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Nonparametric discrimination among distributions on Euclidean space with continuous distribution functions by tolerance regions method, emphasizing errors probability distribution control
Statistical tools of uncertainty quantification can be used to assess the information content of measured observables with respect to present-day theoretical models, to estimate model errors and thereby improve predictive capability, to extrapolate beyond the regions reached by experiment, and to provide meaningful input to applications and planned measurements. To showcase new opportunities offered by such tools, we make a rigorous analysis of theoretical statistical uncertainties in nuclear density functional theory using Bayesian inference methods. By considering the recent mass measurements from the Canadian Penning Trap at Argonne National Laboratory, we demonstrate how the Bayesian analysis and a direct least-squares optimization, combined with high-performance computing, can be used to assess the information content of the new data with respect to a model based on the Skyrme energy density functional approach. Employing the posterior probability distribution computed with a Gaussian process emulator, we apply the Bayesian framework to propagate theoretical statistical uncertainties in predictions of nuclear masses, two-neutron dripline, and fission barriers. Overall, we find that the new mass measurements do not impose a constraint that is strong enough to lead to significant changes in the model parameters. In addition, the example discussed in this study sets the stage for quantifying and maximizing the impact of new measurements with respect to current modeling and guiding future experimental efforts, thus enhancing the experiment-theory cycle in the scientific method.
Conventional methods for uncertainty quantification are generally challenged in the 'tails' of probability distributions. This is specifically an issue for many climate observables since extensive sampling to obtain a reasonable accuracy in tail regions is especially costly in climate models. Moreover, the accuracy of spectral representations of uncertainty is weighted in favor of more probable ranges of the underlying basis variable, which, in conventional bases does not particularly target tail regions. Therefore, what is ideally desired is a methodology that requires only a limited number of full computational model evaluations while remaining accurate enough in the tail region. To develop such a methodology, we explore the use of surrogate models based on non-intrusive Polynomial Chaos expansions and Galerkin projection. We consider non-conventional and custom basis functions, orthogonal with respect to probability distributions that exhibit fat-tailed regions. We illustrate how the use of non-conventional basis functions, and surrogate model analysis, improves the accuracy of the spectral expansions in the tail regions. Finally, we also demonstrate these methodologies using precipitation data from CCSM simulations.
Purpose: To obtain probability distributions of various DNA damage types as a function of the incident electron kinetic energy. Methods: Using Geant4-DNA electron ionization cross-sections, we calculated path length distributions for electrons of energies between 10 eV and 1 MeV, defined as the length between two subsequent interactions. These path lengths were then convolved with probability distributions for the creation of same-strand damage, opposite-strand damage, clustered damage, isolated damage, and same DNA strand target damage. These probability distributions of DNA damage were obtained by a Monte Carlo routine calculating probabilities of interaction in DNA targets inside a nucleosome geometrical model. Results represent the probability of a secondary electron, initially created inside a DNA strand target, of undergoing its next interaction: (1) in the opposite strand (DSB), (2) in the same strand (SSB+), (3) in either the opposite or same-strand (clustered), (4) in the same DNA target (multiple-hit) or (5) more than 10 base pairs away (isolated). Results: Electrons with kinetic energy between 50 and 250 eV have a maximal probability of creating DSB, SSB+, clustered damage and multiple-hits in the same target The probabilities for these damage patterns have values of 2.5%, 4.3%, 6.7% and 5.4%, respectively. Isolated damage is most probable between 700 eV to 900 eV with a probability of 0.2%. Conclusion: We obtained DNA damage probability distributions as a function of electron incident energy. We showed that electrons with kinetic energies between 50 and 250 eV have the highest probability of producing complex forms of DNA damage (DSB, SSB+). We also showed that a double ionization within the same DNA target is the most frequent outcome occurring 5% of the time. It is expected that electron slowing down spectra can be convolved with our formalism to calculate source specific DNA damage patterns. Research grants from governments of Canada and Quebec. PP acknowledges partial support by the CREATE Medical Physics Research Training Network grant of the Natural Sciences and Engineering Research Council (Grant number: 432290)
The YDB impact hypothesis of Firestone et al. (2007) is so extremely improbable it can be considered statistically impossible in addition to being physically impossible. Comets make up only about 1% of the population of Earth-crossing objects. Broken comets are a vanishingly small fraction, and only exist as Earth-sized clusters for a very short period of time. Only a small fraction of impacts occur at angles as shallow as proposed by the YDB impact authors. Events that are exceptionally unlikely to take place in the age of the Universe are 'statistically impossible'. The size distribution of Earth-crossing asteroids is well-constrained by astronomical observations, DoD satellite bolide frequencies, and the cratering record. This distribution can be transformed to a probability density function (PDF) for the largest expected impact of the past 20,000 years. The largest impact of any kind expected over the period of interest is 250 m. Anything larger than 2 km is exceptionally unlikely (probability less than 1%). The impact hypothesis does not rely on any sound physical model. A 4-km diameter comet, even if it fragmented upon entry, would not disperse or explode in the atmosphere. It would generate a crater about 50 km in diameter with a transient cavity as deep as 10 km. There is no evidence for such a large, young crater associated with the YDB. There is no model to suggest that a comet impact of this size is capable of generating continental-wide fires or blast damage, and there is no physical mechanism that could cause a 4-km comet to explode at the optimum height of 500 km. The highest possible altitude for a cometary optimum height is about 15 km, for a 120-m diameter comet. To maximize blast and thermal damage, a 4-km comet would have to break into tens of thousands fragments of this size and spread out over the entire continent, but that would require lateral forces that greatly exceed the drag force, and would not conserve energy. Airbursts are decompression explosions in which projectile material reaches high temperature but not high pressure states. Meteoritic diamonds would be vaporized. Nanodiamonds at the YDB are not evidence for an airburst or for an impact.
Flow and transport through fractured geologic media often leads to anomalous (non-Fickian) transport behavior, the origin of which remains a matter of debate: whether it arises from variability in fracture permeability (velocity distribution), connectedness in the flow paths through fractures (velocity correlation), or interaction between fractures and matrix. Here we show that this uncertainty of distribution- versus correlation-controlled transport can be resolved by combining convergent and push-pull tracer tests because flow reversibility is strongly dependent on velocity correlation, whereas late-time scaling of breakthrough curves is mainly controlled by velocity distribution. We build on this insight, and propose a Lagrangian statistical model that takes the form of a continuous time random walk (CTRW) with correlated particle velocities. In this framework, velocity distribution and velocity correlation are quantified by a Markov process of particle transition times that is characterized by a distribution function and a transition probability. Our transport model accurately captures the anomalous behavior in the breakthrough curves for both push-pull and convergent flow geometries, with the same set of parameters. Thus, the proposed correlated CTRW modeling approach provides a simple yet powerful framework for characterizing the impact of velocity distribution and correlation on transport in fractured media.
Modern electro-optical instruments are typically designed with assemblies of optomechanical members that support optics such that alignment is maintained in service environments that include random vibration loads. This paper presents a nonlinear numerical analysis that calculates statistics for the peak lateral response of optics in an optomechanical sub-assembly subject to random excitation of the housing. The work is unique in that the prior art does not address peak response probability distribution for stationary random vibration in the time domain for a common lens-retainer-housing system with Coulomb damping. Analytical results are validated by using displacement response data from random vibration testing of representative prototype sub-assemblies. A comparison of predictions to experimental results yields reasonable agreement. The Type I Asymptotic form provides the cumulative distribution function for peak response probabilities. Probabilities are calculated for actual lens centration tolerances. The probability that peak response will not exceed the centration tolerance is greater than 80% for prototype configurations where the tolerance is high (on the order of 30 micrometers). Conversely, the probability is low for those where the tolerance is less than 20 micrometers. The analysis suggests a design paradigm based on the influence of lateral stiffness on the magnitude of the response.
The fluence of high-LET particles (HLP) with LET infinity H2O greater than 15 keV micrometers-1 in selected organs and tissues were measured with plastic nuclear track detectors using a life-size human phantom on the 9th Shuttle-Mir Mission (STS-91). The planar-track fluence of HLP during the 9.8-day mission ranged from 1.9 x 10(3) n cm-2 (bladder) to 5.1 x 10(3) n cm-2 (brain) by a factor of 2.7. Based on these data, a probability of HLP hits to a matured cell of each organ or tissue was roughly estimated for a 90-day ISS mission. In the calculation, all cells were assumed to be spheres with a geometric cross-sectional area of 500 micrometers2 and the cell-hit frequency from isotropic space radiation can be described by the Poisson-distribution function. As results, the probability of one or more than 1 hit to a single cell by HLP for 90 days ranged from 17% to 38%; that of two or more than 2 hits was estimated to be 1.3-8.2%. c2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.
Water confined in mesoporous silica plays a central role in its many uses ranging from gas sorption to nanoconfined chemical reactions. Here in this paper, the influence of pore diameter (2.5–5.4 nm) on water hydrogen bond (H-bond) dynamics in MCM41 and SBA15 mesoporous silicas is investigated using femtosecond infrared vibrational spectroscopy and molecular dynamics simulations on selenocyanate (SeCN-) anions dissolved in the pores. As shown recently, SeCN- spectral diffusion is a reliable probe of surrounding water H-bond structural motions. Additionally, the long CN stretch vibrational lifetime facilitates measurement of the full range of confined dynamics, which are much slower than in bulk water. The simulations shed light on quantitative details that are inaccessible from the spatially averaged observables. The dependence of SeCN- orientational relaxation and that of spectral diffusion on the distance from the silica interface are quantitatively described with an exponential decay and a smoothed step-function, respectively. The distance-dependence of both quantities is found to be independent of the diameter of the pores, and the spatial distribution of SeCN- is markedly non-uniform, reaching a maximum between the interface and the pore center. The results indicate that the commonly invoked two-state, or core–shell, model is a more appropriate description of spectral diffusion. Using these insights, we model the full time-dependence of the measured dynamics for all pore sizes and extract the “core” and “shell” dynamical correlation functions and SeCN- spatial probability distributions. The results are critically compared to those for water confined in reverse micelles.
1/s line intensity distribution function for molecular spectra representation using growth curves calculated for Lorentz lines
Continuous probability density functions and discrete probability mass functions are tabulated which maximize the differential entropy or absolute entropy, respectively, among all probability distributions with a given L sub p norm (i.e., a given pth absolute moment when p is a finite integer) and unconstrained or constrained value set. Expressions for the maximum entropy are evaluated as functions of the L sub p norm. The most interesting results are obtained and plotted for unconstrained (real valued) continuous random variables and for integer valued discrete random variables. The maximum entropy expressions are obtained in closed form for unconstrained continuous random variables, and in this case there is a simple straight line relationship between the maximum differential entropy and the logarithm of the L sub p norm. Corresponding expressions for arbitrary discrete and constrained continuous random variables are given parametrically; closed form expressions are available only for special cases. However, simpler alternative bounds on the maximum entropy of integer valued discrete random variables are obtained by applying the differential entropy results to continuous random variables which approximate the integer valued random variables in a natural manner. All the results are presented in an integrated framework that includes continuous and discrete random variables, constraints on the permissible value set, and all possible values of p. Understanding such as this is useful in evaluating the performance of data compression schemes.
Here we present diffraction data that yield the oxygen-oxygen pair distribution function, g{sub OO}(r) over the range 254.2–365.9 K. The running O-O coordination number, which represents the integral of the pair distribution function as a function of radial distance, is found to exhibit an isosbestic point at 3.30(5) Å. The probability of finding an oxygen atom surrounding another oxygen at this distance is therefore shown to be independent of temperature and corresponds to an O-O coordination number of 4.3(2). Moreover, the experimental data also show a continuous transition associated with the second peak position in g{sub OO}(r) concomitant with the compressibility minimum at 319 K.
Differential difference equations involving noncentral chi-square density and distribution functions for solving minimization problem in selection from multivariate normal populations
We recognize that carbon-containing products do not decay and release CO2 to the atmosphere instantaneously, but release that carbon over extended periods of time. For an initial production of a stock of carbon-containing product, we can treat the release as a probability distribution covering the time over which that release occurs. The probability distribution that models the carbon release predicts the amount of carbon that is released as a function of time. The use of a probability distribution in accounting for the release of carbon to the atmosphere realizes a fundamental shift from the idea that all carbon-containing products contribute to a single pool that decays in proportion to the size of the stock. Viewing the release of carbon as a continuous probabilistic process introduces some theoretical opportunities not available in the former paradigm by taking advantage of other fields where the use of probability distributions has been prevalent for many decades. In particular, theories developed in the life insurance industry can guide the development of pricing and payment structures for dealing with the costs associated with the oxidation and release of carbon. These costs can arise from a number of proposed policies (cap and trade, carbon tax, social cost of carbon, etc), but in the end they all result in there being a cost to releasing carbon to the atmosphere. If there is a cost to the emitter for CO2 emissions, payment for that cost will depend on both when the emissions actually occur and how payment is made. Here we outline some of the pricing and payment structures that are possible which result from analogous theories in the life insurance industry. This development not only provides useful constructs for valuing sequestered carbon, but highlights additional motivations for employing a probability distribution approach to unify accounting methodologies for stocks of carbon containing products.
Three-dimensional compressible Direct Numerical Simulation (DNS) data of freely propagating statistically planar and statistically stationary slot-jet turbulent premixed flames has been used to assess the accuracy of the isotropy-derived correction factors, which relate the two-dimensional projections of the different terms of the Favre-averaged scalar dissipation rate transport equation with their corresponding actual three-dimensional counterparts. The accuracy of these correction factors is assessed using both simplified and detailed chemistry-based DNS data, for a range of values of Karlovitz number Ka, heat release parameter tau, and turbulent Reynolds number Re-t. It is shown that the isotropic distribution of the probability density function (pdf) of the angle, phi, between the normal vectors of the measurement plane and of the flame surface provides a simple algebraic relation between the scalar dissipation rates evaluated in two and three dimensions (i.e. (N) over tilde (c) and (N) over tilde (2D)(c)), independent of the considered values of Ka, tau and Re-t. The isotropic relations between two-dimensional and three-dimensional counterparts of the curvature and propagation terms in the transport equation of (N) over tilde (c) are also found to work well for all the values of Ka, tau and Re-t considered here. However, the relation between the value obtained from two-dimensional projection and the true three-dimensional value for the strain rate term in the (N) over tilde (c) transport equation works well only for large values of Re-t and the reasons for this behaviour are explained in detail. It is found that the threshold value of Ret above which the assumption of isotropy yields an accurate relation between two-dimensional projection and three-dimensional values for the strain rate term of the dissipation rate transport equation depends on the regime of the prevailing combustion process.
Tri-n-butyl phosphate (TBP), a representative of neutral organophosphorous ligands, is an important extractant used in solvent extraction process for the recovery of uranium and plutonium from spent nuclear fuel. Microscopic pictures of TBP isomerism and its behavior in n-dodecane diluent were investigated utilizing MD simulations with previously optimized force field parameters for TBP and n-dodecane. Potential Mean Force (PMF) calculations on a single TBP molecule show seven probable TBP isomers. Radial Distribution Functions (RDF) of TBP suggests the existence of TBP trimers at high TBP concentrations in addition to dimers. 2D PMF calculations were performed to determine the angle and distance criteria for TBP trimers. The dimerization and trimerization constants of TBP in n-dodecane were obtained and match our own experimental values using FTIR technique. The new insights into the conformational behaviors of TBP molecule as a monomer and as part of an aggregate could greatly aid the understanding of the complexation between TBP and metal ions in solvent extraction system. The U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences funded the work performed by LXD.
Understanding the properties of random packings of solid objects is of critical importance to a wide variety of fundamental scientific and practical problems. The great majority of the previous works focused, however, on packings of spherical and sphere-like particles. We report the first detailed simulation and characterization of packings of non-overlapping cubic particles. Such packings arise in a variety of problems, ranging from biological materials, to colloids and fabrication of porous scaffolds using salt powders. In addition, packing of cubic salt crystals arise in various problems involving preservation of pavements, paintings, and historical monuments, mineral-fluid interactions, CO 2 sequestration in rock, and intrusion of groundwater aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We have developed a version of the random sequential addition algorithm to generate such packings, and have computed a variety of microstructural descriptors, including the radial distribution function, two-point probability function, orientational correlation function, specific surface, and mean chord length, and have studied the effect of finite system size and porosity on such characteristics. Here, the results indicate the existence of both spatial and orientational long-range order in the packing, which is more distinctive for higher packing densities.
This software generates environmental contours of extreme sea states using buoy observations of significant wave height and energy period or peak period. The code transforms these observations using principal component analysis (PCA) to create an uncorrelated representation of the data. The subsequent components are modeled using probability distributions and parameter fitting functions. The inverse first-order reliability method (I-FORM) is then applied to these models in order to generate an extreme event contour based on a given return period (i.e., 100 years).The subsequent contour is then transformed back into the original input space defined by the variables of interest in order to create an environmental contour of extreme sea states.