Probability Distribution Function Of Parameters Research Articles

Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence

ABSTRACT In this work, we find empirical evidence that the scale-dependent statistical properties of solar wind and magnetohydrodynamic (MHD) turbulence can be described in terms of a family of parametric probability distribution functions (PDFs) known as Normal Inverse Gaussian (NIG). Understanding these PDFs is one of the most important goals in turbulence theory, as they are inherently connected to the intermittent properties of solar wind turbulence. We investigate the properties of PDFs of Elsasser increments based on a large statistical sample from solar wind observations and high-resolution numerical simulations of MHD turbulence. In order to measure the PDFs and their corresponding properties, three experiments are presented: fast and slow solar wind for experimental data and a simulation of reduced MHD (RMHD) turbulence. Conditional statistics on a 23-yr-long sample of WIND data near 1 au and high-resolution pseudo-spectral simulation of steadily driven RMHD turbulence on a $2048^3$ mesh are used to construct scale-dependent PDFs. The empirical PDFs are fitted to NIG distributions, which depend on four free parameters. Our analysis shows that NIG distributions accurately capture the evolution of the PDFs, with scale-dependent parameters, from large scales characterized by a Gaussian distribution, turning to exponential tails within the inertial range and stretched exponentials at dissipative scales. We also show that empirically-measured NIG parameters exhibit well-defined scaling properties that are similar across the three empirical data sets, which may be indicative of universal behaviour.

A Two-Component Probability Distribution Function Describes the Mid-IR Emission from the Disks of Star-forming Galaxies

High-resolution JWST-MIRI images of nearby spiral galaxies reveal emission with complex substructures that trace dust heated both by massive young stars and the diffuse interstellar radiation field. We present high angular (0.″85) and physical resolution (20–80 pc) measurements of the probability distribution function (PDF) of mid-infrared (mid-IR) emission (7.7–21 μm) from 19 nearby star-forming galaxies from the PHANGS-JWST Cycle 1 Treasury. The PDFs of mid-IR emission from the disks of all 19 galaxies consistently show two distinct components: an approximately lognormal distribution at lower intensities and a high-intensity power law component. These two components only emerge once individual star-forming regions are resolved. Comparing with locations of H ii regions identified from Very Large Telescope/MUSE Hα mapping, we infer that the power-law component arises from star-forming regions and thus primarily traces dust heated by young stars. In the continuum-dominated 21 μm band, the power law is more prominent and contains roughly half of the total flux. At 7.7–11.3 μm, the power law is suppressed by the destruction of small grains (including PAHs) close to H ii regions, while the lognormal component tracing the dust column in diffuse regions appears more prominent. The width and shape of the lognormal diffuse emission PDFs in galactic disks remain consistent across our sample, implying a lognormal gas column density N(H) ≈ 1021 cm−2 shaped by supersonic turbulence with typical (isothermal) turbulent Mach numbers ≈5−15. Finally, we describe how the PDFs of galactic disks are assembled from dusty H ii regions and diffuse gas and discuss how the measured PDF parameters correlate with global properties such as star formation rate and gas surface density.

Bayesian calibration and fitting of nuclear thermal–hydraulic models by Markov chain Monte Carlo methods using the Gibbs sampler

This paper applies Bayesian methods to obtain the joint probability distribution function (PDF) of the unknown parameters of some of the correlation types used in thermal hydraulics, which are conditioned to a set of results of separate effect experiments. The problem is solved by Markov chain Monte Carlo methods using the Gibbs sampler. In this paper it is shown how to obtain the conditional PDFs for each one of the parameters, which define two typical correlations that appear in thermal–hydraulic problems, i) the heat transfer in the walls of a pipe and ii) the penetration length of steam discharged in a subcooled pool. The equations for the conditional probabilities of each one of the parameters are deduced and then used to build the Gibbs sampler which is then programmed in R. The PDFs are obtained for each one of the parameters and from these the maximum a posteriori of each one of the parameters is obtained. The new correlations obtained in this way are compared with the experimental data and show satisfactory performance for the case of the penetration of steam in a subcooled pool. Also, we study two cases one with a small number of experimental data and another one with a large number, it is observed that if the number of parameters is small the Markov chain converges fast to the region of high probability of the parameters for both cases with a small number of data and with large number of data, but if the number of parameters increases then the convergence for the case with small number of data becomes poor, however the convergence continuous being very fast for the case with large number of experimental data.

Bridge Damage Detection Using Ambient Loads by Bayesian Hypothesis Testing for a Parametric Subspace of an Autoregressive Model

This study explores a change detection method in modal properties to automate and generalize in-service damage detection for vibration-based structural health monitoring of bridges. The noisy conditions caused by ambient loading pose difficulty for in-service damage detection because the load-induced noise often masks the difference in the modal properties. The proposed method directly converts measured time series into a simplified anomaly indicator robust against load-induced noise. This study adopts a vector autoregressive model to represent the vibration of bridges. Bayesian inference produces a posterior probability distribution function of the model parameters. Principal component analysis extracts a subspace comparable to the modal properties in the model parameters. Bayesian hypothesis testing quantifies anomalies in the extracted subspace. The feasibility of the proposed method is assessed with vibration data from field experiments conducted on an actual steel truss bridge. The field experiment includes damage severing the truss members. The modal frequencies and mode shapes estimated from the principal component analysis correspond well to earlier reported results. The proposed damage detection method successfully indicated all damage considered in the experiment.

Machine Learning Classification and Prediction of Wind Estimation Using Artificial Intelligence Techniques and Normal PDF

Estimating wind energy at a specific wind site depends on how well the real wind data in that area can be represented using an appropriate distribution function. In fact, wind sites differ in the extent to which their wind data can be represented from one region to another, despite the widespread use of the Weibull function in representing the wind speed in various wind locations in the world. In this study, a new probability distribution model (normal PDF) was tested to implement wind speed at several wind locations in Jordan. The results show high compatibility between this model and the wind resources in Jordan. Therefore, this model was used to estimate the values of the wind energy and the extracted energy of wind turbines compared to those obtained by the Weibull PDF. Several artificial intelligence techniques were used (GA, BFOA, SA, and a neuro-fuzzy method) to estimate and predict the parameters of both the normal and Weibull PDFs that were reflected in conjunction with the actual observed data of wind probabilities. Afterward, the goodness of fit was decided with the aid of two performance indicators (RMSE and MAE). Surprisingly, in this study, the normal probability distribution function (PDF) outstripped the Weibull PDF, and interestingly, BFOA and SA were the most accurate methods. In the last stage, machine learning was used to classify and predict the error level between the actual probability and the estimated probability based on the trained and tested data of the PDF parameters. The proposed novel methodology aims to predict the most accurate parameters, as the subsequent energy calculation phases of wind depend on the proper selection of these parameters. Hence, 24 classifier algorithms were used in this study. The medium tree classifier shows the best performance from the accuracy and training time points of view, while the ensemble-boosted trees classifier shows poor performance regarding providing correct predictions.

Precisely modeling offshore jacket structures considering model parameters uncertainty using Bayesian updating

Digital twin (DT) is recognized as a promising tool to perform reliability analysis for offshore platforms. An open issue in developing the DT reliability analysis tool is to establish a high-fidelity highly-precise virtual model for the offshore platform of interest, where the virtual model is able to update its model parameters using actual sensory measurements to eliminate/reduce the effect of parameters uncertainty. To address this challenging issue, this paper proposes a Markov Chain Monte Carlo (MCMC)-based Bayesian updating method to build the requested virtual model for an offshore jacket platform. In this new method, the natural frequencies and mode shapes of the offshore jacket platform, extracted both from the structural dynamic responses of a finite element (FE) model and the sensory measurements in the physical system, are used to construct the likelihood function of the Bayesian inference. Then, the posterior probability distribution function (PDF) of the uncertain parameters is derived under the assumption of uniform prior distribution, which will be used to generate the Markov Chains by implementing the Metropolis-Hastings sampling. Hence, the most probable values of the uncertain parameters can be obtained from the Markov Chains for updating the FE model. A numerical study demonstrates high effectiveness of the proposed model updating method, even with high-level measurement noise and model uncertainty.

The eROSITA Final Equatorial-Depth Survey (eFEDS)

Context.The eROSITA Final Equatorial Depth Survey (eFEDS), observed with eROSITA ahead of its planned 4-yr all-sky survey, is the largest contiguous-field X-ray survey at present. It yielded a large sample of X-ray sources with very rich multiband photometric and spectroscopic coverage.Aims.We present here the eFEDS active galactic nuclei (AGN) catalog and the eROSITA X-ray spectral properties of the eFEDS sources.Methods.Using a Bayesian method, we performed a systematic X-ray spectral analysis for all the eFEDS sources. We adopted multiple spectral models, including single-component power-law or hot-plasma models and double-component models of a power law plus soft excess. We investigated the capacity of eROSITA X-ray spectra for constraining AGN spectral shapes through a detailed analysis of the posterior parameter probability distribution functions. Hierarchical Bayesian modeling was used to recover the spectral parameter distribution of the sample. The source fluxes and luminosities were measured from the posterior of the spectral fitting.Results.The eFEDS AGN catalog (22 079 sources) comprises ~80% of the eFEDS point sources. Despite a large number of faint sources, our spectral fitting provides reasonable measurements of spectral shapes and intrinsic luminosities for a majority of the sources. Because of sample selection bias, this AGN catalog is dominated by X-ray unobscured sources, with an obscured (logNH> 21.5) fraction of 8%; the power-law emission of the hot corona is also relatively soft, with a typical slope of 2.0. For type-I AGN, the X-ray emission is well correlated with the UV emission with the usual anticorrelation between the X-ray to UV spectral slopeαOXand the UV luminosity. The X-ray spectral properties measured with various models are presented for all the eFEDS sources.

SPInS, a pipeline for massive stellar parameter inference

Context. Stellar parameters are required in a variety of contexts, ranging from the characterisation of exoplanets to Galactic archaeology. Among them, the age of stars cannot be directly measured, while the mass and radius can be measured in some particular cases (e.g. binary systems, interferometry). More generally, stellar ages, masses, and radii have to be inferred from stellar evolution models by appropriate techniques. Aims. We have designed a Python tool named SPInS. It takes a set of photometric, spectroscopic, interferometric, and/or asteroseismic observational constraints and, relying on a stellar model grid, provides the age, mass, and radius of a star, among others, as well as error bars and correlations. We make the tool available to the community via a dedicated website. Methods. SPInS uses a Bayesian approach to find the probability distribution function of stellar parameters from a set of classical constraints. At the heart of the code is a Markov chain Monte Carlo solver coupled with interpolation within a pre-computed stellar model grid. Priors can be considered, such as the initial mass function or stellar formation rate. SPInS can characterise single stars or coeval stars, such as members of binary systems or of stellar clusters. Results. We first illustrate the capabilities of SPInS by studying stars that are spread over the Hertzsprung-Russell diagram. We then validate the tool by inferring the ages and masses of stars in several catalogues and by comparing them with literature results. We show that in addition to the age and mass, SPInS can efficiently provide derived quantities, such as the radius, surface gravity, and seismic indices. We demonstrate that SPInS can age-date and characterise coeval stars that share a common age and chemical composition. Conclusions. The SPInS tool will be very helpful in preparing and interpreting the results of large-scale surveys, such as the wealth of data expected or already provided by space missions, such as Gaia, Kepler, TESS, and PLATO.

Halo Counts-in-cells for Cosmological Models with Different Dark Energy

Abstract We examine the counts-in-cells (CiC) probability distribution functions (PDFs) that describe dark matter halos in the Dark Energy Universe Simulations (DEUS). We describe the measurements between redshifts z = 0 to z = 4 on both linear and nonlinear scales. The best fits of the gravitational quasi-equilibrium distribution (GQED), the negative binomial distribution (NBD), the Poisson-Lognormal distribution (PLN), and the Poisson-Lognormal distribution with a bias parameter (PLNB) are compared to simulations. The fits agree reasonably consistently over a range of redshifts and scales. To distinguish quintessence (RPCDM) and phantom (wCDM) dark energy from Λ dark energy, we present a new method that compares the model parameters of the CiC PDFs. We find that the mean and variance of the halo CiC on 2–25h −1 Mpc scales between redshifts 0.65 < z < 4 show significant percentage differences for different dark energy cosmologies. On 15–25 h −1 Mpc scales, the g parameter in NBD, ω parameter in PLN, and b and C b parameters in PLNB show larger percentage differences for different dark energy cosmologies than on smaller scales. On 2–6 h −1 Mpc scales, the kurtosis and the b parameter in the GQED show larger percentage differences for different dark energy cosmologies than on larger scales. For cosmologies explored in the DEUS, the percentage differences between these statistics for the RPCDM and wCDM dark energy cosmologies relative to ΛCDM generally increases with redshift from a few percent to significantly larger percentages at z = 4. Applying our method to simulations and galaxy surveys can provide a useful way to distinguish among dark energy models and cosmologies in general.

A New Distributed Energy Management Strategy for Smart Grid With Stochastic Wind Power

In this article, a distributed energy management strategy is proposed for smart grid with stochastic wind power. The different Weibull probability distribution function (PDF) of wind power during different operation intervals, coupled with ramp rate limits of thermal generators across operation intervals, is considered in this operation cost optimization problem. Under multiagent system communication framework, we first decompose the problem into subproblems associated to operation interval. For each operation interval, parameters of Weibull PDF are estimated by maximum likelihood method according to wind speed obtained by wind speed forecast model of committed wind turbine. Then, a consensus-based alternating direction method of multipliers (ADMM) is proposed to get the optimal operation plan for all committed generators, where Armijo line search algorithm is employed to adaptively obtain the step size in the iterations. The simulation results show that the proposed consensus-based ADMM outperforms distributed projected gradient method in previous study on energy management with wind power; the proposed distributed strategy is valid for short-term energy management for smart grid with stochastic wind power.

Time domain force localization and reconstruction based on hierarchical Bayesian method

In engineering practice, localization and reconstruction of external excitations is a tough problem in absence of prior knowledge. Traditional regularization approaches require assigning regularization parameters and shape parameters before implementing the reconstruction. However, a poor selection of these parameters generally leads to a poor reconstruction. In this paper a time domain hierarchical Bayesian method is proposed to locate and reconstruct the external excitations with automatic selection of parameters. The entire method is performed in two stages. First based on the property of posterior probability distribution function of shape parameters, a novel non-force criterion is proposed to determine the non-force locations quickly. Then based on the determined force locations, a reduced-dimension problem is constructed and the posterior distributions of parameters are sampled by a Metropolis-within-Gibbs sampler with nested blocking technique. Both numerical simulations and laboratory experiment of a cantilever beam under various load conditions are carried out to validate the proposed method.

Bayesian Methods for Radiometric Calibration in Motion Picture Encoding Workflows

The accurate estimation of a camera response function (CRF) allows for proper encoding of camera exposures into motion picture post-production workflows like the Academy Color Encoding System (ACES), helping minimize noncreative manual adjustments. Although there are well-known standard CRFs implemented in typical video camera workflows, motion picture camera workflows and new high dynamic range (HDR) workflows have introduced new standard CRFs, as well as custom and proprietary CRFs. Current methods to estimate this function rely on the use of test charts, restrictive exposure and/or lighting conditions, or assume a simplistic model of the function’s shape. All of these methods become problematic and tough to fit into motion picture production and post-production workflows, where the use of test charts and varying camera or scene setups becomes impractical. We propose a method initially based on the work of Lin, Gu, Yamazaki, and Shum that considers edge color mixtures in an image or image sequence that are affected by the nonlinearity introduced by a CRF. This feature is then used in a Bayesian framework to estimate a posterior probability distribution function of the CRF model parameters approximated by a Markov Chain Monte Carlo (MCMC) algorithm, allowing for a more robust description of the CRF over methods like maximum likelihood (ML) and maximum a posteriori (MAP).

Uncertainty analysis on natural circulation characteristics under ocean conditions

The stability of natural circulation is more susceptible to the uncertainties involved in the thermal-hydraulic process because the driving force is relatively small. Especially in the marine nuclear power plants, the complex ocean motions will introduce more uncertainties in the natural circulation operation. In this work, the joint probability distribution function of wave parameters was investigated, which provided a more realistic simulation of the ocean conditions. On that basis, it was applied to generate ocean motion parameters as uncertain input via the Metropolis-Hastings algorithm. A thermal-hydraulic code that is capable of simulating system behaviors under ocean conditions was developed to propagate the uncertainties. Additionally, the sensitivity analysis was carried out to identify the parameters that have a significant impact on natural circulation characteristics. Results show that the intense ocean motions will introduce large uncertainties in the natural circulation operation, especially at the startup stage. The rolling amplitude contributes the most on the natural circulation flow rate and the coolant temperature, while the heaving motions have less influence. Uncertainties introduced by the ocean motions will damage the natural circulation ability and stability, which need further considerations during the design optimization.

Risk-averse optimal operation of Multiple-Energy Carrier systems considering network constraints

Integrated Multiple-Energy Carrier (MEC) systems with electricity and natural gas infrastructure systems offer distinctive opportunities to enhance the efficiency and the flexibility of energy supply. Efficient operation of these interdependent infrastructure systems faces several challenges corresponding to the risk associated with the uncertainty in the environmental parameters. In this paper, a risk-based framework for energy management of interconnected energy hubs in MEC systems is proposed to minimize the expected system operation cost. The conditional value at risk (CVaR) method is used to quantify the risk associated with uncertainties in the electrical and thermal loads, and the real-time price of electricity. The probability distribution function of the uncertain parameters and auto regressive integrated moving average (ARIMA) models are used to generate scenarios to represent the uncertainty in the system parameters. The proposed model is formulated as a mixed integer linear programming (MILP) problem and solved using CPLEX solver. The developed case studies show the risk-averse and risk-taker operator strategies to demonstrate the merits of the proposed risk-based scheduling over the risk-neutral operation scheme. It is further investigated that how the expected operation cost depends on the risk measures in the risk-averse operation. Also, sensitivity analyses are conducted to assess the energy procurement scheduling based on the proposed approach. Furthermore, the effect of risk parameter on managing the congestion in electricity and natural gas networks is examined.

Markov chain Monte Carlo-based Bayesian method for structural model updating and damage detection

This paper proposes a Bayesian method for structural model updating and damage detection using modal data. A recently developed Markov chain Monte Carlo algorithm is adopted to handle the model updating problem. The proposed Bayesian method focuses on calculation of the posterior probability distribution function of uncertain model parameters. In addition to the most probable values of the uncertain parameters, the associated uncertainties can be calculated with consideration of the effects of both the modeling error and the measurement noise. An experimental case study was carried out with a shear building model under laboratory conditions to study the identifiability of the model-updating problem following the proposed Bayesian method. The results demonstrate the change in the posterior probability distribution function of the uncertain parameters with the amount of measured information. It also demonstrates the ability of the proposed method to handle unidentifiable problems. The proposed Bayesian method is then applied for structural damage detection by calculating the probability distribution of the extent of damage to various structural components. To demonstrate the proposed Bayesian damage-detection method, ambient vibration tests were carried out on a 2-story steel frame with bolted connections. Joint damage was simulated by loosening some bolts at the target beam-column connection. The model-updating results show that the uncertainty associated with the rotational stiffness of the steel joints was very high, rendering the problem almost unidentifiable. Although the problem is almost unidentifiable, the calculated probability distribution of the damage extent can still locate the damaged joint and estimate the damage extent (i.e., the percentage reduction in rotational stiffness) together with the associated uncertainty.

Parametric estimation of dispersive viscoelastic layered media with application to structural health monitoring

We present a sequential Bayesian estimation method to estimate the material parameters that govern the one-dimensional propagation of shear waves through continuous, layered, viscoelastic solids. While the proposed estimation method is generic, namely, can be applied to waveform inversion problems that satisfy the above conditions, we here employ it for system identification of building structures. We approximate the linear-elastic response of building structures subjected to low-amplitude earthquake base excitations by a multilayer dispersive shear beam model with Kelvin-Voigt material subjected to vertically propagating shear waves. Utilizing the proposed sequential Bayesian estimation method, we sequentially update the probability distribution function of the unknown parameters to reduce the discrepancies between the estimated and measured frequency response functions. We next verify and validate the performance of the proposed estimation method and investigate the limitations of the presented structural system identification approach using two case studies. In the first case study, we use the simulated structural response of a three-dimensional 52-story building model subjected to bi-directional low-amplitude ground shakings. We estimate the frequency-dependent phase velocity and damping ratio, as well as the mass distribution along the building height. Then, we verify the structural damage detection and localization capabilities of the presented system identification approach by comparing the wave model parameters estimated from simulated response of undamaged and damaged structural models. In the second case study, we use data measured from a shake table experiment on a full-scale five-story reinforced concrete building specimen, where the estimated wave model parameters capture the progressive structural damage in the test specimen. The validation studies suggest that the sequential Bayesian estimation method based on viscoelastic dispersive wave propagation can be used for system and damage identification of building structures.

Probability-interval hybrid uncertainty analysis for structures with both aleatory and epistemic uncertainties: a review

Traditional structural uncertainty analysis is mainly based on probability models and requires the establishment of accurate parametric probability distribution functions using large numbers of experimental samples. In many actual engineering problems, the probability distributions of some parameters can be established due to sufficient samples available, whereas for some parameters, due to the lack or poor quality of samples, only their variation intervals can be obtained, or their probability distribution types can be determined based on the existing data while some of the distribution parameters such as mean and standard deviation can only be given interval estimations. This thus will constitute an important type of probability-interval hybrid uncertain problem, in which the aleatory and epistemic uncertainties both exist. The probability-interval hybrid uncertainty analysis provides an important mean for reliability analysis and design of many complex structures, and has become one of the research focuses in the field of structural uncertainty analysis over the past decades. This paper reviews the four main research directions in this area, i.e., uncertainty modeling, uncertainty propagation analysis, structural reliability analysis, and reliability-based design optimization. It summarizes the main scientific problems, technical difficulties, and current research status of each direction. Based on the review, this paper also provides an outlook for future research in probability-interval hybrid uncertainty analysis.

Use of the Principles of Maximum Entropy and Maximum Relative Entropy for the Determination of Uncertain Parameter Distributions in Engineering Applications

The determination of the probability distribution function (PDF) of uncertain input and model parameters in engineering application codes is an issue of importance for uncertainty quantification methods. One of the approaches that can be used for the PDF determination of input and model parameters is the application of methods based on the maximum entropy principle (MEP) and the maximum relative entropy (MREP). These methods determine the PDF that maximizes the information entropy when only partial information about the parameter distribution is known, such as some moments of the distribution and its support. In addition, this paper shows the application of the MREP to update the PDF when the parameter must fulfill some technical specifications (TS) imposed by the regulations. Three computer programs have been developed: GEDIPA, which provides the parameter PDF using empirical distribution function (EDF) methods; UNTHERCO, which performs the Monte Carlo sampling on the parameter distribution; and DCP, which updates the PDF considering the TS and the MREP. Finally, the paper displays several applications and examples for the determination of the PDF applying the MEP and the MREP, and the influence of several factors on the PDF.

Towards an accurate real-time locator of infrasonic sources

Infrasonic signals propagate from an atmospheric source via media with stochastic and fast space-varying conditions. Hence, their travel time, the amplitude at sensor recordings and even manifestation in the so-called “shadow zones” are random. Therefore, the traditional least-squares technique for locating infrasonic sources is often not effective, and the problem for the best solution must be formulated in probabilistic terms. Recently, a series of papers has been published about Bayesian Infrasonic Source Localization (BISL) method based on the computation of the posterior probability density function (PPDF) of the source location, as a convolution of a priori probability distribution function (APDF) of the propagation model parameters with likelihood function (LF) of observations. The present study is devoted to the further development of BISL for higher accuracy and stability of the source location results and decreasing of computational load. We critically analyse previous algorithms and propose several new ones. First of all, we describe the general PPDF formulation and demonstrate that this relatively slow algorithm might be among the most accurate algorithms, provided the adequate APDF and LF are used. Then, we suggest using summation instead of integration in a general PPDF calculation for increased robustness, but this leads us to the 3D space-time optimization problem. Two different forms of APDF approximation are considered and applied for the PPDF calculation in our study. One of them is previously suggested, but not yet properly used is the so-called “celerity-range histograms” (CRHs). Another is the outcome from previous findings of linear mean travel time for the four first infrasonic phases in the overlapping consecutive distance ranges. This stochastic model is extended here to the regional distance of 1000 km, and the APDF introduced is the probabilistic form of the junction between this travel time model and range-dependent probability distributions of the phase arrival time picks. To illustrate the improvements in both computation time and location accuracy achieved, we compare location results for the new algorithms, previously published BISL-type algorithms and the least-squares location technique. This comparison is provided via a case study of different typical spatial data distributions and statistical experiment using the database of 36 ground-truth explosions from the Utah Test and Training Range (UTTR) recorded during the US summer season at USArray transportable seismic stations when they were near the site between 2006 and 2008.

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.