Abstract
Pressure balances are known to have a linear straight line equation of the formy = ax + bthat relates the applied pressurexto the effective areay, and recent work has investigated the use of Ordinary Least Squares (OLS), Weighted Least Squares (WLS), and Generalized Least Squares (GLS) regression schemes in order to quantify the expected values of the zero-pressure areaA0 = band distortion coefficientλ = a/bin pressure balance models of the formy = A0(1 + λx). The limitations with conventional OLS, WLS and GLS approaches is that whilst they may be used to quantify the uncertaintiesu(a) andu(b) and the covariancecov(a,b), it is technically challenging to analytically quantify the covariance termcov(A0,λ) without additional Monte Carlo simulations. In this paper, we revisit an earlier Weighted Total Least Squares with Correlation (WTLSC) algorithm to determine the variancesu2(a) andu2(b) along with the covariancecov(a,b), and develop a simple analytical approach to directly infer the corresponding covariancecov(A0,λ) for pressure metrology uncertainty analysis work. Results are compared to OLS, WLS and GLS approaches and indicate that the WTLSC approach may be preferable as it avoids the need for Monte Carlo simulations and additional numerical post-processing to fit and quantify the covariance term, and is thus simpler and more suitable for industrial metrology pressure calibration laboratories. Novel aspects is that a Gnu Octave/Matlab program for easily implementing the WTLSC algorithm to calculate parameter expected values, variances and covariances is also supplied and reported.
Highlights
1.1 Research motivationThe use of least squares based statistical regression analysis is a widely used tool to analyse experimental data when estimating and fitting model parameter values and uncertainties
In order to account for the unavailability of the actual covariance matrix V which may be unknown, or impractical to estimate, one may get a consistent estimate of V usually written as V^ that is termed the feasible generalized least squares (FGLS) estimator using either iteration based commercial routines with software such as Matlab [12], or with newer support vector machines (SVMs) approaches that are currently under development such as recently reported results from Miller and Startz [13]
Limitations of the method are present in that estimates for the correlation terms r(Pi, Si), i = 1, ... , n are technically necessary to utilize the WTLSC algorithm. This issue may be addressed in one of two practical ways, namely (i) setting zero correlations for all of the cross-float data-points such that r(Pi, Si) = 0 for i = 1, ... , n for which the WTLSC algorithm will reproduce results for a Weighted Least Squares (WLS) algorithm as a conservative compromise, or (ii) performing a small number of say 50 or 100 repeat measurements in an Excel spreadsheet which is feasible without the need for any specialist Visual Basic for Applications (VBA) scripting programming within a spreadsheet with appropriate expert physical judgement to generate (Pi, Si) data to fit a Gaussian copula with the 11 lines of RStudio code previously discussed to obtain an approximate estimate for the correlation
Summary
The use of least squares based statistical regression analysis is a widely used tool to analyse experimental data when estimating and fitting model parameter values and uncertainties. The probability density function (PDF) of the output gy(hy) may be determined where hy is a univariate random variable of the model output y, gx(jx) is a univariate random variable of the model input x, and ga,b(ja, jb) is a bivariate random variable of the joint PDF of the model parameters a and b that incorporates the coupling effect.
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