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Various statistical tests that were used to check out random number generators are described. A total of twelve different tests were considered, and from these, six were chosen to be used. The frequency test, max t test, run test, lag product test, gap test, and the matrix test are included.
Two methods are described to simulate, on a digital computer, a set of correlated, stationary, and Gaussian time series with zero mean from the given matrix of power spectral densities and cross spectral densities. The first method is based upon trigonometric series with random amplitudes and deterministic phase angles. The random amplitudes are generated by using a standard random number generator subroutine. An example is given which corresponds to three components of wind velocities at two different spatial locations for a total of six correlated time series. In the second method, the whole process is carried out using the Fast Fourier Transform approach. This method gives more accurate results and works about twenty times faster for a set of six correlated time series.
As many as three iterated statistical model deletion procedures were considered for an experiment. Population model coefficients were chosen to simulate a saturated 2 to the 4th power experiment having an unfavorable distribution of parameter values. Using random number studies, three model selection strategies were developed, namely, (1) a strategy to be used in anticipation of large coefficients of variation, approximately 65 percent, (2) a strategy to be sued in anticipation of small coefficients of variation, 4 percent or less, and (3) a security regret strategy to be used in the absence of such prior knowledge.
A statistical decision procedure called chain pooling had been developed for model selection in fitting the results of a two-level fixed-effects full or fractional factorial experiment not having replication. The basic strategy included the use of one nominal level of significance for a preliminary test and a second nominal level of significance for the final test. The subject has been reexamined from the point of view of using as many as three successive statistical model deletion procedures in fitting the results of a single experiment. The investigation consisted of random number studies intended to simulate the results of a proposed aircraft turbine-engine rotor-burst-protection experiment. As a conservative approach, population model coefficients were chosen to represent a saturated 2 to the 4th power experiment with a distribution of parameter values unfavorable to the decision procedures. Three model selection strategies were developed.
As many as three iterated statistical model deletion procedures are considered for an experiment. Population model coefficients were chosen to simulate a saturated factorial experiment having an unfavorable distribution of parameter values. Using random number studies, three model selection strategies were developed, namely, (1) a strategy to be used in anticipation of large coefficients of variation (neighborhood of 65 percent), (2) strategy to be used in anticipation of small coefficients of variation (4 percent or less), and (3) a security regret strategy to be used in the absence of such prior knowledge.
Population model coefficients were chosen to simulate a saturated 2 to the fourth power fixed effects experiment having an unfavorable distribution of relative values. Using random number studies, deletion strategies were compared that were based on the F distribution, on an order statistics distribution of Cochran's, and on a combination of the two. Results of the comparisons and a recommended strategy are given.
Population model coefficients were chosen to simulate a saturated 2 to the 4th fixed-effects experiment having an unfavorable distribution of relative values. Using random number studies, deletion strategies were compared that were based on the F-distribution, on an order statistics distribution of Cochran's, and on a combination of the two. The strategies were compared under the criterion of minimizing the maximum prediction error, wherever it occurred, among the two-level factorial points. The strategies were evaluated for each of the conditions of 0, 1, 2, 3, 4, 5, or 6 center points. Three classes of strategies were identified as being appropriate, depending on the extent of the experimenter's prior knowledge. In almost every case the best strategy was found to be unique according to the number of center points. Among the three classes of strategies, a security regret class of strategy was demonstrated as being widely useful in that over a range of coefficients of variation from 4 to 65%, the maximum predictive error was never increased by more than 12% over what it would have been if the best strategy had been used for the particular coefficient of variation. The relative efficiency of the experiment, when using the security regret strategy, was examined as a function of the number of center points, and was found to be best when the design used one center point.
Aerosol and cloud measurements have been simulated for a Space Shuttle lidar. Expected errors - in signal, transmission, density, and calibration - are calculated algebraically and checked by simulating measurements and retrievals using random-number generators. By day, vertical structure is retrieved for tenuous clouds, Saharan aerosols, and boundary layer aerosols (at 0.53 and 1.06 micron) as well as strong volcanic stratospheric aerosols (at 0.53 micron). By night, all these constituents are retrieved plus upper tropospheric and stratospheric aerosols (at 1.06 micron), mesospheric aerosols (at 0.53 micron), and noctilucent clouds (at 1.06 and 0.53 micron). The vertical resolution was 0.1-0.5 km in the troposphere, 0.5-2.0 km above, except 0.25-1.0 km in the mesospheric cloud and aerosol layers; horizontal resolution was 100-2000 km.
The interaction of energetic auroral protons with the atmosphere is investigated. The results of a random number algorithm that describes the proton-hydrogen interconversion reactions as the beam loses energy are adopted to construct an energy deposition curve applicable over a wide range of initial proton energies. Ionization rates and production rates of ejected electrons are computed and emission rates of hydrogen Balmer alpha and beta lines are evaluated using recently available low energy cross-sections.
The effects of random member length errors on the surface accuracy, the defocus, and the residual, internal loads of tetrahedral truss antenna reflectors have been studied analytically. The analytical procedure involves performing multiple, deterministic finite element structural analyses for a particular truss. For each analysis, the normally distributed, random member length errors are selected by random number generator. A best fit paraboloid analysis is used to determine a root mean square error and defocus from each analysis. The statistical properties of these quantities as well as the internal loads are calculated from the results of many independent analyses. Results indicate that the number of members in a tetrahedral truss antenna of a given diameter has a significant effect on surface accuracy, defocus and internal loads. It was also found that the member axial stiffnesses and antenna focal length have a very small effect on reflector surface accuracy.
Data obtained from photographs of lightning are presented on the tortuosity of return stroke channels. The data were obtained by making piecewise linear fits to the channels, and recording the cartesian coordinates of the ends of each linear segment. The mean change between ends of the segments was nearly zero in the horizontal direction and was about eight meters in the vertical direction. Histograms of these changes are presented. These data were used to create model lightning channels and to predict the electric fields radiated during return strokes. This was done using a computer generated random walk in which linear segments were placed end-to-end to form a piecewise linear representation of the channel. The computer selected random numbers for the ends of the segments assuming a normal distribution with the measured statistics. Once the channels were simulated, the electric fields radiated during a return stroke were predicted using a transmission line model on each segment. It was found that realistic channels are obtained with this procedure, but only if the model includes two scales of tortuosity: fine scale irregularities corresponding to the local channel tortuosity which are superimposed on large scale horizontal drifts. The two scales of tortuosity are also necessary to obtain agreement between the electric fields computed mathematically from the simulated channels and the electric fields radiated from real return strokes. Without large scale drifts, the computed electric fields do not have the undulations characteristics of the data.
A numerical study was performed on the effects of stochastic geometry and random environmental torques on the pointing accuracy of spinning and three-axis stabilized satellites. The Euler equations for the motions of satellites yielded the stochastic principal moments and a Fokker-Planck analog simulation was employed to model the response vector of a rigid satellite. White noise inputs were obtained from a random number generator. Attention was given to both fast and slow spin satellites. Responses grew with time in all cases. The growth was initially exponential and eventually reached stable values. In comparison with other theoretical techniques, the Fokker-Planck model detailed the most responses and showed that spin-stabilization is preferable for satellites with low inertial noise levels.
Algorithms yield feedback shift registers with maximum regularity. Report provides extensive mathematical treatment of new and previous results related to generation of pseudo-noise binary sequences by feedback shift registers. Generator architectures amenable to efficient implementation in very-large-scale integrated (VLSI) circuits. Report includes literature references to applications of such sequences in random-number generation, radar, VLSI testing, data encryption and decryption, algebraic error-detection and error-correction encoding and decoding, and feedback-shift-register synthesis of sequential machines.
The massively parallel processor is used to perform Monte Carlo simulations for the two dimensional XY model on lattices of sizes up to 128 x 128. A parallel random number generator was constructed, finite size effects were studied, and run times were compared with those on a CRAY X-MP supercomputer.
Self-organization and learning is a distinctive feature of neural nets and processors that sets them apart from conventional approaches to signal processing. It leads to self-programmability which alleviates the problem of programming complexity in artificial neural nets. In this paper architectures for partitioning an optoelectronic analog of a neural net into distinct layers with prescribed interconnectivity pattern to enable stochastic learning by simulated annealing in the context of a Boltzmann machine are presented. Stochastic learning is of interest because of its relevance to the role of noise in biological neural nets. Practical considerations and methodologies for appreciably accelerating stochastic learning in such a multilayered net are described. These include the use of parallel optical computing of the global energy of the net, the use of fast nonvolatile programmable spatial light modulators to realize fast plasticity, optical generation of random number arrays, and an adaptive noisy thresholding scheme that also makes stochastic learning more biologically plausible. The findings reported predict optoelectronic chips that can be used in the realization of optical learning machines.
This report presents work performed between June 1990 and June 1991 and has the following objectives: (1) a comprehensive literature search of imaging technology and coded aperture imaging as well as relevant topics relating to solar flares; (2) an analysis of random number generators; and (3) programming simulation models of hard x ray telescopes. All programs are compatible with NASA/MSFC Space Science LAboratory VAX Cluster and are written in VAX FORTRAN and VAX IDL (Interactive Data Language).
This paper discusses the issue of statistical characterization of life drivers for a probabilistic design analysis (PDA) approach to support the conventional deterministic structural design methods that are currently used. The probabilistic approach takes into consideration the modeling inadequacies and uncertainties in many design variables such as loads, environments, and material properties. The importance of the distributional assumption is motivated by illustrating an example where the results differ substantially due to the distribution selected. Different types of distributions are discussed and techniques for estimating the parameters are given. Given this information, procedures are outlined for selecting the appropriate distribution based on the particular type of variable (i.e., dimensional, performance) as well as the information that is available (i.e., test data, engineering analysis). Finally, techniques are given for generating random numbers from these selected distributions within the PDA process.
The original objectives of this project were to build a test system for the NASA 255/223 Reed/Solomon encoding/decoding chip set and circuit board. This test system was then to be interfaced with a convolutional system at MSFC to examine the performance of the concantinated codes. After considerable work, it was discovered that the convolutional system could not function as needed. This report documents the design, construction, and testing of the test apparatus for the R/S chip set. The approach taken was to verify the error correcting behavior of the chip set by injecting known error patterns onto data and observing the results. Error sequences were generated using pseudo-random number generator programs, with Poisson time distribution between errors and Gaussian burst lengths. Sample means, variances, and number of un-correctable errors were calculated for each data set before testing.