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All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

Discrete Deterministic and Stochastic Petri Nets

Petri nets augmented with timing specifications gained a wide acceptance in the area of performance and reliability evaluation of complex systems exhibiting concurrency, synchronization, and conflicts. The state space of time-extended Petri nets is mapped onto its basic underlying stochastic process, which can be shown to be Markovian under the assumption of exponentially distributed firing times. The integration of exponentially and non-exponentially distributed timing is still one of the major problems for the analysis and was first attacked for continuous time Petri nets at the cost of structural or analytical restrictions. We propose a discrete deterministic and stochastic Petri net (DDSPN) formalism with no imposed structural or analytical restrictions where transitions can fire either in zero time or according to arbitrary firing times that can be represented as the time to absorption in a finite absorbing discrete time Markov chain (DTMC). Exponentially distributed firing times are then approximated arbitrarily well by geometric distributions. Deterministic firing times are a special case of the geometric distribution. The underlying stochastic process of a DDSPN is then also a DTMC, from which the transient and stationary solution can be obtained by standard techniques. A comprehensive algorithm and some state space reduction techniques for the analysis of DDSPNs are presented comprising the automatic detection of conflicts and confusions, which removes a major obstacle for the analysis of discrete time models.

MATHEMATICAL MODELS↗
NASA NTRS1986

Software reliability modeling and analysis

A discrete and, as approximation to it, a continuous model for the software reliability growth process are examined. The discrete model is based on independent multinomial trials and concerns itself with the joint distribution of the first occurrence time of its underlying events (bugs). The continuous model is based on the order statistics of N independent nonidentically distributed exponential random variables. It is shown that the spacings between bugs are not necessarily independent or exponentially (geometrically) distributed. However, there is a statistical rationale for viewing them so conditionally. Some identifiability problems are pointed out and resolved. In particular, it appears that the number of bugs in a program is not identifiable. Estimated upper bounds and confidence bounds for the residual program eror content are given based on the spacings of the first k bugs removed.

Scholz, F.-W.↗
NASA NTRS2012

The Ground Flash Fraction Retrieval Algorithm Employing Differential Evolution: Simulations and Applications

The ability to estimate the fraction of ground flashes in a set of flashes observed by a satellite lightning imager, such as the future GOES-R Geostationary Lightning Mapper (GLM), would likely improve operational and scientific applications (e.g., severe weather warnings, lightning nitrogen oxides studies, and global electric circuit analyses). A Bayesian inversion method, called the Ground Flash Fraction Retrieval Algorithm (GoFFRA), was recently developed for estimating the ground flash fraction. The method uses a constrained mixed exponential distribution model to describe a particular lightning optical measurement called the Maximum Group Area (MGA). To obtain the optimum model parameters (one of which is the desired ground flash fraction), a scalar function must be minimized. This minimization is difficult because of two problems: (1) Label Switching (LS), and (2) Parameter Identity Theft (PIT). The LS problem is well known in the literature on mixed exponential distributions, and the PIT problem was discovered in this study. Each problem occurs when one allows the numerical minimizer to freely roam through the parameter search space; this allows certain solution parameters to interchange roles which leads to fundamental ambiguities, and solution error. A major accomplishment of this study is that we have employed a state-of-the-art genetic-based global optimization algorithm called Differential Evolution (DE) that constrains the parameter search in such a way as to remove both the LS and PIT problems. To test the performance of the GoFFRA when DE is employed, we applied it to analyze simulated MGA datasets that we generated from known mixed exponential distributions. Moreover, we evaluated the GoFFRA/DE method by applying it to analyze actual MGAs derived from low-Earth orbiting lightning imaging sensor data; the actual MGA data were classified as either ground or cloud flash MGAs using National Lightning Detection Network[TM] (NLDN) data. Solution error plots are provided for both the simulations and actual data analyses.

Koshak, William↗
NASA NTRS1974

The abundances of nuclei in the cosmic radiation

The relative abundances are treated as a consequence of processes in cosmic ray transport occurring during passage of the radiation through interstellar material at high velocity. Some of the subjects mentioned are nuclear fragmentation and the production of secondary nuclei, nuclear reactions, energy loss and nuclear decay, ionization, the range-energy relation and propagation variables, capture and loss of electrons, the propagation of nuclei, the transport equation, equilibrium solutions, energy-dependent path length distribution, exponential path length distributions, discrete spectra, sources, supernovae, and the origin of the abundances. The connection between the space-time features of the sources, the material traversed, and the effects of magnetic fields is established by describing the particle-field interaction as a diffusive or random-walk process.

Reames, D. V.↗
NASA NTRS2000

Improved Results for Route Planning in Stochastic Transportation Networks

In the bus network problem, the goal is to generate a plan for getting from point X to point Y within a city using buses in the smallest expected time. Because bus arrival times are not determined by a fixed schedule but instead may be random. the problem requires more than standard shortest path techniques. In recent work, Datar and Ranade provide algorithms in the case where bus arrivals are assumed to be independent and exponentially distributed. We offer solutions to two important generalizations of the problem, answering open questions posed by Datar and Ranade. First, we provide a polynomial time algorithm for a much wider class of arrival distributions, namely those with increasing failure rate. This class includes not only exponential distributions but also uniform, normal, and gamma distributions. Second, in the case where bus arrival times are independent and geometric discrete random variable,. we provide an algorithm for transportation networks of buses and trains, where trains run according to a fixed schedule.

Boyan, Justin↗
NASA NTRS1993

Venus small volcano classification and description

The high resolution and global coverage of the Magellan radar image data set allows detailed study of the smallest volcanoes on the planet. A modified classification scheme for volcanoes less than 20 km in diameter is shown and described. It is based on observations of all members of the 556 significant clusters or fields of small volcanoes located and described by this author during data collection for the Magellan Volcanic and Magmatic Feature Catalog. This global study of approximately 10 exp 4 volcanoes provides new information for refining small volcano classification based on individual characteristics. Total number of these volcanoes was estimated to be 10 exp 5 to 10 exp 6 planetwide based on pre-Magellan analysis of Venera 15/16, and during preparation of the global catalog, small volcanoes were identified individually or in clusters in every C1-MIDR mosaic of the Magellan data set. Basal diameter (based on 1000 measured edifices) generally ranges from 2 to 12 km with a mode of 34 km, and follows an exponential distribution similar to the size frequency distribution of seamounts as measured from GLORIA sonar images. This is a typical distribution for most size-limited natural phenomena unlike impact craters which follow a power law distribution and continue to infinitely increase in number with decreasing size. Using an exponential distribution calculated from measured small volcanoes selected globally at random, we can calculate total number possible given a minimum size. The paucity of edifice diameters less than 2 km may be due to inability to identify very small volcanic edifices in this data set; however, summit pits are recognizable at smaller diameters, and 2 km may represent a significant minimum diameter related to style of volcanic eruption. Guest, et al, discussed four general types of small volcanic edifices on Venus: (1) small lava shields; (2) small volcanic cones; (3) small volcanic domes; and (4) scalloped margin domes ('ticks'). Steep-sided domes or 'pancake domes', larger than 20 km in diameter, were included with the small volcanic domes. For the purposes of this study, only volcanic edifices less than 20 km in diameter are discussed. This forms a convenient cutoff since most of the steep-sided domes ('pancake domes') and scalloped margin domes ('ticks') are 20 to 100 km in diameter, are much less numerous globally than are the smaller diameter volcanic edifices (2 to 3 orders of magnitude lower in total global number), and do not commonly occur in large clusters or fields of large numbers of edifices.

Aubele, J. C.↗
NASA NTRS1994

Redshift data and statistical inference

Frequency histograms and the 'power spectrum analysis' (PSA) method, the latter developed by Yu & Peebles (1969), have been widely employed as techniques for establishing the existence of periodicities. We provide a formal analysis of these two classes of methods, including controlled numerical experiments, to better understand their proper use and application. In particular, we note that typical published applications of frequency histograms commonly employ far greater numbers of class intervals or bins than is advisable by statistical theory sometimes giving rise to the appearance of spurious patterns. The PSA method generates a sequence of random numbers from observational data which, it is claimed, is exponentially distributed with unit mean and variance, essentially independent of the distribution of the original data. We show that the derived random processes is nonstationary and produces a small but systematic bias in the usual estimate of the mean and variance. Although the derived variable may be reasonably described by an exponential distribution, the tail of the distribution is far removed from that of an exponential, thereby rendering statistical inference and confidence testing based on the tail of the distribution completely unreliable. Finally, we examine a number of astronomical examples wherein these methods have been used giving rise to widespread acceptance of statistically unconfirmed conclusions.

Newman, William I.↗
NASA NTRS1994

A stochastic precipitation disaggregation scheme for GCM applications

In the surface hydrologic parameterization of general circulation models (GCMs), it is commonly assumed that the precipitation processes are homogeneous over a GCM grid square and that the precipitation intensity is uniformly distributed. Based on evidence that the spatial distribution of precipitation within a GCM grid square is crucial for the land surface hydrology parameterization, a few researchers have explored the impacts of assuming that the precipitation is exponentially distributed. This paper explores the suitability of the aforementioned assumptions. First, a statistical analysis is conducted of historical precipitation data for three GCM grids in different regions of the United States. The analysis suggests that neither the uniform nor the exponential distribution assumption may be suitable at the GCM grid scale and, that instead, the spatial variability in precipitation is characterized by statistical patterns that are inhomogeneous. These patterns vary from grid to grid and are induced by the interaction between atmospheric conditions and various land surface characteristics, such as topographical features, surface properties, etc. Within the same grid square, however, the statistical patterns are generally constant from year to year. Based on this analysis, a computationally viable (i.e., usable with GCMs) stochastic precipitation disaggregation scheme that utilizes these stable statistical patterns is proposed. The method was used to generate spatially distributed hourly rainfall for a summer season in the southwestern region of the continental United States. Analysis of the results shows that the methodology preserves the seasonal characteristics of spatial variability in precipitation that is observed in the long-term historical data.

Gao, Xiaogang↗
NASA NTRS2011

A Grobner Basis Solution for Lightning Ground Flash Fraction Retrieval

A Bayesian inversion method was previously introduced for retrieving the fraction of ground flashes in a set of flashes observed from a (low earth orbiting or geostationary) satellite lightning imager. The method employed a constrained mixed exponential distribution model to describe the lightning optical measurements. To obtain the optimum model parameters, a scalar function was minimized by a numerical method. In order to improve this optimization, we introduce a Grobner basis solution to obtain analytic representations of the model parameters that serve as a refined initialization scheme to the numerical optimization. Using the Grobner basis, we show that there are exactly 2 solutions involving the first 3 moments of the (exponentially distributed) data. When the mean of the ground flash optical characteristic (e.g., such as the Maximum Group Area, MGA) is larger than that for cloud flashes, then a unique solution can be obtained.

Solakiewicz, Richard↗
NASA NTRS2005

A Methodology for Quantifying Certain Design Requirements During the Design Phase

A methodology for developing and balancing quantitative design requirements for safety, reliability, and maintainability has been proposed. Conceived as the basis of a more rational approach to the design of spacecraft, the methodology would also be applicable to the design of automobiles, washing machines, television receivers, or almost any other commercial product. Heretofore, it has been common practice to start by determining the requirements for reliability of elements of a spacecraft or other system to ensure a given design life for the system. Next, safety requirements are determined by assessing the total reliability of the system and adding redundant components and subsystems necessary to attain safety goals. As thus described, common practice leaves the maintainability burden to fall to chance; therefore, there is no control of recurring costs or of the responsiveness of the system. The means that have been used in assessing maintainability have been oriented toward determining the logistical sparing of components so that the components are available when needed. The process established for developing and balancing quantitative requirements for safety (S), reliability (R), and maintainability (M) derives and integrates NASA s top-level safety requirements and the controls needed to obtain program key objectives for safety and recurring cost (see figure). Being quantitative, the process conveniently uses common mathematical models. Even though the process is shown as being worked from the top down, it can also be worked from the bottom up. This process uses three math models: (1) the binomial distribution (greaterthan- or-equal-to case), (2) reliability for a series system, and (3) the Poisson distribution (less-than-or-equal-to case). The zero-fail case for the binomial distribution approximates the commonly known exponential distribution or "constant failure rate" distribution. Either model can be used. The binomial distribution was selected for modeling flexibility because it conveniently addresses both the zero-fail and failure cases. The failure case is typically used for unmanned spacecraft as with missiles.

Adams, Timothy↗
NASA NTRS1986

The radial distribution of galactic gamma rays. III - The distribution of cosmic rays in the Galaxy and the CO-H2 calibration

The cosmic ray (CR) distribution in the Galaxy inside the solar circle is determined by using the kinematics of both H I and CO to ascertain the spatial distribution of the interstellar gas. The conversion factor between integrated CO line intensity and the molecular hydrogen column density is found, and possible large-scale variations of this relationship throughout the Galaxy are investigated. The galactocentric CR distributions are shown to be described satisfactorily by exponential distributions for radii larger than about three, with a radial scale length of 4-11 kpc for electrons and a scale length greater than 18 kpc for nuclei.

Bloemen, J. B. G. M.↗
NASA NTRS1996

Statistical Models of Landscape Pattern and the Effects of Coarse Resolution of Satellite Imagery on Estimation of Area

Analysis of classified satellite imagery was conducted to characterize errors in estimates of area based on coarse resolution satellite imagery which are due to distortions in sizes of small fragments, and to explore the feasibility of correcting for these errors using a statistical modeling approach. Sizes of bodies of open water on ERS-1 SAR and fire scars on Landsat MSS imagery were measured. Statistical analysis of the smaller scars and ponds as observed with this imagery of relatively fine resolution demonstrated that the distribution of the sizes could be modeled by either of two types of statistical distributions - a power distribution related to fractal processes or a simple exponential distribution. Comparison of the distribution of small bum scars as observed with Landsat to the distribution observed with AVHRR showed distortions due to the coarse spatial resolution of AVHRR caused a net overestimation of bum area. This bias was primarily caused by detection in 2 or 3 AVHRR pixels of bums whose true size was on the order of an AVHRR pixel.

Hlavka, Christine A.↗