Monte Carlo Statistical Method Research Articles

TH‐E‐BRE‐01: A 3D Solver of Linear Boltzmann Transport Equation Based On a New Angular Discretization Method with Positivity for Photon Dose Calculation Benchmarked with Geant4

Purpose:The Linear Boltzmann Transport Equation (LBTE) solved through statistical Monte Carlo (MC) method provides the accurate dose calculation in radiotherapy. This work is to investigate the alternative way for accurately solving LBTE using deterministic numerical method due to its possible advantage in computational speed from MC.Methods:Instead of using traditional spherical harmonics to approximate angular scattering kernel, our deterministic numerical method directly computes angular scattering weights, based on a new angular discretization method that utilizes linear finite element method on the local triangulation of unit angular sphere. As a Result, our angular discretization method has the unique advantage in positivity, i.e., to maintain all scattering weights nonnegative all the time, which is physically correct. Moreover, our method is local in angular space, and therefore handles the anisotropic scattering well, such as the forward‐peaking scattering. To be compatible with image‐guided radiotherapy, the spatial variables are discretized on the structured grid with the standard diamond scheme. After discretization, the improved sourceiteration method is utilized for solving the linear system without saving the linear system to memory. The accuracy of our 3D solver is validated using analytic solutions and benchmarked with Geant4, a popular MC solver.Results:The differences between Geant4 solutions and our solutions were less than 1.5% for various testing cases that mimic the practical cases. More details are available in the supporting document.Conclusion:We have developed a 3D LBTE solver based on a new angular discretization method that guarantees the positivity of scattering weights for physical correctness, and it has been benchmarked with Geant4 for photon dose calculation.

Influence of the optical system and anatomic points on computer-assisted total knee arthroplasty

BackgroundFor over a decade, computer-assisted orthopaedic surgery for total knee arthroplasty has been accepted as ensuring accurate implant alignment in the coronal plane. HypothesisWe hypothesised that lack of accuracy in skeletal landmark identification during the acquisition phase and/or measurement variability of the infrared optical system may limit the validity of the numerical information used to guide the surgical procedure. MethodsWe built a geometric model of a navigation system, with no preoperative image acquisition, to simulate the stages of the acquisition process. Random positions of each optical reflector center and anatomic acquisition point were generated within a sphere of predefined diameter. Based on the virtual geometric model and navigation process, we obtained 30,000 simulations using the Monte Carlo statistical method then computed the variability of the anatomic reference frames used to guide the bone cuts. Rotational variability (α, β, γ) of the femoral and tibial landmarks reflected implant positioning errors in flexion-extension, valgus-varus, and rotation, respectively. ResultsTaking into account the uncertainties pertaining to the 3D infrared optical measurement system and to anatomic point acquisition, the femoral and tibial landmarks exhibited maximal alpha (flexion-extension), beta (valgus-varus), and gamma (axial rotation) errors of 1.65° (0.9°); 1.51° (0,98°), and 2.37° (3.84°), respectively. Variability of the infrared optical measurement system had no significant influence on femoro-tibial alignment angles. ConclusionThe results of a Monte Carlo simulation indicate a certain level of vulnerability of navigation systems for guiding position in rotation, contrasting with robustness for guiding sagittal and coronal alignments. Level of evidenceLevel IV.

Size distribution of supernova remnants and the interstellar medium: the case of M 33

The size distribution of supernova remnants (SNRs) can help to clarify the various aspects of their evolution and interaction with the interstellar medium (ISM). Since the observed samples of SNRs are a collection of objects with very different ages and origin that evolve in different conditions of the ISM, statistical Monte Carlo methods can be used to model their statistical distributions. Based on very general assumptions on the evolution, we have modeled samples of SNRs at various initial and environmental conditions, which were then compared with observed collections of SNRs. In the evolution of SNRs the pressure of the ISM is taken into account, which determines their maximum sizes and lifetimes. When comparing the modeled and observed distributions, it is very important to have homogeneous observational data free from selection effects. We found that a recently published collection of SNRs in M33 (Long et al. 2010, ApJS,187,495) satisfies this requirement if we select the X-ray SNRs with hardness ratios in a limited range of values. An excellent agreement between distributions of this subset of SNRs and the subset of modeled SNRs was reached for a volume filling-factor of the warm phase of the ISM (partly ionized gas with $n_{\rm H}\sim 0.2-0.5~ \rm {cm}^{-3}; T \sim 8000-10000~K $) in M33 of $\sim\ 90%$. The statistical distributions constructed in this way, which reproduce practically all the statistical properties of observed SNRs, allowed us to obtain one of the important parameters of M33: the birthrate is one SNR every $ {140} - {150}$ yr, and the total number of SNRs with a shock Mach number $M_{s} \geq 2$ is larger than $\sim 1000$.

A hierarchical conditional autoregressive model for colorectal cancer survival data

In this article, we propose a Bayesian hierarchical linear mixed model to cancer data with geographical characteristics. The spatial effects are captured via a conditional autoregressive (CAR) model. The survival model is introduced to analyze the survival pattern of colorectal cancer based on geographical factors and patients' disease conditions. We propose a special case of the classic Cox model with Weibull hazard as well as a cure rate model. A CAR prior is used to capture the spatial effects, which are determined based on counties the survival subjects belong to. The computation is done via Gibbs sampling. The ratio‐of‐uniforms method is used to sample from a nonstandard conditional posterior density, and the adaptive rejection method is used to sample from log‐concave densities. The model is applied to the Colon & Rectum Cancer incidences in Iowa from the Surveillance, Epidemiology, and End Results (SEER) database. Computation is implemented in Intel Fortran on Linux platform. WIREs Comput Stat 2014, 6:37–44. doi: 10.1002/wics.1283This article is categorized under: Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC) Statistical and Graphical Methods of Data Analysis > Reliability, Survivability, and Quality Control

A Framework for Quantitative Assessment of Impacts Related to Energy and Mineral Resource Development

Natural resource planning at all scales demands methods for assessing the impacts of resource development and use, and in particular it requires standardized methods that yield robust and unbiased results. Building from existing probabilistic methods for assessing the volumes of energy and mineral resources, we provide an algorithm for consistent, reproducible, quantitative assessment of resource development impacts. The approach combines probabilistic input data with Monte Carlo statistical methods to determine probabilistic outputs that convey the uncertainties inherent in the data. For example, one can utilize our algorithm to combine data from a natural gas resource assessment with maps of sage grouse leks and piñon-juniper woodlands in the same area to estimate possible future habitat impacts due to possible future gas development. As another example: one could combine geochemical data and maps of lynx habitat with data from a mineral deposit assessment in the same area to determine possible future mining impacts on water resources and lynx habitat. The approach can be applied to a broad range of positive and negative resource development impacts, such as water quantity or quality, economic benefits, or air quality, limited only by the availability of necessary input data and quantified relationships among geologic resources, development alternatives, and impacts. The framework enables quantitative evaluation of the trade-offs inherent in resource management decision-making, including cumulative impacts, to address societal concerns and policy aspects of resource development.

Discrete shape modeling for skin model representation

Nowadays, the management of product geometrical variations during the whole product development process is an important issue for companies’ competitiveness. During the design phase, geometric functional requirements and tolerances are derived from the design intent. Furthermore, the manufacturing and measurement stages are two main geometric variations generators according to the two well-known axioms of manufacturing imprecision and measurement uncertainty. GeoSpelling as the basis of the geometrical product specification standard enables a comprehensive modeling framework and an unambiguous language to describe geometric variations covering the overall product lifecycle thanks to a set of concepts and operations based on the fundamental concept of the “Skin Model.” In contrast, only few research studies have focused on the skin model representation and simulation. The skin model as a discrete shape model is the main focus of this work. We investigate here discrete shape and variability modeling fundamentals, Markov Chain Monte Carlo simulation techniques and statistical shape analysis methods to represent, simulate, and analyze skin models. By means of a case study based on a cross-shaped sheet metal part, the results of the skin model simulations are shown here, and the performances of the simulations are described.

Performance-Based Reliability Design for Deep Foundations Using Monte Carlo Statistical Methods (DFI Student Paper Competition 2012)

Deep foundation designs for service limit state are still deterministic in the current AASHTO LRFD Specifications. To address this deficiency, a performance-based reliability design methodology is developed using the Monte Carlo statistical techniques. In the proposed methodology, the design criteria are defined in terms of the allowable displacement. The spatial variability of soil parameters is considered in the proposed methodology by modeling soil parameters as random fields. Failure is defined as the event that the induced displacement exceeds the limiting displacement. The probability of failure by Monte Carlo approach is the ratio of the number of unsatisfactory performance events to the sample size. Three numerical examples are given to illustrate the application of the proposed methodology for laterally loaded and axially loaded drilled shafts, respectively. The spatial variability and correlation of soil properties were shown to exert significant influences on the foundation design.

The convolution statistical model of the breakdown voltage in nitrogen at 20 mbar

Results of the breakdown voltage density distribution of the diode filled with nitrogen at 20 mbar pressure are presented in this paper. These distributions are determined experimentally with the linearly rising voltage applied to the gas diode. The voltage increasing rate is determined in range of 0.1 up to 100 V/s. Every particular distribution is established on the base of the 1000 successive and independent measurements of the breakdown voltage under the identical experimental conditions. These distributions are then compared with the theoretical convolution statistical model. The convolution model is based on the presumption that the time delay consists of the statistical time delay, characterized with the Rayleigh distribution and the breakdown formative time, with the Gaussian distribution. The parameters of the theoretical distributions are determined applying the statistical Monte Carlo method to obtain the best-fitting of experimental distributions. The results show the good agreement of the theoretically determined density distributions with the experimental ones. The improvement is achieved especially on the “left tail” of the distribution, on the side of the smallest detected breakdown voltage, for which, the formative time standard deviation is responsible. This confirmed the applicability of the theoretical convolution statistical model for the description of breakdown voltage distributions.

Prediction Reliability of a Statistical Methodology for Gas Turbine Prognostics

The performance of gas turbines degrades over time and, as a consequence, a decrease in gas turbine performance parameters also occurs, so that they may fall below a given threshold value. Therefore, corrective maintenance actions are required to bring the system back to an acceptable operating condition. In today’s competitive market, the prognosis of the time evolution of system performance is also recommended, in such a manner as to take appropriate action before any serious malfunctioning has occurred and, as a consequence, to improve system reliability and availability. Successful prognostics should be as accurate as possible, because false alarms cause unnecessary maintenance and nonprofitable stops. For these reasons, a prognostic methodology, developed by the authors, is applied in this paper to assess its prediction reliability for several degradation scenarios typical of gas turbine performance deterioration. The methodology makes use of the Monte Carlo statistical method to provide, on the basis of the recordings of past behavior, a prediction of future availability, i.e., the probability that the considered machine or component can be found in the operational state at a given time in the future. The analyses carried out in this paper aim to assess the influence of the degradation scenario on methodology prediction reliability, as a function of a user-defined threshold and minimum value allowed for the parameter under consideration. A technique is also presented and discussed, in order to improve methodology prediction reliability by means a correction factor applied to the time points used for methodology calibration. The results presented in this paper show that, for all the considered degradation scenarios, the prediction error is lower than 4% (in most cases, it is even lower than 2%), if the availability is estimated for the next trend, while it is not higher than 12%, if the availability is estimated five trends ahead. The application of a proper correction factor allows the prediction errors after five trends to be reduced to approximately 5%.

Reliability‐based design of axially loaded drilled shafts using Monte Carlo method

SUMMARYThis paper presents a performance‐based reliability design methodology for axially loaded drilled shaft foundations using the Monte Carlo statistical methods. The performance criteria for an axially loaded drilled shaft are defined in terms of the drilled shaft head displacement. The load transfer method is used as the computational model for solving the nonlinear soil–foundation interaction problem and to predict the load–settlement curve at the drilled shaft head. The input to the computational model and the model error are treated as random variables. Particularly, soil properties such as soil strength parameters are modeled as random fields to account for soil spatial variability. Random field samples are generated using local averaging subdivision according to the prescribed statistical descriptors, including mean, variance, and correlation length. Given performance‐based acceptance criteria, the probability of failure can be evaluated for the prescribed load effects. A computer program is developed to facilitate the computation of the load–displacement curves of the drilled shaft head by using the commonly adopted load transfer concepts (t–z curves and q–w curves for shaft side and toe, respectively). Two design examples are presented, one for uplift and the other for compression, to illustrate the application of the developed performance‐based reliability design methodology. One of the important observations from the examples is that the computed probability of failure can be sensitive to the spatial variations of soil strengths characterized by the correlation structures. It is strongly recommended that spatial variation of soil properties be considered in the reliability‐based foundation design. Copyright © 2012 John Wiley & Sons, Ltd.

A semiparametric Bayesian approach to Wiener system identification

We consider a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process (GP) model for the static nonlinearity. The GP model is a flexible model that can describe different types of nonlinearities while avoiding making strong assumptions such as monotonicity. We derive an inferential method based on recent advances in Monte Carlo statistical methods, known as Particle Markov Chain Monte Carlo (PMCMC). The idea underlying PMCMC is to use a particle filter (PF) to generate a sample state trajectory in a Markov chain Monte Carlo sampler. We use a recently proposed PMCMC sampler, denoted particle Gibbs with backward simulation, which has been shown to be efficient even when we use very few particles in the PF. The resulting method is used in a simulation study to identify two different Wiener systems with non-invertible nonlinearities.

Development of a Statistical Methodology for Gas Turbine Prognostics

To optimize both production and maintenance, from both a technical and an economical point of view, it would be advisable to predict the future health condition of a system and of its components, starting from field measurements taken in the past. For this purpose, this paper presents a methodology, based on the Monte Carlo statistical method, which aims to determine the future operating state of a gas turbine. The methodology allows the system future availability to be estimated, to support a prognostic process based on past historical data trends. One of the most innovative features is that the prognostic methodology can be applied to both global and local performance parameters, as, for instance, machine specific fuel consumption or local temperatures. First, the theoretical background for developing the prognostic methodology is outlined. Then, the procedure for implementing the methodology is developed and a simulation model is set up. Finally, different degradation-over-time scenarios for a gas turbine are simulated and a sensitivity analysis on methodology response is carried out, to assess the capability and the reliability of the prognostic methodology. The methodology proves robust and reliable, with a prediction error lower than 2%, for the availability associated with the next future data trend.

Experimental estimation of the surface roughness distribution parameters in nanochannels

A statistical model of surface roughness is developed to calculate the molecular flows in nanosystems. In this model, surface asperities are represented by a set of flat microareas connected by edges with each other and having normals that differ from the normal to the mean level. A Solver PRO-M atomic force microscope is used to measure the following two parameters of the microscopic roughness of a hard disk: the slope along a scan line and the asperity height. A large experimental sample from the measured values of these parameters is used to obtain a distribution function density for the angle of inclination and conditional distributions (with parameters dependent on this angle) for the asperity height and the area of the triangle formed by the height and the sides of the angle. The latter conditional exponential distribution turns out to be more convenient for calculating random quantities. The results can be employed to simulate boundary conditions when calculating molecular flows by statistical Monte Carlo methods and to estimate the properties of new materials for protective surface coatings in the nanosystems containing gas flows.

Taxonomic analysis of marine phytoplankton

Samples of sea water contain phytoplankton taxa in varying amounts, and marine scientists are interested in the relative abundance of each taxa. Their relative biomass can be ascertained indirectly by measuring the quantity of various pigments using high performance liquid chromatography. However, the conversion from pigment to taxa is mathematically non trivial as it is a positive matrix factorisation problem where both matrices are unknown beyond the level of initial estimates. The prior information on the pigment to taxa conversion matrix is used to give the problem a unique solution. An iteration of two non-negative least squares algorithms gives satisfactory results. Some sample analysis of data indicates prospects for this type of analysis. An alternative more computationally intensive approach using Bayesian methods is discussed. References Efron, B., and Tibirani, R. J., 1993. An introduction to the Bootstrap, Chapman and Hall, New York. Hastings, W. K., 1970. Monte Carlo Sampling Methods Using Markov Chains and Their Applications. Biometrika, 57:97--109. doi:10.1093/biomet/57.1.97 Lawson C. L., and Hanson R. L. 1995. Solving Least Squares Problems, SIAM, Philadelphia, Ch 23. Lee D. D., and Seung H. S. 2001. Algorithms for Non-negative Matrix Factorization, Advances in Neural Information Processing Systems, 13:556--562. http://luthuli.cs.uiuc.edu/ daf/courses/Optimization/Papers/lee01algorithms.pdf Mackey, M. D., Mackey, D. J., Higgins, H. W. and Wright, S. W., 1996. CHEMTAX---A program for estimating class abundances from chemical markers: application to HPLC measurement of phytoplankton, Marine Ecology - Progress Series, 144:266--283. doi:10.3354/meps144265 van den Meersche, K., Soetaert, K., and Middleburg, J. J., 2008. A Bayesian computational estimator for microbial taxonomy based on biomarkers, Limnol. Oceanogr. Methods 6: 190--199. doi:10.4319/lom.2008.6.190 Robert, C. P. and Casella, G., 2004. Monte Carlo statistical methods, Springer, New York. Roberts, G. O. and Rosenthal, J. S., 2001. Optimal Scaling for Various Metropolis--Hastings Algorithms, Statistical Science 16:351--367. doi:10.1214/ss/1015346320 Wikipedia, 2011. Basic linear algebra subprograms, http://en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprograms Wright, S. W., van den Enden, R. L., Pearce, I., Davidson, A. T. Scott, F. J., and Westwood, K. J., 2010. Phytoplankton community structure and stocks in the Southern Ocean (30-80E) determined by CHEMTAX analysis of HPLC pigment signatures Deep-Sea Research II 57:758--778. doi:10.1016/j.dsr2.2009.06.015

Radio frequency identification anti-collision test based on Markov chain model

The anti-collision theory of adaptive Q algorithm and radio frequency identification(RFID) communication timing sequence were analyzed. Then a Markov chain model for simulating multi-tag identification process was established based on the following three definitions. The definitions are identifying efficiency, identifying speed and data state(Q, n), where n is the total remanent tag number, and Q is the slot count. The identifying efficiency and speed were obtained from the model using Monte Carlo statistical method. A software-defined radio program was built to simulate the collision of RFID which could quantify the identifying efficiency and speed effectively. The consistency of model simulation result and test data proves the validity of the model and test method.

Measuring Order in Contact-Poled Organic Electrooptic Materials with Variable-Angle Polarization-Referenced Absorption Spectroscopy (VAPRAS)

Organic nonlinear electrooptical (ONLO) chromophores must be acentrically ordered for the ONLO material to have electrooptic (EO) activity. The magnitude of the order is characterized by the acentric order parameter, <cos(3) β>, where β is the major Euler angle between the main axis of the chromophore and the poling field which imposes the acentric order. The acentric order parameter, which is difficult to measure directly, is related to the centrosymmetric order parameter, defined as <P(2)> = ½(3<cos(2) β>-1), through the underlying statistical distribution. We have developed a method to determine centrosymmetric order of the ONLO chromophores when the order is low (i.e., <P(2)> < 0.1). We have extended the method (begun by Graf et al. J. Appl. Phys. 1994, 75, 3335.) based on the absorption of light to determine the centrosymmetric order parameter induced by a poling field on a thin film sample of ONLO material. We find that the order parameters, analyzed by two different methods, are similar and also consistent with theoretical estimates from modeling of the system using coarse-grained Monte Carlo statistical mechanical methods.

Contrast improvement by selecting ballistic-photons using polarization gating

In this paper a new approach to improve contrast in optical subsurface imaging is presented. The method is based on time-resolved reflectance and selection of ballistic photons using polarization gating. Numerical studies with a statistical Monte Carlo method also reveal that weakly scattered diffuse photons can be eliminated by employing a small aperture and that the contrast improvement strongly depends on the single-scattering phase function. A possible experimental setup is discussed in the conclusions.

Development of robust fixture locating layout for machining workpieces

A fixture is a device that secures and holds a workpiece to maintain its orientation and position during a manufacturing process, so that the desired workpiece quality can be achieved. Although many researchers have worked on locating accuracy analysis and tolerance analysis towards the automated generation of locating schemes for fixtures, most of them have focused on part localization, while feature errors due to geometrical inaccuracies and dimensional deviations during the mass production are usually neglected. In the current paper, a study on fixture locating layout is carried out with the robust design approach by combining the Taguchi method and the Monte-Carlo statistical method in order to increase the quality of final machining workpieces, so that the layout can be robust and insensitive to the errors. The Taguchi method is employed to study the locating effect of the locator's position at different levels and the Monte-Carlo method is applied to simulate the variation of coordinates of the locating contact points. A case study comprising drilling a hole on a prismatic workpiece is also carried out. The paper shows that the errors on the locators and workpiece have significant effects on the features to be machined and thus these variations should not be ignored.

Modeling of optical radiation energy distribution in plant tissue

A three-dimensional mathematical model of interactions of optical radiation with plant tissue taking into account its structural inhomogeneity, spectral properties, and the effects of fluorescence is constructed. The developed model is implemented using the statistical Monte Carlo method for the Henyey-Greenstein phase function. The dependence of differential backscattering and fluorescence coefficients on the concentration of photosynthetic pigments (chlorophylls) is numerically studied. It is demonstrated that numerical characteristics agree with results of physical experiment. The approximate solution based on the expansion of the diffusion and fluorescence radiation fluxes into a series in terms of a small parameter is found. This expansion makes it possible to calculate the field of backscattered radiation with satisfactory accuracy and to qualitatively correctly describe the experimentally observed dependences of the fluorescence coefficient in the region of high chlorophyll concentration.

Theory-Inspired Development of Organic Electro-optic Materials

Correlated time-dependent density functional theory (TDDFT) quantum mechanical and pseudo-atomistic Monte Carlo (PAMC) statistical mechanical methods have been used to assist in the understanding of and to guide the improvement of organic electro-optic (OEO) materials, prepared by electric field poling of pi-electron chromophore-containing materials near their glass transition temperature. Theoretical treatment of the effects of dielectric permittivity and optical frequency on molecular (chromophore) first hyperpolarizabilities has been carried out as well as the analysis of the influence of spatially anisotropic intermolecular electrostatic interactions on the poling-induced noncentrosymmetric order of chromophores. Three classes of OEO materials have been considered in correlated theoretical and experimental investigations: (1) traditional chromophore/polymer composite materials, (2) chromophores covalently incorporated into polymers, dendrimers, and dendronized polymers, and (3) recently discovered materials consisting of chromophores incorporated into chromophore-containing host materials. This latter class of materials is referred to as binary chromophore organic glasses (BCOGs). These BCOGs exhibit exceptional electro-optic activity because of a combination of high chromophore number density, the effect of high dielectric permittivity on molecular first hyperpolarizability, and improved acentric order arising from the intermolecular electrostatic interactions among the two types of chromophores. The electrical conductivity of materials can also influence achievable electro-optic activity, and thin metal oxide buffer layers, introduced to limit charge injection, can significantly improve poling efficiency. Chromophore order can also be influenced, in some cases, by novel processing techniques, such as laser-assisted electric field poling. Thermal and photostability are important parameters for practical application of materials and have been improved dramatically in recent times. Diels-Alder and fluorovinyl ether cycloaddition reactions have been used to elevate final material glass transition temperatures to above 200 degrees C. Photostability is dominated by the photoactivation of singlet oxygen and subsequent attack on electro-optic chromophores. Photostability can be improved by more than 4 orders of magnitude by chromophore modification and material packaging.