Standard Probability Distributions Research Articles

A Wigner Quasiprobability Distribution of Work.

In this article, we introduce a quasiprobability distribution of work that is based on the Wigner function. This proposal rests on the idea that the work conducted on an isolated system can be coherently measured by coupling the system to a quantum measurement apparatus. In this way, a quasiprobability distribution of work can be defined in terms of the Wigner function of the apparatus. This quasidistribution contains the information of the work statistics and also holds a clear operational definition that can be directly measured in a real experiment. Moreover, it is shown that the presence of quantum coherence in the energy eigenbasis is related with the appearance of features related to non-classicality in the Wigner function such as negativity and interference fringes. On the other hand, from this quasiprobability distribution, it is straightforward to obtain the standard two-point measurement probability distribution of work and also the difference in average energy for initial states with coherences.

Development of water re-allocation policy under uncertainty conditions in the inflow to reservoir and demands parameters: a case study of Karaj AmirKabir dam

The process of optimal operation of multipurpose reservoirs is accompanied by large dimensions of decision variables and the uncertainty of hydrological parameters and water demands. Therefore, in determining the optimal operation policies (OOPs), the decision making for water allocation is faced with problems and complexities. One of the effective approaches for sustainable management and optimal allocation from water resources is the multi-objective structural development based on the uncertainty of input parameters. The purpose of this study is to provide OOPs from Karaj AmirKabir multi-purpose reservoir with applying uncertainty in the inflow to reservoir and downstream water demand. The proposed approach has been investigated in two certain and uncertain models, and three objective functions of the system including maximizing hydropower generation, water supply demands, and flood control have been considered to formulate OOPs. Non-dominated sorting genetic algorithm-II (NSGA-II) was performed to optimize the three proposed objective functions and by applying multi-criteria decision-making (MCDM) methods, the best operation scenario was selected. In the uncertainty model, using the interval method and repeated implementation of the deterministic model for completely random scenarios that generated based on the variation interval of the uncertain parameters, the non-deterministic optimal allocation values were produced. Based on these optimal allocation values and the fitting of the standard probability distribution on it, the probability of occurrence of the deterministic allocation values was determined. Production of optimal probabilistic allocation policies can be very useful and efficient in providing real vision to managers to select appropriate policies in different conditions and rare hydrological events. The results obtained from the certain model shows that as a result of optimal allocation to demands, the fuzzy reliability, resiliency, and system stability indexes were improved to 67.81, 21.99, and 24.98 percentage, respectively. Also, in an uncertain model, applying changes of 48% and 22%, respectively, for the inflow and downstream demand has led to changes of 23%, 55%, and 18%, respectively, in the first, second, and third objective functions. The highest impact from uncertain conditions, has been related to the water supply demands with 55% of the range of variations So, the water supply demands, has a higher sensitivity and priority than other reservoir objective functions under uncertain conditions. Another important result extracted from this study is to determine the monthly probability of optimal allocations achievement. Accordingly, in the warm seasons and years in which the reservoir is facing drought, the occurrence probability of the optimal allocations decreases. Given the comprehensiveness of the proposed methodology, this approach is a very suitable tool for determining the optimal water allocations as probabilistic based on the scenarios desired by managers and reservoir operators.

Accurate and efficient forecasted wind energy using selected temporal metrological variables and wind direction

The aim of this work is to find the most efficient and suitable input features to be selected for forecasting monthly wind energy accurately. Machine learning is employed for a modular pipelined neural network, composed of time-delayed and feedforward networks with features of metrological variables such as atmospheric temperature, humidity, wind direction, and wind speed frequency distribution parameters. Logged data over a year’s period at a UAE site are analyzed on daily and monthly bases depending on their variation characteristics, in which standard Weibull probability distribution function is used for the feedforward neural network together with wind direction data, while daily average ambient temperature and humidity are attempted for the composite time delay networks. Different network abstractions of input features are compared, and it is found that wind direction data offer a better wind speed forecast. Wind energy is calculated based on monthly forecasting. A detailed adaptive probabilistic analysis is conducted to predict thresholds in variations of the forecast analysis. Error estimation tools are performed for adopting this method

Estimating the cumulative distribution function of lead-time demand using bootstrapping with and without replacement

Forecasting of the cumulative distribution function (CDF) of demand over lead time is a standard requirement for effective inventory replenishment. In practice, while the demand for some items conforms to standard probability distributions, the demand for others does not, thus making it challenging to estimate the CDF of lead-time demand. Distribution-free methods have been proposed, including resampling of demand from previous individual periods of the demand history, often referred to as bootstrapping in the inventory forecasting literature. There has been a lack of theoretical research on this form of resampling. In this paper, we analyse the bias and variance of CDF estimates obtained by resampling, both with and without replacement. Counterintuitively, we find that the ‘with replacement’ approach does not always dominate ‘without replacement’ in terms of mean square error of CDF estimates. Closed-form expressions are given for the components of Mean Square Error, with and without replacement. For shorter lead times, of two or three periods, these may be used directly to identify series that may benefit from resampling without replacement. Inventory performance implications are evaluated on simulated and empirical data. It is found that marked differences may arise between ‘with replacement’ and ‘without replacement’ bootstrapping approaches. The latter method can be more beneficial for lower target Cycle Service Levels, longer lead times and shorter demand histories.

Ensuring power balance in a hybrid power system with a standby generator

The combination of several non-guaranteed random energy sources (RES), conventional sources, and nonconstant consumer loads in a local system leads to stochastic power imbalances. This study objective consists in determining the possibilities of ensuring the power balance in a hybrid power generation system with a standby generator and a search for the methods of calculating the optimal parameters to achieve energy balance. This objective is achieved by simulating the processes inherent in wind and solar power engineering and the regimes of energy consumption through a combination of random functions with a standard probability distribution. Aggregated data on weather factors for several years in a region with a high renewable energy potential which can be used to describe the behavior of wind and solar energy over time were used as experimental data. The use of multiple simulations of random processes with calculated parameters has made it possible to draw conclusions about the presence of certain ratios of power and the generator control modes. These ratios can determine minimum energy and consumption losses, reduce the likelihood of energy imbalance, more efficiently use the reserved power. Specific features of the stochastic nature of RES related to the presence of trends and random fluctuations at short hourly intervals were additionally taken into account. Possibilities of varying the conditions of and switching on and off of the standby generator were provided. The existence of some ranges was established for the installed power of the generator outside which its use becomes inefficient. The proposed approach makes it possible to find the probability of various system states, assess the reliability of energy supply, and minimize unproductive losses.

Placement and sizing of EESS bundled with uncertainty modeling by two-stage stochastic search based on improved shark smell optimization algorithm in micro-grids

Microgrids are low voltage small-scale grids which are used for supplying the thermal and electric loads in small areas or locations without access to the main grid. A microgrids can be a proper solution for power generation considering the environmental and economic problems of these locations. In this paper, the optimum size and placement of electric energy storage is presented considering the uncertainties associated with renewable energy resources and fault in the microgrid lines as well as impact of responsive loads in a microgrid. The studied hybrid system is composed of wind turbines and fuel cells. Basically, the wind energy is not certainly accessible and the newly suggested method is based on a two-stage stochastic search (TSSS) model for modeling the impact of uncertainty in the power generated by the wind generation resources by considering the wind energy prediction error as the main stochastic parameter in the planning. The standard probability distribution function is used for showing the error variation. With respect to continuity of this function, the prediction error probability function is discretized using a new method and a certain number of scenarios with specific probability are extracted. Finally, the studied problem is converted to an optimization problem and the developed shark smell optimization (SSO) algorithm is used to solve it. Numerous case studies show and approve the performance of the proposed model.

Chance-constrained path planning in narrow spaces for a dubins vehicle

The problem of optimal path planning through narrow spaces in an unstructured environment is considered. The optimal path planning problem for a Dubins agent is formulated as a chance-constrained optimal control problem (CCOCP), wherein the uncertainty in obstacle boundaries is modelled using standard probability distributions. The chance constraints are transformed to deterministic equivalents using the inverse cumulative distribution function and subsequently incorporated into a deterministic optimal control problem. Due to multiple convex sub-regions introduced by the obstacles, the initial guess provided to optimal control solver is crucial for computation time and optimality of the solution. A constrained Delaunay triangulation mesh based approach is developed that ensures the initial guess to lie in the optimal sub-convex region. Finally, off-the-shelf software is used to transcribe the optimal control problem to a nonlinear program (NLP) using Gaussian quadrature orthogonal collocation and solved to obtain an optimal path that conforms to system dynamics. By varying the upper bound on probability of obstacle collision, a family of solutions is generated, parameterized by the risk associated with each solution. This approach enables discovery of special “keyhole trajectories” that can provide significant cost savings in a tightly-spaced obstacle field. Merits of this approach are illustrated by comparing it with the traditional bounded uncertainty approach.

Distribution identification and information loss in a measurement uncertainty network

Measurement uncertainty is an increasingly important consideration in many applications demanding extreme performance levels. In the era of the internet of things and 5G connectivity we can learn more about device performance by utilising the increasing amount of data produced. These data require appropriate information infrastructure to facilitate continuous updating of device performance knowledge. This paper presents the results of a study which NPL undertook with a leading test and measurement device manufacturer to examine how measurement uncertainty propagates through the data traceability chain from national standards to end devices. A hierarchy of siloed calculations and heuristics did not enable a satisfactory metadata exchange within the dataflow to ensure an internally consistent calculation of measurement uncertainty. We therefore propose a novel measurement uncertainty network which contains a set of internally consistent measurement models, traceable to national standards and connected through common quantities. The network facilitates sharing and programmatic processing of measurement data with due regard to timeliness, privacy preservation and adherence to FAIR principles in measurement data exchange. An illustrative example of this network is presented with techniques to determine the best-fitting standard probability distribution for a given dataset and the resulting change in information content.

Implicitly adaptive importance sampling

Adaptive importance sampling is a class of techniques for finding good proposal distributions for importance sampling. Often the proposal distributions are standard probability distributions whose parameters are adapted based on the mismatch between the current proposal and a target distribution. In this work, we present an implicit adaptive importance sampling method that applies to complicated distributions which are not available in closed form. The method iteratively matches the moments of a set of Monte Carlo draws to weighted moments based on importance weights. We apply the method to Bayesian leave-one-out cross-validation and show that it performs better than many existing parametric adaptive importance sampling methods while being computationally inexpensive.

Strategic formulation of industrial maintenance based on equipment reliability in a sugar and ethanol production plant

Maintenance and management have substantial importance in the search of the company’s competitive advantages. In this direction, a reliability analysis of equipments is very important for the definition of the most suitable maintenance strategy. The main goal of this paper is to assess the reliability-centered maintenance of the industrial reliability curve of a sugar cane department of an industry located in São Paulo State, Brazil. The proposed research method was based on the application of existing statistical modeling for the times between failures (TBF) of all reported equipment failures or the interruption times in the production line. These times were modeled by standard lifetime probability distributions as log-normal and Weibull distributions. The results showed that the company's strategy for preventive maintenance during the off-season is not adequate and the statistical analysis also identified important factors that affect the company's maintenance strategy. These results could be of great interest for the company and for engineering applications in general.

A new one parameter Rayleigh Maxwell distribution

In this article, a new one-parameter lifetime distribution has been suggested. The distribution is a two-component finite mixture of Rayleigh and Maxwell distribution. The various statistical properties of the distribution such as moments, skewness, kurtosis, moment generating function, and characteristics function have been discussed. Survival function and hazard function are also studied. The maximum likelihood estimate for the unknown parameter under the proposed model is derived. Finally, the model is fitted to a real-life failure data, and outcomes were compared with some standard statistical probability distributions.

Integral cross section of isomeric state formation in (neutron,nucleon) reactions using an Am–Be source

The present work targeted the study of the isomeric state formation in (n,n′) and (n,p) reactions using the spectral emission of an 241Am–Be neutron source which favors the (neutron, nucleon) reactions over the (n, γ) and (n,2n) ones. The integral cross section of the isomeric nuclear states in the neutron-induced nucleon-emission reactions of fast neutrons on 115In, 199Hg, 58Ni, and 195Pt was measured experimentally. The monitor reaction of neutrons was the 115In(n,n′)115mIn using the evaluated integral cross section of 269−15+13 mb, which was confirmed to have a constant isomeric ratio in the energy range between ∼1.6 MeV and ∼10 MeV. Integral cross sections for the 199Hg(n,n′)199mHg, 58Ni(n,p)58mCo, 58Ni(n,p)58m+gCo and 195Pt(n,n′)195mPt reactions were found to be 449 ± 13 mb, 112 ± 34 mb, 343 ± 72 mb and 740 ± 55 mb, respectively. Each experimental value was based on independent measurements of samples irradiated with a constant neutron flux from an 241Am–Be source (having standard emission probability). The experimental results are compared with the integrated cross sections from a new evaluation for neutron induced reactions weighted to standard neutron emission probability distribution. Integral values estimated from theoretical calculations and evaluated libraries were used for the examination of our results. Results confirmed that formation of isomeric states is sensitive to the neutron energy distribution. The behavior of the isomeric ratio of neutron reaction leading to isomeric state formation was found to be useful in order to distinguish between reactions suitable for use as monitor ones (yielding a constant isomeric ratio throughout the emission energy range) and those used to probe the disturbance in the neutron spectral distribution.

Using Chebyshev’s inequality to predict a random determinant for i.i.d. Gamma and Weibull elements

This paper gives a probabilistic analysis of a determinant D of order 2 and 3 in which the elements are i.i.d.Gamma and Weibull variates. Using Chebyshev’s inequality, fiducial limits of D are obtained for order 2 and 3. The results are then compared with those obtained for other standard probability distributions.

Structural damage detection based on variational Bayesian inference and delayed rejection adaptive Metropolis algorithm

Existing studies on sparse Bayesian learning for structural damage detection usually assume that the posterior probability density functions follow standard distributions which facilitate to circumvent the intractable integration problem of the evidence by means of numerical sampling or analytical derivation. Moreover, the uncertainties of each mode are usually quantified as a common parameter to simplify the calculation. These assumptions may not be realistic in practice. This study proposes a sparse Bayesian method for structural damage detection suitable for standard and nonstandard probability distributions. The uncertainty corresponding to each mode is assumed as different. Variational Bayesian inference is developed and the posterior probability density functions of each unknown are individually derived. The parameters are found to follow the gamma distribution, whereas the distribution of the damage index cannot be directly obtained because of the nonlinear relationship in its posterior probability density function. The delayed rejection adaptive Metropolis algorithm is then adopted to generate numerical samples of the damage index. The coupled damage index and parameters in the variational Bayesian inference are successively calculated via an iterative process. A laboratory-tested frame is utilised to verify the effectiveness of the proposed method. The results indicate that the sparse damage can be accurately detected. The proposed method has the advantage of high accuracy and broad applicability.

The new distribution (Topp Leone Marshall Olkin-Weibull) properties with an application

The few standard probability distributions available are insufficient for modelling naturally occurring events. Most especially, normal distribution is not ideal for modelling asymmetrical data. Generalizing a new distribution is an important area in probability theory. It is known that we can improve the performance of the distributions by adding a new parameters. So, in this paper, we introduce Topp Leone Marshall Olkin-Weibull distribution as new method with four parameters based on recently proposed family (Topp Leone Marshall-Olkin of distributions). Accordingly, mathematical properties such as quantile function, moments generating functions, entropy, and order statistics have been investigated. The estimation of four parameters by maximum likelihood method was implemented. Application for real data based on the proposed method can be employed to show the best fitting as compared with the other models.

Improved gray wolf optimization algorithm for solving placement and sizing of electrical energy storage system in micro-grids

Micro-grids consist of distributed power generation systems (DGs), distributed energy storage devices (DSs), and loads. Micro-grids are small-scale networks at low voltage levels that are use to provide thermal and electrical loads of small locations where there is no access to the main electrical grid. Given the environmental and economic issues for these areas, micro-grids can be a good solution for energy production. In this paper, determining the size and location of optimal electrical energy storage systems is presented. In other side, a new method based on the cost benefit analysis for optimal sizing of an energy storage system in a microgrid (MG) is proposed. The uncertainties associated with renewable energy sources and the occurrence of defects in the grid connection network and the effect of the contribution of load responses in a micro-grid are taken into account. The combined system consists of wind turbines and fuel cells. Basically, wind power is not definitively available. The new proposed method is based on two-stage randomization design (TSRD) for modeling the effect of wind power uncertainty so that the predicted wind energy error is considered as the main random parameter in the model. A standard probability distribution function is used to represent the error variations. Given the continuity of the mentioned function, the probability error function is extracted using the new discrete method and a certain number of scenarios with a certain probability. Finally, the problem has been transformed into an optimization problem, and a gray wolf optimization (GWO) algorithm has been used to solve it. In the proposed developed model based on local and global search, the algorithm tries to reach the final result in the shortest possible time and with the most precision. The results of the simulation show the efficiency of the proposed method in solving the micro-grid problem.

A new one parameter Rayleigh Maxwell distribution

In this article, a new one-parameter lifetime distribution has been suggested. The distribution is a two-component finite mixture of Rayleigh and Maxwell distribution. The various statistical properties of the distribution such as moments, skewness, kurtosis, moment generating function, and characteristics function have been discussed. Survival function and hazard function are also studied. The maximum likelihood estimate for the unknown parameter under the proposed model is derived. Finally, the model is fitted to a real-life failure data, and outcomes were compared with some standard statistical probability distributions.

Performance evaluation of web service response time probability distribution models for business process cycle time simulation

ContextThe adoption of Business Process Management (BPM) is enabling companies to improve the pace of building new capabilities, enhancing existing ones, and measuring process performance to identify bottlenecks. It is essential to compute the cycle time of the process to assess the performance of a business process. The cycle time typically forms part of service level agreements (SLAs) and is a crucial contributor to the overall user experience and productivity. The simulation technique is versatile and has broad applicability for determining realistic cycle time using historical data of web service response time. BPM tools offer inadequate support for modeling input data used in simulation in the form of descriptive statistics or standard probability distributions like normal, lognormal, which results in inaccurate simulation results. ObjectiveWe evaluate the effectiveness of different parametric and non-parametric probability distributions for modeling data of web service response time. We further assess how the choice of probability distribution impacts the accuracy of the simulated cycle time of a business process. The work is the first of such a study using real-world data for encouraging Business Process Simulation Specification (BPSim) standard setters and BPM tools to enhance their support for such distributions in their simulation engine. MethodWe consider several parametric and non-parametric distributions and explore how well these distributions fit web service response time from extensive public and a real-world dataset. The cycle time of the business process of a real-world system is simulated using the identified distributions to model the underlying web service data. ResultsOur results show that kernel distribution is the most suitable choice, followed by Burr. Kernel outperforms Burr by 86.63% for the public and 84.21% for the real-world dataset. The choice of distribution affects the percentile ranks like 90 and above than the median. The use of single-point values underestimates cycle time values at higher percentiles. ConclusionBased on our empirical results, we recommend the addition of kernel and Burr to the current list of distributions supported by BPSim and BPM tools.

Probability Representation of Photon States and Tomography†

We give a review of the tomographic probability representation of quantum mechanics. We present the formalism of quantum states and quantum observables using the formalism of standard probability distributions and classical-like random variables. We study the coherent and number states of photons in the probability representation and obtain the evolution equation and energy spectra in the form of equations for probability distributions.

On the number of Monte Carlo runs in comparative probabilistic LCA

IntroductionThe Monte Carlo technique is widely used and recommended for including uncertainties LCA. Typically, 1000 or 10,000 runs are done, but a clear argument for that number is not available, and with the growing size of LCA databases, an excessively high number of runs may be a time-consuming thing. We therefore investigate if a large number of runs are useful, or if it might be unnecessary or even harmful.Probability theoryWe review the standard theory or probability distributions for describing stochastic variables, including the combination of different stochastic variables into a calculation. We also review the standard theory of inferential statistics for estimating a probability distribution, given a sample of values. For estimating the distribution of a function of probability distributions, two major techniques are available, analytical, applying probability theory and numerical, using Monte Carlo simulation. Because the analytical technique is often unavailable, the obvious way-out is Monte Carlo. However, we demonstrate and illustrate that it leads to overly precise conclusions on the values of estimated parameters, and to incorrect hypothesis tests.Numerical illustrationWe demonstrate the effect for two simple cases: one system in a stand-alone analysis and a comparative analysis of two alternative systems. Both cases illustrate that statistical hypotheses that should not be rejected in fact are rejected in a highly convincing way, thus pointing out a fundamental flaw.Discussion and conclusionsApart form the obvious recommendation to use larger samples for estimating input distributions, we suggest to restrict the number of Monte Carlo runs to a number not greater than the sample sizes used for the input parameters. As a final note, when the input parameters are not estimated using samples, but through a procedure, such as the popular pedigree approach, the Monte Carlo approach should not be used at all.