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NASA NTRS1975

Distribution and moments of radial error

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 n-dimensions.

White, R. G.↗

NASA NTRS1994

Understanding software faults and their role in software reliability modeling

This study is a direct result of an on-going project to model the reliability of a large real-time control avionics system. In previous modeling efforts with this system, hardware reliability models were applied in modeling the reliability behavior of this system. In an attempt to enhance the performance of the adapted reliability models, certain software attributes were introduced in these models to control for differences between programs and also sequential executions of the same program. As the basic nature of the software attributes that affect software reliability become better understood in the modeling process, this information begins to have important implications on the software development process. A significant problem arises when raw attribute measures are to be used in statistical models as predictors, for example, of measures of software quality. This is because many of the metrics are highly correlated. Consider the two attributes: lines of code, LOC, and number of program statements, Stmts. In this case, it is quite obvious that a program with a high value of LOC probably will also have a relatively high value of Stmts. In the case of low level languages, such as assembly language programs, there might be a one-to-one relationship between the statement count and the lines of code. When there is a complete absence of linear relationship among the metrics, they are said to be orthogonal or uncorrelated. Usually the lack of orthogonality is not serious enough to affect a statistical analysis. However, for the purposes of some statistical analysis such as multiple regression, the software metrics are so strongly interrelated that the regression results may be ambiguous and possibly even misleading. Typically, it is difficult to estimate the unique effects of individual software metrics in the regression equation. The estimated values of the coefficients are very sensitive to slight changes in the data and to the addition or deletion of variables in the regression equation. Since most of the existing metrics have common elements and are linear combinations of these common elements, it seems reasonable to investigate the structure of the underlying common factors or components that make up the raw metrics. The technique we have chosen to use to explore this structure is a procedure called principal components analysis. Principal components analysis is a decomposition technique that may be used to detect and analyze collinearity in software metrics. When confronted with a large number of metrics measuring a single construct, it may be desirable to represent the set by some smaller number of variables that convey all, or most, of the information in the original set. Principal components are linear transformations of a set of random variables that summarize the information contained in the variables. The transformations are chosen so that the first component accounts for the maximal amount of variation of the measures of any possible linear transform; the second component accounts for the maximal amount of residual variation; and so on. The principal components are constructed so that they represent transformed scores on dimensions that are orthogonal. Through the use of principal components analysis, it is possible to have a set of highly related software attributes mapped into a small number of uncorrelated attribute domains. This definitively solves the problem of multi-collinearity in subsequent regression analysis. There are many software metrics in the literature, but principal component analysis reveals that there are few distinct sources of variation, i.e. dimensions, in this set of metrics. It would appear perfectly reasonable to characterize the measurable attributes of a program with a simple function of a small number of orthogonal metrics each of which represents a distinct software attribute domain.

Munson, John C.↗

NASA NTRS1990

Is the Sun a Long Period Variable

The inventory of atmospheric radiocarbon exhibits quasi-periodic variations of mean period of bar-lambda=269 years over the entire 9000 year record. But the period is inconstant and subject to random variability (sigma m exp. 1/2 = 119 years). The radiocarbon maxima correspond to the quasiperiodic extension of the Maunder minimum throughout the Holocene and resolve the long-standing issue of Maunder cyclicity. The radiocarbon maxima are amplitude modulated by the approx. 2300 year period and thus vary significantly in peak value. The approx. 2300 year period in turn appears to not be modulated by the secular geomagnetic variation. Detection of a Maunder-like sequence of minima in tree ring growth of Bristlecone pine and its correlation with the Maunder (1890, 1922) cyclicity in the radiocarbon record supports the inference that solar forcing of the radiocarbon record is accompanied by a corresponding forcing of growth of timberline Bristlecone pine. Because of the random component of the Maunder period, prediction of climate, if tied to the Maunder cycle other than probabilistically, is significantly hindered. For the mean Maunder period of 269 years, the probability is 67 percent that a given climatic maximum lies anywhere between 150 and 388 years.

Sonett, Charles P.↗

NASA NTRS1996

Probabilistic Component Mode Synthesis of Nondeterministic Substructures

Standard methods of structural dynamic analysis assume that the structural characteristics are deterministic. Recognizing that these characteristics are actually statistical in nature researchers have recently developed a variety of methods that use this information to determine probabilities of a desired response characteristic, such as natural frequency, without using expensive Monte Carlo simulations. One of the problems in these methods is correctly identifying the statistical properties of primitive variables such as geometry, stiffness, and mass. We present a method where the measured dynamic properties of substructures are used instead as the random variables. The residual flexibility method of component mode synthesis is combined with the probabilistic methods to determine the cumulative distribution function of the system eigenvalues. A simple cantilever beam test problem is presented that illustrates the theory.

Brown, Andrew M.↗

NASA NTRS1995

Development of a Probabilistic Component Mode Synthesis Method for the Analysis of Non-Deterministic Substructures

Standard methods of structural dynamic analysis assume that the structural characteristics are deterministic. Recognizing that these characteristics are actually statistical in nature, researchers have recently developed a variety of methods that use this information to determine probabilities of a desired response characteristic, such as natural frequency, without using expensive Monte Carlo simulations. One of the problems in these methods is correctly identifying the statistical properties of primitive variables such as geometry, stiffness, and mass. This paper presents a method where the measured dynamic properties of substructures are used instead as the random variables. The residual flexibility method of component mode synthesis is combined with the probabilistic methods to determine the cumulative distribution function of the system eigenvalues. A simple cantilever beam test problem is presented that illustrates the theory.

Brown, Andrew M.↗

NASA NTRS1976

Methods for combining payload parameter variations with input environment

Methods are presented for calculating design limit loads compatible with probabilistic structural design criteria. The approach is based on the concept that the desired limit load, defined as the largest load occurring in a mission, is a random variable having a specific probability distribution which may be determined from extreme-value theory. The design limit load, defined as a particular of this random limit load, is the value conventionally used in structural design. Methods are presented for determining the limit load probability distributions from both time-domain and frequency-domain dynamic load simulations. Numerical demonstrations of the method are also presented.

Merchant, D. H.↗

NASA NTRS1975

Methods for Combining Payload Parameter Variations with Input Environment

Methods are presented for calculating design limit loads compatible with probabilistic structural design criteria. The approach is based on the concept that the desired limit load, defined as the largest load occuring in a mission, is a random variable having a specific probability distribution which may be determined from extreme-value theory. The design limit load, defined as a particular value of this random limit load, is the value conventionally used in structural design. Methods are presented for determining the limit load probability distributions from both time-domain and frequency-domain dynamic load simulations. Numerical demonstrations of the methods are also presented.

Merchant, D. H.↗

NASA NTRS1982

A unified method for evaluating real-time computer controllers: A case study

A real time control system consists of a synergistic pair, that is, a controlled process and a controller computer. Performance measures for real time controller computers are defined on the basis of the nature of this synergistic pair. A case study of a typical critical controlled process is presented in the context of new performance measures that express the performance of both controlled processes and real time controllers (taken as a unit) on the basis of a single variable: controller response time. Controller response time is a function of current system state, system failure rate, electrical and/or magnetic interference, etc., and is therefore a random variable. Control overhead is expressed as a monotonically nondecreasing function of the response time and the system suffers catastrophic failure, or dynamic failure, if the response time for a control task exceeds the corresponding system hard deadline, if any. A rigorous probabilistic approach is used to estimate the performance measures. The controlled process chosen for study is an aircraft in the final stages of descent, just prior to landing. First, the performance measures for the controller are presented. Secondly, control algorithms for solving the landing problem are discussed and finally the impact of the performance measures on the problem is analyzed.

Shin, K. G.↗

NASA NTRS1985

A unified method for evaluating real-time computer controllers and its application

Shin, K. G.↗

NASA NTRS1979

Soft X-ray observations of degenerate dwarfs and cataclysmic variables

Observations in outburst of two dwarf novae, SS Cyg and U Gem, reveal soft, pulsed X-ray emission greatly enhanced over the emission present during the quiescent periods. In order to characterize the nature of the X-ray pulsations a random phase analysis was carried out, which indicated the phase of SS Cyg behaved like a random variable with zero mean and an r.m.s. value that increased with the square root of the time. Attention is also given to the properties of the detected dwarf novae during optical quiescences, to isolated white dwarfs, and to optical observations of AM Her. It is concluded that the dwarf novae, which occur in binary systems, appear to be controlled by the varying accretion of matter generated by the main sequence companion, and that the X-ray emission during outburst appears to consist of two separate components: a periodically varying (on about a 10 s timescale) soft component, together with a more slowly randomly varying hard component.

Garmire, G.↗

NASA NTRS2000

Demonstration of the Application of Composite Load Spectra (CLS) and Probabilistic Structural Analysis (PSAM) Codes to SSME Heat Exchanger Turnaround Vane

This report describes a probabilistic structural analysis performed to determine the probabilistic structural response under fluctuating random pressure loads for the Space Shuttle Main Engine (SSME) turnaround vane. It uses a newly developed frequency and distance dependent correlation model that has features to model the decay phenomena along the flow and across the flow with the capability to introduce a phase delay. The analytical results are compared using two computer codes SAFER (Spectral Analysis of Finite Element Responses) and NESSUS (Numerical Evaluation of Stochastic Structures Under Stress) and with experimentally observed strain gage data. The computer code NESSUS with an interface to a sub set of Composite Load Spectra (CLS) code is used for the probabilistic analysis. A Fatigue code was used to calculate fatigue damage due to the random pressure excitation. The random variables modeled include engine system primitive variables that influence the operating conditions, convection velocity coefficient, stress concentration factor, structural damping, and thickness of the inner and outer vanes. The need for an appropriate correlation model in addition to magnitude of the PSD is emphasized. The study demonstrates that correlation characteristics even under random pressure loads are capable of causing resonance like effects for some modes. The study identifies the important variables that contribute to structural alternate stress response and drive the fatigue damage for the new design. Since the alternate stress for the new redesign is less than the endurance limit for the material, the damage due high cycle fatigue is negligible.

Rajagopal, Kadambi R.↗

NASA NTRS2010

Notes on SAW Tag Interrogation Techniques

We consider the problem of interrogating a single SAW RFID tag with a known ID and known range in the presence of multiple interfering tags under the following assumptions: (1) The RF propagation environment is well approximated as a simple delay channel with geometric power-decay constant alpha >/= 2. (2) The interfering tag IDs are unknown but well approximated as independent, identically distributed random samples from a probability distribution of tag ID waveforms with known second-order properties, and the tag of interest is drawn independently from the same distribution. (3) The ranges of the interfering tags are unknown but well approximated as independent, identically distributed realizations of a random variable rho with a known probability distribution f(sub rho) , and the tag ranges are independent of the tag ID waveforms. In particular, we model the tag waveforms as random impulse responses from a wide-sense-stationary, uncorrelated-scattering (WSSUS) fading channel with known bandwidth and scattering function. A brief discussion of the properties of such channels and the notation used to describe them in this document is given in the Appendix. Under these assumptions, we derive the expression for the output signal-to-noise ratio (SNR) for an arbitrary combination of transmitted interrogation signal and linear receiver filter. Based on this expression, we derive the optimal interrogator configuration (i.e., transmitted signal/receiver filter combination) in the two extreme noise/interference regimes, i.e., noise-limited and interference-limited, under the additional assumption that the coherence bandwidth of the tags is much smaller than the total tag bandwidth. Finally, we evaluate the performance of both optimal interrogators over a broad range of operating scenarios using both numerical simulation based on the assumed model and Monte Carlo simulation based on a small sample of measured tag waveforms. The performance evaluation results not only provide guidelines for proper interrogator design, but also provide some insight on the validity of the assumed signal model. It should be noted that the assumption that the impulse response of the tag of interest is known precisely implies that the temperature and range of the tag are also known precisely, which is generally not the case in practice. However, analyzing interrogator performance under this simplifying assumption is much more straightforward and still provides a great deal of insight into the nature of the problem.

Barton, Richard J.↗

NASA NTRS2005

Probabilistic Reasoning for Plan Robustness

A planning system must reason about the uncertainty of continuous variables in order to accurately project the possible system state over time. A method is devised for directly reasoning about the uncertainty in continuous activity duration and resource usage for planning problems. By representing random variables as parametric distributions, computing projected system state can be simplified in some cases. Common approximation and novel methods are compared for over-constrained and lightly constrained domains. The system compares a few common approximation methods for an iterative repair planner. Results show improvements in robustness over the conventional non-probabilistic representation by reducing the number of constraint violations witnessed by execution. The improvement is more significant for larger problems and problems with higher resource subscription levels but diminishes as the system is allowed to accept higher risk levels.

autonomous planning↗

NASA NTRS2002

A statistical model for interpreting computerized dynamic posturography data

Computerized dynamic posturography (CDP) is widely used for assessment of altered balance control. CDP trials are quantified using the equilibrium score (ES), which ranges from zero to 100, as a decreasing function of peak sway angle. The problem of how best to model and analyze ESs from a controlled study is considered. The ES often exhibits a skewed distribution in repeated trials, which can lead to incorrect inference when applying standard regression or analysis of variance models. Furthermore, CDP trials are terminated when a patient loses balance. In these situations, the ES is not observable, but is assigned the lowest possible score--zero. As a result, the response variable has a mixed discrete-continuous distribution, further compromising inference obtained by standard statistical methods. Here, we develop alternative methodology for analyzing ESs under a stochastic model extending the ES to a continuous latent random variable that always exists, but is unobserved in the event of a fall. Loss of balance occurs conditionally, with probability depending on the realized latent ES. After fitting the model by a form of quasi-maximum-likelihood, one may perform statistical inference to assess the effects of explanatory variables. An example is provided, using data from the NIH/NIA Baltimore Longitudinal Study on Aging.

NASA Discipline Neuroscience↗

NASA NTRS1989

A probabilistic formulation for fracture energy of continuous fibre-matrix composites

Since the fracture energy is a key design variable in brittle-matrix composites, the calculation of the energy and its probabilistic distribution becomes a prominent issue. A demonstration is presented of the way in which the statistical moments of the fracture-energy random variable may be directly calculated. The case considered is that in which continuous fibers possess periodic points of weakness along their length; the shear strength at the interface caused by the reentry of the fibers into the matrix will produce fiber-debonding.

Wetherhold, Robert C.↗

NASA NTRS1998

Development of a Probabilistic Dynamic Synthesis Method for the Analysis of Nondeterministic Structures

Accounting for the statistical geometric and material variability of structures in analysis has been a topic of considerable research for the last 30 years. The determination of quantifiable measures of statistical probability of a desired response variable, such as natural frequency, maximum displacement, or stress, to replace experience-based "safety factors" has been a primary goal of these studies. There are, however, several problems associated with their satisfactory application to realistic structures, such as bladed disks in turbomachinery. These include the accurate definition of the input random variables (rv's), the large size of the finite element models frequently used to simulate these structures, which makes even a single deterministic analysis expensive, and accurate generation of the cumulative distribution function (CDF) necessary to obtain the probability of the desired response variables. The research presented here applies a methodology called probabilistic dynamic synthesis (PDS) to solve these problems. The PDS method uses dynamic characteristics of substructures measured from modal test as the input rv's, rather than "primitive" rv's such as material or geometric uncertainties. These dynamic characteristics, which are the free-free eigenvalues, eigenvectors, and residual flexibility (RF), are readily measured and for many substructures, a reasonable sample set of these measurements can be obtained. The statistics for these rv's accurately account for the entire random character of the substructure. Using the RF method of component mode synthesis, these dynamic characteristics are used to generate reduced-size sample models of the substructures, which are then coupled to form system models. These sample models are used to obtain the CDF of the response variable by either applying Monte Carlo simulation or by generating data points for use in the response surface reliability method, which can perform the probabilistic analysis with an order of magnitude less computational effort. Both free- and forced-response analyses have been performed, and the results indicate that, while there is considerable room for improvement, the method produces usable and more representative solutions for the design of realistic structures with a substantial savings in computer time.

Brown, A. M.↗

NASA NTRS1981

Reliability-based optimization under random vibration environment

A methodology of formulating the optimum design problem for structural systems with random parameters and subjected to random vibration as a mathematical programming problem is presented. The proposed method is applied to the optimum design of a cantilever beam with a tip mass and a truss structure supporting a water tank. The excitations are assumed to be Gaussian processes and the geometric and material properties are taken to be normally distributed random variables. The probabilistic constraints are specified for individual failure modes since it is easier to specify the reliability level for each failure mode keeping in view the consequences of failure in that particular mode. The time parameter appearing in the random vibration based constraints is eliminated by replacing the probabilities of failure by suitable upper bounds. The numerical results demonstrate the feasibility and effectiveness of applying the reliability-based design concepts to structures with random parameters and operating in random vibration environment.

Rao, S. S.↗

NASA NTRS1978

Random motion analysis of flexible satellite structures

A singular perturbation formulation is used to study the responses of a flexible satellite when random measurement errors can occur. The random variables, at different instants of time, are assumed to be uncorrelated. Procedures for obtaining maxima and minima are described, and a variation of the linear method is developed for the formal solution of the two-point boundary-value problems represented by the variational equations. Random and deterministic solutions for the structural position coordinates are studied, and an analytic algorithm for treating the force equation of motion is developed. Since the random system indicated by the variational equation will always be asymptotically unstable, any analysis of stability must be based on the deterministic system.

Huang, T. C.↗