Type Of Probability Density Function Research Articles

Effects of NO Models on the Prediction of NO Formation in a Regenerative Furnace

A study of the effects of nitric oxide (NO) models on the prediction of NO formation in a gas-fired regenerative furnace with highly preheated air was undertaken. Three chemical kinetic processes for NO formation/depletion, i.e., thermal NO, prompt NO, and NO reburning, are included. In the thermal NO model, the sensitivity encountered when using two different approaches, namely the equilibrium approach and the partial equilibrium approach, for determining the O radical concentration was studied. The effects of the third reaction in the thermal NO mechanism, NO reduction (reburning) mechanism, and different types of probability density functions (PDFs) on the NO predictions have also been tested. The sensitivity of the excess air ratio on the NO generation rate in the furnace has been investigated. Finally, the impact of the temperature on the NO formation rate in the regenerative furnace was discussed. [S0195-0738(00)00304-6]

Understanding of Relationship Between the Average Mass Transport Rate And The Moments Of Permeability

AbstractTo estimate the transport rate of radionuclides in the geosphere, we must consider the spatial variability of permeability. However, the borehole data of permeability are limited and we can not determine the type of probability density function, though the measurement data reflect the most significant hydraulic properties about geologic media including innumerable cracks or fast flow paths. While the recent models describing radioactive nuclide transport in near/far-field have assumed a certain probability density function (typically a lognormal distribution) as a permeability distribution, we cannot always obtain sufficient measurement data to define the function. However, the available data of permeability at least give us the moments such as the arithmetic mean, the standard deviation and the skewness for the distribution.The purpose of this paper is to get an understanding of the general relationship between the average mass transport rates and the moments. Using various types of probability density functions and pseudo random-numbers, hypothetical permeability distributions are generated. With these distributions, this paper obtains the average transport rates described as the numerical impulseresponse based on the advection-dispersion model for a two-dimensional region. The calculated results show that, for the dimensionless standard deviation up to around 1, the three moments are enough to characterize the permeability distribution for the purposes of the nuclide transport prediction.In this work, for five specified probability density functions, the upper and lower bounds of skewness are derived as a function of the dimensionless arithmetic mean and standard deviation. The obtained upper and lower bounds explicitly show that the Bernoulli trials (a discrete probability density function) yield the widest range in the skewness against the standard deviation. Since the response has lower peak and longer tail as the skewness goes to the lower bound value, we can predict the shape of the breakthrough curve from the three moments of the borehole data.

Statistical attributes of the power learning curve model

A learning curve is a well known tool which describes the relation between the performance of a task and the number of repetitions of that task. A learning curve model may be used as a prediction tool in various applications of operations planning and control. Previous models have treated learning as a deterministic phenomenon, whereas this paper addresses its stochastic nature and treats learning as a random process. Exact expressions of the variance, of the coefficient of variation and of the probability density function of the performance, expressed as a function of the number of repetitions of a task, are derived for the power model. Mathematicalanalysis reveals that the coefficient of variation is independent of the number of repetitions. The probability density function type was also found to be independent of the number of repetitions. Analysis of actual learning data validated the theoretical findings. These findings enable improved forecasting of future performance by using a stochastic rather than a deterministic estimation.

Probabilistic evaluation of the economical damage due to harmonic losses in industrial energy system

The problem of harmonic loss evaluation is of growing interest among power system engineers, both for the increase of operating costs and the decrease of the useful life of the system components. The paper addresses this problem with a probabilistic approach to take into account the randomness of current and voltage harmonics; a method for evaluating the expected value of the operating costs and of the premature aging costs is presented. A numerical application to a test system is also discussed, showing the viability of the proposed approach and the significance of the results; in particular, the aging costs turn out to be unaffected by the type of probability density function describing the harmonics.

Estimation of the Radionuclide Transport by Applying the Mean, the Standard Deviation and the Skewness of Permeability

ABSTRACTA new method for estimating the mass transport by using the stochastic values (the arithmetic mean, the standard deviation and the skewness) of permeability is presented. Generally, detail of permeability distribution cannot be obtained except for moments of the distribution. Also, measurement results of permeability for the rock matrix including cracks or fast flowpaths do not always follow the log-normal distribution frequently applied. In such a situation, we must evaluate the characteristic permeabilities for the whole or some regions of the disposal site including the accessible environment.The authors have investigated the characteristic permeability on the basis of some probability density functions of permeability, applying the Monte Carlo method and FEM. It was found that its value does not depend on type of probability density function of permeability, but on the arithmetic mean, the standard deviation and the skewness of permeability [1].This paper describes the use of the stochastic values of permeability for estimating the rate of radioactivity release to the accessible environment, applying the advection-dispersion model to two-dimensional, heterogeneous media. When a discrete probability density function (referred to as ‘the Bernoulli trials’) and the lognormal distribution have common values for the arithmetic mean, the standard deviation and the skewness of permeability, the calculated transport rates (described as the pseudo impulse responses) show good agreements for Peclet number around 10 and the dimensionless standard deviation around 1. Further, it is found that the transport rates apparently depends not only on the arithmetic mean and the standard deviation, but also on the skewness of permeability. When the value of skewness dose not follow the lognormal distribution which has only two independent parameters (the mean and the standard deviation), we can replicate the three moments estimated from an observed distribution of permeability, by using the Bernoulli trials having three independent parameters.

Parameter Uncertainty Propagation Analysis for Urban Rainfall Runoff Modelling

The paper proposes a strategy for model uncertainty propagation analysis. As an example, parameter uncertainty propagation analysis in the runoff block of the HYSTEM-EXTRAN model is carried out. The model is a modification of the well-known SWMM (Storm Water Management Model). Uncertainty propagation methods such as first-order analysis, sensitivity analysis, statistical linearization and Monte-Carlo analysis are discussed and applied. A pathway of parameter uncertainty propagation analysis is given based on validity, simplicity, and computational requirements. The pathway starts with sensitivity analysis which may help to reduce the dimensions of a multidimensional model by discarding insensitive parameters. This is to obtain a mathematically tractable uncertainty propagation problem for a complicated model. Then, the nonlinearity of the model must be quantified to check the validity of first-order analysis. If first-order analysis is not valid, and if components of model output uncertainty need to be known, the application of statistical linearization is the only analytical alternative. Monte-carlo analysis can always be applied and taken as a reference as long as the components of the model output uncertainty are not of interest. The parameter sensitivity is characterized by its sensitivity coefficient which is defined as the ratio of the coefficient of variance of a model output to the coefficient of variance of the model parameter itself. A nonlinear rainfall runoff model usually results in a variable parameter sensitivity. Hence, recommendations about parameter sensitivity cannot be generalized for a given rainfall-runoff model, but depend on the type and the range of the model output variable. It is shown that the type of probability density function describing the parameter uncertainty with known mean and variance has only a small effect on the results of the model output uncertainty.

Use of multiple vector quantisation for semicontinuous-HMM speech recognition

Although the continuous hidden Markov model (CHMM) technique seems to be the most flexible and complete tool for speech modelling. It is not always used for the implementation of speech recognition systems because of several problems related to training and computational complexity. Thus, other simpler types of HMMs, such as discrete (DHMM) or semicontinuous (SCHMM) models, are commonly utilised with very acceptable results. Also, the superiority of continuous models over these types of HMMs is not clear. The authors' group has previously introduced the multiple vector quantisation (MVQ) technique, the main feature of which is the use of one separated VQ codebook for each recognition unit. The MVQ technique applied to DHMM models generates a new HMM modelling (basic MVQ models) that allows incorporation into the recognition dynamics of the input sequence information wasted by the discrete models in the VQ process. The authors propose a new variant of HMM models that arises from the idea of applying MVQ to SCHMM models. These are SCMVQ-HMM (semicontinuous multiple vector quantisation HMM) models that use one VQ codebook per recognition unit and several quantisation candidates for each input vector. It is shown that SCMVQ modelling is formally the closest one to CHMM, although requiring even less computation than SCHMMs. After studying several implementation issues of the MVQ technique. Such as which type of probability density function should be used, the authors show the superiority of SCMVQ models over other types of HMM models such as DHMMs, SCHMMs or the basic MVQs.

Range estimation statistics utilizing Beta density functions

Traditional analysis of the triangulation ranging problem utilizes Gaussian distributions to represent angle or time‐delay measurement errors. This approach leads to mathematically intractable solutions due to infinite limits on the Gaussian distribution. To resolve these problems, this analysis can be carried out by replacing the Gaussian measurement error statistics by an error model with finite limits (Beta density function). This type of probability density function presents a credible error distribution over a finite range that maps into an estimated range density function from which the range error moments can be generated. This approach is intuitively useful because not only can the validity of the linearized model be evaluated, but this analysis also yields realistic bounds on ranging performance.

A burn-in program for wearout unaffected equipments

This paper describes a method useful for the evaluation of the burn-in time for repairable equipments in order to lower their failure rate as much as possible and bring them out of their early failure period. Reference is made to repairable equipments whose useful lifetime is very long, such as electronic telecommunication equipments, and whose wearout period does not take place before they are removed from operation field, because obsolete. The improvement of the equipment average Mean Time Between Failure is calculated as a function of the burn-in time. The mathematical formalism applies to any type of probability density function during early failures period. The method for solving the problem is explained and specific examples, under the hypothesis of exponential, lognormal, Weibull distribution, are given to illustrate the concepts.

A computer program for the graphical and iterative fitting of probability density functions to biological data

A tutorially-assisted, interactive program, written for a Digital Equipment Corporation LAB-11 minicomputer (PDP-11/20), is described which allows a user to fit (with or without automatic estimation of initial parameter values), by a method of nonlinear least squares, any one of seven different types of probability density functions (p.d.f.'s) to an empirical frequency distribution; the latter of which may be input to the program or formed by the program whenever it is furnished a series of times between events. The iteratively-obtained, “best fit” p.d.f. is displayed on a two color, point-plot display against the background of a point-plot histogram. By selecting any one of nine output modes, the user is allowed: (1) to view histograms successively on the point-plot display, (2) to generate selected p.d.f.'s, (3) to “force” p.d.f.'s having known parameters through the histogram data, (4) to obtain Chi-square ( χ 2) and Kolmogorov-Smirnov estimates of the goodness of fit to the data, and (5) to apply a special test [Williams and Kloot, 1953] in order to determine whether the least squares estimates of two candidate models are statistically different. The resident driver program and the four overlayable program segments are written in standard FORTRAN IV; except for two plot routines, which are written in PDP-11 assembly language.