Numerous models of physical systems contain parameters whose values are not known exactly. The physical and mathematical complexities arising in the prediction of the statistical behavior of such systems are discussed. Although the discussions are far from providing a satisfactory solution to such problems, they perhaps, by utilization of simple examples, will create a greater awareness of the statistical effect of random parameters.

Topics: NASA Technical Reports Server (NTRS), DYNAMIC RESPONSE, LINEAR SYSTEMS, RANDOM VARIABLES,...

Determining the distribution of statistics by sampling was investigated. Characteristic functions, the quadratic regression problem, and the differential equations for the characteristic functions are analyzed.

Topics: NASA Technical Reports Server (NTRS), DATA SAMPLING, STATISTICAL DISTRIBUTIONS, DIFFERENTIAL...

Under the assumptions that 1) the search region can be divided up into N non-overlapping sub-regions that are searched sequentially, 2) the probability of detection is unity if a sub-region is selected, and 3) no information is available to guide the search, there are two extreme case models. The search can be done perfectly, leading to a uniform distribution over the number of searches required, or the search can be done with no memory, leading to a geometric distribution for the number of...

Topics: NASA Technical Reports Server (NTRS), PROBABILITY THEORY, TARGETS, MATHEMATICAL MODELS, DETECTION,...

A stochastic approximation algorithm for estimating the proportions in a mixture of normal densities is presented. The algorithm is shown to converge to the true proportions in the case of a mixture of two normal densities.

Topics: NASA Technical Reports Server (NTRS), ALGORITHMS, APPROXIMATION, MIXTURES, STOCHASTIC PROCESSES,...

A beta distribution is specified by range parameters a b, and two shape parameters alpha and beta 0. The computer program presented calculates any desired probability and/or fractile point for specified values of a, b, alpha, and beta. This program additionally computes gamma function values for integer and noninteger arguments.

Topics: NASA Technical Reports Server (NTRS), COMPUTER PROGRAMS, PROBABILITY DISTRIBUTION FUNCTIONS,...

An algorithm is presented to relabel automatically the nodes of an arbitrary finite-element mesh. The purpose of such relabeling is to reduce the bandwidth of the master stiffness matrix produced by the finite-element method. The algorithm uses a random process for the relabeling. Computing time is reduced substantially, compared to systematic methods.

Topics: NASA Technical Reports Server (NTRS), ALGORITHMS, COMPUTER TECHNIQUES, FINITE ELEMENT METHOD,...

The optimal partitioning of random distributed programs is discussed. It is concluded that the optimal partitioning of a homogeneous random program over a homogeneous distributed system either assigns all modules to a single processor, or distributes the modules as evenly as possible among all processors. The analysis rests heavily on the approximation which equates the expected maximum of a set of independent random variables with the set's maximum expectation. The results are strengthened by...

Topics: NASA Technical Reports Server (NTRS), COMPUTER NETWORKS, DISTRIBUTED PROCESSING, MULTIPROCESSING...

Previously, the performance of the split-symbol moments estimator (SSME) of signal-to-noise ratio (SNR) has been evaluated by means of approximate expressions for the estimator mean and variance. These are asymptotic formulas in the sense that they become accurate as the number of estimator samples gets large. Here, exact closed-form expressions are obtained for the same quantities. These expressions confirm the accuracy of the previously derived asymptotic results, and, unlike the asymptotic...

Topics: NASA Technical Reports Server (NTRS), ESTIMATING, SIGNAL TO NOISE RATIOS, STATISTICAL ANALYSIS,...

Recent advances in observational techniques related to geodetic work (VLBI, laser ranging) make it imperative that more consideration should be given to modeling problems. Uncertainties in the effect of atmospheric refraction, polar motion and precession-nutation parameters, cannot be dispensed with in the context of centimeter level geodesy. Even physical processes that have generally been previously altogether neglected (station motions) must now be taken into consideration. The problem of...

Topics: NASA Technical Reports Server (NTRS), GEODESY, PROBABILITY THEORY, STATISTICAL ANALYSIS, STOCHASTIC...

An examination of the ages and sizes of 114 terrestrial impact craters shows that their aging kinetics can be described by the diffusion laws. The macrodiffusion coefficient which determines random displacements of mineral masses on the Earth has a mean value of 0.02 sq m/year. The amount of matter in a crater that contains information about the impact event decreases with time according to the 1/T law. The basic characteristic parameter of a crater is its initial area, inasmuch as sufficiently...

Topics: NASA Technical Reports Server (NTRS), AGING (MATERIALS), CRATERS, EARTH (PLANET), IMPACT DAMAGE,...

Methods for obtaining the probability density function of random signal combustions are discussed. These methods provide a realistic criteria for the design of control systems subjected to external noise with several important applications for aerospace problems.

Topics: NASA Technical Reports Server (NTRS), CONVOLUTION INTEGRALS, PROBABILITY DENSITY FUNCTIONS, RANDOM...

These slides describe the data analysis methods that are used to determine inputs for probabilistic risk models supporting the Space Shuttle Program. Other applications can follow a similar path probably using different data sources. Statistical approaches are different and not addressed here. Topics included here: 1) Prior Distribution; 2) Likelihood Data; 3) Bayesian Updating; and 4) Uncertainty and Error. Note: This is a high-level discussion and is not intended to be a tutorial.

Topics: NASA Technical Reports Server (NTRS), SPACE SHUTTLES, PROBABILITY THEORY, DATA PROCESSING,...

This report contains a complete list of EOS/AMSU-A commands for all modes of operation and testing with a description of their effects. There are no EOS/AMSU-A critical commands i.e., commands that could damage the instrument in any situation.

Topics: NASA Technical Reports Server (NTRS), COMPUTER PROGRAMS, EARTH OBSERVING SYSTEM (EOS), MICROWAVE...

Standard statistical processing of anomaly data in the screwworm eradication data system is possible from data compiled on magnetic tapes with the Univac 1108 computer. The format and organization of the data in the data base, which is also available on dedicated disc storage, are described.

Topics: NASA Technical Reports Server (NTRS), ANOMALIES, DATA BASES, LARVAE, RANDOM VARIABLES, COMPUTER...

The NESSUS and NASTRAN computer codes were successfully integrated. The enhanced NESSUS code will use NASTRAN for the structural Analysis and NESSUS for the probabilistic analysis. Any quantities in the NASTRAN bulk data input can be random variables. Any NASTRAN result that is written to the output2 file can be returned to NESSUS as the finite element result. The interfacing between NESSUS and NASTRAN is handled automatically by NESSUS. NESSUS and NASTRAN can be run on different machines using...

Topics: NASA Technical Reports Server (NTRS), FINITE ELEMENT METHOD, PROBABILITY DISTRIBUTION FUNCTIONS,...

Failure times of software undergoing random debugging can be modelled as order statistics of independent but nonidentically distributed exponential random variables. Using this model inferences can be made about current reliability and, if debugging continues, future reliability. This model also shows the difficulty inherent in statistical verification of very highly reliable software such as that used by digital avionics in commercial aircraft.

Topics: NASA Technical Reports Server (NTRS), APPLICATIONS PROGRAMS (COMPUTERS), FAULT TOLERANCE, QUALITY...

At the NASA Glenn Research Center, the primal and dual integrated force methods are being extended for the stochastic analysis of structures. The stochastic simulation can be used to quantify the consequence of scatter in stress and displacement response because of a specified variation in input parameters such as load (mechanical, thermal, and support settling loads), material properties (strength, modulus, density, etc.), and sizing design variables (depth, thickness, etc.). All the...

Topics: NASA Technical Reports Server (NTRS), STOCHASTIC PROCESSES, PROBABILITY DISTRIBUTION FUNCTIONS,...

The problem of estimating parameters of dynamic systems is addressed in order to present the theoretical basis of system identification and parameter estimation in a manner that is complete and rigorous, yet understandable with minimal prerequisites. Maximum likelihood and related estimators are highlighted. The approach used requires familiarity with calculus, linear algebra, and probability, but does not require knowledge of stochastic processes or functional analysis. The treatment...

Topics: NASA Technical Reports Server (NTRS), MAXIMUM LIKELIHOOD ESTIMATES, OPTIMIZATION, PARAMETER...

A FORTRAN 4 routine is given which may be used to generate random observations of a continuous real valued random variable. Normal distribution of F(x), X, E(akimas), and E(linear) is presented in tabular form.

Topics: NASA Technical Reports Server (NTRS), FORTRAN, RANDOM VARIABLES, INPUT/OUTPUT ROUTINES, STATISTICAL...

A model of the petroleum exploration process that tests empirically the hypothesis that at an early stage in the exploration of a basin, the process behaves like sampling without replacement is proposed along with a model of the spatial distribution of petroleum reserviors that conforms to observed facts. In developing the model of discovery, the following topics are discussed: probabilitistic proportionality, likelihood function, and maximum likelihood estimation. In addition, the spatial...

Topics: NASA Technical Reports Server (NTRS), CRUDE OIL, MATHEMATICAL MODELS, OIL EXPLORATION, FREQUENCY...

In this paper we develop the mathematical theory of proportional and scale change models to perform reliability analysis. The results obtained will be applied for the Reaction Control System (RCS) thruster valves on an orbiter. With the advent of extended EVA's associated with PROX OPS (ISSA & MIR), and docking, the loss of a thruster valve now takes on an expanded safety significance. Previous studies assume a homogeneous population of components with each component having the same failure...

Topics: NASA Technical Reports Server (NTRS), MATHEMATICAL MODELS, RELIABILITY ANALYSIS, VALVES, ROCKET...

The NESSUS probabilistic analysis computer program has been developed with a built-in finite element analysis program NESSUS/FEM. However, the NESSUS/FEM program is specialized for engine structures and may not contain sufficient features for other applications. In addition, users often become well acquainted with a particular finite element code and want to use that code for probabilistic structural analysis. For these reasons, this work was undertaken to develop an interface between NESSUS...

Topics: NASA Technical Reports Server (NTRS), COMPUTER PROGRAMS, FINITE ELEMENT METHOD, INTERFACES,...

Starting with an n-dimensional variable whose density function is a convex combination of normal densities, the equations for a maximum-likelihood estimate are obtained. Iterative procedures for obtaining solutions to the likelihood equations are discussed along with conditions for local convergence.

Topics: NASA Technical Reports Server (NTRS), NUMERICAL ANALYSIS, PROBABILITY THEORY, ESTIMATES, ITERATION,...

The cause of the long-suspected excessive conservatism in the prevailing structural deterministic safety factor has been identified as an inherent violation of the error propagation laws when reducing statistical data to deterministic values and then combining them algebraically through successive structural computational processes. These errors are restricted to the applied stress computations, and because mean and variations of the tolerance limit format are added, the errors are positive,...

Topics: NASA Technical Reports Server (NTRS), STRUCTURAL ANALYSIS, SAFETY FACTORS, STRUCTURAL RELIABILITY,...

Predictions about the future are contingent on future usage, but also on the quality of the models employed and the assessment of the current health state. These factors, amongst others, need to be considered to arrive at a prediction that is conducted through a rigorous method but where the confidence bounds are not prohibitively large.

Topics: NASA Technical Reports Server (NTRS), INVENTORY MANAGEMENT, DECISION MAKING, PROBABILITY THEORY,...

For univariate random sequences, the power spectral density acts like a probability density function of the frequencies present in the sequence. This dissertation extends that concept to bivariate random sequences. For this purpose, a function called the joint spectral density is defined that represents a joint probability weighing of the frequency content of pairs of random sequences. Given a pair of random sequences, the joint spectral density is not uniquely determined in the absence of any...

Topics: NASA Technical Reports Server (NTRS), BIVARIATE ANALYSIS, DENSITY DISTRIBUTION, POWER SPECTRA,...

The possibility of using on-line signal statistics to detect electronic equipment nonlinearities is discussed. The results of an investigation using Gaussian statistics are presented, and a nonlinearity test that uses ratios of the moments of a Gaussian random variable is developed and discussed. An outline for further investigation is presented.

Topics: NASA Technical Reports Server (NTRS), FAILURE, NONLINEAR SYSTEMS, NORMAL DENSITY FUNCTIONS, RANDOM...

This paper presents a reliability- and robustness-based formulation for robust control synthesis for systems with probabilistic uncertainty. In a reliability-based formulation, the probability of violating design requirements prescribed by inequality constraints is minimized. In a robustness-based formulation, a metric which measures the tendency of a random variable/process to cluster close to a target scalar/function is minimized. A multi-objective optimization procedure, which combines...

Topics: NASA Technical Reports Server (NTRS), ROBUSTNESS (MATHEMATICS), CONTROLLERS, PROBABILITY THEORY,...

We provide a framework to bound the probability that accumulated errors were never above a given threshold on numerical algorithms. Such algorithms are used for example in aircraft and nuclear power plants. This report contains simple formulas based on Levy's and Markov's inequalities and it presents a formal theory of random variables with a special focus on producing concrete results. We selected four very common applications that fit in our framework and cover the common practices of systems...

Topics: NASA Technical Reports Server (NTRS), ALGORITHMS, PROBABILITY THEORY, RANDOM VARIABLES, HYPOTHESES,...

Let X be a two-dimensional random variable distributed according to N2(mu,Sigma) and let bar-X and S be the respective sample mean and covariance matrix calculated from N observations of X. Given a containment probability beta and a level of confidence gamma, we seek a number c, depending only on N, beta, and gamma such that the ellipsoid R = (x: (x - bar-X)'S(exp -1) (x - bar-X) less than or = c) is a tolerance region of content beta and level gamma; i.e., R has probability gamma of containing...

Topics: NASA Technical Reports Server (NTRS), APPROXIMATION, BIVARIATE ANALYSIS, MONTE CARLO METHOD,...

A set of variables which are crucial to the economic outcome of petroleum exploration are discussed. These are treated as random variables; the values they assume indicate the number of successes that occur in a drilling program and determine, for a particular discovery, the unit production cost and net economic return if that reservoir is developed. In specifying the joint probability law for those variables, extreme and probably unrealistic assumptions are made. In particular, the different...

Topics: NASA Technical Reports Server (NTRS), CRUDE OIL, OIL EXPLORATION, COSTS, HISTOGRAMS, MONTE CARLO...

NASA regulations require that formal risk analysis be performed on a program at each of several milestones as it moves toward full-scale development. Program risk analysis is discussed as a systems analysis approach, an iterative process (identification, assessment, management), and a collection of techniques. These techniques, which range from simple to complex network-based simulation were surveyed. A Program Risk Analysis Handbook was prepared in order to provide both analyst and manager...

Topics: NASA Technical Reports Server (NTRS), COST EFFECTIVENESS, NASA PROGRAMS, PROJECT MANAGEMENT, RISK,...

The Probabilistic Dynamic Synthesis method is a new technique for obtaining the statistics of a desired response engineering quantity for a structure with non-deterministic parameters. The method uses measured data from modal testing of the structure as the input random variables, rather than more "primitive" quantities like geometry or material variation. This modal information is much more comprehensive and easily measured than the "primitive" information. The...

Topics: NASA Technical Reports Server (NTRS), RANDOM VARIABLES, SUBSTRUCTURES, DEGREES OF FREEDOM, MONTE...

This paper presents a robust control design methodology based on the estimation of the first two order moments of the random variables and processes that describe the controlled response. Synthesis is performed by solving an multi-objective optimization problem where stability and performance requirements in time- and frequency domains are integrated. The use of the first two order moments allows for the efficient estimation of the cost function thus for a faster synthesis algorithm. While...

Topics: NASA Technical Reports Server (NTRS), PROBABILITY THEORY, ALGORITHMS, SYNTHESIS, MATHEMATICAL...

The goal of any modeling methodology is to develop a mathematical description of a system that is accurate in its representation and also permits analysis of structural and/or performance properties. Inherently, trade-offs exist between the level detail in the model and the ease with which analysis can be performed. Petri nets (PN's), a highly graphical modeling methodology for Discrete Event Dynamic Systems, permit representation of shared resources, finite capacities, conflict,...

Topics: NASA Technical Reports Server (NTRS), ANALYSIS (MATHEMATICS), EXPONENTIAL FUNCTIONS, MATHEMATICAL...

Population models for dependence between two angular measurements and for dependence between an angular and a linear observation are proposed. The method of canonical correlations first leads to new population and sample measures of dependence in this latter situation. An example relating wind direction to the level of a pollutant is given. Next, applied to pairs of angular measurements, the method yields previously proposed sample measures in some special cases and a new sample measure in...

Topics: NASA Technical Reports Server (NTRS), ANGULAR CORRELATION, LINEAR SYSTEMS, RANDOM VARIABLES,...

A Robbins-Monro type multidimensional stochastic approximation algorithm which converges in mean square and with probability one to the fixed point of a locally contractive regression function is developed. The algorithm is applied to obtain maximum likelihood estimates of the parameters for a mixture of multivariate normal distributions.

Topics: NASA Technical Reports Server (NTRS), APPROXIMATION, MAXIMUM LIKELIHOOD ESTIMATES, STOCHASTIC...

Viewgraphs of probabilistic structural analysis for nuclear thermal propulsion are presented. The objective of the study was to develop a methodology to certify Space Nuclear Propulsion System (SNPS) Nozzle with assured reliability. Topics covered include: advantage of probabilistic structural analysis; space nuclear propulsion system nozzle uncertainties in the random variables; SNPS nozzle natural frequency; and sensitivity of primitive variable uncertainties SNPS nozzle natural frequency and...

Topics: NASA Technical Reports Server (NTRS), COMPONENT RELIABILITY, NUCLEAR PROPULSION, PROBABILITY...

The present analysis uses the Monte Carlo method to solve the problem of rarefied diffusion between parallel walls. The diffusing molecules are evaporated or emitted from one of two parallel walls and diffused through another molecular species. The analysis treats the diffusing molecule as undergoing a Markov random walk and the local macroscopic properties are found as the expected value of the random variable, the random walk payoff. By biasing the transition probabilities and changing the...

Topics: NASA Technical Reports Server (NTRS), GASEOUS DIFFUSION, MOLECULAR FLOW, MONTE CARLO METHOD,...

Two-dimensional (2-D) homogeneous magnetohydrodynamic (MHD) turbulence has many of the same qualitative features as three-dimensional (3-D) homogeneous MHD turbulence.The se features include several ideal invariants, along with the phenomenon of broken ergodicity. Broken ergodicity appears when certain modes act like random variables with mean values that are large compared to their standard deviations, indicating a coherent structure or dynamo.Recently, the origin of broken ergodicity in 3-D...

Topics: NASA Technical Reports Server (NTRS), MAGNETOHYDRODYNAMIC TURBULENCE, MAGNETOHYDRODYNAMIC WAVES,...

An obvious goal of reliability analysis is the avoidance of system failure. However, it is generally recognized that it is often not feasible to design a practical or useful system for which failure is impossible. Thus it is necessary to use techniques that estimate the likelihood of failure based on modeling the uncertainty about such items as the demands on and capacities of various elements in the system. This usually involves the use of probability theory, and a design is considered...

Topics: NASA Technical Reports Server (NTRS), RELIABILITY ANALYSIS, PROBABILITY THEORY, STOCHASTIC...

It is shown that all combinations of symmetrical input distributions with difference distortion measures give a bit assignment rule identical to the well-known rule for a Gaussian input distribution with mean-square error. Published work is examined to show that the bit assignment rule is useful for transforms of full pictures, but subjective bit assignments for transform picture coding using small block sizes are significantly different from the theoretical bit assignment rule. An intuitive...

Topics: NASA Technical Reports Server (NTRS), LINEAR TRANSFORMATIONS, OPTICAL DATA PROCESSING, ALGORITHMS,...

An investigation of the moments and probability distribution of the resultant of two normally distributed random variables is presented. This is the so-called generalized Rayleigh distribution which has many applications in the study of wind shear, random noise, and radar. The most general formula was derived, and two special cases were considered for which tables of the moments and probability distribution functions are included as an appendix. One of the special cases was generalized to...

Topics: NASA Technical Reports Server (NTRS), DISTRIBUTION MOMENTS, RADIAL DISTRIBUTION, RANDOM VARIABLES,...

This and a companion paper propose techniques for constructing parametric mathematical models describing key features of the distribution of an output variable given input-output data. By contrast to standard models, which yield a single output value at each value of the input, Random Predictors Models (RPMs) yield a random variable at each value of the input. Optimization-based strategies for calculating RPMs having a polynomial dependency on the input and a linear dependency on the parameters...

Topics: NASA Technical Reports Server (NTRS), RANDOM VARIABLES, MATHEMATICAL MODELS, PREDICTIONS,...

Performability modeling and evaluation techniques are applied to the SIFT computer as it might operate in the computational evironment of an air transport mission. User-visible performance of the total system (SIFT plus its environment) is modeled as a random variable taking values in a set of levels of accomplishment. These levels are defined in terms of four attributes of total system behavior: safety, no change in mission profile, no operational penalties, and no economic process whose...

Topics: NASA Technical Reports Server (NTRS), COMPUTATION, COMPUTERS, MATHEMATICAL MODELS, PERFORMANCE...

Much work has been performed in developing empirical global cloud cover models. This investigation was made to determine an underlying theoretical statistical distribution to represent worldwide cloud cover. The beta distribution with probability density function is given to represent the variability of this random variable. It is shown that the beta distribution possesses the versatile statistical characteristics necessary to assume the wide variety of shapes exhibited by cloud cover. A total...

Topics: NASA Technical Reports Server (NTRS), CLOUD COVER, MATHEMATICAL MODELS, STATISTICAL DISTRIBUTIONS,...

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...

Topics: NASA Technical Reports Server (NTRS), COMPUTERIZED SIMULATION, PROBABILITY THEORY, REGRESSION...

An adaptive filter which uses a minimum variance criteria to estimate state noise covariance is presented. It is not necessary to assume white state noise in order to implement the filter. Simulation results are given which demonstrate that the filter tracks a satellite in the presence of modeling errors better than a conventional minimum variance filter with state noise. It is also shown that the propagated convariance matrix of the filter is an accurate indicator of the filter's performance.

Topics: NASA Technical Reports Server (NTRS), ADAPTIVE FILTERS, MINIMUM VARIANCE ORBIT DETERMINATION,...

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...

Topics: NASA Technical Reports Server (NTRS), TRANSPORTATION NETWORKS, ROUTES, ALGORITHMS, SCHEDULES,...

We investigate the performance of a generalized linear mixed model in predicting the Probabilities of Collision (Pc) for conjunction events. Specifically, we apply this model to the log(sub 10) transformation of these probabilities and argue that this transformation yields values that can be considered bounded in practice. Additionally, this bounded random variable, after scaling, is zero-inflated. Consequently, we model these values using the zero-inflated Beta distribution, and utilize the...

Topics: NASA Technical Reports Server (NTRS), COLLISIONS, PROBABILITY THEORY, BAYES THEOREM, PREDICTIONS,...