Principal component regression complexity

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

N.B Vogt, Polynomial principal component regression an approach to analysis and interpretation of complex mixture relationships in multivariate environmental data, Chemom. Intell. Lab. Syst, 7, 119-130 (1989). 

The simplest definition of model complexity is based on the number of terms in the model or, in other words, the model complexity is made up by the number of model variables from Ordinary Least Squares regression cpx = p), the number M of significant principal components from Principal Component Regression (cpx = M), and the number of significant latent variables from Partial Least Squares regression (cpx = M)... 

The fourth-derivative spectra of molybdenum complexes of tetramethyldithiocarbamate (tiram) fungicide were used for its quantification in commercial samples and in wheat grains [41], Atrazine and cyanazine were assayed in food samples by first- derivative spectrophotometry [42]. In order to improve results of assay, the first-derivative spectra of the binary mixture were subjected to chemometiic treatment (classical least squares, CLS principal component regression, PCR and p>artial least squares, PLS). A combination of first-derivative with PCR and PLS models were applied for determination of both herbicides in biological samples [42]. A first-derivative spectrophotometry was used as a reference method for simultaneous determination BriUant Blue, Simset Yellow and Tartrazine in food [43]. 

The improvement in computer technology associated with spectroscopy has led to the expansion of quantitative infrared spectroscopy. The application of statistical methods to the analysis of experimental data is known as chemometrics [5-9]. A detailed description of this subject is beyond the scope of this present text, although several multivariate data analytical methods which are used for the analysis of FTIR spectroscopic data will be outlined here, without detailing the mathematics associated with these methods. The most conunonly used analytical methods in infrared spectroscopy are classical least-squares (CLS), inverse least-squares (ILS), partial least-squares (PLS), and principal component regression (PCR). CLS (also known as K-matrix methods) and PLS (also known as P-matrix methods) are least-squares methods involving matrix operations. These methods can be limited when very complex mixtures are investigated and factor analysis methods, such as PLS and PCR, can be more useful. The factor analysis methods use functions to model the variance in a data set. 

Complex chemical systems are composed of one or more components in a mixture with a significant degree of spectral interference, or of several components with a large amount of mutual physical and/or chemical interaction. In these cases, quantitative analysis is best performed by statistical methods such as principal component regression (PCR) or partial least squares (PLS) [36] these are offered in the software packages of instrument manufacturers and software suppliers. Artificial neural networks (ANNs) should be primarily used when a data set is nonlinear [37]. 

In some cases, the primary quantitative information is simply not available to allow building a principal component regression (PCR) or partial least squares model. There may not be a primary calibration method available for the constituent of interest, or the samples may simply be too complex. However, the spectrum of a sample is unique to the composition of its constituents. Samples of the same or similar composition quality should have spectra that are very similar as well. Theoretically, it should be possible to tell the difference between a good sample and a bad one by comparing their spectra. 

Since most quantitative applications are on mixtures of materials, complex mathematical treatments have been developed. The most common programs are Multiple Linear Regression (MLR), Partial Least Squares (PLS), and Principal Component Analyses (PCA). While these are described in detail in another chapter, they will be described briefly here. 

The data obtained from NMR spectroscopy are generally complex, and thus multivariate statistical methods are often used to extract the maximum information from the large datasets. Of the multivariate methods, the most frequently used are principal component analysis (PCA), logistic regression,... 

It is clear that for an unsymmetrical data matrix that contains more variables (the field descriptors at each point of the grid for each probe used for calculation) than observables (the biological activity values), classical correlation analysis as multilinear regression analysis would fail. All 3D QSAR methods benefit from the development of PLS analysis, a statistical technique that aims to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the F space. PLS is related to principal component analysis (PCA)." ° However, instead of finding the hyperplanes of maximum variance, it finds a linear model describing some predicted variables in terms of other observable variables and therefore can be used directly for prediction. Complexity reduction and data... 

To establish a correlation between the concentrations of different kinds of nucleosides in a complex metabolic system and normal or abnormal states of human bodies, computer-aided pattern recognition methods are required (15, 16). Different kinds of pattern recognition methods based on multivariate data analysis such as principal component analysis (PCA) (8), partial least squares (16), stepwise discriminant analysis, and canonical discriminant analysis (10, 11) have been reported. Linear discriminant analysis (17, 18) and cluster analysis were also investigated (19,20). Artificial neural network (ANN) is a branch of chemometrics that resolves regression or classification problems. The applications of ANN in separation science and chemistry have been reported widely (21-23). For pattern recognition analysis in clinical study, ANN was also proven to be a promising method (8). 

Water is the most important chemical constituent of fruits and vegetables and water highly absorbs NIR radiation, so the NIR spectrum of such materials is dominated by water. Further, the NIR spectrum is essentially composed of a large set of overtones and combination bands. This, in combination with the complex chemical composition of a typical fruit or vegetable causes the NIR spectrum to be highly convoluted. Multivariate statistical techniques are required to extract the information about quality attributes which is buried in the NIR spectrum Developments in multivariate statistical techniques such as partial least squares (PLS) regression and principal component analysis (PCA) are then applied to extract the required information from such convoluted spectra (Cozzolino et al., 2006b McClure, 2003 Naes et al., 2004 Nicolai et al., 2007 ). 

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