Multivariate nonlinear calibration

To take account of interactions between individual components (association, nonlinearities), calibration using multivariate data analysis is often also carried out with mixtures rather than pure substances. Despite this fact, limitations to this method of assessment are encountered quickly. Therefore, the so-called inverse method using the g-matrix is employed, and either principal component regression (PCR) or the partial least squares (PLS) method is used [6, [114], [116]. In both methods, calibration is carried out not with pure substances, but with various mixtures, which must cover the expected concentration range of all components. Within limits, this can allow for non-linearities ... 

S. Sekulics, B.R. Kowalski, Z.Y. Wang, et al., Nonlinear multivariate calibration methods in analytical chemistry. Anal. Chem., 65 (1993) A835-A845. 

Nonlinearity is a subject the specifics of which are not prolifically or extensively discussed as a specific topic in the multivariate calibration literature, to say the least. Textbooks routinely cover the issues of multiple linear regression and nonlinearity, but do not cover the issue with full-spectrum methods such as PCR and PLS. Some discussion does exist relative to multiple linear regression, for example in Chemometrics A Textbook by D.L. Massart et al. [6], see Section 2.1, Linear Regression (pp. 167-175) and Section 2.2, Non-linear Regression, (pp. 175-181). The authors state,... 

P.J. Gemperhne, J.R. Long and V.G. Gregoriov, Nonlinear multivariate calibration using principal components regression and artificial neural networks. Anal Chem., 63, 2313-2323 (1991). 

In traditional method validation, assessment of the calibration has been discussed in terms of linear calibration models for univariate systems, with an emphasis on the range of concentrations that conform to a linear model (linearity and the linear range). With modern methods of analysis that may use nonlinear models or may be multivariate, it is better to look at the wider picture of calibration and decide what needs to be validated. Of course, if the analysis uses a method that does conform to a linear calibration model and is univariate, then describing the linearity and linear range is entirely appropriate. Below I describe the linear case, as this is still the most prevalent mode of calibration, but where different approaches are required this is indicated. 

Luo et al. [83] used an ANN to perform multivariate calibration in the XRF analysis of geological materials and compared its predictive performance with cross-validated PLS. The ANN model yielded the highest accuracy when a nonlinear relationship between the characteristic X-ray line intensity and the concentration existed. As expected, they also found that the prediction accuracy outside the range of the training set was bad. 

Physical and chemical effects can be combined for identification as sample matrix effects. Matrix effects alter the slope of calibration curves, while spectral interferences cause parallel shifts in the calibration curve. The water-methanol data set contains matrix effects stemming from chemical interferences. As already noted in Section 5.2, using the univariate calibration defined in Equation 5.4 requires an interference-free wavelength. Going to multivariate models can correct for spectral interferences and some matrix effects. The standard addition method described in Section 5.7 can be used in some cases to correct for matrix effects. Severe matrix effects can cause nonlinear responses requiring a nonlinear modeling method. 

Using artificial neural networks to develop calibration models is also possible. The reader is referred to the literature [68-70] for further information. Neural networks are commonly utilized when the data set maintains a large degree of nonlinearity. Additional multivariate approaches for nonlinear data are described in the literature [71, 72],... 

A very important and interesting field of chemometrics is the neural networks field, which can be successfully employed in nonlinear mapping, in nonlinear multivariate calibration, and in linear and nonlinear pattern classification. Also, confidence intervals are determined using calibration... 

Normally these scaled X and y variables are centered around their means x, y prior to the calibration analysis. The main reason is that centering normally gives better linear approximation of possible nonlinear stmctures. Multivariate calibration, after all, concerns how to make good local approximations to the true but unknown nonlinear X-y relationship for a certain type of samples... 

IR and Raman spectroscopy are both quantitative techniques, in that both the IR absorbance and the Raman scattered intensity are, to a first approximation, linearly proportional to the number density of vibrating species. Although quantitative information can be extracted from a single absorbance or scattered intensity (after calibration against absolute standards), it is more usual to measure the ratio of the analyte band of interest against an independent internal standard, in order to allow for variations in pathlength, deviations from ideality, detector nonlinearity, etc. The reader is referred elsewhere for a full discussion of quantitative methods (including multivariate techniques) [9, 10]. 

In chemometrics, multivariate calibration methods provide a convenient way to determine several components in a mixture within one experimental step, without the tedious operation of separation of these components. The method of calculation usually used is PLS method. Artificial neural network is also often used especially when the data set exhibits obvious nonlinearity. But it is prone to overfitting. Therefore, several types of techniques have been developed to prevent overfitting. At the same time, support vector regression, as a method suitable to treat nonlinear data without serious overfitting, can be used as a new method of computation in multivariate calibration. An example of using SVR in multivariate calibration will be described as follows. 

Quantitative analysis for one or more analytes through the simultaneous measurement of experimental parameters such as molecular UV or infrared absorbance at multiple wavelengths can be achieved even where clearly defined spectral bands are not discernible. Standards of known composition are used to compute and refine quantitative calibration data assuming linear or nonlinear models. Principal component regression (PCR) and partial least squares (PLS) regression are two multivariate regression techniques developed from linear regression (Topic B4) to optimize the data. 

---