiting joint distribution p(g, t) in the limit of long CTRW, . Our exact analytical results show a very rich behavior of this joint PDF in the plane, which we study in great detail. Our main results are verified by numerical simulations. This work provides a non trivial extension to CTRWs of the recent study in the discrete time setting by Majumdar et al (2014 J. Stat. Mech. P09013).

Probabilistic undervoltage load shedding using point estimate method

Undervoltage load shedding (UVLS) is executed as a last resort to avoid voltage instability in critical situations. In this study, a robust UVLS scheme based on the point estimate method (PEM) is developed taking load uncertainties into consideration. The utilised PEM converts the continuous probability distribution functions (PDFs) of load uncertainty to discrete statistical moments. The extracted moments could represent the behaviours of original continuous PDFs. The proposed stochastic UVLS strategy is expressed as a non-linear programming problem, which is optimised using sequential quadratic programming technique. The fulfilment of the suggested scheme is compared with Monte Carlo simulation technique for two IEEE 14-bus and IEEE 118-bus test grids.

Stochastic Water Quality Modeling of an Impaired River Impacted by Climate Change

AbstractA new stochastic water quality modeling tool was applied to quantify potential climate change effects on a nutrient impaired reach in the Pacific Northwest. This tool allows for multiple stochastic inputs for steady state river water quality simulations. A previously published calibrated deterministic model of the targeted reach was adapted for this study. This model simulates steady-state nutrient, algae, and dissolved oxygen dynamics with both point and nonpoint pollutant loadings. It also includes simulation of diurnally varying water temperature, calculated as a function of air temperature, shading, and streamflow using heat budget equations. For this study, local summer air temperature and critical low flows were treated stochastically in the model. Parameterization of these inputs was based on analysis of multiple global climate model (GCM) projections for the study area corresponding to a 2060 planning horizon. Continuous probability distribution functions were fitted to ensemble GCM data s...

Enhancement of bio-oil derived chemicals in aqueous phase using ionic liquids: Experimental and COSMO-SAC predictions using a modified hydrogen bonding expression

In this work the commercial hydrophobic imidazolium based ionic liquid, 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([BMIM][TF2N]) was investigated for the extraction of acetic acid and furfural from aqueous solution. The liquid–liquid equilibrium (LLE) data for the ternary mixture of [BMIM][TF2N] (1)+acetic acid (2)+water (3) and [BMIM][TF2N] (1)+furfural (2)+water (3) were experimentally determined at T=298.15K and atmospheric pressure. The two phase region for acetic acid system was smaller as compared to the furfural system. The distribution co-efficient and selectivity values were greater than unity which indicates easier extraction of acetic acid and furfural from aqueous phase. The selectivity values for furfural were of two orders of magnitude greater than acetic acid. The experimental results were then correlated with the thermodynamic activity coefficient models namely NRTL and UNIQUAC models. The root mean square deviation (RMSD) obtained in NRTL and UNIQUAC models for both the systems were in the range of 0.26–0.75%. Thereafter a continuous probability distribution function for screening charge density has been used to obtain the interaction due to the hydrogen bonding term in conductor like screening model-segment activity coefficient (COSMO-SAC). This implementation gave RMSD values of 2.9% and 2.2% for IL–acetic acid–water and IL–furfural–water systems, respectively.