A weighted total least-squares algorithm for any fitting model with correlated variables

Abstract

An algorithm able to deal with any desired fitting model was developed for regression problems with uncertain and correlated variables. A typical application concerns the determination of calibration curves, especially (i) in those cases in which the uncertainties on the independent variables xi cannot be considered negligible with respect to those associated with the dependent variables yi, and (ii) when correlations exist among xi and yi. In the metrological field, several types of software have already been dedicated to the determination of calibration curves, some being focused just on problem (i) and a few others considering also problem (ii) but only for a straight-line fitting model. The proposed algorithm is able to deal with problems (i) and (ii) at the same time, for a generic fitting model. The tool was developed in the MATLAB® environment and validated on several benchmark data sets, fitted with linear and non-linear regression models. A review of the most commonly applied approximations to the parameter uncertainty is also presented, together with a Monte Carlo method proposed for comparison purposes with the results provided by the formula for the uncertainty evaluation which is implemented in the software.

Highly accurate intensity factors of pure CO2 lines near 2 μm

“…Each point results from the statistical analysis of 100 values, yielding the mean value and the standard deviation. The experimental points were fitted to a straight line by using a weighted total least-squares linear regression, 22 in order to consider the uncertainties on both the absorbance and the number density. A good agreement is found between the experimental points and the best-fit line, the root-mean square (rms) value of the relative residuals being 0.6%.…”

Traceable measurement of specific organic species at industrially relevant surface by infrared spectroscopy

“…the sum of the weighted squared residuals normalized by the number of degree of freedom) showed a high goodness of fit for all the contaminants, being much lower than the expected unit value. Estimates of parameters a , b and c and associated covariance matrix were calculated according to the WTLS procedure . From such results, it was straightforward to obtain the analysis curve by inverting the calibration curve.…”

“…The least‐squares method implemented for the fitting was a weighted total least‐squares (WTLS) regression, which is able to deal with uncertainty (and correlation) in both the dependent (average intensities) and independent (mass values) variables. The algorithm used for the WTLS regression is a matlab ‐based tool for calibration problems, recently developed at INRIM.…”

A new photometric ozone reference in the Huggins bands: the absolute ozone absorption cross section at the 325 nm HeCd laser wavelength

“…It seems, however, that fewer studies (e.g. Amiri-Simkooei et al, 2014;Bremser and Hässel-barth, 1998;Bremser et al, 2007;Malengo and Pennecchi, 2013) are devoted to the problem of when the structure of the covariance matrix is more complex and when correlations exist between uncertainties in different values of x and/or y. This type of question arises in chemometric or metrological applications when calibration lines need to be used or when instruments are to be compared.…”