Regression on principal components

The first step in carrying out principal component regression (PCR) is, 

Having carried out PCA, what comes next The principal component scores are treated as any other variables would be in a multiple regression analysis and MLR models are constructed as shown in eqn (7.4) 

A number of things may be seen from these equations. The first component to be incorporated in the regression models was indeed the first PC and this, combined with a constant, accounted for half of the variance in the dependent variable set (i =0.5). The next component to enter the model, however, was PC4 despite the fact that the variables in the set of 11 had been chosen for their individual ability to describe k. Clearly, the linear combinations imposed by PCA, combined with the requirements of orthogonality, did not produce new variables in PC2 and PC3 which were useful in the description of k. The third PC to be included has an eigenvalue of less than one and yet it is seen to be significant in eqn (7.8). If the eigenvalue cut-off of less than one had been imposed on this data set, eqn (7.8) would not have been found. 

As extra terms are added to the regression model it can be seen that the regression coefficients do not change, unlike the case for MLR with untransformed variables where collinearity and multicollinearity amongst the descriptors can lead to instability in the regression coefficients. The regression coefficients in eqns (7.6) to (7.8) remain constant because the 

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Principal Component Principal Component Analysis Polychlorinated Biphenyl Regression on Principal Components P-Glycoprotein... 

Regression on principal components (PCR) is another from of regression modeling that may be used for continuous response data. Here, the independent variables (the x set) are computed from the descriptor variables using PC A as shown in Equation 7.1. These are the principal component scores and they have several advantages ... 

Figure 14.10 Drivers of liking/disliking obtained by regression on principal component.

Regression on principal components and partial least squares 149... 

In a general way, we can state that the projection of a pattern of points on an axis produces a point which is imaged in the dual space. The matrix-to-vector product can thus be seen as a device for passing from one space to another. This property of swapping between spaces provides a geometrical interpretation of many procedures in data analysis such as multiple linear regression and principal components analysis, among many others [12] (see Chapters 10 and 17). 

Table 5. Linear regression between principal component scores antibiotic resistance MPNs and soil physico-chemical characteristics based on the...

The diffusion of correlation methods and related software packages, such as partial-least-squares regression (PLS), canonical correlation on principal components, target factor analysis and non-linear PLS, will open up new horizons to food research. 

The methods of data analysis depend on the nature of the final output. If the problem is one of classification, a number of multivariate classifiers are available such as those based on principal components analysis (SIMCA), cluster analysis and discriminant analysis, or non-linear artificial neural networks. If the required output is a continuous variable, such as a concentration, then partial least squares regression or principal component regression are often used [20]. 

Sousa, S.I.V., Martins, F.G., Alvim-Ferraz, M.C.M and Pereira, M.C., 2007. Multiple linear regression and artificial neural networks based on principal components to predict ozone concentrations. Environmental Modelling Software 22, p.97-103. 

On the other hand, atomic emission spectra are inherently well suited for multivariate analysis due to the fact that the intensity data can be easily recorded at multiple wavelengths. The only prerequisite is that the cahbration set encompasses all likely constituents encountered in the real sample matrix. Calibration data are therefore acquired by a suitable experimental design. Not surprisingly, many of the present analytical schemes are based on multivariate calibration techniques such as multiple linear regression (MLR), principal components regression (PCR), and partial least squares regression (PLS), which have emerged as attractive alternatives. 

The effect of sugar and acid contents in fruit on the transmitted output power was fully examined and used to estimate the optical parameter preferable for detecting internal quality. The performances of multiple Unear regression (MLR), principal component regression (PCR), and partial least-squares regression (PLS) analysis by TOF-NIRS were also compared to those from normal methods using reflectance data. 

The best conditions to operate an axially viewed ICP were set employing experimental designs. The multivariate effect of carrier gas flow and RF power on several analytical figures of merit was studied. Multivariate regression and principal component analysis were used as well to model the system. 

Hasegawa, T. (2006) Spectral simulation study on the influence of the principal component step on principal component regression. Appl Spectrosc., 60, 95-98. 

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