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The maximum likelihood estimation is a general estimation procedure. It is often compared to estimation procedures like the ordinary least squares regression or generalized method of moments, to name a few. We discuss some basics about the maximum likelihood estimation, its advantages and disadvantages, and provide an example application to a gamma distribution function.
The maximum likelihood estimation is a general estimation procedure. It is often compared to estimation procedures like the ordinary least squares regression or generalized method of moments, to name a few. We discuss some basics about the maximum likelihood estimation, its advantages and disadvantages, and provide an example application to a gamma distribution function.
The switching kinetics of RF-magnetron reactively sputtered ~200 nm thick Zn 1-x Mg x O (ZMO) ferroelectric thin films with x = 0.41 and x = 0.27 prepared on Pt/Ti/SiO 2 /Si substrates were studied at applied fields near the coercive fields, ranging from 3.4 to 5.1 MV cm -1 , and at temperatures ranging from room temperature to 100 °C. Polarization reversal in ZMO followss the Kolmogorov-Avrami-Ishibashi kinetics model for nucleation and growth at applied fields near the coercive field, and obeys an individual column switching (ICS) model at higher applied fields required to switch most of the spontaneous polarization. Switching in the high-applied field regime can be described using either the Gaussian or inverse gamma distribution functions depending on the mole fraction of magnesium in the film. The switching current transients of the high-Mg content ZMO film (x = 0.41) are always bi-modal, whereas the low Mg composition film (x = 0.27) is described by the Gaussian distribution initially following wake-up, and becomes bimodal with continued cycling. Rayleigh-like behavior of the dielectric constant revealed a simultaneous increase of the irreversible and decrease of the reversible contributions to the dielectric constant, which was ascribed to an increase in the density of mobile domain walls with cycling and resulted in faster switching.
In the cold dark matter scenario, galactic dark matter halos are populated with a large number of smaller subhalos. Previous work has shown that dark matter annihilations in subhalos can generate a distinctive, non-Poisson signal in the gamma-ray photon counts probability distribution function (PDF). Here we show that the gamma-ray PDF also carries information about the velocity dependence of the dark matter annihilation cross section. After calculating the PDF assuming s-wave and Sommerfeld-enhanced annihilation, we perform a mock data analysis to illustrate how current and future observations can constrain the microphysics of the dark matter annihilation. We find that, with current Fermi data, and assuming a dark matter annihilation cross section roughly at the limit of current bounds from annihilation in dwarf spheroidal galaxies, one can potentially distinguish the non-Poissonian fluctuations expected from dark matter annihilation in subhalos from Poisson sources, as well as from dark matter models with an incorrect velocity-dependence. Here we explore how robust these results are to assumptions about the modeling of the galactic gamma-ray background, but further work is needed to determine the impact of realistic astrophysical source populations on our results. We also point out a four-parameter degeneracy between the velocity dependence of the dark matter annihilation, the minimum subhalo mass, the power law index of the subhalo mass function, and the normalization of the dark matter signal. This degeneracy can be broken with priors from N-body simulations or from observational constraints on the subhalo mass function.
Precipitation is a key process within the shallow cloud lifecycle. The Cloud System Evolution in the Trades (CSET) campaign included the first deployment of a 94 GHz Doppler radar and 532 nm lidar. Despite a larger sampling volume, initial mean radar/lidar retrieved rain rates (Schwartz et al. 2019) based on the upward-pointing remote sensor datasets are systematically less than those measured by in-situ precipitation probes in the cumulus regime. Subsequent retrieval improvements produce rainrates that compare better to in-situ values, but still underestimate. Retrieved shallow cumulus drop sizes can remain too small and too few, with an overestimated shape parameter narrowing the raindrop size distribution too much. Three potential causes for the discrepancy are explored: the gamma functional fit to the dropsize distribution, attenuation by rain and cloud water, and an underaccounting of Mie dampening of the reflectivity. A truncated exponential fit may represent the dropsizes below a showering cumulus cloud more realistically, although further work would be needed to fully evaluate the impact of a different dropsize representation upon the retrieval. The rain attenuation is within the measurement uncertainty of the radar. Mie dampening of the reflectivity is shown to be significant, in contrast to previous stratocumulus campaigns with lighter rain rates, and may be difficult to constrain well with the remote measurements. An alternative approach combines an a priori determination of the dropsize distribution width based on the in-situ data with the mean radar Doppler velocity and reflectivity. This can produce realistic retrievals, although a more comprehensive assessment is needed to better characterize the retrieval errors.
Abstract Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge–Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.
We consider statistical characteristics of laser interference recorded in the focal region of an IR optoelectronic observation system under field and out-of-field illumination by laser radiation. The applicability of the analytical description of experimental histograms of the laser interference signal distribution at the photodetector array output using the probability density approximations in the form of gamma distribution and the Gaussian function is substantiated. It is shown that the specific form of these functions is determined both by the mean value (mathematical expectation) of the interference signal and by the characteristic parameter M, which depends on the statistical properties of the laser interference. (laser applications and other issues in quantum electronics)
The candidate An(III)/Ln(III) separation ligand hexa-n-octylnitrilo-triacetamide (HONTA) was irradiated under envisioned SELECT (Solvent Extraction from Liquid waste using Extractants of CHON-type for Transmutation) process conditions (n-dodecane/0.1 M HNO 3 ) using a solvent test loop in conjunction with cobalt-60 gamma irradiation. The extent of HONTA radiolysis and complimentary degradation product formation was quantified by HPLC-ESI-MS/MS. Further, the impact of HONTA radiolysis on process performance was evaluated by measuring the change in 243 Am and 154 Eu distribution ratios as a function of absorbed gamma dose. HONTA was found to decay exponentially with increasing dose, affording a dose coefficient of d = (4.48 ± 0.19) × 10 –3 kGy –1 . Multiple degradation products were detected by HPLC-ESI-MS/MS with dioctylamine being the dominant quantifiable species. Both 243 Am and 154 Eu distribution ratios exhibited an induction period of ~70 kGy for extraction (0.1 M HNO 3 ) and back-extraction (4.0 M HNO 3 ) conditions, after which both values decreased with absorbed dose. The decrease in distribution ratios was attributed to a combination of the destruction of HONTA and ingrowth of dioctylamine, which is capable of interfering in metal ion complexation. Furthermore, the loss of HONTA with absorbed gamma dose was predominantly attributed to its reaction with the n-dodecane radical cation (R˙ + ). These R˙ + reaction kinetics were measured for HONTA and its 241 Am and 154 Eu complexes using picosecond pulsed electron radiolysis techniques. All three second-order rate coefficients (k) were essentially diffusion limited in n-dodecane indicating a significant reaction pathway: k(HONTA + R˙ + ) = (7.6 ± 0.8) × 10 9 M –1 s –1 , k(Am(HONTA) 2 + R˙ + ) = (7.1 ± 0.7) × 10 10 M –1 s –1 , and k(Eu(HONTA) 2 + R˙ + ) = (9.5 ± 0.5) × 10 10 M –1 s –1 . HONTA-metal ion complexation afforded an order-of-magnitude increase in rate coefficient. Additional nanosecond time-resolved measurements showed that both direct and indirect HONTA radiolysis yielded the short-lived (< 100 ns) HONTA radical cation and a second long-lived (µs) species identified as the HONTA triplet excited state. The latter was confirmed by a series of oxygen quenching picosecond pulsed electron measurements, affording a quenching rate coefficient of k( 3 [HONTA]* + O 2 ) = 2.2 × 10 8 M –1 s –1 . Overall, both the HONTA radical cation and triplet excited state are important precursors to the suite of measured HONTA degradation products.
Context. Because Cepheid variable stars have long been used as a cosmic benchmark for scaling distances in our Galaxy and beyond, the accuracy of stellar evolution models for Cepheids have wide-reaching effects. However, our understanding of the dynamics in the interiors of these physically complex stars is limited. Aims. Our goal is to provide a detailed multi-dimensional picture of hydrodynamic convection and convective boundary mixing in the interior of Cepheids. Methods. Using the Modules for Experiments in Stellar Astrophysics (MESA), we studied the structure of intermediate-mass stars that cross the instability strip. Then, we performed two-dimensional hydrodynamic simulations of six stars with the fully compressible Multidimensional Stellar Implicit Code (MUSIC). Our simulations did not model the radial pulsations but focused on the interior structure of this family of stars. We developed and applied a new statistical analysis to examine convection and convective boundary mixing in the interior of these stellar simulations. Results. Based on a grid of MESA models, we demonstrated that a common structure for intermediate mass Cepheids includes an interior convective shell as well as a thin outer convective envelope. Using the extreme value theory approach to analyze our MUSIC simulation data, we found that overshooting above the convective shell fills the space between these convectively unstable layers. We developed a new statistical analysis that provides a clearer picture of how overshooting fills this layer; it also allowed us to formulate a detailed comparison between overshooting above and below the convective shell. Our analysis effectively decomposes the overshooting layer into two layers: a weak overshooting layer and a strong overshooting layer. Statistically, this is accomplished by decomposing the strongly non-Gaussian probability density function into a mixture of gamma distributions. Using our mixture model, we showed that the ratio of overshooting lengths above and below the convective shell depends directly on the radial extent of the convective shell as well as its depth in the star. We proposed a new form for the diffusion coefficient that addresses the need for overlapping overshooting layers between convective shells. We introduced the idea of a “super-mixing layer” where overshooting from both the convective shell and the convective envelope results in efficient mixing and could be viewed as merging the two adjacent convective zones.
Here, the validity of the gamma distribution in describing the neutron number probability distribution function for both isolated and coupled multiplying assemblies when constrained to reproduce the true mean and variance is investigated in lumped geometry by numerical comparison with kinetic Monte Carlo simulations. The mean and variance are obtained from numerical solution of moment equations constructed from the relevant forward Master equation with assembly coupling coefficients obtained from a view factor method. Numerical results for a two-group, two coupled assemblies model, with static and dynamic reactivity insertion, show that except for subcritical assemblies, the gamma distribution well-approximates the number distribution. Differences in the fast and thermal neutron population shapes are explained in terms of effective source strengths due to downscatter and coupling.
This project aimed to investigate a new approach to bulk microphysics schemes. We identified the need to move beyond the fixed structural assumptions and approximation of existing schemes. For example, most schemes assume some functional form for the rain drop size distribution (e.g. a gamma or exponential distribution). Most schemes then evolve some number of statistical moments of that distribution via various processes, such as evaporation, sedimentation, collision-coalescence, and collisional breakup. These microphysical processes, in turn, are typically some fixed functional form derived from either some other (more detailed) model, or using some single-particle rates that are then integrated over the size distribution. While these process rate functions may have some free parameters to adjust, in other cases doing so is impossible (e.g. when the process rates are analytical or piecewise solutions to some target function). Our goal was to build and test a scheme that assumed no size distribution form, and used a series of power laws as the basis for the process rates. Moments of the size distribution would be predicted, but no underlying size distribution would be specified. Any number or choice of size distribution moments could be used, and any number of power laws could be employed to model the process rates. Thus, our approach is flexible, and can seamlessly scale across levels of complexity, as demanded by the data. Bayesian inference then would provide the formalism to estimate the model parameters and structure, allowing for robust uncertainty quantification. We proposed testing this framework in idealized simulations using bin microphysical schemes as a data source. We also proposed using real data to inform our microphysics scheme, and also that we would integrate our scheme into WRF.
While the total cover of broken cloud fields can in principle be obtained from one-dimensional measurements, the cloud size distribution normally differs between two-dimensional (area) and one-dimensional retrieval (chord length) methods. In this study, we use output from high-resolution Large Eddy Simulations to generate a transfer function between the two. We retrieve chord lengths and areas for many clouds, and plot the one as a function of the other, and vice versa. We find that the cloud area distribution conditional on the chord length behaves like a gamma distribution with well-behaved parameters, with a mean μ=1.1L and a shape parameter β=L−0.645. Using this information, we are able to generate a transfer function that can adjust the chord length distribution so that it comes much closer to the cloud area distribution. Our transfer function improves the error in predicting the mean cloud size, and is performs without strong biases for smaller sample sizes. However, we find that the method is still has difficulties in accurately predicting the frequency of occurrence of the largest cloud sizes.
We present a framework for $q_T$ resummation at N 3 LL+NNLO accuracy for arbitrary color-singlet processes based on a factorization theorem in SCET. Our implementation CuTe-MCFM is fully differential in the Born kinematics and matches to large-$q_T$ fixed-order predictions at relative order $α^2_s$. It provides an efficient way to estimate uncertainties from fixed-order truncation, resummation, and parton distribution functions. In addition to W ± , Z and H production, also the diboson processes $\gamma\gamma$, Z$\gamma$, ZH and W ± H are available, including decays. We discuss and exemplify the framework with several direct comparisons to experimental measurements as well as inclusive benchmark results. In particular, we present novel results for $\gamma\gamma$ and Z$\gamma\gamma$ at N 3 LL+NNLO and discuss in detail the power corrections induced by photon isolation requirements.
Jet fragmentation transverse momentum ( j T ) distributions are measured in proton-proton (pp) and proton-lead (p-Pb) collisions at s NN = 5 . 02 TeV with the ALICE experiment at the LHC. Jets are reconstructed with the ALICE tracking detectors and electromagnetic calorimeter using the anti- k T algorithm with resolution parameter R = 0 . 4 in the pseudorapidity range |η| < 0 . 25. The j T values are calculated for charged particles inside a fixed cone with a radius R = 0 . 4 around the reconstructed jet axis. The measured j T distributions are compared with a variety of parton-shower models. Herwig and P ythia 8 based models describe the data well for the higher j T region, while they underestimate the lower j T region. The j T distributions are further characterised by fitting them with a function composed of an inverse gamma function for higher j T values (called the “wide component”), related to the perturbative component of the fragmentation process, and with a Gaussian for lower j T values (called the “narrow component”), predominantly connected to the hadronisation process. The width of the Gaussian has only a weak dependence on jet transverse momentum, while that of the inverse gamma function increases with increasing jet transverse momentum. For the narrow component, the measured trends are successfully described by all models except for Herwig. For the wide component, Herwig and PYTHIA 8 based models slightly underestimate the data for the higher jet transverse momentum region. These measurements set constraints on models of jet fragmentation and hadronisation.
Multi-modal Free-moving Data Fusion (MFDF) is a python package that implements low and high-level functionality for performing qualitative and quantitative gamma-ray imaging analyses. Such analyses enable the use of gamma-ray spectrometers and/or imagers to estimate the distribution and quantity of radiological materials in an environment. Free-moving 3D imaging requires additional information about the 3D trajectory and orientation of the system, however all of these methods can be applied to 2D static gamma-ray imaging as well. The package includes methods (MLEM and MAP) for distributed source reconstruction, methods (PSL) for point source reconstruction, the ability to ingest and utilize quantitative detector response functions (for absolute analyses), and the ability to perform 3D estimation of dose-rates from quantitative reconstructions.
Transverse single-spin asymmetries in the semi-inclusive deep-inelastic production of isolated photons ($\gamma$SIDIS), $A_{UT}^{\gamma {\rm SIDIS}}$, provide an unprecedented opportunity to extract the quark-gluon-quark correlators $F_{FT}(x,x')$ and $G_{FT}(x,x')$ point-by-point in their full support $x,x'$. We utilize realistic models for these functions, based on input from the Sivers transverse momentum dependent parton distribution function and imposing constraints from the $d_2$ matrix element calculated in lattice QCD, in order to provide numerical estimates for $A_{UT}^{\gamma {\rm SIDIS}}$ at the Electron-Ion Collider (EIC). We thoroughly explore the EIC phase space in order to isolate in which regions the asymmetry can be sizable, finding it can be as much as $10\%$ or larger for certain kinematics. Given that $F_{FT}(x,x')$ and $G_{FT}(x,x')$ are basically unknown, $A_{UT}^{\gamma {\rm SIDIS}}$ will be an important future measurement to learn about multi-parton correlations in the nucleon.
The radiolytic stability of the highly selective ligands CyMe 4 BTBP and CyMe 4 BTPhen against ionizing gamma radiation was studied in 1-octanol solution. CyMe 4 BTBP and CyMe 4 BTPhen are important extractants for a potential treatment of used nuclear fuel. They were studied under identical experimental conditions to directly compare the effects of gamma and pulsed electron radiolysis on the ligands and systematically study the influence of structural changes in the ligand backbone. Distribution ratios of Am 3+ , Cm 3+ and Eu 3+ , the residual concentration of CyMe 4 BTBP and CyMe 4 BTPhen in solution, and the formation of radiolysis products were studied as a function of absorbed gamma dose and presence of an acidic aqueous phase during irradiation. Quantitative and semi-quantitative analyses were used to elucidate the radiolysis mechanism for both ligands. Addition products of alpha-hydroxyoctyl radicals formed through radiolysis of the 1-octanol diluent to the ligand molecules were identified as the predominant radiolysis products. These addition products also extract trivalent metal ions, as distribution ratios remained high although the parent molecule concentrations decreased. Therefore, the utilization time of a solvent using these extractants under the harsh conditions of used nuclear fuel treatment could be considerably longer than expected. Understanding the radiolysis mechanism is crucial for designing more radiation resistant extractants.
In the machine learning problem of multilabel classification, the objective is to determine for each test instance which classes the instance belongs to. In this work, we consider an extension of multilabel classification, called multilabel proportion prediction, in the context of radioisotope identification (RIID) using gamma spectra data. We aim to not only predict radioisotope proportions, but also identify out-of-distribution (OOD) spectra. We achieve this goal by viewing gamma spectra as discrete probability distributions, and based on this perspective, we develop a custom semi-supervised loss function that combines a traditional supervised loss with an unsupervised reconstruction error function. Our approach was motivated by its application to the analysis of short-lived fission products from spent nuclear fuel. In particular, we demonstrate that a neural network model trained with our loss function can successfully predict the relative proportions of 37 radioisotopes simultaneously. The model trained with synthetic data was then applied to measurements taken by Pacific Northwest National Laboratory (PNNL) to conduct analysis typically done by subject-matter experts. Here, we also extend our approach to successfully identify when measurements are OOD, and thus should not be trusted, whether due to the presence of a novel source or novel proportions.