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Articles on theoretical and applied mathematics are introduced. The articles cover information that might be of interest to workers in statistics and information theory, computational aids that could be used by scientists and engineers, and mathematical techniques for design and control.
The Monte Carlo procedure is a model sampling technique. A model is established, and the behavior of sample units in this model is followed. A sufficient number of sample units are followed to obtain a statistical average or macroscopic quantities, which are the quantities of interest. This technique was used in crude form by Fermi in connection with the building of the first atomic pile. Later, Von Neumann and Ulam developed and used the Monte Carlo procedure extensively in developing the atomic bomb. Since then this technique has gained considerable use in nuclear reactor problems (refs. 1 and 2), and we at the Lewis Research Center have been extending it to thermal radiation (refs. 3 and 4), rarefied gas flows, and plasma flow problems (ref. 5). This technique, which requires a large number of sample histories to obtain solutions with small variances, is receiving greater use because of the development of the high-speed electronic computers.
Any process which may be modeled statistically for a simple plasma particle may be modeled by the particle in cell technique. The success of the calculation is then dependent on having a large enough number of particles that the statistical treatment is meaningful. Thus it is possible to include the effects of secondary emission, backscattering, charge sticking and possibly dielectric breakdown, photoemission, and spallation. The first plasma dielectric interaction included in the computer code for simulating plasma insulator interactions is secondary electron emission. A calculated current density vs. voltage curve is presented and compared to an experimental curve.
Type Ia PSCs are believed to be composed of nitric acid hydrate particles. Recent results from the SOLVE/THESEO 2000 campaign showed evidence that this type of PSC was composed of a small number of very large particles capable of sedimentary denitrification of regions of the stratosphere. It is unknown whether homogeneous or heterogeneous nucleation is responsible for the formation of these PSCs. Arctic winters are tending to be colder in response to global tropospheric warming. The degree to which this influences ozone depletion will depend on the freezing mechanism of nitric acid hydrate particles. If nucleation is homogeneous it implies that the freezing process is an inherent property of the particle, while heterogeneous freezing means that the extent of PSCs will depend in part on the number of nuclei available. The Polar Ozone and Aerosol Measurement (POAM)II and III satellites have been making observations of stratospheric aerosols and Polar Stratospheric Clouds (PSCs) since 1994. Recently, we have developed a technique that can discriminate between Type Ia and Ib PSCs using these observations. A statistical approach is employed to demonstrate the robustness of this approach and results are compared with lidar measurements. The technique is used to analyze observations from POAM II and II during Northern Hemisphere winters where significant PSC formation occurred with the objective of exploring Type I PSC formation mechanisms. The different PSCs identified using this method exhibit different growth curve as expressed as extinction versus temperature.
An automated technique is presented for designing effective decision tree classifiers predicated only on a priori class statistics. The procedure relies on linear feature extractions and Bayes table look-up decision rules. Associated error matrices are computed and utilized to provide an optimal design of the decision tree at each so-called 'node'. A by-product of this procedure is a simple algorithm for computing the global probability of correct classification assuming the statistical independence of the decision rules. Attention is given to a more precise definition of decision tree classification, the mathematical details on the technique for automated decision tree design, and an example of a simple application of the procedure using class statistics acquired from an actual Landsat scene.
Analytical Mechanics Associates, Inc., has developed a software toolkit that filters and processes navigational data from multiple sensor sources. A key component of the toolkit is a trajectory optimization technique that reduces the sensitivity of Kalman filters with respect to model parameter uncertainties. The sensor fusion toolkit also integrates recent advances in adaptive Kalman and sigma-point filters for non-Gaussian problems with error statistics. This Phase II effort provides new filtering and sensor fusion techniques in a convenient package that can be used as a stand-alone application for ground support and/or onboard use. Its modular architecture enables ready integration with existing tools. A suite of sensor models and noise distribution as well as Monte Carlo analysis capability are included to enable statistical performance evaluations.
We present results from the first lattice QCD calculation of the two-pion contributions to the light-quark connected vector-current correlation function obtained from staggered-quark operators. We employ the MILC Collaboration’s gauge-field ensemble with 2 + 1 + 1 flavors of highly improved staggered sea quarks at a lattice spacing of a ≈ 0.15 fm with a light sea-quark mass at its physical value. The two-pion contributions allow for a refined determination of the noisy long-distance tail of the vector-current correlation function, which we use to compute the light-quark connected contribution to hadronic vacuum polarization (HVP) with improved statistical precision. We compare our results with traditional noise-reduction techniques used in lattice QCD calculations of the light-quark connected HVP, namely, the so-called fit and bounding methods. We observe a factor of roughly 3 improvement in the statistical precision in the determination of the HVP contribution to the muon’s anomalous magnetic moment over these approaches. We also lay the group theoretical groundwork for extending this calculation to finer lattice spacings with increased numbers of staggered two-pion taste states.
Investigation of the effects of sensor spatial resolution on estimating the amount of clouds covering the earth by simulating various cloud distributions and sizes, and measuring the known cloud amount with resolution of different sizes using a cloud-no cloud threshold technique often applied in automatic data processing. Cloud amount statistics have been tabulated for a three-orders-of-magnitude range in the ratio (R) of areal cloud size to areal resolution size for seven cloud amounts between 6 and 90%. Three different cloud patterns were used. These were 1) a regularly spaced pattern of homogeneous dots arranged in rows and columns (to simulate cloud streets), 2) a randomly spaced pattern of the same dots (to simulate randomly oriented cumulus clouds), and 3) a heterogeneous cloud size distribution irregularly spaced (to simulate a view of different cloud types and sizes). Two cloud amount estimation techniques were tested. Cloud amounts of 100% (method 1) and 50% (method 2) were assigned to partially filled resolution elements. Using criteria applicable to some studies carried out in the past, it is shown that cloud amount estimations can be in error by as much as 86 and 38%, respectively, for the two methods. Nomograms have been developed which subtantially improve the estimate of the true cloud cover for R less than 100, provided that R can be determined.
We describe a simple image processing technique that is useful for the visualization and depiction of gradually evolving or intermittent structures in solar physics extreme-ultraviolet imagery. The technique is an application of image segmentation, which we call "Persistence Mapping," to isolate extreme values in a data set, and is particularly useful for the problem of capturing phenomena that are evolving in both space and time. While integration or "time-lapse" imaging uses the full sample (of size N ), Persistence Mapping rejects (N - 1)/N of the data set and identifies the most relevant 1/N values using the following rule: if a pixel reaches an extreme value, it retains that value until that value is exceeded. The simplest examples isolate minima and maxima, but any quantile or statistic can be used. This paper demonstrates how the technique has been used to extract the dynamics in long-term evolution of comet tails, erupting material, and EUV dimming regions.
The investigators under this grant studied ways to improve the statistical analysis of astronomical data. They looked at existing techniques, the development of new techniques, and the production and distribution of specialized software to the astronomical community. Abstracts of nine papers that were produced are included, as well as brief descriptions of four software packages. The articles that are abstracted discuss analytical and Monte Carlo comparisons of six different linear least squares fits, a (second) paper on linear regression in astronomy, two reviews of public domain software for the astronomer, subsample and half-sample methods for estimating sampling distributions, a nonparametric estimation of survival functions under dependent competing risks, censoring in astronomical data due to nondetections, an astronomy survival analysis computer package called ASURV, and improving the statistical methodology of astronomical data analysis.
In this study, a technique for estimating vertical profiles of precipitation from multifrequency, multiresolution active and passive microwave observations is investigated. The technique is applicable to the Tropical Rainfall Measuring Mission (TRMM) observations and it is based on models that simulate high-resolution brightness temperatures as functions of observed reflectivity profiles and a parameter related to the rain drop-size-distribution. The modeled high-resolution brightness temperatures are used to determine normalized brightness temperature polarizations at the microwave radiometer resolution. An optimal estimation procedure is employed to minimize the differences between the simulated and observed normalized polarizations by adjusting the drop-size-distribution parameter. The impact of other unknowns that are not independent variables in the optimal estimation but affect the retrievals is minimized through statistical parameterizations derived from cloud model simulations. The retrieval technique is investigated using TRMM observations collected during the Kwajalein Experiment (KWAJEX). These observations cover an area extending from 5 deg to deg N latitude and 166 deg to 172 deg E longitude from July to September 1999, and are coincident with various ground-based observations, facilitating a detailed analysis of the retrieved precipitation. Using the method developed in this study, precipitation estimates consistent with both the passive and active TRMM observations are obtained. Various parameters characterizing these estimates, i.e. the rain rate, the precipitation water content, the drop-size-distribution intercept, and the mass weighted mean drop diameter, are in good qualitative agreement with independent experimental and theoretical estimates. Combined rain estimates are in general higher than the official TRMM Precipitation Radar (PR) only estimates for the area and the period considered in the study. Ground-based precipitation estimates, derived from an analysis of rain gauge and ground radar data, are in better agreement with the combined estimates than with the TRMM PR-only estimates, which suggests that information useful for improving the radar-only estimates is contained in the brightness temperature data.
Turbulent flows preferentially concentrate inertial particles depending on their stopping time or Stokes number, which can lead to significant spatial variations in the particle concentration. Cascade models are one way to describe this process in statistical terms. Here, we use a direct numerical simulation (DNS) dataset of homogeneous, isotropic turbulence to determine probability distribution functions (PDFs) for cascade multipliers, which determine the ratio by which a property is partitioned into subvolumes as an eddy is envisioned to decay into smaller eddies. We present a technique for correcting effects of small particle numbers in the statistics. We determine multiplier PDFs for particle number, flow dissipation, and enstrophy, all of which are shown to be scale dependent. However, the particle multiplier PDFs collapse when scaled with an appropriately defined local Stokes number. As anticipated from earlier works, dissipation and enstrophy multiplier PDFs reach an asymptote for sufficiently small spatial scales. From the DNS measurements, we derive a cascade model that is used it to make predictions for the radial distribution function (RDF) for arbitrarily high Reynolds numbers, Re, finding good agreement with the asymptotic, infinite Re inertial range theory of Zaichik and Alipchenkov [New J. Phys. 11, 103018 (2009)]. We discuss implications of these results for the statistical modeling of the turbulent clustering process in the inertial range for high Reynolds numbers inaccessible to numerical simulations.
Machine learning technology spans many areas and today plays a significant role in addressing a wide range of problems in critical domains,i.e., healthcare, autonomous driving, finance, manufacturing, cybersecurity,etc. Metamorphic testing (MT) is considered a simple but very powerful approach in testing such computationally complex systems for which either an oracle is not available or is available but difficult to apply. Conventional metamorphic testing techniques have certain limitations in verifying deep learning-based models (i.e., convolutional neural networks (CNNs)) that have a stochastic nature (because of randomly initializing the network weights) in their training. In this article, we attempt to address this problem by using a statistical metamorphic testing (SMT) technique that does not require software testers to worry about fixing the random seeds (to get deterministic results) to verify the metamorphic relations (MRs). We propose seven MRs combined with different statistical methods to statistically verify whether the program under test adheres to the relation(s) specified in the MR(s). We further use mutation testing techniques to show the usefulness of the proposed approach in the healthcare space and test two CNN-based deep learning models (used for pneumonia detection among patients). The empirical results show that our proposed approach uncovers 85.71% of the implementation faults in the classifiers under test (CUT). Furthermore, we also propose an MRs minimization algorithm for the CUT, thus saving computational costs and organizational testing resources.
We analyze narrow-band eclipse light curves of the nova-like cataclysmic variable UX UMa, obtained from low-resolution spectra spanning lambda lambda 3600-9800 A . The light curves for narrow bands in the continuum as well as those for individual spectral lines are treated independently, and are used to construct images of the accretion disk's brightness distribution using the maximum-entropy eclipse-mapping technique. Particular attention is paid to the propagation of statistical uncertainties in the data and to how the analysis may introduce systematic errors in the final result. From the many narrrow band images we have reconstructed the spectra from isolated parts of the accretion disk. These spectra reveal that the inner disk radiates a continuum spectrum which peaks in the near UV and has the hydrogen Balmer lines in absorption (with the exception of H-alpha), whereas the outer disk is much fainter, has a much redder spectrum, and has Balmer emission lines. Our analysis reveals the presence of an uneclipsed component of the total light, whose spectrum is very red and has Balmer lines in emission. This unexpected feature of the eclipse mapping technique offers a new tool for an independent assessment of the secondary star's spectrum in eclipsing cataclysmic variables.
Two techniques proposed for the recovery of the crater production functions on densely cratered surfaces were appraised. The first technique classifies craters according to their degree of rim preservation, then interprets the statistics of the freshest craters as closely mimicing the full population's production function. This method is misleading for densely cratered surfaces. The second technique measures and weighs each crater diameter by the fraction of the crater that remains unoverlapped by subsequent impacts. The method is theoretically sound even at saturation crater densities, though it too fails at high densities.
A generalized covariance analysis technique which permits the study of the sensitivity of linear estimation algorithms to errors in a priori statistics has been developed and programed. Several sample cases are presented to illustrate the use of this technique. Modifications to the Simulated Trajectories Error Analysis Program (STEAP) to enable targeting a multiprobe mission of the Planetary Explorer type are discussed. The logic for the mini-probe targeting is presented. Finally, the initial phases of the conversion of the Viking mission Lander Trajectory Reconstruction (LTR) program for use on Venus missions is discussed. An integrator instability problem is discussed and a solution proposed.
Historically, microbiological spaceflight requirements have been established in a subjective manner based upon expert opinion of both environmental and clinical monitoring results and the incidence of disease. The limited amount of data, especially from long-duration missions, has created very conservative requirements based primarily on the concentration of microorganisms. Periodic reevaluations of new data from later missions have allowed some relaxation of these stringent requirements. However, the requirements remain very conservative and subjective in nature, and the risk of crew illness due to infectious microorganisms is not well defined. The use of modeling techniques for microbial risk has been applied in the food and potable water industries and has exceptional potential for spaceflight applications. From a productivity standpoint, this type of modeling can (1) decrease unnecessary costs and resource usage and (2) prevent inadequate or inappropriate data for health assessment. In addition, a quantitative model has several advantages for risk management and communication. By identifying the variable components of the model and the knowledge associated with each component, this type of modeling can: (1) Systematically identify and close knowledge gaps, (2) Systematically identify acceptable and unacceptable risks, (3) Improve communication with stakeholders as to the reasons for resource use, and (4) Facilitate external scientific approval of the NASA requirements. The modeling of microbial risk involves the evaluation of several key factors including hazard identification, crew exposure assessment, dose-response assessment, and risk characterization. Many of these factors are similar to conditions found on Earth; however, the spaceflight environment is very specialized as the inhabitants live in a small, semi-closed environment that is often dependent on regenerative life support systems. To further complicate modeling efforts, microbial dose-response characteristics may be affected by a potentially dysfunctional crew immune system during a mission. In addition, microbial virulence has been shown to change under certain conditions during spaceflight, further complicating dose-response characterization. An initial study of the applicability of microbial risk assessment techniques was performed using Crew Health Care System (CHeCS) operational data from the International Space Station potable water systems. The risk of infection from potable water was selected as the flight systems and microbial ecology are well defined. This initial study confirmed the feasibility of using microbial risk assessment modeling for spaceflight systems. While no immediate threat was detected, the study identified several medically significant microorganisms that could pose a health risk if uncontrolled. The study also identified several specific knowledge gaps in making a risk assessment and noted that filling these knowledge gaps is essential as the risk estimates may change by orders of magnitude depending on the answers. The current phase of the microbial risk assessment studies focuses on the dose-response relationship of specific infectious agents, focusing on Salmonella enterica Typhimurium, Pseudomonas spp., and Escherichia coli, as their evaluation will provide a better baseline for determining the overall hazard characterization. The organisms were chosen as they either have been isolated on spacecraft or have an identified route of infection during a mission. The characterization will utilize dose-response models selected either from the peer-reviewed literature and/or by using statistical approaches. Development of these modeling and risk assessment techniques will help to optimize flight requirements and to protect the safety, health, and performance of the crew.
Atmospheric data assimilation is a class of techniques used for producing descriptions of fields of air temperature, pressure, humidity, wind, etc. on a spatial grid or in terms of a finite functional representation. These are then used to initialize numerical weather forecasts or to analyze the atmosphere for other purposes. The techniques combine past, present, and even future observations in an approximate statistically optimal way. Various types of statistical or physically-based models and their corresponding adjoints are employed to relate diverse fields in both time and space and to relate what is observed to what is being analyzed. Computationally, the problem is very demanding and onstraining on the techniques that can be employed on a routine basis.