SEARCH · Search NASA
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
We develop a method to estimate the sum of conditional means of a sequence of random variables given our access to only a subsequence by spot-checking. The method works with non-independent-and-identically-distributed (non-i.i.d.) random variables and can be applied for certifying ongoing quantum information tasks.
Statistical procedures taken from the field of survival analysis have been adapted to astronomical usage and have been applied to a sample of stars in the B-V color range between 0.1 and 0.5 with measured soft X-ray luminosities and projected equatorial velocities. The two-sample problem and linear regression problem with arbitrarily censored data were studied. A new method for determining the linear regression between two random variables in the presence of arbitrary censoring has been developed which can also be used for a likelihood-ratio test for the independence of two random variables and for principal-component analysis in the presence of arbitrary censoring. The required numerical computations can be carried out straightforwardly and rapidly.
Stochastic modelling approaches are presented to capture random effects at multiple time and length scales. Random processes that occur at the microscale produce nondeterministic effects at the macroscale. Here we present three stochastic modeling approaches that describe random processes at microscopic length scales and map these processes to the macroscopic length scale. The first stochastic modeling approach is based upon a particle based numerical technique to solve a Stochastic Differential Equation (SDE) using an arbitrary diffusion process to capture random processes at the microstructural level. The second approach prescribes a Probability Density Function (PDF) for the drift and diffusion of the random variable derived using the forward and backward Kolmogorov equations. This method requires mean and drift evolution PDF transport equations. The third approach is the coupling of multiple random variables which are dependent on each other. The relationship of the PDFs and a coupling function, known as a copula, produces a Joint Probability Density Function (JPDF). These stochastic modeling approaches are implemented into a Multiple Component (MC) shock physics computational code and used to model statistical fracture and reactive flow applications.
This paper proposes formulations and algorithms for design optimization under both aleatory (i.e., natural or physical variability) and epistemic uncertainty (i.e., imprecise probabilistic information), from the perspective of system robustness. The proposed formulations deal with epistemic uncertainty arising from both sparse and interval data without any assumption about the probability distributions of the random variables. A decoupled approach is proposed in this paper to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency. The proposed methods are illustrated for the upper stage design problem of a two-stage-to-orbit (TSTO) vehicle, where the information on the random design inputs are only available as sparse point and/or interval data. As collecting more data reduces uncertainty but increases cost, the effect of sample size on the optimality and robustness of the solution is also studied. A method is developed to determine the optimal sample size for sparse point data that leads to the solutions of the design problem that are least sensitive to variations in the input random variables.
The equations for processing ground tracking data during a space shuttle ascent or entry, or any nonfree flight phase of a shuttle mission are given. The resulting computer program processes data from up to three stations simultaneously: C-band station number 1; C-band station number 2; and an S-band station. The C-band data consists of range, azimuth, and elevation angle measurements. The S-band data consists of range, two angles, and integrated Doppler data in the form of cycle counts. A nineteen element state vector is used in Kalman filter to process the measurements. The acceleration components of the shuttle are taken to be independent exponentially-correlated random variables. Nine elements of the state vector are the measurement bias errors associated with range and two angles for each tracking station. The biases are all modeled as exponentially-correlated random variables with a typical time constant of 108 seconds. All time constants are taken to be the same for all nine state variables. This simplifies the logic in propagating the state error covariance matrix ahead in time.
The reliability of two graphite-epoxy stiffened panels that contain uncertainties is examined. For one panel, the effect of an overall bow-type initial imperfection is studied. The size of the bow is assumed to be a random variable. The failure mode is buckling. The benefits of quality control are explored by using truncated distributions. For the other panel, the effect of uncertainties in a strain-based failure criterion is studied. The allowable strains are assumed to be random variables. A geometrically nonlinear analysis is used to calculate a detailed strain distribution near an elliptical access hole in a wing panel that was tested to failure. Calculated strains are used to predict failure. Results are compared with the experimental failure load of the panel.
A mathematical procedure is presented for the calculation of exact cumulative distribution statistics for a viscosity-free variant of Burgers nonlinear partial differential equation (PDE) in one space dimension and time subject to sinusoidal initial data with uncertain (random variable) amplitude or phase shift. Analytical solutions of nonlinear PDEs with uncertain initial and/or boundary data are invaluable benchmarks in assessing approximate uncertainty quantification techniques. The Burgers equation solution with uncertain initial data results in nonsmooth solution behavior in both physical and random variable dimensions which provides a severe test for approximate uncertainty quantification techniques. Mathematical proofs are provided to verify that exact cumulative distribution statistics can be systematically and robustly obtained for all forward time.
Radiometers must be periodically calibrated because the receiver response fluctuates. Many techniques exist to correct for the time varying response of a radiometer receiver. An analytical technique has been developed that uses generalized least squares regression (LSR) to predict the performance of a wide variety of calibration algorithms. The total measurement uncertainty including the uncertainty of the calibration can be computed using LSR. The uncertainties of the calibration samples used in the regression are based upon treating the receiver fluctuations as non-stationary processes. Signals originating from the different sources of emission are treated as simultaneously existing random processes. Thus, the radiometer output is a series of samples obtained from these random processes. The samples are treated as random variables but because the underlying processes are non-stationary the statistics of the samples are treated as non-stationary. The statistics of the calibration samples depend upon the time for which the samples are to be applied. The statistics of the random variables are equated to the mean statistics of the non-stationary processes over the interval defined by the time of calibration sample and when it is applied. This analysis opens the opportunity for experimental investigation into the underlying properties of receiver non stationarity through the use of multiple calibration references. In this presentation we will discuss the application of LSR to the analysis of various calibration algorithms, requirements for experimental verification of the theory, and preliminary results from analyzing experiment measurements.
A reliability analysis of structures subjected to random service loads and periodic proof tests treats gust loads and maneuver loads as random processes. Crack initiation, crack propagation, and strength degradation are treated as the fatigue process. The time to fatigue crack initiation and ultimate strength are random variables. Residual strength decreases during crack propagation, so that failure rate increases with time. When a structure fails under periodic proof testing, a new structure is built and proof-tested. The probability of structural failure in service is derived from treatment of all the random variables, strength degradations, service loads, proof tests, and the renewal of failed structures. Some numerical examples are worked out.
Traditional design approaches for composite materials have employed deterministic criteria for failure analysis. New approaches are required to predict the reliability of composite structures since strengths and stresses may be random variables. This report will examine and compare methods used to evaluate the reliability of composite laminae. The two types of methods that will be evaluated are fast probability integration (FPI) methods and Monte Carlo methods. In these methods, reliability is formulated as the probability that an explicit function of random variables is less than a given constant. Using failure criteria developed for composite materials, a function of design variables can be generated which defines a 'failure surface' in probability space. A number of methods are available to evaluate the integration over the probability space bounded by this surface; this integration delivers the required reliability. The methods which will be evaluated are: the first order, second moment FPI methods; second order, second moment FPI methods; the simple Monte Carlo; and an advanced Monte Carlo technique which utilizes importance sampling. The methods are compared for accuracy, efficiency, and for the conservativism of the reliability estimation. The methodology involved in determining the sensitivity of the reliability estimate to the design variables (strength distributions) and importance factors is also presented.
The paper presents a methodology for quantifying, propagating, and managing the uncertainty in the data required to initialize computer simulations of complex phenomena. The purpose of the methodology is to permit the quantitative assessment of a certification level to be associated with the predictions from the simulations, as well as the design of a data acquisition strategy to achieve a target level of certification. The value of a methodology that can address the above issues is obvious, specially in light of the trend in the availability of computational resources, as well as the trend in sensor technology. These two trends make it possible to probe physical phenomena both with physical sensors, as well as with complex models, at previously inconceivable levels. With these new abilities arises the need to develop the knowledge to integrate the information from sensors and computer simulations. This is achieved in the present work by tracing both activities back to a level of abstraction that highlights their commonalities, thus allowing them to be manipulated in a mathematically consistent fashion. In particular, the mathematical theory underlying computer simulations has long been associated with partial differential equations and functional analysis concepts such as Hilbert spares and orthogonal projections. By relying on a probabilistic framework for the modeling of data, a Hilbert space framework emerges that permits the modeling of coefficients in the governing equations as random variables, or equivalently, as elements in a Hilbert space. This permits the development of an approximation theory for probabilistic problems that parallels that of deterministic approximation theory. According to this formalism, the solution of the problem is identified by its projection on a basis in the Hilbert space of random variables, as opposed to more traditional techniques where the solution is approximated by its first or second-order statistics. The present representation, in addition to capturing significantly more information than the traditional approach, facilitates the linkage between different interacting stochastic systems as is typically observed in real-life situations.
This paper is a further development of the results of [20] where, based on the negative binomial model for the duration of wet periods measured in days [16], an asymptotic approximation was proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Fr´echet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor–Fisher distribution. Here we extend this result to the mth extremes, m ∈ ℕ, and sample quantiles. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution [17] and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions [10]. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution of the maximum precipitation is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions this limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. It is also shown that the limit distribution of sample quantiles is the well-known Student distribution. Several methods are proposed for the estimation of the parameters of the asymptotic distributions. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability (“Bayesian”) models.
A methodology is developed to computationally simulate the uncertain behavior of composite structures. The uncertain behavior includes buckling loads, natural frequencies, displacements, stress/strain etc., which are the consequences of the random variation (scatter) of the primitive (independent random) variables in the constituent, ply, laminate and structural levels. This methodology is implemented in the IPACS (Integrated Probabilistic Assessment of Composite Structures) computer code. A fuselage-type composite structure is analyzed to demonstrate the code's capability. The probability distribution functions of the buckling loads, natural frequency, displacement, strain and stress are computed. The sensitivity of each primitive (independent random) variable to a given structural response is also identified from the analyses.
A methodology is developed to simulate computationally the uncertain behavior of composite structures. The uncertain behavior includes buckling loads, natural frequencies, displacements, stress/strain, etc., which are the consequences of the random variation (scatter) of the primitive (independent random) variables in the constituent, plv. laminate and structural levels. This methodology is implemented in a computer code integrated probabilistic assessment of composite structures (IPACS). A fuselage-type composite structure is analyzed to demonstrate the code's capability . The probability distribution functions of the buckling loads, natural frequency, displacement, strain and stress are computed. The sensitivity of each primitive (independent random) variable to a given structural response is also identified from the analyses.
Inference and learning of graphical models are both well-studied problems in statistics and machine learning that have found many applications in science and engineering. However, exact inference is intractable in general graphical models, which suggests the problem of seeking the best approximation to a collection of random variables within some tractable family of graphical models. In this paper, we focus on the class of planar Ising models, for which exact inference is tractable using techniques of statistical physics. Based on these techniques and recent methods for planarity testing and planar embedding, we propose a greedy algorithm for learning the best planar Ising model to approximate an arbitrary collection of binary random variables (possibly from sample data). Given the set of all pairwise correlations among variables, we select a planar graph and optimal planar Ising model defined on this graph to best approximate that set of correlations. Finally, we demonstrate our method in simulations and for two applications: modeling senate voting records and identifying geo-chemical depth trends from Mars rover data.
Let X be a K-dimensional random variable serving as input for a system with output Y (not necessarily of dimension k). given X, an outcome Y or a distribution of outcomes G(Y/X) may be obtained either explicitly or implicity. The situation is considered in which there is a real world data set X sub j sub = 1 (n) and a means of simulating an outcome Y. A method for empirical random number generation based on the sample of observations of the random variable X without estimating the underlying density is discussed.
This paper presents recent developments in efficient structural reliability analysis methods. The paper proposes an efficient, adaptive importance sampling (AIS) method that can be used to compute reliability and reliability sensitivities. The AIS approach uses a sampling density that is proportional to the joint PDF of the random variables. Starting from an initial approximate failure domain, sampling proceeds adaptively and incrementally with the goal of reaching a sampling domain that is slightly greater than the failure domain to minimize over-sampling in the safe region. Several reliability sensitivity coefficients are proposed that can be computed directly and easily from the above AIS-based failure points. These probability sensitivities can be used for identifying key random variables and for adjusting design to achieve reliability-based objectives. The proposed AIS methodology is demonstrated using a turbine blade reliability analysis problem.
We study extreme value statistics of multiple sequences of random variables. For each sequence with N variables, independently drawn from the same distribution, the running maximum is defined as the largest variable to date. We compare the running maxima of m independent sequences and investigate the probability S N that the maxima are perfectly ordered, that is, the running maximum of the first sequence is always larger than that of the second sequence, which is always larger than the running maximum of the third sequence, and so on. The probability S N is universal: it does not depend on the distribution from which the random variables are drawn. For two sequences, S N ~N –1/2 , and in general, the decay is algebraic, S N ~N –σm , for large N. We analytically obtain the exponent σ 3 ≅1.302931 as root of a transcendental equation. Moreover, the exponents σ m grow with m, and we show that σ m ~m for large m.