Abstract
We present an analytical formula that estimates the uncertainty in concentrations predicted by linear multivariate calibration, particularly partial least-squares (PLS). We emphasize the analysis of spectroscopic data. The derivation addresses the important limit in which calibration error is negligible in comparison to noise in the prediction spectra. The formula is expressed in terms of standard PLS calibration parameters and the amplitude of spectral noise; it is therefore straightforward to evaluate. To test the formula, we performed PLS analysis upon simulated spectra and upon experimental Raman spectra of dissolved biological analytes in water. In each instance, the root-mean-squared error of prediction was compared to the estimate from the formula. Accurate uncertainty estimates were obtained in cases where calibration noise was lower than prediction noise, and surprisingly good estimates were obtained even when the noise levels were equal. By comparing measured and estimated uncertainties, we assessed the robustness of each PLS calibration model. The scaling of prediction uncertainty with the spectral signal-to-noise ratio is also discussed.
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