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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Numerical methods and hypoexponential approximations for gamma distributed delay differential equations

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.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

An Analysis of the Johnson-Lindenstrauss Lemma with the Bivariate Gamma Distribution

Probabilistic proofs of the Johnson-Lindenstrauss lemma imply that random projection can reduce the dimension of a data set and approximately preserve pairwise distances. If a distance being approximately preserved is called a success, and the complement of this event is called a failure, then such a random projection likely results in no failures. Assuming a Gaussian random projection, the lemma is proved by showing that the no-failure probability is positive using a combination of Bonferroni's inequality and Markov's inequality. This paper modifies this proof in two ways to obtain a greater lower bound on the no-failure probability. First, Bonferroni's inequality is applied to pairs of failures instead of individual failures. Second, since a pair of projection errors has a bivariate gamma distribution, this probability of a pair of successes is bounded using an inequality from [Jensen, 1969]. If n is the number of points to be embedded and μ is the probability of success, then this leads to an increase in the lower bound on the no-failure probability of $\frac{1}{2}$ ($\genfrac{}{}{0pt}{}{n}{2}$) (1- μ ) 2 is ($\genfrac{}{}{0pt}{}{n}{2}$) is even and $\frac{1}{2}$ (($\genfrac{}{}{0pt}{}{n}{2}$)-1) (1- μ ) 2 if ($\genfrac{}{}{0pt}{}{n}{2}$) is odd. For example, if n =10 5 points are to be embedded in k =10 4 dimensions with a tolerance of ϵ=0.1, then the improvement in the lower bound is on the order of 10 -14 . We also show that further improvement is possible if the inequality in [Jensen, 1969] extends to three successes, though we do not have a proof of this result.

96 KNOWLEDGE MANAGEMENT AND PRESERVATION↗
NASA NTRS1984

The galactic gamma-ray distribution: Implications for galactic structure and the radial cosmic ray gradient

The radial distribution of gamma ray emissivity in the Galaxy was derived from flux longitude profiles, using both the final SAS-2 results and the recently corrected COS-B results and analyzing the northern and southern galactic regions separately. The recent CO surveys of the Southern Hemisphere, were used in conjunction with the Northern Hemisphere data, to derive the radial distribution of cosmic rays on both sides of the galactic plane. In addition to the 5 kpc ring, there is evidence from the radial asymmetry for spiral features which are consistent with those derived from the distribution of bright HII regions. Positive evidence was also found for a strong increase in the cosmic ray flux in the inner Galaxy, particularly in the 5 kpc region in both halves of the plane.

Harding, A. K.↗
NASA NTRS1985

The galactic gamma-ray distribution - Implications for galactic structure and the radial cosmic-ray gradient

The radial distribution of gamma ray emissivity in the Galaxy was derived from flux longitude profiles, using both the final SAS-2 results and the recently corrected COS-B results and analyzing the northern and southern galactic regions separately. The recent CO surveys of the Southern Hemisphere were used in conjunction with the Northern Hemisphere data, to derive the radial distribution of cosmic rays on both sides of the galactic plane. In addition to the 5 kpc ring, there is evidence from the radial asymmetry for spiral features which are consistent with those derived from the distribution of bright H II regions. Positive evidence was also found for a strong increase in the cosmic ray flux in the inner Galaxy, particularly in the 5 kpc region in both halves of the plane.

Harding, A. K.↗
NASA NTRS1992

Statistics of backscatter radar return from vegetation

The statistical characteristics of radar return from vegetation targets are investigated through a simulation study based upon the first-order scattered field. For simulation purposes, the vegetation targets are modeled as a layer of randomly oriented and spaced finite cylinders, needles, or discs, or a combination of them. The finite cylinder is used to represent a branch or a trunk, the needle for a stem or a coniferous leaf, and the disc for a decidous leaf. For a plane wave illuminating a vegetation canopy, simulation results show that the signal returned from a layer of disc- or needle-shaped leaves follows the Gamma distribution, and that the signal returned from a layer of branches resembles the log normal distribution. The Gamma distribution also represents the signal returned from a layer of a mixture of branches and leaves regardless of the leaf shapes. Results also indicate that the polarization state does not have a significant impact on signal distribution.

Karam, M. A.↗
NASA NTRS1983

Some properties of a 5-parameter bivariate probability distribution

A five-parameter bivariate gamma distribution having two shape parameters, two location parameters and a correlation parameter was developed. This more general bivariate gamma distribution reduces to the known four-parameter distribution. The five-parameter distribution gives a better fit to the gust data. The statistical properties of this general bivariate gamma distribution and a hypothesis test were investigated. Although these developments have come too late in the Shuttle program to be used directly as design criteria for ascent wind gust loads, the new wind gust model has helped to explain the wind profile conditions which cause large dynamic loads. Other potential applications of the newly developed five-parameter bivariate gamma distribution are in the areas of reliability theory, signal noise, and vibration mechanics.

Tubbs, J. D.↗
NASA NTRS1982

A bivariate gamma probability distribution with application to gust modeling

A five-parameter gamma distribution (BGD) having two shape parameters, two location parameters, and a correlation parameter is investigated. This general BGD is expressed as a double series and as a single series of the modified Bessel function. It reduces to the known special case for equal shape parameters. Practical functions for computer evaluations for the general BGD and for special cases are presented. Applications to wind gust modeling for the ascent flight of the space shuttle are illustrated.

Smith, O. E.↗
NASA NTRS1989

Monte Carlo models and analysis of galactic disk gamma-ray burst distributions

Gamma-ray bursts are transient astronomical phenomena which have no quiescent counterparts in any region of the electromagnetic spectrum. Although temporal and spectral properties indicate that these events are likely energetic, their unknown spatial distribution complicates astrophysical interpretation. Monte Carlo samples of gamma-ray burst sources are created which belong to Galactic disk populations. Spatial analysis techniques are used to compare these samples to the observed distribution. From this, both quantitative and qualitative conclusions are drawn concerning allowed luminosity and spatial distributions of the actual sample. Although the Burst and Transient Source Experiment (BATSE) experiment on Gamma Ray Observatory (GRO) will significantly improve knowledge of the gamma-ray burst source spatial characteristics within only a few months of launch, the analysis techniques described herein will not be superceded. Rather, they may be used with BATSE results to obtain detailed information about both the luminosity and spatial distributions of the sources.

Hakkila, Jon↗
NASA NTRS1985

The hump in the Cerenkov lateral distribution of gamma ray showers

The lateral distribution of atmospheric Cerenkov photons emitted by gamma ray showers of energy 100 GeV is calculated. The lateral distribution shows a characteristic hump at a distance of approx. 135 meter from the core. The hump is shown to be due to electrons of threshold energy 1 GeV, above which the mean scattering angle becomes smaller than the Cerenkov angle.

Sinha, S.↗
NASA NTRS1981

Vector wind profile gust model

The development of a vector wind gust model that is suitable for orbital flight test operations and trade studies was studied. Verification of the hypothesis that gust component variables are gamma distributed, gust modulus is approximately Weibull distributed, and zonal and meridional gust components are bivariate gamma distributed is emphasized. A method of testing for bivariate gamma distributed variables, and two distributions for gust modulus are described. The results of extensive hypothesis testing of one of the distributions are presented, and the validity of the gamma distribution for representation of gust component variables is established.

Adelfang, S. I.↗
NASA NTRS1981

Vector wind profile gust model

To enable development of a vector wind gust model suitable for orbital flight test operations and trade studies, hypotheses concerning the distributions of gust component variables were verified. Methods for verification of hypotheses that observed gust variables, including gust component magnitude, gust length, u range, and L range, are gamma distributed and presented. Observed gust modulus has been drawn from a bivariate gamma distribution that can be approximated with a Weibull distribution. Zonal and meridional gust components are bivariate gamma distributed. An analytical method for testing for bivariate gamma distributed variables is presented. Two distributions for gust modulus are described and the results of extensive hypothesis testing of one of the distributions are presented. The validity of the gamma distribution for representation of gust component variables is established.

Adelfang, S. I.↗
NASA NTRS1974

High energy gamma ray astronomy

The SAS-2 gamma ray experiment and its detection of celestial gamma rays are described. Data also cover intensity of high energy gamma rays, gamma ray distribution, gamma ray origin, and diffuse radiation.

Fichtel, C. E.↗
NASA NTRS1985

The galactic gamma-ray distribution and the radial cosmic ray gradient

The radial distribution is derived of gamma ray emissivity and the cosmic ray intensity in the Galaxy using the final SAS-2 results, the recently corrected COS-B results, and recent CO surveys of the Northern and Southern Galactic Hemispheres. In addition to the 5 kpc ring of enhanced emission, there is evidence for spiral features. A strong increase in the cosmic ray flux is found in the inner Galaxy, particularly in the 5 kpc region, in both halves of the plane.

Harding, A. K.↗
NASA NTRS2016

Polarized View of Supercooled Liquid Water Clouds

Supercooled liquid water (SLW) clouds, where liquid droplets exist at temperatures below 0 C present a well known aviation hazard through aircraft icing, in which SLW accretes on the airframe. SLW clouds are common over the Southern Ocean, and climate-induced changes in their occurrence is thought to constitute a strong cloud feedback on global climate. The two recent NASA field campaigns POlarimeter Definition EXperiment (PODEX, based in Palmdale, California, January-February 2013) and Studies of Emissions and Atmospheric Composition, Clouds and Climate Coupling by Regional Surveys (SEAC4RS, based in Houston, Texas in August- September 2013) provided a unique opportunity to observe SLW clouds from the high-altitude airborne platform of NASA's ER-2 aircraft. We present an analysis of measurements made by the Research Scanning Polarimeter (RSP) during these experiments accompanied by correlative retrievals from other sensors. The RSP measures both polarized and total reflectance in 9 spectral channels with wavelengths ranging from 410 to 2250 nm. It is a scanning sensor taking samples at 0.8deg intervals within 60deg from nadir in both forward and backward directions. This unique angular resolution allows for characterization of liquid water droplet size using the rainbow structure observed in the polarized reflectances in the scattering angle range between 135deg and 165deg. Simple parametric fitting algorithms applied to the polarized reflectance provide retrievals of the droplet effective radius and variance assuming a prescribed size distribution shape (gamma distribution). In addition to this, we use a non-parametric method, Rainbow Fourier Transform (RFT),which allows retrieval of the droplet size distribution without assuming a size distribution shape. We present an overview of the RSP campaign datasets available from the NASA GISS website, as well as two detailed examples of the retrievals. In these case studies we focus on cloud fields with spatial features varying between glaciated and liquid phases at altitudes as high as 10 km, which correspond to temperatures close to the homogeneous freezing temperature of pure water drops (about -35 C or colder). The multimodal droplet size distributions retrieved from RSP data in these cases are consistent with the multi-layer cloud structure observed by correlative Cloud Physics Lidar (CPL) measurements.

Clouds↗
NASA NTRS2016

Retrievals of Cloud Droplet Size from the RSP Data: Validation Using in Situ Measurements

We present comparisons of cloud droplet size distributions retrieved from the Research Scanning Polarimeter (RSP) data with correlative in situ measurements made during the North Atlantic Aerosols and Marine Ecosystems Study (NAAMES). This field experiment was based at St. Johns airport, Newfoundland, Canada with the latest deployment in May - June 2016. RSP was onboard the NASA C-130 aircraft together with an array of in situ and other remote sensing instrumentation. The RSP is an along-track scanner measuring polarized and total reflectances in9 spectral channels. Its unique high angular resolution allows for characterization of liquid water droplet size using the rainbow structure observed in the polarized reflectances in the scattering angle range between 135 and 165 degrees. A parametric fitting algorithm applied to the polarized reflectances provides retrievals of the droplet effective radius and variance assuming a prescribed size distribution shape (gamma distribution). In addition to this, we use a non-parametric method, Rainbow Fourier Transform (RFT), which allows us to retrieve the droplet size distribution (DSD) itself. The latter is important in the case of clouds with complex structure, which results in multi-modal DSDs. During NAAMES the aircraft performed a number of flight patterns specifically designed for comparison of remote sensing retrievals and in situ measurements. These patterns consisted of two flight segments above the same straight ground track. One of these segments was flown above clouds allowing for remote sensing measurements, while the other was at the cloud top where cloud droplets were sampled. We compare the DSDs retrieved from the RSP data with in situ measurements made by the Cloud Droplet Probe (CDP). The comparisons show generally good agreement with deviations explainable by the position of the aircraft within cloud and by presence of additional cloud layers in RSP view that do not contribute to the in situ DSDs. In the latter case the distributions retrieved from the RSP data were consistent with the multi-layer cloud structures observed in the correlative High Spectral Resolution Lidar (HSRL) profiles. The comparison results provide a rare validation of polarimetric droplet size retrieval techniques, which can be used for analysis of satellite data on global scale.

cloud droplets↗