Principal component regression dimensionality

Factor spaces are a mystery no more We now understand that eigenvectors simply provide us with an optimal way to reduce the dimensionality of our spectra without degrading them. We ve seen that, in the process, our data are unchanged except for the beneficial removal of some noise. Now, we are ready to use this technique on our realistic simulated data. PCA will serve as a pre-processing step prior to ILS. The combination of Principal Component Analysis with ILS is called Principal Component Regression, or PCR. 

Factor The result of a transformation of a data matrix where the goal is to reduce the dimensionality of the data set. Estimating factors is necessary to construct principal component regression and partial least-squares models, as discussed in Section 5.3.2. (See also Principal Component.)... 

The usual objective of PC A is to reduce the dimensionality of a data matrix, or determine its intrinsic dimensionality. The PCs can also be used in other QSAR methods including linear regression models (termed principal component regression, PCR). However, PLS gives similar results and is generally preferred to PCR. 

When describing the PCA, it has been noticed that the components are orthogonal (i.e., uncorrelated) and that the dimensionality of the resulting space (i.e., the number of significant components) is much lower than the dimensionality of the original space. Therefore, it can be seen that both the aforementioned limitations have been overcome. As a consequence, it is possible to apply OLS to the scores originated by PCA. This technique is Principal Component Regression (PCR). 

In the case of multivariate modeling, several independent as well as several dependent variables may operate. Out of the many regression methods, we will learn about the conventional method of ordinary least squares (OLS) as well as methods that are based on biased parameter estimations reducing simultaneously the dimensionality of the regression problem, that is, principal component regression (PCR) and the partial least squares (PLS) method. 

Partial least squares and principal components regression can be used to reduce the dimensionality of the input space, in this case attempting to reduce the degrees of freedom of the models to 14 or less without losing the most important information in the input data. 

Although colorimetric methods were the earliest to be used for pesticide analysis [203], competitive spectroscopic methodologies for the determination of these pollutants were not developed until the last decade. The spectroscopic determination of several pesticides in mixtures has been the major hindrance, especially when their analytical characteristics are similar and their signals overlap as a result. Multivariate calibration has proved effective with a view to developing models for qualitative and quantitative prediction from spectroscopic data. Thus, partial least squares (PLS) and principal component regression (PCR) have been used as calibration models for the spectrofluorimetric determination of three pesticides (carbendazim, fuberidazole, and thiabendazole) [204]. A three-dimensional excitation-emission matrix fluorescence method has also been used for this purpose (Table 18.3) [205]. 

Marhaba, T.F., Borgaonkar, A.D., and Punburananon, K. (2009). Principal component regression model applied to dimensionally reduced spectral fluorescent signature for the determination of organic character and THM formation potential of source water. 

An alternative and illuminating explanation of reduced rank regression is through a principal component analysis of Y, the set of fitted F-variables resulting from an unrestricted multivariate multiple regression. This interpretation reveals the two least-squares approximations involved projection (regression) of Y onto X, followed by a further projection (PCA) onto a lower dimensional subspace. 

In QSAR and QSPR studies, the standard ways of removing redundancy from large numbers of topological and topographical indices include principal component analysis, chi-squared analysis, and multiple regression analysis (MRA). Most QSAR and QSPR applications deal with very small datasets, and so the dimensionality does not cause a problem for PCA or chi-squared analysis. MRA does not impose any restrictions on the type and number of descriptors. The selection process is based on two principles, namely, to cover as much of parametric space as possible (principle of variance) while choosing independent descriptors (principle of orthogonality). 

Infrared, near-infrared, and Raman spectroscopy were used to study high density, linear low density (LLDPE), and low density polyethylene. Overlapping spectral bands were separated using the second derivative, principal component analysis, and two-dimensional correlation analysis. A model was developed, using partial least squares regression, to calculate the density of LLDPE. 1 ref. 

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