## 1 Introduction
\begin{pmatrix}s_{11}&\cdots&s_{1m}\\ \vdots&\ddots&\vdots\\ s_{n1}&\cdots&s_{nm}\end{pmatrix} .\]  
Let \(c=(c_{1},\ldots,c_{m})^{T}\)--the vector of unknown concentrations, \(b=(b_{1},\ldots,b_{m})^{T}\)--the measured spectrum of gas composition of known substances with unknown concentrations. This implies that concentrations can be found by solving the system of linear algebraic equations in the matrix form:  
\[Sc=b. \tag{2}\]  
Iterative algorithms like Multivariate Curve Resolution (MCR) [18; 19; 20] and Univariate Calibration [21] combined with least squares [22] and Levenberg-Marquardt methods of extreme search [23; 24; 25] were the first methods applied for Equation (2) solution. All these techniques are based on a priori knowledge of all \(s_{i}(\lambda)\) relevant to a studied gas sample. The major problem of such approaches in content analysis of exhaled air or other gas mixtures of natural origin spectra is that their composition is unknown. In this case, \(S=\overset{\sim}{S}+S_{L}\), and \(c=\hat{c}+c_{L}\), where \(\overset{\sim}{S}\)--the matrix of individual spectra of components, which are definitely present in the studied gas mixture; \(S_{L}\)--the matrix of unaccounted (latent) components, \(\hat{c}\)--vector of unknown concentrations of accounted components, and \(c_{L}\)--the vector of corresponding concentrations of unaccounted components. Here, \(\hat{c}\equiv(c_{1},\ldots,c_{m})^{T}\), a symbol with tilde means known data. The experimentally measured matrix \(b\) containsrandom additive part \(R\) associated with noise, measurement error, etc. Therefore, \(b=\overset{\sim}{b}+R\). Then Equation (2) takes form:  
\[\overset{\sim}{S}\mathcal{E}=\overset{\sim}{b}+R-S_{L}(\mathcal{E}+c_{L})- \overset{\sim}{S}c_{L} \tag{3}\]  
Opposite to Equation (2), the system (3) is ill-posed. There are two reasons for the latter: the presence of random noise and unaccounted (latent) components. Regarding random noise, according to the criterion proposed by [PERSON], the problem is well-posed if three conditions are satisfied: the solution exists, is unique, and depends continuously on the initial conditions. Otherwise, the problem is ill-posed. In the case of Equation (3), a solution exists, but other conditions are not met; such a task requires using special methods. The random noise \(R\) can be reduced by preliminary noise filtration. When continuous dependence of the solution on the initial conditions is absent, regularization methods and iterative procedures of \(\mathcal{E}\) calculation can be used [26]. The problem of the presence of unaccounted (latent) components is related to so-called \"grey analytical systems\" for which their qualitative chemical composition is incomplete [27]. Regression methods like principal component analysis based on the transformation of spectral responses into orthogonal latent variables (principal components) allow estimating the number of components in a studied spectrum of a gas mixture. It can be completed by estimating the described variance in reduced feature space or using Kaiser's rule [28]. However, there is no strict rule for the described variance threshold selection; [PERSON]'s rule works well when there are several principal components with eigenvalues being much higher than their average value and the remaining principal components have lesser eigenvalues. Another drawback is that the achieved solution of Equation (1) in principal component space of less dimensionality compared to the initial one is not unique. The task of qualitative and quantitative analysis in molecular absorption spectroscopy is closely related to the machine learning (ML) field [29]. The qualitative and quantitative absorption spectra analysis based on ML corresponds to classification and regression problems [30]. The difference is that the ML algorithm predicts the value of a logical (presence/absence of specific molecular components in the mixture) [29; 31; 32] or a real variable (like \(c_{i}\)) [33; 34].