## 1 Introduction  
Measuring the elemental composition of samples is an indispensable part of modern science. The preference here is given to instrumental methods capable of simultaneous multielement analysis. One of the most popular methods for this purpose is X-ray fluorescence spectrometry (XRF) due to its reasonable price, nondestructive analysis procedure, simple sample preparation, and applicability to a broad variety of objects in different aggregate states. There are two main varieties of XRF spectrometer: wavelength dispersive (WDXRF) and energy dispersive (EDXRF). WDXRF offers high sensitivity and good resolution (selectivity) but requires a long measuring procedure in multielement mode; therefore, it is quite expensive and cannot be employed for in-field measurements. In spite of the fact that accuracy and selectivity of EDXRF are not that high compared to WDXRF, EDXRF is a more popular variety as it provides for simple, inexpensive, and fast measurements and can be miniaturized for in-field applications. Due to these features, EDXRF has found numerous applications in various fields of applied science: geochemistry [1], material science [2], pharmacology [3], archeology [4], environmental studies [5], etc. [6]. While qualitative analysis (elemental identification) with EDXRF is usually straightforward, the quantitative determination of element content may often pose certain problems due to so-called matrix effects. These mainly come from the absorption and scattering of X-ray radiation (both primary and fluorescence) by the elements of the sample and lead to the skewing of fluorescence line intensities and thus to the incorrect quantification of target elements. For instance, if Fe is determined in the presence of Mg, then the signal of the former will be higher than it would be without this matrix element. In contrast, determination of Fe in the presence of Nb or W leads to a decrease in the iron's signal. Inaccurate quantification can also come from the overlapping of fluorescence lines, e.g., when Mn lines overlap with Cr and Fe [7]. Several methods have been developed to take into account these matrix effects. Traditional methods include the fundamental parameters (FP) and influence coefficients (IC) approaches [7]. FP is based on the fundamental model of X-ray radiation interaction with matter and allows one to quantify the elements in the sample without using standard samples. Among the limitations of FP is the fact that it may yield inaccurate results when the sample under study contains a significant quantity of light elements (typically Z \(<\) 11). FP is typically considered as a semiquantitative method. IC is based on the construction of a regression model relating the intensities of certain XRF lines (from both target and matrix elements) to the content of the target element. IC can offer higher accuracy, but it requires a sufficient quantity of standard samples with known elemental composition and the construction of an individual regression equation for each target element. In recent years, attempts have been made to apply chemometric techniques for solving the matrix effects problem [8; 9; 10]. However, it was demonstrated that in most cases IC outperforms traditional chemometric multivariate regression tools [11]. This is because chemometric tools cannot take into account the specific nature of matrix effects in XRF. This issue was successfully circumvented in [12], where IC was combined with partial least squares regression (PLS). Such a combination allows for a more effective accounting of matrix effects and eliminates the need for separate regression equations for each analyte. Since the matrix effects depend on the particular elemental composition of the sample, they will strongly depend on a particular sample type. Thus, successful calibration requires standard samples with a composition similar to that of the object under analysis. In this way, a separate independent calibration model is required for each type of sample even if the analyzed element is the same.