Probability Distribution Function Of Parameters Research Articles

Minimum zone evaluation of roundness using hybrid global search approach

Engineering parts usually deviate from their intended shapes during their manufacturing due to the inaccuracies of machine tools, deformation of various elements of machine tool, cutting tools and workpiece. Resulting geometric errors affect the functionality and assembly of the parts significantly. This demands a reliable strategy to accurately assess the errors during the final part quality inspection. Among all the geometric features, circular feature is very common on the majority of the engineering parts. Hence, the measurement and evaluation of circularity with a high degree of accuracy are of utmost importance. In the present work, a hybrid approach is proposed to accurately evaluate the circularity error. This approach comprises a least squares method (LSM) and a novel probabilistic global search Lausanne (PGSL) technique. The LSM is used to reduce the search space initially. Within the reduced search space, the PGSL performs efficient, fine and global search. In the process of establishing minimum zone circles to the measured data of the circular features, the proposed approach dynamically updates the probability distribution function of the circularity parameters continuously. The update procedure ensures that the probability of moving towards the potential optimal solutions is increased. The algorithm has been tested using several benchmark datasets for its generalization capability and robustness. The proposed strategy is found to be efficient in yielding accurate results. Therefore, the algorithm can be implemented in computer-aided circularity measuring instruments in order to minimize acceptance of bad parts and rejection of good parts.

A Statistical Framework for Generalized Linear Image Processing Systems

Generalized linear image processing systems have been developed from physical image formation models, human visual perception models, and mathematical models. Although there have been many papers on the extension, parameterization, and symmetrization of some of these systems, what is lacking is a unified framework such that the development and study of such systems can be performed based on a common ground. In this paper, we propose a conceptual image sensor which models how the light energy is converted into the sensor data. In the proposed sensor model, the input energy is regarded as a random variable and the conversion is through the cumulative distribution function. Based on the sensor model, we suggest a statistical framework by which new systems can be derived, and existing and seemingly unrelated systems can be studied from a unified perspective. The proposed statistical framework not only provides a principled way to symmetrizing systems through the even extension of the probability distribution function (PDF) and a natural way for the parameterization of systems through parameters of PDF. In this paper, we demonstrate new applications of the statistical framework through a numerical approximation of the lower incomplete gamma function, through the enhancement of the dynamic range and manipulation of the sharpness of images by using the scalar multiplication operation of the parametric system, through an application of a new system in fusion of multi-exposure images, and through an application of the three new systems for the correction of incorrect exposure.

A comparison of five forest interception models using global sensitivity and uncertainty analysis

Summary Interception by the forest canopy plays a critical role in the hydrologic cycle by removing a significant portion of incoming precipitation from the terrestrial component. While there are a number of existing physical models of forest interception, few studies have summarized or compared these models. The objective of this work is to use global sensitivity and uncertainty analysis to compare five mechanistic interception models including the Rutter, Rutter Sparse, Gash, Sparse Gash, and Liu models. Using parameter probability distribution functions of values from the literature, our results show that on average storm duration [Dur], gross precipitation [PG], canopy storage [S] and solar radiation [Rn] are the most important model parameters. On the other hand, empirical parameters used in calculating evaporation and drip (i.e. trunk evaporation as a proportion of evaporation from the saturated canopy [∊], the empirical drainage parameter [b], the drainage partitioning coefficient [pd], and the rate of water dripping from the canopy when canopy storage has been reached [Ds]) have relatively low levels of importance in interception modeling. As such, future modeling efforts should aim to decompose parameters that are the most influential in determining model outputs into easily measurable physical components. Because this study compares models, the choices regarding the parameter probability distribution functions are applied across models, which enables a more definitive ranking of model uncertainty.

Statistical Analysis of the Effect of Nanoparticles Volume Fraction on Turbulent Forced Convective Heat Transfer Coefficient of Nanofluid in a Circular Tube

In this paper, the statistical analysis of the effect of nanoparticles volume fraction on one of the most important thermal characteristics turbulent flow of nanofluid i.e. convection heat transfer coefficient, inside a circular tube with uniform wall heat flux is investigated numerically. Also, water as a base fluid and Al2O3 as suspended particles with a diameter of 36 nm are considered. Heat transfer characteristics are computed using the solution of elliptic equations based on discrete the finite volume method and the second order upwind. The relationship between pressure and velocity using SIMPLEC algorithm is established. In this study, the variation of volume fraction of nanoparticles is assumed in the range of 0 to 6%. The best probability distribution function of the heat transfer parameters are selected using chi square test that various probability distribution such as: Gamma, Normal, Lognormal, Gumbel, and Frechet are evaluated based on numerical analysis of tube flow. After reviewing the results, it was found that with increasing volume fraction of nanoparticles, the convective heat transfer coefficient increases. On the other hand, the convective heat transfer coefficients with regard to variation of volume fraction of nanoparticles follow Gumbel Max probability distribution function.

Automated Airborne EM and borehole data integration for depth to bedrock extraction

Airborne electromagnetic (AEM) was used to supplement a geotechnical investigation for a highway construction project in Norway. Variable bedrock threshold resistivity hindered efforts to track depth to bedrock, motivating us to develop an automated algorithm to extract depth to bedrock from both boreholes and AEM data. We developed two variations of this algorithm: one using simple Gaussian or inverse distance weighting interpolators, and another using ordinary kriging and combined parameter probability distribution functions. Evaluation shows that for preliminary surveys, significant savings in boreholes required can be made without sacrificing bedrock model accuracy. However, issues with AEM noise and data quality likely reduced the comparative advantage that including AEM provided. Moreover, AEM cannot supersede direct sampling where the model accuracy required exceed the resolution possible with the geophysical method. Nevertheless, using AEM in the way can still reduce the number of required boreholes and hence reduce site investigation costs because we can identify high probability zones for shallow bedrock, identify steep or anomalous bedrock topography, and estimate the spatial variability of depth

Predictions in ungauged basins: an approach for regionalization of hydrological models considering the probability distribution of model parameters

Regionalization of model parameters by developing appropriate functional relationship between the parameters and basin characteristics is one of the potential approaches to employ hydrological models in ungauged basins. While this is a widely accepted procedure, the uniqueness of the watersheds and the equifinality of parameters bring lot of uncertainty in the simulations in ungauged basins. This study proposes a method of regionalization based on the probability distribution function of model parameters, which accounts the variability in the catchment characteristics. It is envisaged that the probability distribution function represents the characteristics of the model parameter, and when regionalized the earlier concerns can be addressed appropriately. The method employs probability distribution of parameters, derived from gauged basins, to regionalize by regressing them against the catchment attributes. These regional functions are used to develop the parameter characteristics in ungauged basins based on the catchment attributes. The proposed method is illustrated using soil water assessment tool model for an ungauged basin prediction. For this numerical exercise, eight different watersheds spanning across different climatic settings in the USA are considered. While all the basins considered in this study were gauged, one of them was assumed to be ungauged (pseudo-ungauged) in order to evaluate the effectiveness of the proposed methodology in ungauged basin simulation. The process was repeated by considering representative basins from different climatic and landuse scenarios as pseudo-ungauged. The results of the study indicated that the ensemble simulations in the ungauged basins were closely matching with the observed streamflow. The simulation efficiency varied between 57 and 61 % in ungauged basins. The regional function was able to generate the parameter characteristics that were closely matching with the original probability distribution derived from observed streamflow data.

An intelligent approach based on GSO for optimal management of renewable-based MGs

This article suggests a probabilistic framework based on Scenario production to take account the uncertainties in the optimal operation management of Micro Grids (MGs). The MG includes different renewable energy resources such as Wind Turbine (WT), Micro Turbine (MT), Photovoltaic (PV), Fuel Cell ( FC) and one battery as the storage device. The suggested framework is based on scenario generation and Roulette wheel mechanism to generate different scenarios for handling the uncertainties of different parameters. It uses normal distribution function as a probability distribution function of random parameters. The uncertainties which are considered in this paper are grid bid changes, load demand forecasting error and PV and WT output power productions. It is worthy to say that solving the MG problem for 24 hours of a day by considering different uncertainties and different constraints needs one powerful optimization method that can converge fast when it doesn’t fall in local optimum point. As a result, one Group Search Optimization (GSO) algorithm is introduced to prospect the total search space globally. The GSO algorithm is originated from group living of animals. Besides the GSO algorithm, one modification is also proposed for this algorithm. The proposed framework and method is implemented o one test grid-connected MG as a typical grid.

Numerical detection of the Gardner transition in a mean-field glass former.

Recent theoretical advances predict the existence, deep into the glass phase, of a novel phase transition, the so-called Gardner transition. This transition is associated with the emergence of a complex free energy landscape composed of many marginally stable sub-basins within a glass metabasin. In this study, we explore several methods to detect numerically the Gardner transition in a simple structural glass former, the infinite-range Mari-Kurchan model. The transition point is robustly located from three independent approaches: (i) the divergence of the characteristic relaxation time, (ii) the divergence of the caging susceptibility, and (iii) the abnormal tail in the probability distribution function of cage order parameters. We show that the numerical results are fully consistent with the theoretical expectation. The methods we propose may also be generalized to more realistic numerical models as well as to experimental systems.

A phenomenological contact model: Understanding the graft–tunnel interaction in anterior cruciate ligament reconstructive surgery

In this paper, we sought to expand the fidelity of a validated model of the anterior cruciate ligament reconstruction (ACL-R) procedure by incorporating a stick–slip contact model with linear pressure–overclosure relationship at the interface. The suggested model is characterized by three unknown parameters, friction coefficient, shear stress softening and contact stiffness. In the absence of any isolated experiments exploring the graft–tunnel interactions during an aggregate joint load, the calibration data used in this study are derived from a reported biomechanical study. A Bayesian calibration procedure was employed to find the unknown probability distribution function (PDF) of these contact parameters. Initially, the response surface approximations of the predicted graft forces from laxity test simulations was adopted to estimate the likelihood of noisy experimental data reported in the literature. Then, the wide domain of contact parameters was sampled sequentially based on the Marcov Chain Monte Carlo (MCMC) method with acceptance–rejection criteria to search for population of samples in significantly narrower domain of unknown parameters that are associated with the highest occurrence likelihood of noisy experimental data. Our simulations with calibrated contact parameters indicate that pre-tensioning applied at 30° of flexion leads to larger graft force after the joint is fully extended compared to the graft force when the same pre-tensioning force is applied at full extension. Moreover, regardless of the pre-tensioning force, the graft–tunnel contact pressure is larger when the fixation of the graft is performed at full extension, increasing with the pre-tensioning force.

Reducing uncertainty based on model fitness: Application to a reservoir model

Sensitivity and uncertainty analysis are important tools in the modelling process: they assign confidence to model results, can aid in focusing monitoring and preservation efforts, and can be used in model simplification. A weakness of global sensitivity and uncertainty analysis methodologies is the often subjective definition of prior parameter probability distributions, especially in data-poor areas. We apply Monte Carlo filtering in conjunction with quantitative variance-based global sensitivity and uncertainty analysis techniques to address this weakness and define parameter probability distributions in the absence of measured data. This general methodology is applied to a reservoir model of the Okavango Delta, Botswana. In addition to providing a methodology for setting prior parameter distributions, results show that the use of Monte Carlo filtering reduces model uncertainty and produces simulations that better represent the calibrated ranges. Thus, Monte Carlo filtering increases the accuracy and precision of parametric model uncertainty. Results also show that the most important parameters in our model are the volume thresholds, the reservoir area/volume coefficient, floodplain porosity, and the island extinction coefficient. The reservoir representing the central part of the wetland, where flood waters separate into several independent distributaries, is a keystone area within the model. These results identify critical areas and parameters for monitoring and managing, refine and reduce input/output uncertainty, and present a transferable methodology for developing parameter probability distribution functions, especially when using empirical models in data-scarce areas. Keywords : sensitivity analysis, uncertainty analysis, Monte Carlo filtering, reservoir model, Okavango Delta

A parallel gradient method for the determination of the design point in probabilistic analysis

In the probabilistic analysis of engineering systems, the design point denotes a particular set of input parameters where the system response is most likely to take a given design value. It provides important information on the system behaviour and its sensitivity to input parameters. The design point is determined from the joint probability distribution function (pdf) of input parameters. Mathematically, the problem is equivalent to an isoperimetric problem: find a stationary point of the joint pdf subject to the given value of the system response. The proposed method depends on the response and the joint pdf being parallel at the stationary point. This requires the projection of the pdf gradient to be zero on the hyperplane orthogonal to the response gradient. Therefore, the design point is found by moving iteratively on the response surface in the direction of a non-zero projection vector until this vector vanishes. Convergence of this process is intuitively guaranteed. The method can be implemented for any number of input parameters. An example application is presented which demonstrates finding the most probable design metocean conditions for a floating structure. Such a problem is part of the response based analysis of offshore systems, which provided the initial motivation for this work. References H. O. Madsen, S. Krenk and N. C. Lind. Methods of structural safety . Dover, 1986. http://store.doverpublications.com/0486445976.html . S. R. Winterstein. An introduction to structural reliability, FORM, and LRFD design. Technical report , 2011. https://dl.dropboxusercontent.com/u/3121325/lrfd/lrfd.pdf . S. Winterstein, T. C. Ude, C. A. Cornell, P. Bjerager and S. Haver. Environmental parameters for extreme response: inverse FORM with omission sensitivity. Structural Safety and Reliability, Proc., ICOSSAR `93 , A. A. Balkema, 1994. R. G. Standing, R. Eichaker, H. D. Lawes, B. Campbell and R. B. Corr. Benefits of applying response based design methods to deepwater FPSOs. Offshore Tech. Conf. , 2002. doi:10.4043/14232-MS . P. S. Tromans and L. Vanderschuren. Response based design conditions in the North Sea: application of a new method. Offshore Tech. Conf. , 1995. doi:10.4043/7683-MS . I. M. Gefland and S. V. Fomin. Calculus of variations . Dover, 2000. http://store.doverpublications.com/0486414485.html . J. Mathews and R. L. Walkers. Mathematical Methods of Physics . W. A. Benjamin, 1964. E. M. Bitner-Gregersen. Joint probabilistic description of combined seas. ASME Proc., Safety and Reliability , OMAE2005–67382:169–180, 2005. doi:10.1115/OMAE2005-67382 . N. Morgan. Marine technology reference book . Butterworth–Heinemann, 1990.

A method to reduce the computational requirement while assessing uncertainty of complex hydrological models

The quantification of uncertainty in the simulations from complex physically based distributed hydrologic models is important for developing reliable applications. The generalized likelihood uncertainty estimation method (GLUE) is one of the most commonly used methods in the field of hydrology. The GLUE helps reduce the parametric uncertainty by deriving the probability distribution function of parameters, and help analyze the uncertainty in model output. In the GLUE, the uncertainty of model output is analyzed through Monte Carlo simulations, which require large number of model runs. This induces high computational demand for the GLUE to characterize multi-dimensional parameter space, especially in the case of complex hydrologic models with large number of parameters. While there are a lot of variants of GLUE that derive the probability distribution of parameters, none of them have addressed the computational requirement in the analysis. A method to reduce such computational requirement for GLUE is proposed in this study. It is envisaged that conditional sampling, while generating ensembles for the GLUE, can help reduce the number of model simulations. The mutual relationship between the parameters was used for conditional sampling in this study. The method is illustrated using a case study of Soil and Water Assessment Tool (SWAT) model on a watershed in the USA. The number of simulations required for the uncertainty analysis was reduced by 90 % in the proposed method compared to existing methods. The proposed method also resulted in an uncertainty reduction in terms of reduced average band width and high containing ratio.

Testing the universality of the IMF with Bayesian statistics: young clusters

The universality of the stellar initial mass function (IMF) is tested using Bayesian statistics with a sample of eight young Galactic stellar clusters (IC 348, ONC, NGC 2024, NGC 6611, NGC 2264, $\rho$ Ophiuchi, Chameleon I, and Taurus). We infer the posterior probability distribution function (pPDF) of the IMF parameters when the likelihood function is described by a tapered power law function, a lognormal distribution at low masses coupled to a power law at higher masses, and a multi-component power law function. The inter-cluster comparison of the pPDFs of the IMF parameters for each likelihood function shows that these distributions do not overlap within the $1\sigma$ uncertainty level. Furthermore, the most probable values of the IMF parameters for most of the clusters deviate substantially from their values for the Galactic field stellar IMF. We also quantify the effects of taking into account the completeness correction as well as the uncertainties on the measured masses. The inclusion of the former affects the inferred pPDFs of the slope of the IMF at the low mass end while considering the latter affects the pPDFs of the slope of the IMF in the intermediate- to high mass regime. As variations are observed in all of the IMF parameters at once and for each of the considered likelihood functions, even for completeness corrected samples, we argue that the observed variations are real and significant, at least for the sample of eight clusters considered in this work. The results presented here clearly show that the IMF is not universal.

Bayesian System Reliability Assessment Method with Maximum Entropy as Prior Distribution

Bayesian system reliability assessment method with maximum entropy as prior distribution is developed in the present analysis. The system reliability assessment model is established, and the method and process of system reliability assessment are put forward. The method is based on the theory of information entropy. The probability density function and distribution function of system reliability parameters are given by using maximum entropy method. Prior distribution of Bayesian analysis method is the distribution function. Likelihood function is given by a small amount of field test data. According to Bayesian methods, posterior distribution is determined by using prior distribution and likelihood function. The point estimation and interval estimation of system reliability parameters are calculated in order to evaluate the system reliability. The method solves better the problem of system reliability assessment for small sample cases, and is an optimum method for reliability assessment of large and complex systems.

Probabilistic Optimal Power Flow with Weibull Probability Distribution Function of System Loading Using Percentiles Estimation

This article proposes a probabilistic optimal power flow for hour-ahead scheduling in a power system considering the statistical distribution function of system loading using the Weibull probability distribution function. In the proposed probabilistic optimal power flow, the deterministic optimal power flow problem is solved as the sub-problem in probabilistic optimal power flow and decomposed into a total operating cost minimization sub-objective, which is solved by successive quadratic programming and the real power loss minimization sub-objective, which is solved by successive linear programming. In the proposed method, the Weibull probability distribution function parameters of the optimal power flow variables are estimated from percentile values and evaluated by Akaike information criteria. The proposed probabilistic optimal power flow algorithm is tested on the IEEE 30-bus and IEEE 300-bus systems and compared to Monte Carlo simulation. The investigations show that the proposed probabilistic optimal power flow can successfully estimate the probability distribution function parameters of optimal power flow output variables considering the Weibull probability distribution function of system load with simple and minimal computational procedure. With the proposed estimation of the Weibull parameters method, the number of optimal power flow runs can be reduced substantially in the probabilistic optimal power flow process.

Optimal design of stimulus experiments for robust discrimination of biochemical reaction networks

Motivation: Biochemical reaction networks in the form of coupled ordinary differential equations (ODEs) provide a powerful modeling tool for understanding the dynamics of biochemical processes. During the early phase of modeling, scientists have to deal with a large pool of competing nonlinear models. At this point, discrimination experiments can be designed and conducted to obtain optimal data for selecting the most plausible model. Since biological ODE models have widely distributed parameters due to, e.g. biologic variability or experimental variations, model responses become distributed. Therefore, a robust optimal experimental design (OED) for model discrimination can be used to discriminate models based on their response probability distribution functions (PDFs).Results: In this work, we present an optimal control-based methodology for designing optimal stimulus experiments aimed at robust model discrimination. For estimating the time-varying model response PDF, which results from the nonlinear propagation of the parameter PDF under the ODE dynamics, we suggest using the sigma-point approach. Using the model overlap (expected likelihood) as a robust discrimination criterion to measure dissimilarities between expected model response PDFs, we benchmark the proposed nonlinear design approach against linearization with respect to prediction accuracy and design quality for two nonlinear biological reaction networks. As shown, the sigma-point outperforms the linearization approach in the case of widely distributed parameter sets and/or existing multiple steady states. Since the sigma-point approach scales linearly with the number of model parameter, it can be applied to large systems for robust experimental planning.Availability: An implementation of the method in MATLAB/AMPL is available at http://www.uni-magdeburg.de/ivt/svt/person/rf/roed.html.Contact: flassig@mpi-magdeburg.mpg.deSupplementary information: Supplementary data are are available at Bioinformatics online.

Assessing the impact of uncertainty on flood risk estimates with reliability analysis using 1-D and 2-D hydraulic models

Abstract. This paper addresses the use of reliability techniques such as Rosenblueth's Point-Estimate Method (PEM) as a practical alternative to more precise Monte Carlo approaches to get estimates of the mean and variance of uncertain flood parameters water depth and velocity. These parameters define the flood severity, which is a concept used for decision-making in the context of flood risk assessment. The method proposed is particularly useful when the degree of complexity of the hydraulic models makes Monte Carlo inapplicable in terms of computing time, but when a measure of the variability of these parameters is still needed. The capacity of PEM, which is a special case of numerical quadrature based on orthogonal polynomials, to evaluate the first two moments of performance functions such as the water depth and velocity is demonstrated in the case of a single river reach using a 1-D HEC-RAS model. It is shown that in some cases, using a simple variable transformation, statistical distributions of both water depth and velocity approximate the lognormal. As this distribution is fully defined by its mean and variance, PEM can be used to define the full probability distribution function of these flood parameters and so allowing for probability estimations of flood severity. Then, an application of the method to the same river reach using a 2-D Shallow Water Equations (SWE) model is performed. Flood maps of mean and standard deviation of water depth and velocity are obtained, and uncertainty in the extension of flooded areas with different severity levels is assessed. It is recognized, though, that whenever application of Monte Carlo method is practically feasible, it is a preferred approach.

Prediction of Georeferencing Precision of Pushbroom Scanner Images

The precision of georeferencing of pushbroom satellite images without using ground control points is estimated. The study includes what-if scenarios for various precisions of star trackers, GPS receivers, image acquisition timing, interior camera parameters, and digital elevation model (DEM). Moreover, the effects of satellite altitude and sensor positions on the pushbroom array are also taken into account. The probability distribution function of the input parameters is determined by considering the sensor characteristics, environmental effects, and assumptions. The precision of the georeferenced coordinates is estimated by Monte Carlo simulation and differential sensitivity analysis. The analyses provide the most probable georeferenced coordinates and their precision for a certain precision of sensors and imaging geometry. The study is conducted based on the BilSAT orbit and SPOT 5 and Landsat optical system characteristics. An Shuttle Radar Topography Mission (SRTM) DEM and a DEM obtained from stereo images are used in georeferencing.

Probabilistic Multicompartmental Model for Interpreting DGT Kinetics in Sediments

Extensive research has been performed on the use of the DIFS (DGT-Induced Fluxes in Soils and Sediments) model to interpret diffusive gradients in thin-film, or DGT, measurements in soils and sediments. The current report identifies some areas where the DIFS model has been shown to yield poor results and proposes a model to address weaknesses. In particular, two major flaws in the current approaches are considered: (i) many studies of accumulation kinetics in DGT exhibit multiple kinetic stages and (ii) several combinations of the two fitted DIFS parameters can yield identical results, leaving the question of how to select the 'best' combination. Previously, problem (i) has been addressed by separating the experimental data sets into distinct time segments. To overcome these problems, a model considering two types of particulate binding sites is proposed, instead of the DIFS model which assumed one single particulate pool. A probabilistic approach is proposed to fit experimental data and to determine the range of possible physical parameters using Probability Distribution Functions (PDFs), as opposed to single values without any indication of their uncertainty. The new probabilistic model, called DGT-PROFS, was tested on three different formulated sediments which mainly differ in the presence or absence of iron oxides. It was shown that a good fit can be obtained for the complete set of data (instead of DIFS-2D) and that a range of uncertainty values for each modeling parameter can be obtained. The interpretation of parameter PDFs allows one to distinguish between a variety of geochemical behaviors, providing useful information on metal dynamics in sediments.

Bayesian Nonparametric Estimation of Continuous Monotone Functions with Applications to Dose–Response Analysis

In this article, we consider monotone nonparametric regression in a Bayesian framework. The monotone function is modeled as a mixture of shifted and scaled parametric probability distribution functions, and a general random probability measure is assumed as the prior for the mixing distribution. We investigate the choice of the underlying parametric distribution function and find that the two-sided power distribution function is well suited both from a computational and mathematical point of view. The model is motivated by traditional nonlinear models for dose-response analysis, and provides possibilities to elicitate informative prior distributions on different aspects of the curve. The method is compared with other recent approaches to monotone nonparametric regression in a simulation study and is illustrated on a data set from dose-response analysis.