Gaussian Probability Distribution Function Research Articles

Assessing Wet and Dry Periods Using Standardized Precipitation Index Fractal (SPIF) and Polygons: A Novel Approach

In the open literature, there are numerous studies on the normal and extreme (flood and drought) behavior of wet and dry periods based on the understanding of the standard precipitation index (SPI), which provides a series of categorizations by considering the standard normal (Gaussian) probability distribution function (PDF). The numerical meaning of each categorization assessment is quite lacking in terms of future predictions of wet and dry period duration based on historical records. This paper presents a new approach for calculating possible formations of future wet and dry period durations based on historical records through an effective fractal geometric forecasting approach. The essence of the proposed methodology is based on the number of dry periods (steps) of non-overlapping monthly duration along consecutive broken line paths in the SPI classification for wet and dry period durations. It has been observed that the plot of periods on double logarithmic paper falls along a straight line against the number of such periods, implying a power function, which is the essence of fractal geometry. Extending the empirically derived straight line provides the number of periods that may occur in the future over a range of SPI levels. This methodology is referred to as SPI fractal (SPIF), and the classic SPI classification is converted into SPIF wet and dry polygons, which provide additional information about the drought period number within a valid polygonal area, compared to the classic SPI results. The wet and dry period features of any hydro-meteorology time series are constrained in SPIF polygons. The application of the methodology was carried out on monthly rainfall records on the European side of the Istanbul Florya meteorological station in Turkey.

Riemannian Manifold-Based Feature Space and Corresponding Image Clustering Algorithms.

Image feature representation is a key factor influencing the accuracy of clustering. Traditional point-based feature spaces represent spectral features of an image independently and introduce spatial relationships of pixels in the image domain to enhance the contextual information expression ability. Mapping-based feature spaces aim to preserve the structure information, but the complex computation and the unexplainability of image features have a great impact on their applications. To this end, we propose an explicit feature space called Riemannian manifold feature space (RMFS) to present the contextual information in a unified way. First, the Gaussian probability distribution function (pdf) is introduced to characterize the features of a pixel in its neighborhood system in the image domain. Then, the feature-related pdfs are mapped to a Riemannian manifold, which constructs the proposed RMFS. In RMFS, a point can express the complex contextual information of corresponding pixel in the image domain, and pixels representing the same object are linearly distributed. This gives us a chance to convert nonlinear image segmentation problems to linear computation. To verify the superiority of the expression ability of the proposed RMFS, a linear clustering algorithm and a fuzzy linear clustering algorithm are proposed. Experimental results show that the proposed RMFS-based algorithms outperform their counterparts in the spectral feature space and the RMFS-based ones without the linear distribution characteristics. This indicates that the RMFS can better express features of an image than spectral feature space, and the expressed features can be easily used to construct linear segmentation models.

Study on radiological environmental impact assessment of nuclear fuel cycle facilities

The radiological assessment model REIA 1.0 is established to assess the radio-logical environmental impact of radioactive airborne effluents released from nuclear fuel cycle facilities. It is based on the double Gaussian probability distribution function model and considers the wind pendulum effect of the long-term diffusion factor. The simulation results of the REIA 1.0 model are compared with those of the AERMOD-VIEW model and radiation environmental monitoring results of nuclear fuel cycle facilities. The results are: using the REIA 1.0 and AERMOD-VIEW models, the relative deviations of simulation results are less than 33.33% in 192 subregions. Except for three points, the relative deviations are less than 20%, the simulated results are consistent with the radiation environmental monitoring results in seven sites using the REIA 1.0 model, and the REIA 1.0 model can characterize the incremental contribution of radionuclide concentration in the air caused by the radioactive airborne effluents from nuclear fuel cycle facilities. It can effectively assess the radiological environmental impact of nuclear fuel cycle facilities.

Cyber-secure energy and flexibility scheduling of interconnected local energy networks with introducing an XGBoost-assisted false data detection and correction method

The growing penetration of renewable energy resources (RERs) and the ever-increasing implementation of information and communication technologies in smart grids necessitate the utilization of fast and safe operation methods. These schemes should be able to cope with the intermittency of uncertain RERs and fully explore the capabilities of new technologies while considering the vulnerabilities as a result of digitalization. Here, the integrity of the energy and flexibility scheduling and trading heavily depends on the security of the cyber-physical system. Local energy networks (LENs) are introduced to easily incorporate and utilize distributed energy resources, improving electrical power reliability and efficiency. This study develops a cyber-secure energy and flexibility scheduling platform for interconnected local energy networks. Hence, it introduces a novel false data detection and correction method based on the XGBoost machine learning method and investigates the performance of the proposed algorithm in the context of an interconnected local energy network and the corresponding energy and flexibility transactions. In this regard, this paper initially presents a bilevel energy and flexibility scheduling problem for interconnected local energy networks (ILENs) considering cyber-attacks (BLEFSCAILEN), which tries to satisfy the energy and flexibility requirements of the ILEN guaranteeing a resilient decision against false data injection cyber-attacks. Within the proposed BLEFSCAILEN a bi-level multi-objective optimization problem is defined where the operator of ILEN optimizes both energy and flexibility trading among LENs in the upper level to maximize its revenue. In the proposed structure, the operator of each LEN is situated at the lower level and tries to minimize the scheduling cost and simultaneously maximize the local flexibility index. Furthermore, in order to evaluate and improve the proposed system’s cyber-security, a false data injection attack is constructed using the Gaussian probability distribution function. Herein, the scheduling of ILEN is optimized considering vulnerabilities of electricity consumption data as a consequence of cyber-attack. Finally, the novel XGBoost-assisted deviation bound-based false data detection/correction (XGBFDD) algorithm is suggested to mitigate the above-mentioned cyber-attack and attain near real-world conditions. Simulation results demonstrate that the proposed false data detection/correction strategy exhibits a maximum accuracy of 91.67 precent in mitigating the attack effects. This is because the acquired post-correction results exhibit minor deviations in comparison to the original optimization output that is calculated with the pre-attack consumption data.

Orbital Uncertainty Propagation Based on Adaptive Gaussian Mixture Model under Generalized Equinoctial Orbital Elements

The number of resident space objects (RSOs) has been steadily increasing over time, posing significant risks to the safe operation of on-orbit assets. The accurate prediction of potential collision events and implementation of effective and nonredundant avoidance maneuvers require the precise estimation of the orbit positions of objects of interest and propagation of their associated uncertainties. Previous research mainly focuses on striking a balance between accurate propagation and efficient computation. A recently proposed approach that integrates uncertainty propagation with different coordinate representations has the potential to achieve such a balance. This paper proposes combining the generalized equinoctial orbital elements (GEqOE) representation with an adaptive Gaussian mixture model (GMM) for uncertainty propagation. Specifically, we implement a reformulation for the orbital dynamics so that the underlying state and the moment feature of the GMM are propagated under the GEqOE coordinates. Starting from an initial Gaussian probability distribution function (PDF), the algorithm iteratively propagates the uncertainty distribution using a detection-splitting module. A differential entropy-based nonlinear detector and a splitting library are utilized to adjust the number of GMM components dynamically. Component splitting is triggered when a predefined threshold of differential entropy is violated, generating several GMM components. The final probability density function (PDF) is obtained by a weighted summation of the component distributions at the target time. Benefiting from the nonlinearity reduction caused by the GEqOE representation, the number of triggered events largely decreases, causing the necessary number of components to maintain uncertainty realism also to decrease, which enables the proposed approach to achieve good performance with much more efficiency. As demonstrated by the results of propagation in three scenarios with different degrees of complexity, compared with the Cartesian-based approach, the proposed approach achieves comparable accuracy to the Monte Carlo method while largely reducing the number of components generated during propagation. Our results confirm that a judicious choice of coordinate representation can significantly improve the performance of uncertainty propagation methods in terms of accuracy and computational efficiency.

A stochastic generative model for citation networks among academic papers.

We propose a stochastic generative model to represent a directed graph constructed by citations among academic papers, where nodes and directed edges represent papers with discrete publication time and citations respectively. The proposed model assumes that a citation between two papers occurs with a probability based on the type of the citing paper, the importance of cited paper, and the difference between their publication times, like the existing models. We consider the out-degrees of citing paper as its type, because, for example, survey paper cites many papers. We approximate the importance of a cited paper by its in-degrees. In our model, we adopt three functions: a logistic function for illustrating the numbers of papers published in discrete time, an inverse Gaussian probability distribution function to express the aging effect based on the difference between publication times, and an exponential distribution (or a generalized Pareto distribution) for describing the out-degree distribution. We consider that our model is a more reasonable and appropriate stochastic model than other existing models and can perform complete simulations without using original data. In this paper, we first use the Web of Science database and see the features used in our model. By using the proposed model, we can generate simulated graphs and demonstrate that they are similar to the original data concerning the in- and out-degree distributions, and node triangle participation. In addition, we analyze two other citation networks derived from physics papers in the arXiv database and verify the effectiveness of the model.

Innovative standard degree-day indicator (SDI) concept and application for monthly energy consumption control

In practice, monthly temperature records play a major role calculating the heating and cooling degree-days of the air conditioner management in buildings. Classically, heating and cooling degree calculations are based on the comparison of outdoor temperature measurements with a standard level called the base temperature. Truncating the temperature series at a constant base temperature causes heating (deficits) and cooling (surpluses) degrees. Since outdoor temperature records are of random character, future predictions with probabilistic and statistical methods are convenient means for calculations. All previous methodologies in the literature are based on crisp logic with simple mathematical calculations. This paper proposes a new methodological approach by converting the actual temperature record probability distribution function (PDF) into a probabilistic standardized degree indicator (SDI) in normal (Gaussian) PDF form with zero mean and unit standard deviation. Furthermore, as a new evaluation methodology fuzzy logic inference calculations are provided. SDI is divided into seven classes, three for cooling, heating, and a single class for uncooled-unheated states. Heating and cooling degree changes according to SDI values are presented with crisp and fuzzy sets. The application of the methodology is made for Kadıköy urban area on the Asian side of Istanbul, Turkey. In the case of fuzzy set SDI air conditioned operation, it has been determined that there is cooling energy saving of approximately 25% compared to the crisp set. The same is true for the heating degree-day side.

Remote sensing image segmentation by combining manifold projection and persistent homology

This paper presents an image segmentation algorithm by combining Manifold Projection and Persistent Homology (MP_PH). First, for a given image, the spectral measures of each pixel and its neighbor pixels are modeled with Gaussian Probability Distribution Function (GPDF) in an exponential family fashion. The Riemannian manifold, i.e. the data sub-manifold for the pixel, is built by taking the parameters of the GPDF exponential family model as its coordinates to depict the statistical characteristics of the original image. By Legendre transformation, the data sub-manifold is transformed into a parameter sub-manifold to depict all possible segmentation results. Only points representing classes of current segmentation results are activated on the parameter sub-manifold. Then, simplicial complexes constructed from the original image are used to compute persistent homology. The optimal scale can be obtained from persistent homology to compute the optimal homology group generated by homology generators which are referred to some pixels belonging to the same class. Finally, the segmentation is performed by projecting points of the data sub-manifold belonging to the same homology generator to the nearest activated point of the parameter sub-manifold, and updating all the activated points according to the projection results. As a result, all the activated points tend to be optimal segmentation. The experiments for synthetic and real images show that the proposed algorithm has high segmentation accuracy.

K-Means Clustering Based High Order Weighted Probabilistic Fuzzy Time Series Forecasting Method

In the present study, we propose a novel high-order weighted fuzzy time series (FTS) forecasting method using k-mean clustering, weighted fuzzy logical relations and probabilistic fuzzy set (PFS). Objective of proposed forecasting method is to handle occurrence of recurrence of fuzzy logical relations and both non-probabilistic and probabilistic uncertainties in assigning membership grades to time series datum. The proposed PFS-based forecasting method uses Gaussian probability distribution function to assign probabilities to membership grades. Proposed FTS forecasting method uses high-order weighted fuzzy logical relation in which each fuzzy logical relation uses the weight in ascending order. Superiority of proposed method is shown by implementing it on SBI share price at BSE, India and University of Alabama enrollments. Error measures and statistical parameters, for example, coefficient of correlation, coefficient of determination, performance parameter, evaluation parameter and tracking signal are also used to confirm the outperformance and validity of the proposed PFS-based forecasting method.

Models and Monte Carlo Simulations of the Mean Sinuosity of Major Meandering Rivers

The purpose of this research is to investigate the sinuosity of major rivers in the United States and the world, and to compare them to that predicted by the existing theories. It is shown that the average sinuosity of meandering rivers deviates considerably from what has been reported previously as π. Calculations of the mean value of actual sinuosities of major rivers in the United States and in the World show that this average is very close to 2. Exact models as well as a Monte Carlo simulation for meandering rivers that is based on Gaussian probability distribution function are also presented, and the possibility of composite meandering is discussed.

Gaussian Random Number Generator With Reconfigurable Mean and Variance Using Stochastic Magnetic Tunnel Junctions

Generating high-quality random numbers with a Gaussian probability distribution function is an important and resource-consuming computational task for many applications in the fields of machine learning and Monte Carlo algorithms. Recently, complementary metal–oxide–semiconductor (CMOS)-based digital hardware architectures have been explored as specialized Gaussian random-number generators (GRNGs). These CMOS-based GRNGs have a large area and require entropy sources at their input that increase the computing cost. In this letter we present a GRNG that works on the principle of the Boltzmann law in a physical system made from an interconnected network of thermally unstable magnetic tunnel junctions. The presented hardware can produce multibit Gaussian random numbers at gigahertz speed and can be configured to generate distributions with a desired mean and variance. An analytical derivation of the required interconnection and bias strengths is provided, followed by numerical simulations to demonstrate the functionalities of the GRNG.

A novel two-stage optimization method for the reliability based security constraints unit commitment in presence of wind units

In the restructured power system environment, independent system operator solves the Reliability-security constraint unit commitment (R–SCUC) problem that included reliability, economic issue and security. Therefore, in the modern bulk of power systems, the existing centralized methods cannot solve the complex issues such as large power system complexity. In this paper, a novel two-level optimization approach for a reliable, secure, and optimal generation scheduling is proposed. The power system is divided into several zones, in which each one solves its own R–SCUC problem. In the face of different uncertainties and time-dependent sources such as wind units there is new challenges in power systems reliability. In the proposed approach, load forecasting and wind units simulated based on the Gaussian probability distribution function and weibull probability distribution function (PDF), respectively. Energy bid modeling is formulated to fit the R–SCUC problem. In order to take into account reliability model, intelligent search techniques, i.e., TEENS, LOLP, are implemented in the whole power system. The proposed methods are performed in a 6-bus network, IEEE 118-bus and IEEE 4672-bus test systems. The presented approach has been programed by GAMS, and the results show the effectiveness of the proposed methodology to enhance security and reliability of the power system.

A machine-learning based quantitative evaluation of the fluid components on T2-D spectrum

Estimating the saturation of different fluids in reservoirs is key to determine the reservoir quality and to calculate reserves. Due to the overlap of different fluids on the inverted T2-D spectrum, estimating the saturation and evaluating the fluid components proves to be a difficult task. In this paper, we propose a machine learning method that integrates GABP neural network and Gaussian Mixture Model (GMM) to quantitatively evaluate the fluid components on T2-D spectrum. Distinct from other machine learning approaches in oil and gas exploration, the proposed method accounts for the physical significance between the T2-D spectrum and fluid saturation to determine a new saturation formula. We perform numerical simulations and subsequently process the simulated T2-D data. The results demonstrate the greater accuracy and stability of the GABP predictions compared to the BP neural network. However, both sets of results have no physical significance. The GMM is able to determine different Gaussian probability distribution functions (PDFs), which are then used to characterize different fluids on T2-D spectrum. The saturation determined via the Gaussian PDFs is more accurate than those of the GABP and BP. Our work reveals the ability of the proposed method to quantitatively evaluate the fluid components on T2-D spectrum.

Lifetime Extension of Three-Dimensional Wireless Sensor Networks, Based on Gaussian Coverage Probability

The aim of this work is to evaluate the nth joined probability of three-dimensional wireless sensor networks, and to extend the lifetime of these networks. A Gaussian probability distribution function is assumed for the power coverage probability for each sensor in the 3-dimensional cartesian and spherical coordinates. The overall joint probability is evaluated from each sensor to a target in the network, and then the network lifetime of sensors power sensing a number of targets, is extended based on removing redundancies of powering all sensors at the same time. Proportional to the evaluated probabilities, sensors are energized during slots of periodic time. The formulated probabilities are assumed to be uncorrelated among each sensor to any target zone. A case study is introduced to demonstrate extending the lifetime of a network comprising 7 sensors targeting two uncorrelated zones, in which 8 different cases of subsets are formed, when a minimum threshold of overall power coverage probability of 35% is assumed. Network lifetime is extended more than 70%, with some sensors reaching more than 90% power saving. This work can be extended to deal with other types of probabilities, as well as with cases of correlated sensor-target coverages.

Molecular dynamics lattice gas equilibrium distribution function for Lennard-Jones particles.

The molecular dynamics lattice gas (MDLG) method maps a molecular dynamics (MD) simulation onto a lattice gas using a coarse-graining procedure. This is a novel fundamental approach to derive the lattice Boltzmann method (LBM) by taking a Boltzmann average over the MDLG. A key property of the LBM is the equilibrium distribution function, which was originally derived by assuming that the particle displacements in the MD simulation are Boltzmann distributed. However, we recently discovered that a single Gaussian distribution function is not sufficient to describe the particle displacements in a broad transition regime between free particles and particles undergoing many collisions in one time step. In a recent publication, we proposed a Poisson weighted sum of Gaussians which shows better agreement with the MD data. We derive a lattice Boltzmann equilibrium distribution function from the Poisson weighted sum of Gaussians model and compare it to a measured equilibrium distribution function from MD data and to an analytical approximation of the equilibrium distribution function from a single Gaussian probability distribution function. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Thermodynamics of a Brownian particle in a nonconfining potential.

We consider the overdamped Brownian dynamics of a particle starting inside a square potential well which, upon exiting the well, experiences a flat potential where it is free to diffuse. We calculate the particle's probability distribution function (PDF) at coordinate x and time t,P(x,t), by solving the corresponding Smoluchowski equation. The solution is expressed by a multipole expansion, with each term decaying t^{1/2} faster than the previous one. At asymptotically large times, the PDF outside the well converges to the Gaussian PDF of a free Brownian particle. The average energy, which is proportional to the probability of finding the particle inside the well, diminishes as E∼1/t^{1/2}. Interestingly, we find that the free energy of the particle, F, approaches the free energy of a freely diffusing particle, F_{0}, as δF=F-F_{0}∼1/t, i.e., at a rate faster than E. We provide analytical and computational evidence that this scaling behavior of δF is a general feature of Brownian dynamics in nonconfining potential fields. Furthermore, we argue that δF represents a diminishing entropic component which is localized in the region of the potential, and which diffuses away with the spreading particle without being transferred to the heat bath.

Temporal Trend Possibilities of Annual Rainfall and Standardized Precipitation Index in the Central Highlands, Vietnam

This paper employed the Şen’s innovative trend analysis (ITA) method to identify overall and partial trends in annual rainfall records over the Central Highlands of Vietnam. Additionally, the lowest values of Standardized Precipitation Index (SPI) for 3 and 12 months were subjected to the analysis to reveal potential changes in the greatest intensity of meteorological drought. The outcomes from a preliminary assessment by applying the normal (Gaussian) probability distribution functions (PDFs) indicate significant increases in standard deviation values of the second half subseries (current climate), thereby implying greater level of rainfall variability. The results from temporal trend analysis show that there is high domination of neutral and decreasing trends in annual rainfall records, with the estimated ITA slopes varying mainly from − 10 to − 14 mm/year. Furthermore, the low value sub-groups are mainly characterized by significant decreasing trends, which is an evident indication of possible increases in drought intensity. These results were further demonstrated by evaluating the ITA plots for the lowest values of SPI in both 3-month and 12-month timescales. The lowest values of SPI are mainly lower than − 1.5 and − 2.0 corresponding to severe and extreme drought categories, and most of the scatter points in low SPI cluster are accumulated in the lower triangular area of 1:1 line. Generally, it is expected that these findings will contribute various well-founded foundations to manage water–use activities and natural hazards associated with rainfall shortages.

Jump point identification in hydro-meteorological time series by crossing methodology

The climate change impact appears as a decreasing or increasing monotonic trend in hydro-meteorology time series records due to greenhouse gas (GHG) emissions causing to global warming and climate change impacts. On the other hand, there may be abrupt changes in the form of jumps in these series due to natural and engineering activities. Although trend identification methods are rather common in the literature, jump determination conventional methodologies are rather rare and their applications present some restrictive asumptions like the serial independence and normal (Gaussian) probability distribution function (PDF). The methodology presented in this paper is away from each of such assumptions and it depicts the minimum number of upcrossing along horizontal truncation levels within the time series variation domain. The applications of the methodology are given for annual Danube River discharge records, Romania; New Jersey rainfall and temperature records, USA; monthly rainfall records and Van Lake level fluctuations, Turkey.

Viscuit and the fluctuation theorem investigation of shear viscosity by molecular dynamics simulations: The information and the noise.

The shear viscosity, η, of model liquids and solids is investigated within the framework of the viscuit and Fluctuation Theorem (FT) probability distribution function (PDF) theories, following Heyes et al. [J. Chem. Phys. 152, 194504 (2020)] using equilibrium molecular dynamics (MD) simulations on Lennard-Jones and Weeks-Chandler-Andersen model systems. The viscosity can be obtained in equilibrium MD simulation from the first moment of the viscuit PDF, which is shown for finite simulation lengths to give a less noisy plateau region than the Green-Kubo method. Two other formulas for the shear viscosity in terms of the viscuit and PDF analysis are also derived. A separation of the time-dependent average negative and positive viscuits extrapolated from the noise dominated region to zero time provides another route to η. The third method involves the relative number of positive and negative viscuits and their PDF standard deviations on the two sides for an equilibrium system. For the FT and finite shear rates, accurate analytic expressions for the relative number of positive to negative block average shear stresses is derived assuming a shifted Gaussian PDF, which is shown to agree well with non-equilibrium molecular dynamics simulations. A similar treatment of the positive and negative block average contributions to the viscosity is also shown to match the simulation data very well.

Effect of a magnetic field in an S = 1 antiferromagnetic chain with single-ion anisotropy and alternating bonds with disorder

In this study, we investigated the quantization of magnetization at low temperatures in a spin-1 antiferromagnetic chain with single-ion anisotropy by considering the quenched disorder in alternating bonds. Each spin i interacts with its nearest neighbors via Ji and αJi couplings, where Ji is considered to obey a Gaussian or Bernoulli probability distribution function, with width σ. Zero temperature phase diagrams were obtained by minimizing the energy of the Hamiltonian. For finite temperatures, a transfer matrix technique was applied in order to treat the partition function. Consequently, the magnetization curve calculated by deriving the quenched average of the free energy exhibited interesting plateaus, which were modified when randomness was considered. Experimentally, these plateaus appear when some chain structures are subjected to a magnetic field.