Quantitative modeling and uncertainty estimation for small-sample LIBS using Gaussian negative log-likelihood and monte carlo dropout methods

Abstract

Laser-induced breakdown spectroscopy (LIBS) has emerged as a powerful analytical technique widely applied in materials science, environmental monitoring, and other fields. In traditional quantitative analysis, models are established for the sample-to-element content based on acquired characteristic spectra combined with specific modeling methods. Small-sample LIBS poses significant challenges to researchers due to its high dimensionality and limited sample size, making it difficult for traditional modeling methods to establish effectively generalized and robust models. Addressing the issues of overfitting during training and the uncertainty of LIBS data, we developed an artificial neural network (ANN) model. Unlike traditional ANN which uses mean squared error functions as training losses, Gaussian negative log-likelihood (GLL) functions were used to train the model. This approach allows for simultaneous consideration of the quantitative prediction of sample element concentrations and the uncertainty of the quantification model prediction. Moreover, the Monte Carlo Dropout (MCDropout) method was applied to reduce prediction bias, resulting in a more robust, generalizable, and uncertainty-aware quantitative model. Compared to traditional quantitative methods, the method proposed in this paper achieved quantitative accuracies of 0.9877, 0.9939, 0.9876, and 0.9899 for four elements (Mn, Mo, Cr, Cu) in stainless steel, respectively. The uncertainty output effectively informs data comprehension and spectral analysis. To the best of our knowledge, this study is the first to propose utilizing model-quantified spectral uncertainty in LIBS quantitative tasks. This innovative study not only provides solutions for small-sample LIBS learning but also enhances the stability and accuracy of quantitative modeling by addressing the impact of uncertainty on training and reducing uncertainty. It is crucial for overcoming the limitations of traditional methods in small-sample LIBS quantification and achieving reliable real-time, high-precision LIBS detection, taking LIBS technology a step further towards practical reliability and precision.