Form Of Probability Distribution Function Research Articles

Online aggregation of probability forecasts with confidence

The paper presents numerical experiments and some theoretical developments in prediction with expert advice (PEA). One experiment deals with predicting electricity consumption depending on temperature and uses real data. As the pattern of dependence can change with season and time of the day, the domain naturally admits PEA formulation with experts having different “areas of expertise”. We consider the case where several competing methods produce online predictions in the form of probability distribution functions. The dissimilarity between a probability forecast and an outcome is measured by a loss function (scoring rule). A popular example of scoring rule for continuous outcomes is Continuous Ranked Probability Score (CRPS). In this paper the problem of combining probabilistic forecasts is considered in the PEA framework. We show that CRPS is a mixable loss function and then the time-independent upper bound for the regret of the Vovk aggregating algorithm using CRPS as a loss function can be obtained. Also, we incorporate a “smooth” version of the method of specialized experts in this scheme which allows us to combine the probabilistic predictions of the specialized experts with overlapping domains of their competence.

Verification and Validation and Uncertainty Quantification of Code Models

The input uncertainties propagation methods are the most frequently applied statistical methods in uncertainty analyses. Among them, particularly popular are the methods based on Wilks’ formula. Numerous studies on uncertainty analyses show that the identification and quantification of input uncertainties is a major problem with uncertainty analyses. Among input uncertainties evaluation, the identification and quantification of physical model uncertainties in thermal-hydraulic codes appear to be particularly difficult.This paper deals with this problem by proposing inherent model uncertainties quantification by code developers in the frame of code development and validation. The introduction of the extended code validation would not only contribute to potential uncertainty analyses, solving to a large degree the problem of model uncertainties quantification, but also contribute to code validation, and as a consequence, improve the safety issues. A not-negligible factor is also better management of the resources. Instead of uncertainty quantification repeatedly performed by each user, the quantification could be performed once and, in addition, by experts having the required know-how.Introducing this new standard in code validation would require additional effort from the code developers but integral quantification of the model uncertainties would be profitable also for code development. In fact, by code development, in particular if the model is own development of the team, such an accuracy (or uncertainty) evaluation is usually performed. The additional effort, in this case, would be to describe the present information in the form of probability distribution functions or at least in the form of ranges.

Simulating turbulence–radiation interactions using a presumed probability density function method

In turbulent combustion, the turbulent fluctuations of temperature and species concentrations have strong effects on chemical and radiative heat sources. Turbulence–chemistry interactions (TCI) and turbulence–radiation interactions (TRI) create a set of “closure” problems when the governing partial differential equations are averaged. The presumed probability distribution function (presumed-PDF) method assumes a form of probability distribution function to close the chemical source term. The emphasis of this work is developing a high-fidelity radiation model that works in tandem with combustion models that use the presumed-PDF method to close the turbulent source terms. A finite volume based photon Monte Carlo method with a line-by-line spectral model is applied with the presumed-PDFs of mixture fraction, scalar dissipation rate and enthalpy defect to account for TRI effects. An efficient wavenumber selection scheme is proposed for the line-by-line photon Monte Carlo method considering TRI. The model is validated with one-dimensional exact line-by-line solutions for different TRI treatments and with a coupled combustion simulation for an open jet flame.

A risk-based multi-objective model for optimal placement of sensors in water distribution system

In this study, a new stochastic model based on Conditional Value at Risk (CVaR) and multi-objective optimization methods is developed for optimal placement of sensors in water distribution system (WDS). This model determines minimization of risk which is caused by simultaneous multi-point contamination injection in WDS using CVaR approach. The CVaR considers uncertainties of contamination injection in the form of probability distribution function and calculates low-probability extreme events. In this approach, extreme losses occur at tail of the losses distribution function. Four-objective optimization model based on NSGA-II algorithm is developed to minimize losses of contamination injection (through CVaR of affected population and detection time) and also minimize the two other main criteria of optimal placement of sensors including probability of undetected events and cost. Finally, to determine the best solution, Preference Ranking Organization METHod for Enrichment Evaluation (PROMETHEE), as a subgroup of Multi Criteria Decision Making (MCDM) approach, is utilized to rank the alternatives on the trade-off curve among objective functions. Also, sensitivity analysis is done to investigate the importance of each criterion on PROMETHEE results considering three relative weighting scenarios. The effectiveness of the proposed methodology is examined through applying it to Lamerd WDS in the southwestern part of Iran. The PROMETHEE suggests 6 sensors with suitable distribution that approximately cover all regions of WDS. Optimal values related to CVaR of affected population and detection time as well as probability of undetected events for the best optimal solution are equal to 17,055 persons, 31 mins and 0.045%, respectively. The obtained results of the proposed methodology in Lamerd WDS show applicability of CVaR-based multi-objective simulation–optimization model for incorporating the main uncertainties of contamination injection in order to evaluate extreme value of losses in WDS.

Application of IoT for the Maintaining Rolling Stocks

<p><strong>Purpose:</strong> This paper presents a framework for simulation on IoT based CBM (condition based monitoring) for rolling stocks. This enables to allocate maintenance resources effectively while satisfying preventive maintenance requirements.</p><p><strong>Methodology/Approach:</strong> We exploits Reliability centered maintenance (RCM) based on KTX (Korea Tran eXpress, Korea’s high-speed rail system) motor reduction unit failure data for three years by utilising the internet of things (IoT) and RAMS (Reliability, Availability, Maintainability, Safety) methods.</p><p><strong>Findings:</strong> We come up with the predictive maintenance indicator; reliability functions as to the desired service level; and the failure and defect prediction indicator takes the form of cumulative failure function in the form of probability distribution function, which aim to realise the real-time condition monitoring and maintaining technical support services. Internet of Things (IoT) has been an important apparatus to improve the maintenance efficiency.</p><p><strong>Research Limitation/implication:</strong> This paper has limitations that the data are collected from references, not actual data; the detailed descriptions of IoT application to the railway rolling stocks are omitted, and it is not dealt in depth how maintenance efforts and performance are improved through the suggested reliability centered maintenance.</p><strong>Originality/Value of paper:</strong> This study has the academic importance in a sense that it integrates RAMS based maintenance methods and IoT. RAMS centered maintenance provides powerful rules for deciding a failure management policy; when it is technically appropriate; and for providing precise criteria for deciding how often routine tasks should be carried out. It will lead to the improved cost efficiency, sustainability and maintainability of railway maintenance system since the staff do not have to visit installation sites frequently. Lately, there is general agreement that prevention was better than inspection and that an increase in preventive cost was the means of reducing total quality costs. In connection with this issue, we will address the way of reducing failure costs and prevention costs with IoT: new appraisal method.

Input Controls for Entering Uncertain Data

Although more and more data is collected automatically, many interfaces still require manual input. When we, for example, enter our daily calorie intake or calculate our ecological footprint, we often have to guess the weight of the food or what distance we have covered with our car. In this paper, we propose a solution to overcome the problem of forcing users to enter a single value when they are unsure about the actual input. On the basis of a slider, we designed four input controls which allow the input of uncertain data in the form of probability distribution functions. To evaluate our input controls, we conducted two studies collecting subjective and objective feedback. Based on the evaluation, we derived implications for their usage. We additionally provide an open-source toolkit with the evaluated input controls that can be included in web applications and customized for different contexts and tasks.

Winners and losers from climate change in agriculture: Insights from a case study in the Mediterranean basin

The Mediterranean region has always shown a marked inter-annual variability in seasonal weather, creating uncertainty in decisional processes of cultivation and livestock breeding that should not be neglected when modeling farmers' adaptive responses. This is especially relevant when assessing the impact of climate change (CC), which modifies the atmospheric variability and generates new uncertainty conditions, and the possibility of adaptation of agriculture. Our analysis examines this aspect reconstructing the effects of inter-annual climate variability in a diversified farming district that well represents a wide range of rainfed and irrigated agricultural systems in the Mediterranean area. We used a Regional Atmospheric Modelling System and a weather generator to generate 150 stochastic years of the present and near future climate. Then, we implemented calibrated crop and livestock models to estimate the corresponding productive responses in the form of probability distribution functions (PDFs) under the two climatic conditions. We assumed these PDFs able to represent the expectations of farmers in a discrete stochastic programming (DSP) model that reproduced their economic behaviour under uncertainty conditions. The comparison of the results in the two scenarios provided an assessment of the impact of CC, also taking into account the possibility of adjustment allowed by present technologies and price regimes. The DSP model is built in blocks that represent the farm typologies operating in the study area, each one with its own resource endowment, decisional constraints and economic response. Under this latter aspect, major differences emerged among farm typologies and sub-zones of the study area. A crucial element of differentiation was water availability, since only irrigated C3 crops took full advantage from the fertilization effect of increasing atmospheric CO2 concentration. Rainfed crop production was depressed by the expected reduction of spring rainfall associated to the higher temperatures. So, a dualism emerges between the smaller impact on crop production in the irrigated plain sub-zone, equipped with collective water networks and abundant irrigation resources, and the major negative impact in the hilly area, where these facilities and resources are absent. However intensive dairy farming was also negatively affected in terms of milk production and quality, and cattle mortality because of the increasing summer temperatures. This provides explicit guidance for addressing strategic adaptation policies and for framing farmers' perception of CC, in order to help them to develop an awareness of the phenomena that are already in progress, which is a prerequisite for effective adaptation responses.

A probability estimation method for reliability analysis using mapped Gegenbauer polynomials

Based on the theory of Gegenbauer polynomial approximation, this article presents a probability estimation method for reliability analysis of mechanical systems. Analytical evaluation or statistical analysis is used to compute statistical moments. Then, the well-known Gegenbauer series approach is applied to approximate the probability distribution functions and cumulative distribution functions. The proposed method is completely distribution-free due to the fact that no classical theoretical distributions are assumed in the whole process, and the approximate analytic description provides a universal form of probability distribution function and cumulative distribution function. Five numerical examples involving mathematic functions and mechanical engineering problems are applied to illustrate the application of the proposed method. The results show that the probability distribution functions and cumulative distribution functions of random variables and responses can be accurately and efficiently derived from Gegenbauer polynomial approximation method with high-order statistical moments.

A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach

We present a new stochastic analysis for steady and transient one-dimensional heat conduction problem based on the homogenization approach. Thermal conductivity is assumed to be a random field K consisting of random variables of a total number N. Both steady and transient solutions T are expressed in terms of the homogenized solution T∼ and its spatial derivatives T(x,t)=T∼+∑n=1∞Ln(x)∂nT∼/∂xn, where homogenized solution T∼ is obtained by solving the homogenized equation with effective thermal conductivity. Both mean and variance of stochastic solutions can be obtained analytically for K field consisting of independent identically distributed (i.i.d) random variables. The mean and variance of T are shown to be dependent only on the mean and variance of these i.i.d variables, not the particular form of probability distribution function of i.i.d variables. Variance of temperature field T can be separated into two contributions: the ensemble contribution (through the homogenized temperature T∼); and the configurational contribution (through the random variable Ln(x)). The configurational contribution is shown to be proportional to the local gradient of T∼. Large uncertainty of T field was found at locations with large gradient of T∼ due to the significant configurational contributions at these locations. Numerical simulations were implemented based on a direct Monte Carlo method and good agreement is obtained between numerical Monte Carlo results and the proposed stochastic analysis.

Reliability-based approach to the geotechnical design of tailings dams

Mine tailings dams are geotechnical structures that are designed to provide adequate and safe storage of tailings materials both during and after the end of mine life. The design of such structures is thus of utmost importance not only to the success of the mining operation, but more importantly to the safety of the surrounding environment such as freshwater resources, wildlife and community developments. This paper presents a reliability-based approach for the geotechnical design of mine tailings dams through a real life case study of a water retention tailings dam. The stability analysis is conducted with a hydromechanical, elasto-plastic finite difference model. The reliability analysis is carried out for three stochastic parameters namely the cohesion and friction angle of the dam core material as well as its hydraulic conductivity. The results are presented in the form of probability distribution functions of the generated factor of safety. The generated reliability indices are compared to target reliability indices to help evaluate the geotechnical design of the dam.

Probabilistic decision model of wind power investment and influence of green power market

This paper presents results from a model of a representative wind power investor's decision making process using a Monte Carlo simulation of a project financial analysis. Data, in the form of probability distribution functions (PDFs) for key input variables were collected from interviews with investors and other professionals active in the U.S. wind power industry using a formal expert elicitation protocol. This study presents the first quantitative estimates of the effect of the U.S. voluntary Renewable Energy Certificate (REC) market on renewable energy generation. The results indicate that the investment decisions of wind power project developers in the United States are unlikely to have been altered by the voluntary REC market. The problem with the current voluntary REC market is that it does not offer developers a reliable risk-adjusted revenue stream. Consequently, the claims by U.S. green power retailers and promoters that voluntary market RECs result in additional wind power projects lack credibility. Even dramatic increases in voluntary market REC prices, in the absence of long-term contracts, were found to have only a small effect on investor behavior.

Stochastic modelling of soil water and plant water stress using cumulant expansion theory

AbstractIn this study, a dynamic modelling scheme is presented to derive the probabilistic structure of soil water and plant water stress when subject to stochastic precipitation conditions. The newly developed model has the form of the Fokker–Planck equation, and its applicability as a model for the probabilistic evolution of soil water and plant water stress is investigated under climate change scenarios. This model is based on the cumulant expansion theory, and has the advantage of providing the probabilistic solution in the form of probability distribution function (PDF), from which one can obtain the ensemble average behaviour of the dynamics. The simulation result of soil water confirms that the proposed soil water model can properly reproduce the results obtained from observations, and it also proves that soil water behaves with consistent cycle based on the precipitation pattern. The plant water stress simulation shows two different PDF patterns according to the precipitation. Moreover, with all the simulation results with climate change scenarios, it can be concluded that the future soil water and plant water stress dynamics will differently behave with different climate change scenarios. Copyright © 2010 John Wiley & Sons, Ltd.

Least-Cost Design of Water Distribution Networks under Demand Uncertainty

Due to inherent variability in instantaneous water consumption levels, values of demands at nodes in a water distribution system remain one of the major sources of uncertainty in the network design process. Uncertainty in demand leads to uncertainty in head at the nodes, which, in turn, affects the system performance and has to be taken into account when designing new water distribution systems or extending/rehabilitating existing ones. One approach to dealing with this difficulty is to formulate and solve the stochastic optimization problem providing a robust, cost-effective solution. However, stochastic formulation usually requires Monte Carlo simulation, which involves calculation of a large number of state estimates, even for relatively simple networks. This renders the approach intractable when combined with heuristic adaptive search techniques, such as genetic algorithms (GAs) or simulated annealing. These methodologies require the fitness function to be evaluated for thousands of possible network configurations in the course of the search process. In this paper a new approach to quantifying the influence of demand uncertainty on nodal heads is proposed. The original stochastic model is reformulated as a deterministic one, which uses standard deviation as a natural measure of variability. Such an approach allows the use of effective numerical methods to quantify the influence of uncertainty on the robustness of water distribution system solutions. The deterministic equivalent is then coupled with an efficient GA solver to find robust and economic solutions. The proposed methodology was tested on the New York tunnels and Anytown problems. A number of low cost network solutions were found for different levels of reliability and different forms of probability distribution function for demands. The robustness of the solutions found was compared to known solutions for deterministic formulations, whose results were postprocessed using full Monte Carlo simulation.

On-line novelty detection for artefact identification in automatic anaesthesia record keeping

We report the design of a kernel-based on-line novelty detector (ADDaM — Automatic Dynamic Data Mapper) and its use in the detection of artefacts in an automatic anaesthesia record keeper (AARK). ADDaM produces a partitioned history of any ordered data stream and constructs a probability distribution function (PDF) from that history using Gaussian kernels. Two forms of PDF are possible: a s tatic PDF where the prior probability of each kernel is determined by the number of observations it represents and a temporal PDF where more recent observations have a higher prior probability. Testing against the current PDF assesses the novelty of the next point entering the stream. The performance of this method for artefact detection in heart rate data was compared to Kalman, ARIMA and moving mean filters using receiver operator characteristic (ROC) curves. Performance was measured using the area under the curves (AUC), and the false positive rate (FPR) and positive predictive value (PPV) calculated at the optimal cost-point on the curves. The results obtained were: ADDaM (Static PDF) AUC 0.92, FPR 0.12, PPV 0.12 and ADDaM (Temporal PDF) AUC 0.97, FPR 0.12, PPV 0.15. Both ADDaM-based methods out performed all other on-line methods tested.

Simple Probabilistic and Statistical Risk Calculations in an Aquifer

An objective methodology is proposed for assessing potential ground water yield on the basis of the randomness inherent in hydrogeological parameters as a result of uncertainty. Randomness is used here in the statistical sense to describe the ground water phenomenon on a regional basis. A systematic approach to selecting aquifer parameters is presented that leads to reliable ground water storage and subsurface flow estimates by incorporating a certain risk level. In addition, the averages of the specific ground water capacity and subsurface flow rate are calculated through a perturbation approach. This study demonstrates that cross-correlations between basic hydrogeologic variables might underestimate or overestimate model parameter values. Furthermore, the results of aquifer storage and subsurface flow estimates are obtained in the form of probability distribution functions. It is proved that the surface flow rates are presented by a log-normal distribution. Contrary to conventional point estimate, a simple and objective methodology is proposed in order to obtain the most probable interval estimates. Accordingly, projections of likely future ground water storage and flow rates can be readily calculated.

Properties of the cosmological density distribution function

The properties of the probability distribution function (PDF) of the cosmological continuous density field are studied. We focus our analysis on the quasi-linear regime where various calculations, based on dynamically motivated methods, have been presented: either by using the Zel'dovich approximation (ZA) or by using the perturbation theory to evaluate the behavior of the moments of the distribution function. We show how these two approaches are related to each other and that they can be used in a complementary way. For that respect the one-dimensional dynamics, where the ZA is exact solution, has first been used as a testing ground. In particular we show that, when the density PDF obtained with the ZA is regularized, its various moments exhibit the behavior expected by the perturbation theory applied to the ZA. We show that ZA approach can be used for arbitrary initial conditions (not only Gaussian) and that the nonlinear evolution of the moments can be obtained. The perturbation theory can be used for the exact dynamics. We take into account the final filtering of the density field both for ZA and perturbation theory. Applying these techniques we got the generating function of the moments for the one-dimensional dynamics, the three- dimensional ZA, with and without smoothing effects. We also suggest methods to build PDFs. One is based on the Laplace inverse transform of the moment generating function. The other, the Edgeworth expansion, is obtained when the previous generating function is truncated at a given order and allows to evaluate the PDF out of limited number of moments. It provides insight on the relationship between the moments and the shape of the density PDF. In particular it provides an alternative method to evaluate the skewness and kurtosis by measuring the PDF around its maximum. Eventually, results obtained from a numerical simulation with CDM initial conditions have been used to validate the accuracy of the considered approximations. We explain the successful log- normal fit of the PDF from that simulation at moderate σ as mere fortune, but not as a universal form of density PDF in general.

On the probability of the longest run length in an independent series

Various problems concerning the analysis of droughts lead to the problem of finding the probability distribution function (PDF), of the longest run lengths in a series of finite length. Unfortunately, the exact PDF of such runs is not available in the literature. This paper is concerned with the derivation of the exact PDF of the longest run length to be expected in an independent series. In the derivation of the PDF, different definitions of runs encountered in the hydrology literature have been considered. The analytical derivations are solved numerically on digital computers and the results are presented in forms of cumulative PDF and the expected maximum run length changes with the sample lengths. The application of the methodology developed herein is presented for three annual flow series.

Physical model of intermittency in turbulence: Inertial-range non-Gaussian statistics.

We present a physical model to describe the equilibrium probability distribution function (PDF) of velocity differences across an inertial-range distance in 3D isotropic turbulence. The form of the non-Gaussian PDF agrees well with data from direct numerical simulations. It is shown that these PDF's obey a self-similar property, and the resulting inertial-range exponents of high-order velocity structure functions are in agreement with both experimental and numerical data. The model suggests a physical explanation for the phenomenon of intermittency and the nature of multifractality in fully developed turbulence, namely, local self-distortion of turbulent structures.