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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.↗

NASA NTRS2005

Confidence Testing of Shell 405 and S-405 Catalysts in a Monopropellant Hydrazine Thruster

As part of the transfer of catalyst manufacturing technology from Shell Chemical Company (Shell 405 catalyst manufactured in Houston, Texas) to Aerojet (S-405 manufactured in Redmond, Washington), Aerojet demonstrated the equivalence of S-405 and Shell 405 at beginning of life. Some US aerospace users expressed a desire to conduct a preliminary confidence test to assess end-of-life characteristics for S-405. NASA Marshall Space Flight Center (MSFC) and Aerojet entered a contractual agreement in 2004 to conduct a confidence test using a pair of 0.2-lbf MR-103G monopropellant hydrazine thrusters, comparing S-405 and Shell 405 side by side. This paper summarizes the formulation of this test program, explains the test matrix, describes the progress of the test, and analyzes the test results. This paper also includes a discussion of the limitations of this test and the ramifications of the test results for assessing the need for future qualification testing in particular hydrazine thruster applications.

McRight, Patrick↗

NASA NTRS2006

An Overview of Space Exploration Simulation (Basis of Confidence) Documentation

Models and simulations (M&S) are critical resources in the exploration of space. They support program management, systems engineering, integration, analysis, test, and operations by providing critical information that supports key analyses and decisions (technical, cost and schedule). Consequently, there is a clear need to establish a solid understanding of M&S strengths and weaknesses, and the bounds within which they can credibly support decision making. In this presentation we will describe how development of simulation capability documentation will be used to form a Basis of Confidence (Basis of Confidence) for National Aeronautics and Space Administration (NASA) M&S. The process by which BOC documentation is developed will be addressed, as well as the structure and critical concepts that are essential for establishing credibility of NASA's Exploration Systems Mission Directorate (ESMD) legacy M&S. We will illustrate the significance of BOC documentation in supporting decision makers and Accreditation Authorities in M&S risk management.

Bray, Alleen↗

NASA NTRS2011

On the Use of Experimental Methods to Improve Confidence in Educed Impedance

Results from impedance eduction methods developed by NASA Langley Research Center are used throughout the acoustic liner community. In spite of recent enhancements, occasional anomalies persist with these methods, generally at frequencies where the liner produces minimal attenuation. This investigation demonstrates an experimental approach to educe impedance with increased confidence over a desired frequency range, by combining results from successive tests with different cavity depths. A series of tests is conducted with three wire-mesh facesheets, for which the results should be weakly dependent on source sound pressure level and mean grazing flow speed. First, a raylometer is used to measure the DC flow resistance of each facesheet. These facesheets are then mounted onto a frame and a normal incidence tube is used to determine their respective acoustic impedance spectra. A comparison of the acoustic resistance component with the DC flow resistance for each facesheet is used to validate the measurement process. Next, each facesheet is successively mounted onto three frames with different cavity depths, and a grazing flow impedance tube is used to educe their respective acoustic impedance spectra with and without mean flow. The no-flow results are compared with those measured in the normal incidence tube to validate the impedance eduction method. Since the anti-resonance frequency varies with cavity depth, each sample provides robust results over a different frequency range. Hence, a combination of results can be used to determine the facesheet acoustic resistance. When combined with the acoustic reactance, observed to be weakly dependent on the source sound pressure level and grazing flow Mach number, the acoustic impedance can be educed with increased confidence. Representative results of these tests are discussed, and the complete database is available in electronic format upon request.

Jones, Michael G.↗

NASA NTRS2011

Towards Measurement of Confidence in Safety Cases

Arguments in safety cases are predominantly qualitative. This is partly attributed to the lack of sufficient design and operational data necessary to measure the achievement of high-dependability targets, particularly for safety-critical functions implemented in software. The subjective nature of many forms of evidence, such as expert judgment and process maturity, also contributes to the overwhelming dependence on qualitative arguments. However, where data for quantitative measurements is systematically collected, quantitative arguments provide far more benefits over qualitative arguments, in assessing confidence in the safety case. In this paper, we propose a basis for developing and evaluating integrated qualitative and quantitative safety arguments based on the Goal Structuring Notation (GSN) and Bayesian Networks (BN). The approach we propose identifies structures within GSN-based arguments where uncertainties can be quantified. BN are then used to provide a means to reason about confidence in a probabilistic way. We illustrate our approach using a fragment of a safety case for an unmanned aerial system and conclude with some preliminary observations

Denney, Ewen↗

NASA NTRS2012

Assessing Confidence in Pliocene Sea Surface Temperatures to Evaluate Predictive Models

In light of mounting empirical evidence that planetary warming is well underway, the climate research community looks to palaeoclimate research for a ground-truthing measure with which to test the accuracy of future climate simulations. Model experiments that attempt to simulate climates of the past serve to identify both similarities and differences between two climate states and, when compared with simulations run by other models and with geological data, to identify model-specific biases. Uncertainties associated with both the data and the models must be considered in such an exercise. The most recent period of sustained global warmth similar to what is projected for the near future occurred about 3.33.0 million years ago, during the Pliocene epoch. Here, we present Pliocene sea surface temperature data, newly characterized in terms of level of confidence, along with initial experimental results from four climate models. We conclude that, in terms of sea surface temperature, models are in good agreement with estimates of Pliocene sea surface temperature in most regions except the North Atlantic. Our analysis indicates that the discrepancy between the Pliocene proxy data and model simulations in the mid-latitudes of the North Atlantic, where models underestimate warming shown by our highest-confidence data, may provide a new perspective and insight into the predictive abilities of these models in simulating a past warm interval in Earth history.This is important because the Pliocene has a number of parallels to present predictions of late twenty-first century climate.

simulation↗

NASA NTRS2015

Confidence Hills Mineralogy and Chemin Results from Base of Mt. Sharp, Pahrump Hills, Gale Crater, Mars

The Mars Science Laboratory (MSL) rover Curiosity recently completed its fourth drill sampling of sediments on Mars. The Confidence Hills (CH) sample was drilled from a rock located in the Pahrump Hills region at the base of Mt. Sharp in Gale Crater. The CheMin X-ray diffractometer completed five nights of analysis on the sample, more than previously executed for a drill sample, and the data have been analyzed using Rietveld refinement and full-pattern fitting to determine quantitative mineralogy. Confidence Hills mineralogy has several important characteristics: 1) abundant hematite and lesser magnetite; 2) a 10 angstrom phyllosilicate; 3) multiple feldspars including plagioclase and alkali feldspar; 4) mafic silicates including forsterite, orthopyroxene, and two types of clinopyroxene (Ca-rich and Ca-poor), consistent with a basaltic source; and 5) minor contributions from sulfur-bearing species including jarosite.

Cavanagh, P. D.↗

NASA NTRS2022

Sharing Operational Intent with Containment Confidence Level for Negotiating Deconfliction in Upper Class E Airspace

Community-based Cooperative Separation Management (CSM) is expected to provide separation services in Upper Class E airspace (near and above FL600). Under CSM, operators are responsible for maintaining separation. The CSM concept is enabled by sharing Operational Intent (OI) among the operators to ensure common situation awareness. The OI is represented as four-dimensional (time and space) information that indicates where an aircraft would be contained within the space and time, with a known level of confidence. However, each vehicle’s ability to stay within its region of OI may differ based on each vehicle’s performance characteristics, resulting in varying OI sizes among the vehicles. Such varying OI size could adversely affect efficient and fair access to the airspace. In this paper, an OI-generation algorithm under varying OI size restriction with Containment Confidence Level (CCL) is presented. High-Altitude Long Endurance (HALE) balloon operations are used as an example application. A framework is presented by which CCL information is used in the deconfliction process. A fast-time simulation experiment is conducted to evaluate the feasibility of the proposed framework. The simulation results show a reduced number of unnecessary deconfliction actions.

Upper Class E Traffic Management↗

NASA NTRS2022

Sharing Operational Intent with Containment Confidence Level for Negotiating Deconfliction in Upper Class E Airspace

Upper Class E Traffic Management, ETM, Cooperative↗

NASA NTRS2023

Aggregating Confidence: A Problem and A Solution

This paper has two main purposes: to articulate a concern about assessing arguments based on confidence; and to provide an assessment procedure which takes this concern into account. The concern is that confidence is a tricky notion for non-deductive Arguments: doubts inherent in Reasoning coalesces with doubts inherent in Premises. And, within a Compound Argument, this may happen many times over, thus accumulating in unpredictable ways. We explain two procedures we have developed to help address the concern in a methodical way.

language↗

DOE OSTI2026

Bounded-Confidence Models of Multidimensional Opinions with Topic-Weighted Discordance

People’s opinions on a wide range of topics often evolve over time through their interactions with others. Models of opinion dynamics primarily focus on one-dimensional opinions, which represent opinions on one topic. However, opinions on various topics are rarely isolated; instead, they can be interdependent and correlated. In a bounded-confidence model (BCM) of opinion dynamics, agents are receptive to each other only if their opinions are sufficiently similar. Here, we extend classical agent-based BCMs—namely, the Hegselmann–Krause BCM, which has synchronous interactions, and the Deffuant–Weisbuch BCM, which has asynchronous interactions—to a multidimensional setting, in which the opinions are multidimensional vectors representing opinions of different topics and opinions on different topics are interdependent. To measure opinion differences between agents, we introduce topic-weighted discordance functions that account for opinion differences in all topics. We define regions of receptiveness for our models, and we use them to characterize the steady-state opinion clusters and provide an analytical approach to compute these regions. In addition, we numerically simulate our models on various networks with initial opinions drawn from a variety of distributions. When initial opinions are correlated across different topics, our topic-weighted BCMs yield significantly different results in both transient and steady states compared to baseline models, where the dynamics of each opinion topic are independent.

Mathematics and Computing↗