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NASA NTRS1992

Minimax confidence intervals in geomagnetism

The present paper uses theory of Donoho (1989) to find lower bounds on the lengths of optimally short fixed-length confidence intervals (minimax confidence intervals) for Gauss coefficients of the field of degree 1-12 using the heat flow constraint. The bounds on optimal minimax intervals are about 40 percent shorter than Backus' intervals: no procedure for producing fixed-length confidence intervals, linear or nonlinear, can give intervals shorter than about 60 percent the length of Backus' in this problem. While both methods rigorously account for the fact that core field models are infinite-dimensional, the application of the techniques to the geomagnetic problem involves approximations and counterfactual assumptions about the data errors, and so these results are likely to be extremely optimistic estimates of the actual uncertainty in Gauss coefficients.

Stark, Philip B.↗

NASA NTRS2003

On the Confidence Limit of Hilbert Spectrum

Confidence limit is a routine requirement for Fourier spectral analysis. But this confidence limit is established based on ergodic theory: For stationary process, temporal average equals the ensemble average. Therefore, one can divide the data into n-sections and treat each section as independent realization. Most natural processes in general, and climate data in particular, are not stationary; therefore, there is a need for the Hilbert Spectral analysis for such processes. Here ergodic theory is no longer applicable. We propose to use various adjustable parameters in the shifting processes of the Empirical Mode Decomposition (EMD) method to obtain an ensemble of Intrinsic Mode Function 0 sets. Based on such an ensemble, we introduce a statistical measure in. a form of confidence limits for the Intrinsic Mode Functions, and consequently, the Hilbert spectra. The criterion of selecting the various adjustable parameters is based on the orthogonality test of the resulting M F sets. Length-of-day data from 1962 to 2001 will be used to illustrate this new approach. Its implication in climate data analysis will also be discussed.

Huang, Norden↗

NASA NTRS2007

A Comparison of Various Stress Rupture Life Models for Orbiter Composite Pressure Vessels and Confidence Intervals

In conjunction with a recent NASA Engineering and Safety Center (NESC) investigation of flight worthiness of Kevlar Overwrapped Composite Pressure Vessels (COPVs) on board the Orbiter, two stress rupture life prediction models were proposed independently by Phoenix and by Glaser. In this paper, the use of these models to determine the system reliability of 24 COPVs currently in service on board the Orbiter is discussed. The models are briefly described, compared to each other, and model parameters and parameter uncertainties are also reviewed to understand confidence in reliability estimation as well as the sensitivities of these parameters in influencing overall predicted reliability levels. Differences and similarities in the various models will be compared via stress rupture reliability curves (stress ratio vs. lifetime plots). Also outlined will be the differences in the underlying model premises, and predictive outcomes. Sources of error and sensitivities in the models will be examined and discussed based on sensitivity analysis and confidence interval determination. Confidence interval results and their implications will be discussed for the models by Phoenix and Glaser.

Grimes-Ledesma, Lorie↗

NASA NTRS2010

Confidence-Based Feature Acquisition

Confidence-based Feature Acquisition (CFA) is a novel, supervised learning method for acquiring missing feature values when there is missing data at both training (learning) and test (deployment) time. To train a machine learning classifier, data is encoded with a series of input features describing each item. In some applications, the training data may have missing values for some of the features, which can be acquired at a given cost. A relevant JPL example is that of the Mars rover exploration in which the features are obtained from a variety of different instruments, with different power consumption and integration time costs. The challenge is to decide which features will lead to increased classification performance and are therefore worth acquiring (paying the cost). To solve this problem, CFA, which is made up of two algorithms (CFA-train and CFA-predict), has been designed to greedily minimize total acquisition cost (during training and testing) while aiming for a specific accuracy level (specified as a confidence threshold). With this method, it is assumed that there is a nonempty subset of features that are free; that is, every instance in the data set includes these features initially for zero cost. It is also assumed that the feature acquisition (FA) cost associated with each feature is known in advance, and that the FA cost for a given feature is the same for all instances. Finally, CFA requires that the base-level classifiers produce not only a classification, but also a confidence (or posterior probability).

Wagstaff, Kiri L.↗

NASA NTRS2006

A Comparison of Various Stress Rupture Life Models for Orbiter Composite Pressure Vessels and Confidence Intervals

In conjunction with a recent NASA Engineering and Safety Center (NESC) investigation of flight worthiness of Kevlar Ovenvrapped Composite Pressure Vessels (COPVs) on board the Orbiter, two stress rupture life prediction models were proposed independently by Phoenix and by Glaser. In this paper, the use of these models to determine the system reliability of 24 COPVs currently in service on board the Orbiter is discussed. The models are briefly described, compared to each other, and model parameters and parameter error are also reviewed to understand confidence in reliability estimation as well as the sensitivities of these parameters in influencing overall predicted reliability levels. Differences and similarities in the various models will be compared via stress rupture reliability curves (stress ratio vs. lifetime plots). Also outlined will be the differences in the underlying model premises, and predictive outcomes. Sources of error and sensitivities in the models will be examined and discussed based on sensitivity analysis and confidence interval determination. Confidence interval results and their implications will be discussed for the models by Phoenix and Glaser.

stress ratios↗

NASA NTRS2023

A Confidence-Based Approach to Including Survivors in a Probabilistic TID Failure Assessment

A probabilistic total ionizing dose (TID) failure assessment is extended to include survivor data, enabling the bounding of failure probability to a desired confidence level (CL) without failure data. The extension provides an avenue for analyzing microelectronics tested for TID without reaching a failure mode, a scenario often encountered by missions utilizing commercial-off-the-shelf (COTS) technologies. Using the type-I censored likelihood formulation and a realistic upper bound on expected device performance, the failure probability space is bounded by confidence contours within the context of a variable environment. The framework accommodates any type of distribution assumed for the part failure or the environment under consideration. Furthermore, the framework can be utilized pre-emptively to plan future device TID tests, minimizing costs while meeting survival requirements. Heritage data may also be used as survivors to further minimize testing costs when parts are from the same lot, but the amount of constraint derived from heritage is limited. Altogether, the framework enables a formal, mathematically rigorous analysis of radiation tolerant devices tested to a maximum dose, as well as flight heritage, in a hardness assurance methodology.

confidence↗

NASA NTRS1980

Confidence regions of planar cardiac vectors

A method for plotting the confidence regions of vectorial data obtained in electrocardiology is presented. The 90%, 95% and 99% confidence regions of cardiac vectors represented in a plane are obtained in the form of an ellipse centered at coordinates corresponding to the means of a sample selected at random from a bivariate normal distribution. An example of such a plot for the frontal plane QRS mean electrical axis for 80 horses is also presented.

Dubin, S.↗

NASA NTRS1984

Maximum likelihood algorithm using an efficient scheme for computing sensitivities and parameter confidence intervals

Improved techniques for estimating airplane stability and control derivatives and their standard errors are presented. A maximum likelihood estimation algorithm is developed which relies on an optimization scheme referred to as a modified Newton-Raphson scheme with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. The fitted surface allows sensitivity information to be updated at each iteration with a significant reduction in computational effort compared to integrating the analytically-determined sensitivity equations or using a finite difference scheme. An aircraft estimation problem is solved using real flight data to compare MNRES with the commonly used modified Newton-Raphson technique; MNRES is found to be faster and more generally applicable. Parameter standard errors are determined using a random search technique. The confidence intervals obtained are compared with Cramer-Rao lower bounds at the same confidence level. It is observed that the nonlinearity of the cost function is an important factor in the relationship between Cramer-Rao bounds and the error bounds determined by the search technique.

Murphy, P. C.↗

NASA NTRS1984

NASA's approach to flight confidence

NASA's confidence in the flight readiness of aerospace hardware and software is achieved by a thorough integration of safety activities into every program facet, from concept through the mission. This involves technical and administrative personnel, organizations that specify requirements, design, manufacturing, test and the operators. Reviews by inhouse and external specialists form an integral part of the assurance process. Examples of safety issues and their resolution for some power and propulsion functions are given (lithium cells, autoignition/fretting in high pressure oxygen environments, ignition sources from auxiliary power unit, and low thrust rocket engines). Finally, some comments on NASA's integrated safety activities and the Aerospace Safety Advisory Panel's role in the NASA review and assessment process all of which provides added confidence in achieving a high level of mission safety and success.

Roth, G. L.↗

NASA NTRS1986

Test program to provide confidence in liquid oxygen cooling of hydrocarbon fueled rocket thrust chambers

An experimental program has been planned at the NASA Lewis Research Center to build confidence in the feasibility of liquid oxygen cooling for hydrocarbon fueled rocket engines. Although liquid oxygen cooling has previously been incorporated in test hardware, more runtime is necessary to gain confidence in this concept. In the previous tests, small oxygen leaks developed at the throat of the thrust chamber and film cooled the hot-gas side of the chamber wall without resulting in catastrophic failure. However, more testing is necessary to demonstrate that a catastrophic failure would not occur if cracks developed further upstream between the injector and the throat, where the boundary layer has not been established. Since under normal conditions cracks are expected to form in the throat region of the thrust chamber, cracks must be initiated artificially in order to control their location. Several methods of crack initiation are discussed in this report. Four thrust chambers, three with cracks and one without, should be tested. The axial location of the cracks should be varied parametrically. Each chamber should be instrumented to determine the effects of the cracks, as well as the overall performance and durability of the chambers.

Armstrong, E. S.↗

NASA NTRS1986

Confidence limits for small numbers of events in astrophysical data

The calculation of limits for small numbers of astronomical counts is based on standard equations derived from Poisson and binomial statistics; although the equations are straightforward, their direct use is cumbersome and involves both table-interpolations and several mathematical operations. Convenient tables and approximate formulae are here presented for confidence limits which are based on such Poisson and binomial statistics. The limits in the tables are given for all confidence levels commonly used in astrophysics.

Gehrels, N.↗

NASA NTRS1987

Development of confidence limits by pivotal functions for estimating software reliability

The utility of pivotal functions is established for assessing software reliability. Based on the Moranda geometric de-eutrophication model of reliability growth, confidence limits for attained reliability and prediction limits for the time to the next failure are derived using a pivotal function approach. Asymptotic approximations to the confidence and prediction limits are considered and are shown to be inadequate in cases where only a few bugs are found in the software. Departures from the assumed exponentially distributed interfailure times in the model are also investigated. The effect of these departures is discussed relative to restricting the use of the Moranda model.

Dotson, Kelly J.↗

NASA NTRS1987

Efficient computation of parameter confidence intervals

An important step in system identification of aircraft is the estimation of stability and control derivatives from flight data along with an assessment of parameter accuracy. When the maximum likelihood estimation technique is used, parameter accuracy is commonly assessed by the Cramer-Rao lower bound. It is known, however, that in some cases the lower bound can be substantially different from the parameter variance. Under these circumstances the Cramer-Rao bounds may be misleading as an accuracy measure. This paper discusses the confidence interval estimation problem based on likelihood ratios, which offers a more general estimate of the error bounds. Four approaches are considered for computing confidence intervals of maximum likelihood parameter estimates. Each approach is applied to real flight data and compared.

Murphy, Patrick C.↗

NASA NTRS1989

Confidence set inference with a prior quadratic bound

In the uniqueness part of a geophysical inverse problem, the observer wants to predict all likely values of P unknown numerical properties z=(z sub 1,...,z sub p) of the earth from measurement of D other numerical properties y (sup 0) = (y (sub 1) (sup 0), ..., y (sub D (sup 0)), using full or partial knowledge of the statistical distribution of the random errors in y (sup 0). The data space Y containing y(sup 0) is D-dimensional, so when the model space X is infinite-dimensional the linear uniqueness problem usually is insoluble without prior information about the correct earth model x. If that information is a quadratic bound on x, Bayesian inference (BI) and stochastic inversion (SI) inject spurious structure into x, implied by neither the data nor the quadratic bound. Confidence set inference (CSI) provides an alternative inversion technique free of this objection. Confidence set inference is illustrated in the problem of estimating the geomagnetic field B at the core-mantle boundary (CMB) from components of B measured on or above the earth's surface.

Backus, George E.↗

NASA NTRS1988

A variance based confidence criterion for ERA identified modal parameters

The realization theory is developed in a systematic manner for the Eigensystem Realization Algorithm (ERA) used for system identification. First, perturbation results are obtained which describe the linearized changes in the identified parameters resulting from small change in the data. Formulas are then derived that can be used to evaluate the variance of each of the identified parameters, assuming that the noise level is sufficiently low to allow the application of linearized results. These variances can be converted to give confidence intervals for each of the parameters for any chosen confidence level.

Longman, Richard W.↗

NASA NTRS1989

Confidence set interference with a prior quadratic bound

Neyman's (1937) theory of confidence sets is developed as a replacement for Bayesian interference (BI) and stochastic inversion (SI) when the prior information is a hard quadratic bound. It is recommended that BI and SI be replaced by confidence set interference (CSI) only in certain circumstances. The geomagnetic problem is used to illustrate the general theory of CSI.

Backus, George E.↗

NASA NTRS1991

Developing Confidence Limits For Reliability Of Software

Technique developed for estimating reliability of software by use of Moranda geometric de-eutrophication model. Pivotal method enables straightforward construction of exact bounds with associated degree of statistical confidence about reliability of software. Confidence limits thus derived provide precise means of assessing quality of software. Limits take into account number of bugs found while testing and effects of sampling variation associated with random order of discovering bugs.

Hayhurst, Kelly J.↗

NASA NTRS1991

Determination of confidence limits for experiments with low numbers of counts

Two different methods, classical and Bayesian, for determining confidence intervals involving Poisson-distributed data are compared. Particular consideration is given to cases where the number of counts observed is small and is comparable to the mean number of background counts. Reasons for preferring the Bayesian over the classical method are given. Tables of confidence limits calculated by the Bayesian method are provided for quick reference.

Kraft, Ralph P.↗