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NASA NTRS1976

An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions, 2

The problem of obtaining numerically maximum likelihood estimates of the parameters for a mixture of normal distributions is addressed. In recent literature, a certain successive approximations procedure, based on the likelihood equations, is shown empirically to be effective in numerically approximating such maximum-likelihood estimates; however, the reliability of this procedure was not established theoretically. Here, a general iterative procedure is introduced, of the generalized steepest-ascent (deflected-gradient) type, which is just the procedure known in the literature when the step-size is taken to be 1. With probability 1 as the sample size grows large, it is shown that this procedure converges locally to the strongly consistent maximum-likelihood estimate whenever the step-size lies between 0 and 2. The step-size which yields optimal local convergence rates for large samples is determined in a sense by the separation of the component normal densities and is bounded below by a number between 1 and 2.

Peters, B. C., Jr.↗

NASA NTRS1976

The numerical evaluation of maximum-likelihood estimates of the parameters for a mixture of normal distributions from partially identified samples

Likelihood equations determined by the two types of samples which are necessary conditions for a maximum-likelihood estimate are considered. These equations, suggest certain successive-approximations iterative procedures for obtaining maximum-likelihood estimates. These are generalized steepest ascent (deflected gradient) procedures. It is shown that, with probability 1 as N sub 0 approaches infinity (regardless of the relative sizes of N sub 0 and N sub 1, i=1,...,m), these procedures converge locally to the strongly consistent maximum-likelihood estimates whenever the step size is between 0 and 2. Furthermore, the value of the step size which yields optimal local convergence rates is bounded from below by a number which always lies between 1 and 2.

Walker, H. F.↗

NASA NTRS1978

An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions

This paper addresses the problem of obtaining numerically maximum-likelihood estimates of the parameters for a mixture of normal distributions. In recent literature, a certain successive-approximations procedure, based on the likelihood equations, was shown empirically to be effective in numerically approximating such maximum-likelihood estimates; however, the reliability of this procedure was not established theoretically. Here, we introduce a general iterative procedure, of the generalized steepest-ascent (deflected-gradient) type, which is just the procedure known in the literature when the step-size is taken to be 1. We show that, with probability 1 as the sample size grows large, this procedure converges locally to the strongly consistent maximum-likelihood estimate whenever the step-size lies between 0 and 2. We also show that the step-size which yields optimal local convergence rates for large samples is determined in a sense by the 'separation' of the component normal densities and is bounded below by a number between 1 and 2.

Peters, B. C., Jr.↗

NASA NTRS1992

Methods of periodicity analysis - Relationship between the Rayleigh analysis and a maximum likelihood method

For periodicity analysis of occurrence rates of discrete events, one can use the maximum likelihood method or the 'Rayleigh analysis'. In a maximum likelihood analysis using a sinusoidal distribution, one tries various values of amplitude A and phase angle Theta(0) of the distribution function. We show that these two methods are essentially equivalent to one another in spite of their different mathematical origins. Using the Rayleigh analysis, therefore, we can simply calculate A and Theta(0) which maximize the likelihood. Using the cumulative nature of the logarithmic likelihood, we can identify time intervals during which the periodicity is in operation. When a periodicity operates only in certain time intervals, it is important to identify these intervals. Mathematically, the above technique is applicable only to discrete events. However, with slight modifications we can apply this technique to general cases - periodicity analysis of measurements of continuously varying quantities.

Bai, T.↗

NASA NTRS1999

Likelihood Analysis for Mega Pixel Maps

The derivation of cosmological parameters from astrophysical data sets routinely involves operations counts which scale as O(N(exp 3) where N is the number of data points. Currently planned missions, including MAP and Planck, will generate sky maps with N(sub d) = 10(exp 6) or more pixels. Simple "brute force" analysis, applied to such mega-pixel data, would require years of computing even on the fastest computers. We describe an algorithm which allows estimation of the likelihood function in the direct pixel basis. The algorithm uses a conjugate gradient approach to evaluate X2 and a geometric approximation to evaluate the determinant. Monte Carlo simulations provide a correction to the determinant, yielding an unbiased estimate of the likelihood surface in an arbitrary region surrounding the likelihood peak. The algorithm requires O(N(sub d)(exp 3/2) operations and O(Nd) storage for each likelihood evaluation, and allows for significant parallel computation.

Kogut, Alan J.↗

NASA NTRS2013

The Atacama Cosmology Telescope: Likelihood for Small-Scale CMB Data

The Atacama Cosmology Telescope has measured the angular power spectra of microwave fluctuations to arcminute scales at frequencies of 148 and 218 GHz, from three seasons of data. At small scales the fluctuations in the primordial Cosmic Microwave Background (CMB) become increasingly obscured by extragalactic foregounds and secondary CMB signals. We present results from a nine-parameter model describing these secondary effects, including the thermal and kinematic Sunyaev-Zel'dovich (tSZ and kSZ) power; the clustered and Poisson-like power from Cosmic Infrared Background (CIB) sources, and their frequency scaling; the tSZ-CIB correlation coefficient; the extragalactic radio source power; and thermal dust emission from Galactic cirrus in two different regions of the sky. In order to extract cosmological parameters, we describe a likelihood function for the ACT data, fitting this model to the multi-frequency spectra in the multipole range 500 < l < 10000. We extend the likelihood to include spectra from the South Pole Telescope at frequencies of 95, 150, and 220 GHz. Accounting for different radio source levels and Galactic cirrus emission, the same model provides an excellent fit to both datasets simultaneously, with χ2/dof= 675/697 for ACT, and 96/107 for SPT. We then use the multi-frequency likelihood to estimate the CMB power spectrum from ACT in bandpowers, marginalizing over the secondary parameters. This provides a simplified 'CMB-only' likelihood in the range 500 < l < 3500 for use in cosmological parameter estimation

Cosmology↗

DOE OSTI2015

Applying understanding of likelihood and consequences to demonstrate post-closure criticality safety of a geological disposal facility - 14377

Radioactive Waste Management (RWM) is responsible for implementing geological disposal of the UK's higher activity wastes. These wastes include significant quantities of fissile nuclides and, therefore, demonstration of criticality safety under disposal conditions forms an important component of RWM's disposal system safety case. At the time of disposal, controls (e.g. fissile material limits and/or water barriers) will ensure that waste packages are sub-critical. These controls, plus the containment provided by the engineered barrier system of a geological disposal facility (GDF) will also ensure that criticality is prevented for a long time after disposal. However, over post-closure timescales conditions in the GDF will evolve, and therefore underpinning knowledge is required to demonstrate: under what conditions criticality could occur; what the likelihood of such conditions developing is; and what the consequences would be if a post-closure criticality did occur. These requirements are set out in the UK environment agencies' Guidance on Requirements for Authorisation of a GDF. This paper discusses the disposal of Pressurised Water Reactor fuel, as an example of a waste type that may require disposal. It briefly selects relevant research to combine and summarise results from three recent RWM research projects. The first 'Likelihood of Criticality' considered under what conditions post-closure criticality could occur, whilst a second 'Modelling of Consequences of Hypothetical Criticality' focused on understanding the local physical consequences (temperature rise, power and rock response) of hypothetical 'what-if' post-closure criticality. The paper then explains how this understanding is applied to make safety arguments as to why post-closure criticality is not a significant concern. This is achieved via a third, stream of work that has considered the effects of a potential criticality on pathways that might give rise to a post-closure risk in the future, such as transport of radionuclides in groundwater. Collectively RWMs post-closure criticality safety work allows us to conclude in our generic environmental safety case that a post closure criticality is a low likelihood event, and even if one of these low likelihood events was to occur its consequences would be insignificant to GDF performance. This paper also serves to give context to two companion UK post-closure criticality safety papers which communicate further information on our modelling development and more detailed results underpinning our demonstration of post-closure criticality safety. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2024

Assessment of Bird Strike Likelihood to Refine Bird Strike Risk Models

In its most basic form, bird strike risk is comprised of a frequency component that reflects the likelihood of a collision and a severity component that reflects the cost (monetary or otherwise) of the incident. The bird strike risk model currently used by United State Department of Agriculture (USDA) Wildlife Services to evaluate the risk posed by individual bird species at airports and establish priorities for management was developed in 2018. The model uses airport-specific data on the number of reported strikes for a species recorded in the Federal Aviation Administration (FAA)’s National Wildlife Strike Database as a measure of frequency and the species’ relative hazard score as a measure of severity. The model was tested against independent data, found to perform well overall, and is being implemented widely across the United States. However, the model has limitations, including that species known to pose risk to aircraft locally, but not present in the strike record database, are not reflected as a major component of risk. Standard bird survey methodology commonly used at airports (e.g. point counts or transects) potentially can be used to complement wildlife strike records to calculate frequency or relative abundance of species. However, these methods generally focus on airport-wide population estimation and often ignore vital information that contributes to the true likelihood of a strike, such as use of runway protection zones and other critical areas, and spatial and temporal overlap with departing or approaching aircraft. As such, a more detailed understanding of space use by birds across landcovers and population fluctuations across the year is needed to accurately estimate the likelihood of bird strikes at airports. In this manuscript, we will review the extant risk model, including a discussion on its limitations. We then discuss approaches for refining our understanding of strike likelihood and briefly touch on needs for estimating probability of strike severity (cost).

bird strike, aircraft collision, damage by wildlif↗

DOE OSTI2018

On the application of the expected log-likelihood gain to decision making in molecular replacement

Molecular-replacement phasing of macromolecular crystal structures is often fast, but if a molecular-replacement solution is not immediately obtained the crystallographer must judge whether to pursue molecular replacement or to attempt experimental phasing as the quickest path to structure solution. The introduction of the expected log-likelihood gain has given the crystallographer a powerful new tool to aid in making this decision. The eLLG is the log-likelihood gain on intensity expected from a correctly placed model. It is calculated as a sum over the reflections of a function dependent on the fraction of the scattering for which the model accounts, the estimated model coordinate error and the measurement errors in the data. It is shown how the eLLG may be used to answer the question 'can I solve my structure by molecular replacement?'. However, this is only the most obvious of the applications of the eLLG. It is also discussed how the eLLG may be used to determine the search order and minimal data requirements for obtaining a molecular-replacement solution using a given model, and for decision making in fragment-based molecular replacement, single-atom molecular replacement and likelihood-guided model pruning.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2025

A dynamic likelihood approach to filtering transport processes: advection-diffusion dynamics

A Bayesian data assimilation scheme is formulated for advection-dominated advective and diffusive evolutionary problems, based upon the Dynamic Likelihood (DLF) approach to filtering. The DLF was developed specifically for hyperbolic problems –waves–, and in this paper, it is extended via a split step formulation, to handle advection-diffusion problems. In the dynamic likelihood approach, observations and their statistics are used to propagate probabilities along characteristics, evolving the likelihood in time. The estimate posterior thus inherits phase information. For advection-diffusion the advective part of the time evolution is handled on the basis of observations alone, while the diffusive part is informed through the model as well as observations. We expect, and indeed show here, that in advection-dominated problems, the DLF approach produces better estimates than other assimilation approaches, particularly when the observations are sparse and have low uncertainty. The added computational expense of the method is cubic in the total number of observations over time, which is on the same order of magnitude as a standard Kalman filter and can be mitigated by bounding the number of forward propagated observations, discarding the least informative data.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET

Photon sequences from single-molecule Förster resonance energy transfer (FRET) experiments can be analyzed using a maximum likelihood method. Parameters of the underlying kinetic model (FRET efficiencies of the states and transition rates between conformational states) are obtained by maximizing the appropriate likelihood function. In addition, the errors (uncertainties) of the extracted parameters can be obtained from the curvature of the likelihood function at the maximum. We study the standard deviations of the parameters of a two-state model obtained from photon sequences with recorded colors and arrival times. The standard deviations can be obtained analytically in a special case when the FRET efficiencies of the states are 0 and 1 and in the limiting cases of fast and slow conformational dynamics. These results are compared with the results of numerical simulations. The accuracy and, therefore, the ability to predict model parameters depend on how fast the transition rates are compared to the photon count rate. In the limit of slow transitions, the key parameters that determine the accuracy are the number of transitions between the states and the number of independent photon sequences. In the fast transition limit, the accuracy is determined by the small fraction of photons that are correlated with their neighbors. The relative standard deviation of the relaxation rate has a “chevron” shape as a function of the transition rate in the log-log scale. The location of the minimum of this function dramatically depends on how well the FRET efficiencies of the states are separated.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Likelihood-based docking of models into cryo-EM maps

Optimized docking of models into cryo-EM maps requires exploiting an understanding of the signal expected in the data to minimize the calculation time while maintaining sufficient signal. The likelihood-based rotation function used in crystallography can be employed to establish plausible orientations in a docking search. A phased likelihood translation function yields scores for the placement and rigid-body refinement of oriented models. Optimized strategies for choices of the resolution of data from the cryo-EM maps to use in the calculations and the size of search volumes are based on expected log-likelihood-gain scores computed in advance of the search calculation. Tests demonstrate that the new procedure is fast, robust and effective at placing models into even challenging cryo-EM maps.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2015

Maximum Likelihood Source Localization Using Elpasolite as a Dual Gamma Neutron Directional Detector

The problem of accurately detecting extremely low levels of nuclear radiation is rapidly increasing in importance in nuclear counter-proliferation, verification, and environmental and waste management. Because the 239Pu gamma signature may be weak, for instance, even when compared to the natural terrestrial background, coincidence counting with the 239Pu neutron signature may improve overall 239Pu detection sensitivity. However, systems with sufficient multiple-particle detectors require demonstration that the increased sensitivity be sufficiently high to overcome added cost and weight. We report the results of measurements and calculations to determine sensitivity that can be gained in detecting low levels of nuclear radiation from use of a relatively new detector technology based on elpasolite crystals. We have performed investigations exploring cerium (Ce3+)-doped elpasolites Cs2LiYCl6:Ce3+0.5% (CLYC) and Cs2LiLa(Br6)90%(Cl6)10%:Ce3+0.5% (CLLBC:Ce). These materials can provide energy resolution (r(E) = 2.35σ(E)/E) as good as 2.9% at 662 keV (FWHM). The crystals show an excellent neutron and gamma radiation response. The goals of the investigation were to set up the neutron/gamma pulse shape discrimination electronics for elpasolite detectors; perform limited static source benchmarking, testing, and evaluation to validate system performance; and explore application of a maximum likelihood algorithm for source location. Data were measured and processed through a maximum likelihood estimation algorithm, providing a direction to the radioactive source for each individual position. The estimated directions were good representations for the actual directions to the radioactive source. This paper summarizes the maximum likelihood results for our elpasolite system.

maximum likelihood, radiation detectors, neutron d↗

NASA NTRS2003

Comparison of two weighted integration models for the cueing task: linear and likelihood

In a task in which the observer must detect a signal at two locations, presenting a precue that predicts the location of a signal leads to improved performance with a valid cue (signal location matches the cue), compared to an invalid cue (signal location does not match the cue). The cue validity effect has often been explained with a limited capacity attentional mechanism improving the perceptual quality at the cued location. Alternatively, the cueing effect can also be explained by unlimited capacity models that assume a weighted combination of noisy responses across the two locations. We compare two weighted integration models, a linear model and a sum of weighted likelihoods model based on a Bayesian observer. While qualitatively these models are similar, quantitatively they predict different cue validity effects as the signal-to-noise ratios (SNR) increase. To test these models, 3 observers performed in a cued discrimination task of Gaussian targets with an 80% valid precue across a broad range of SNR's. Analysis of a limited capacity attentional switching model was also included and rejected. The sum of weighted likelihoods model best described the psychophysical results, suggesting that human observers approximate a weighted combination of likelihoods, and not a weighted linear combination.

NASA Program Biomedical Research and Countermeasur↗

DOE OSTI2024

Maximum Likelihood Estimation: Some Basics

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.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

Maximum Likelihood Estimation: Some Basics

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Estimating the Likelihood of GHG Concentration Scenarios From Probabilistic Integrated Assessment Model Simulations

The climate scenarios that form the basis for current climate risk assessments have no assigned probabilities, and this impedes the analysis of future climate risks. This paper proposes an approach to estimate the probability of carbon dioxide (CO 2 ) concentration scenarios used in key climate change modeling experiments. It computes the CO 2 emissions compatible with the concentrations prescribed by Coupled Model Intercomparison Project Phase 5 (CMIP5) and CMIP6 experiments. The distribution of these compatible cumulative emissions is interpreted as the likelihood of future emissions given a concentration pathway. Using Bayesian analysis, the probability of each pathway can be estimated from a probabilistic sample of future emissions. The approach is demonstrated with five probabilistic CO 2 emission simulation ensembles from four Integrated Assessment Models (IAM), leading to independent estimates of the likelihood of the CO 2 concentration of Representative Concentration Pathways (RCP) and Shared Socioeconomic Pathways (SSP). Results suggest that SSP5-8.5 is unlikely for the second half of the 21st century, but offer no clear consensus on which of the remaining scenarios is most likely. Estimates of likelihoods of CO 2 concentrations associated with RCP and SSP scenarios are affected by sampling errors, differences in emission sources simulated by the IAMs, and a lack of a common experimental framework for IAM simulations. These shortcomings, along with a small IAM ensemble size, limit the applicability of the results presented here. Novel joint IAM and the Earth System Model experiments are needed to deliver actionable probabilistic climate risk assessments.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2014

Planck intermediate results: XVI. Profile likelihoods for cosmological parameters

We explore the 2013 Planck likelihood function with a high-precision multi-dimensional minimizer (Minuit). This allows a refinement of the CDM best-fit solution with respect to previously-released results, and the construction of frequentist confidence intervals using profile likelihoods. The agreement with the cosmological results from the Bayesian framework is excellent, demonstrating the robustness of the Planck results to the statistical methodology. We investigate the inclusion of neutrino masses, where more significant differences may appear due to the non-Gaussian nature of the posterior mass distribution. By applying the Feldman-Cousins prescription, we again obtain results very similar to those of the Bayesian methodology. However, the profile-likelihood analysis of the cosmic microwave background (CMB) combination ( Planck +WP+highL) reveals a minimum well within the unphysical negative-mass region. We show that inclusion of the Planck CMB-lensing information regularizes this issue, and provide a robust frequentist upper limit Σ$m_v$ ≤0:26 eV (95% confidence) from the CMB+lensing+BAO data combination.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗