Estimation Errors of Component Spectra Estimated by Means of the Concentration-Spectrum Correlation: Part I

Abstract

It is shown by theoretical analysis and computer simulations that statistical errors accompany the estimates of component spectra of complex mixtures calculated by the concentration-spectrum correlation method. It is revealed that the estimation errors consist of a superposition of other component spectra, each multiplied by a weighting factor. The weighting factor contains three statistical parameters of the sample: two sample standard deviations of both the objective and the interfering components, and the sample correlation coefficient between these two component concentrations. The probability density function of the weighting factor, as well as its ensemble average and standard deviation, is derived. It is shown that the ensemble standard deviation of estimation errors, which is a good measure for the accuracy of the estimated spectrum, can be estimated by the nonparametric method called a bootstrap. The behavior of the estimation errors and the effectiveness of the bootstrap method are demonstrated by computer simulations.