Principle component regression

Principle Component Regression and Partial Least Squares, PCR and PLS... 

Principle components regression (PCR) is one of the supervised methods commonly employed to analyze NMR data. This method is typically used for developing a quantitative model. In simple terms, PCR can be thought of as PCA followed by a regression step. In PCR, the scores matrix (T) obtained in PCA (Section 3.1) is related to an external variable in a least squares sense. Recall that the data matrix can be reconstructed or estimated using a limited number of factors (/ffact), such that only the fc = Mfaet PCA loadings (l fc) are required to describe the data matrix. Eq. (15) can be reconstructed as... 

The principle component regression (PCR), which uses an eigenvector-eigenvalue decomposition or a single value decomposition or a singular value decomposition of the matrix E [38,71,72]. Both require a lot of calculation time. 

In principle both the classical and the inverse approach use a multivariate data set. But in the classical approach the variance is minimised, whereas in the inverse approach one tries to find an equilibrium between bias and variance. Therefore the bias is reduced and by the procedure of predictive receivable error sum of squares either via a singular value decomposition or the bidiagonalisation method estimated values, either according to principle component regression or partial least squares, are found. The multilinear regression on the other hand will find the best linear unbiased estimation as an approach to a true concentration. Besides applications in absorption spectroscopy, fluorescence spectra can also be evaluated [74]. 

R. Marbach, H. M. Heise, Calibration modeling by partial least-squares and principle component regression and its optimization using an improved leverage correction for prediction testing, J. Chemom. Intell. Lab. Syst. 9 (1990) 45. 

This same approach can be used for a mixture of three components. More complex mixtures can be unraveled through computer software that uses an iterative process at multiple wavelengths to calculate the concentrations. Mathematical approaches used include partial least squares, multiple least squares, principle component regression, and other statistical methods. Multicomponent analysis using UV absorption has been used to determine how many and what type of aromatic amino acids are present in a protein and to quantify five different hemoglobins in blood. 

TABLE 4.2.2. Statistical Parameters for Principle Component Regression (PCR) and Partial Least-Squares (PLS) Analysis... 

PCR Principle component regression rms Root mean square... 

If gas selectivity cannot be achieved by improving the sensor setup itself, it is possible to use several nonselective sensors and predict the concentration by model based, such as multilinear regression (MLR), principle component analysis (PCA), principle component regression (PCR), partial least squares (PLS), and multivariate adaptive regression splines (MARS), or data-based algorithms, such as cluster analysis (CA) and artificial neural networks (ANN) (for details see Reference 10) (Figure 22.5). For common applications of pattern recognition and multi component analysis of gas mixtures, arrays of sensors are usually chosen... 

Crawford, N. F. and Hellmuth, W. W., Refinery Octane Blend Modeling Using Principle Components Regression of Gas Chromatographic Data, Fuel, Vol. 69, 1990, pp. 443-447. 

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