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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.
A method first presented by Goodman is used to derive an equation for the statistical effects associated with laser returns from satellites having retroreflecting arrays of cube corners. The effect of the distribution on the returns of a satellite-tracking system is illustrated by a computation based on randomly generated numbers.
A computer program was developed to generate random numbers on the Sigma V computer. A subroutine gave a random real number from a uniform distribution on the interval (0,1). A normal distribution with mean zero and variance 1 was also obtained.
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 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.
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.
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.
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 Linear Autoland Simulink model was created to be a modular test environment for testing of control system components in commercial aircraft. The input variables, physical laws, and referenced frames used are summarized. The state space theory underlying the model is surveyed and the location of the control actuators described. The equations used to realize the Dryden gust model to simulate winds and gusts are derived. A description of the pseudo-random number generation method used in the wind gust model is included. The longitudinal autopilot, lateral autopilot, automatic throttle autopilot, engine model and automatic trim devices are considered as subsystems. The experience in converting the Airlabs FORTRAN aircraft control system simulation to a graphical simulation tool (Matlab/Simulink) is described.
It has been demonstrated that classical probabilities, and in particular, probabilistic Turing machine, can be simulated by combining chaos and non-Lipschitz dynamics, without utilization of any man-made devices(such as random number generators). Self-orgainizing properties of systems coupling simulated and calculated probabilities and their link to quantum computations are discussed. Special attention was focused upon coupled stochastic processes, defined in terms of conditional probabilities, for which joint probability does not exist. Simulations of quantum probabilities are also discussed.
It has been demonstrated that classical probabilities, and in particular, probabilistic Turing machine, can be simulated by combining chaos and non-LIpschitz dynamics, without utilization of any man-made devices (such as random number generators). Self-organizing properties of systems coupling simulated and calculated probabilities and their link to quantum computations are discussed.
A document discusses the emergence of randomness in solutions of coupled, fully deterministic ODE-PDE (ordinary differential equations-partial differential equations) due to failure of the Lipschitz condition as a new phenomenon. It is possible to exploit the special properties of ordinary differential equations (represented by an arbitrarily chosen, dynamical system) coupled with the corresponding Liouville equations (used to describe the evolution of initial uncertainties in terms of joint probability distribution) in order to simulate stochastic processes with the proscribed probability distributions. The important advantage of the proposed approach is that the simulation does not require a random-number generator.
There are complex structures for which the installation and positioning of the lightning protection system (LPS) cannot be done using the lightning protection standard guidelines. As a result, there are some "unprotected" or "exposed" areas. In an effort to quantify the lightning threat to these areas, a Monte Carlo statistical tool has been developed. This statistical tool uses two random number generators: a uniform distribution to generate origins of downward propagating leaders and a lognormal distribution to generate returns stroke peak currents. Downward leaders propagate vertically downward and their striking distances are defined by the polarity and peak current. Following the electrogeometrical concept, we assume that the leader attaches to the closest object within its striking distance. The statistical analysis is run for 10,000 years with an assumed ground flash density and peak current distributions, and the output of the program is the probability of direct attachment to objects of interest with its corresponding peak current distribution.
There are complex structures for which the installation and positioning of the lightning protection system (LPS) cannot be done using the lightning protection standard guidelines. As a result, there are some "unprotected" or "exposed" areas. In an effort to quantify the lightning threat to these areas, a Monte Carlo statistical tool has been developed. This statistical tool uses two random number generators: a uniform distribution to generate the origin of downward propagating leaders and a lognormal distribution to generate the corresponding returns stroke peak currents. Downward leaders propagate vertically downward and their striking distances are defined by the polarity and peak current. Following the electrogeometrical concept, we assume that the leader attaches to the closest object within its striking distance. The statistical analysis is run for N number of years with an assumed ground flash density and the output of the program is the probability of direct attachment to objects of interest with its corresponding peak current distribution.
We have identified a star, EPIC 249706694 (HD 139139), that was observed during K2 Campaign 15 with the Kepler extended mission that appears to exhibit 28 transit-like events over the course of the 87-day observation. The unusual aspect of these dips, all but two of which have depths of 200 ± 80 ppm, is that they exhibit no periodicity, and their arrival times could just as well have been produced by a random number generator. We show that no more than four of the events can be part of a periodic sequence. We have done a number of data quality tests to ascertain that these dips are of astrophysical origin, and while we cannot be absolutely certain that this is so, they have all the hallmarks of astrophysical variability on one of two possible host stars (a likely bound pair) in the photometric aperture. We explore a number of ideas for the origin of these dips, including actual planet transits due to multiple or dust emitting planets, anomalously large TTVs, S- and P-type transits in binary systems, a collection of dust-emitting asteroids, ‘dipper-star’ activity, and short-lived starspots. All transit scenarios that we have been able to conjure up appear to fail, while the intrinsic stellar variability hypothesis would be novel and untested.
In geophysics, we are confronted with many under-determined inverse problems. For example, all of our observations of earthquakes are made at the Earth’s surface. So, when we try to infer how slip during an earthquake evolves in space and time, we find that there are many potential slip histories that are consistent with our limited observations and our understanding of earthquake physics. One way to approach these problems is with Bayesian analysis which allows us to infer the ensemble of all potential slip models that satisfy the observations and our prior knowledge of earthquake physics. In Bayesian analysis, our prior knowledge is known as the prior probability density function or prior PDF, the fit to the data is known as the data likelihood, and the target PDF that satisfies both the prior PDF and data likelihood is known as the posterior PDF. However, simulating the posterior PDF typically requires using Markov Chain Monte Carlo (MCMC) to draw tens of billions of random realizations of earthquake slip models, which may not be computationally feasible. To make this and similar geophysical inversions computationally tractable, we developed the Cascading Adaptive Transitional Metropolis In Parallel (CATMIP) algorithm. CATMIP is an efficient parallel Markov Chain Monte Carlo (MCMC) sampler that is used for model fitting and uncertainty quantification in geophysics. Example use cases are earthquake rupture modeling, determining mineral composition on Mars, reconstructing the history of ocean salinity, and historical earthquake relocation. CATMIP employs many parallel instances of the Metropolis algorithm for sampling in a transitioning framework. Transitioning is a process in which a set of random samples at equilibrium with a known probability density function (PDF) are used as seeds for the Markov chains to sample successive target PDFs that incrementally move the distribution from the starting seeds to the final desired PDF that describes the relative plausibility of potential values for the model parameters. The algorithm is implemented as a Master-Worker model employing MPI for communication. The worker processes are loosely coupled with global parameters periodically optimized by the master process. This provides a very high amount of parallelism with little communication between updates. During the presentation we will discuss the history of the algorithm and elaborate the earthquake rupture modeling use case for the CATMIP package. Our first step toward GPU optimization was to optimize the code for the CPU. CPU profiling revealed that most of the compute time is spent in calls to level 2 BLAS routines and calls to GSL random number generators. We revised the algorithm to employ level 3 BLAS routines instead. In our presentation we will describe how this was accomplished. Adding GPU support to CATMIP consisted mostly of replacing the calls to GSL with calls to GPU vendor-provided library routines. A small number of loops were directly implemented in CUDA. In the presentation will provide implementation details. Finally, we will discuss methods for profiling and opportunities for further optimizing GPU execution. By creating a code with the flexibility to run on either a CPU or GPU architecture, CATMIP can be used on systems ranging from large CPU-based HPC environments to single servers with GPU acceleration and everything in between.
Methods for generating pseudorandom numbers from nonuniform statistical distributions