Robert G. Easterling

Multivariate Least-Squares Methods Applied to the Quantitative Spectral Analysis of Multicomponent Samples

In an extension of earlier work, weighted multivariate least-squares methods of quantitative FT-IR analysis have been developed. A linear least-squares approximation to nonlinearities in the Beer-Lambert law is made by allowing the reference spectra to be a set of known mixtures. The incorporation of nonzero intercepts in the relation between absorbance and concentration further improves the approximation of nonlinearities while simultaneously accounting for nonzero spectral baselines. Pathlength variations are also accommodated in the analysis, and under certain conditions, unknown sample pathlengths can be determined. All spectral data are used to improve the precision and accuracy of the estimated concentrations. During the calibration phase of the analysis, pure component spectra are estimated from the standard mixture spectra. These can be compared with the measured pure component spectra to determine which vibrations experience nonlinear behavior. In the predictive phase of the analysis, the calculated spectra are used in our previous least-squares analysis to estimate sample component concentrations. These methods were applied to the analysis of the IR spectra of binary mixtures of esters. Even with severely overlapping spectral bands and nonlinearities in the Beer-Lambert law, the average relative error in the estimated concentrations was <1%.

Improved sensitivity of infrared spectroscopy by the application of least squares methods

Improved sensitivity and precision in the quantitative analysis of trace gases by Fourier transform infrared spectroscopy have been achieved by the application of new spectral least squares methods. By relating all of the spectral information present in the reference spectrum of a trace gas to that of the unknown sample and by appropriately fitting the baseline, detections of trace gases can be obtained even though the individual spectral features may lie well below the noise level. Four least squares methods incorporating different baseline assumptions were investigated and compared using calibrated gases of CO, N2O, and CO2 in dry air. These methods include: (I) baseline known, (II) baseline linear over the spectral region of interest, (III) baseline linear over each spectral peak, and (IV) negligible baseline shift between successive data points. Methods III and IV were found to be most reliable for the gases studied. When method III is applied to the spectra of these trace gases, detection limits improved by factors of 5 to 7 over conventional methods applied to the same data. “Three sigma” detection limits are equal to 0.6, 0.2, and 0.08 ppm for CO, N2O, and CO2, respectively, when a 10-cm pathlength at a total pressure of 640 Torr is used with a ∼35 min measurement time at 0.06 cm−1 resolution.

Application of New Least-Squares Methods for the Quantitative Infrared Analysis of Multicomponent Samples

Improvements have been made in previous least-squares regression analyses of infrared spectra for the quantitative estimation of concentrations of multicomponent mixtures. Spectral baselines are fitted by least-squares methods, and overlapping spectral features are accounted for in the fitting procedure. Selection of peaks above a threshold value reduces computation time and data storage requirements. Four weighted least-squares methods incorporating different baseline assumptions were investigated using FT-IR spectra of the three pure xylene isomers and their mixtures. By fitting only regions of the spectra that follow Beers Law, accurate results can be obtained using three of the fitting methods even when baselines are not corrected to zero. Accurate results can also be obtained using one of the fits even in the presence of Beers Law deviations. This is a consequence of pooling the weighted results for each spectral peak such that the greatest weighting is automatically given to those peaks that adhere to Beers Law. It has been shown with the xylene spectra that semiquantitative results can be obtained even when all the major components are not known or when expected components are not present. This improvement over previous methods greatly expands the utility of quantitative least-squares analyses.

Approximate Confidence Limits for System Reliability

Abstract A method is presented for obtaining system confidence limits based on component test results. The technique consists of estimating the asymptotic variance of the maximum likelihood estimate of system reliability, equating this to the estimate of the variance in binomial sampling, and solving for and , the pseudo-numbers of system tests and successes. These are then substituted into the incomplete beta function and confidence limits obtained in the usual way for binomial sampling.

System-based component-test plans and operating characteristics: binomial data

Component-test plans are often designed by allocating system reliability among the systems components, then choosing individual component plans suitable for demonstrating achievcment of each components reliability goal. This approach does not consider how much information relative to the system reliability goal is provided by the ensemble of component tests. We consider the notion of system reliability operating characteristic (OC) curves, based on the component tests, and illustrate their use in designing or evaluating an overall test program. By specifying OC values (akin to producers and consumers risks), optimum, system-ortented component-test plans can be derived. These ideas are illustrated for a series system, and for a simple series-parallel system, with binomial data.

Estimation of extreme quantiles based on sensitivity tests : a comparative study

In reliability applications, it is often of interest to estimate extreme quantiles of the distribution of critical stimulus levels based on a limited number of sensitivity tests. To evaluate and compare the effectiveness of available sequential-design sensitivity tests in this setting, several such tests were compared through Monte Carlo simulation. Samples of size 20, 35, and 50 were used to estimate the .99 and, 999 quantiles of the probit model. Each test was evaluated using initial parameter estimates equal to the parameters of the model and then using inaccurate estimates. The effect of assuming the probit model when the true model was logit, generalized logistic, or Type I extreme-value was also studied. Our results show that, when the model is correctly specified, tests designed to estimate the model parameters and subsequently to estimate quantiles as a function of the model parameters provided more accurate quantile estimates than tests designed to estimate a specified quantile. The tests that es...

Statistical Tolerancing Based on Consumer's Risk Considerations

In a production process it is necessary to specify tolerances for characteristics, such as dimensions within which the measured characteristic must fall in order for the part to be acceptable. Such tolerances may often be set informally, but a considera..

Combining Component and System Information

The situation in which results are available on both components and simple systems consisting of m identical, but independent components in series or in parallel is considered. The cases considered are attribute testing both at single and multiple stress levels and life testing. The method of maximum likelihood is used to obtain estimates; and approximate tests of hypotheses and confidence intervals are obtained using the asymptotic variances of the maximum likelihood estimates.

Discrimination Intervals for Percentiles in Regression

Abstract In simple linear regression the 100pth percentile of the distribution of Y for a given value of X is β0 + β1 X + Zp σ, where Zp is the 100pth percentile of the standard normal distribution. The problem considered is to estimate the value of X for which this percentile is equal to some specified value, Yp . Both point and interval estimates are considered. A procedure for obtaining exact confidence intervals is derived, and in addition, two approximate and two conservative procedures for obtaining intervals are presented and compared. An example is given to illustrate these procedures and an application to covariance analysis is given.

A Personal View of the Bayesian Controversy in Reliability and Statistics

There appear to be two important developing trends in reliability. One is away from the use of statistics by reliability engineers; the other is toward increased use of Bayesian techniques. One source of the latter may be disillusionment with what is regarded as classical statistics; however, one source of the former may be dissatisfaction with what has been called Bayesian statistics. The purpose of this paper is to discuss these trends and to present a personal view of the issues between classical and Bayesian statistics. The objective is to show that there is some substance to the classical statisticians opposition to Bayesian inference and that the issues are pertinent and meaningful to the reliability engineer.