PCA and PCR

The difference between PLS and PCR is the manner in which the x data are compressed. Unlike the PCR method, where x data compression is done solely on the basis of explained variance in X followed by subsequent regression of the compressed variables (PCs) to y (a simple two-step process), PLS data compression is done such that the most variance in both x and y is explained. Because the compressed variables obtained in PLS are different from those obtained in PCA and PCR, they are not principal components (or PCs) Instead, they are often referred to as latent variables (or LVs). 

This section is for those who are interested in knowing the mechanics of the PCA and PCR calculations (Refs. 8, 10, 11, 15). There are actually two different methods used to calculate the principal components of a set of data the NIPALS algorithm and decomposition of covariance. The following descriptions of these algorithms assume that the matrices involved have the following dimensions A is an x p matrix of spectral absorbances, S is... 

PCR is a combination of PCA and MLR, which are described in Sections 9.4.4 and 9.4.3 respectively. First, a principal component analysis is carried out which yields a loading matrix P and a scores matrix T as described in Section 9.4.4. For the ensuing MLR only PCA scores are used for modeling Y The PCA scores are inherently imcorrelated, so they can be employed directly for MLR. A more detailed description of PCR is given in Ref. [5. ... 

Samola and Urleb [15] reported qualitative and quantitative analysis of OTC using near-infrared (NIR) spectroscopy. Multivariate calibration was performed on NIR spectral data using principle component analysis (PCA), PLS-1, and PCR. 

Mathematical Description of PLS and PCR In this section, the PCR algorithm is described as a two-step procedure of PCA followed by MLR. Although in practice the steps are combined, we feel tliis is tlie most intuitive approach to understanding the algorithm. This description of PCR is followed by a brief discussion of the differences between PLS and PCR. 

Some understanding of the size of the model that should be used (i.e.. the number of fiiaors to include in the model) can be gained by examining this table. Note that for PLS and PCR each successh e factor does not always have to explain a decreasing amount of concentration variance. For example, in Table 5.15 the second factor explains 26.33% of the concentration variation while the first factor explains only 25.76%. For the measurement data (K). the percent variance explained per factor for PCR always decreases this is a prop-ert> of PCA (see Section 4.2.2). For PLS, a strict decrease in the percent variance explained for R is not always observed. 

Although these factors are fairly similar to those described in Chapter 3, there are some relevant differences. In effect, in PCA (and hence in PCR), the factors explain most of the variance in the X-domain regardless of whether such variance is or is not related to the analyte, whereas in PLS they are calculated so that they explain not only variance in the spectral variables but also variance which is related to the property of interest. A typical example might be a series of graphite furnace measurements where a baseline shift appears due to some... 

Although the shape of the Visual-Empirical Region-of-Influence (VERI) mask is invariable, it size scales automatically according to the properties of the cluster (Fig. 10.10b). The VERI algorithm requires preprocessing of the data and for that purpose PCA or PCR preprocessing is routinely used. It is relatively immune to the presence of unknowns, and nonlinearity and nonadditivity of sensor responses (Osbourn et al., 1998). It has been used successfully to determine the optimum... 

The advent of personal computers greatly facilitated the application of spectroscopic methods for both quantitative and qualitative analysis. It is no longer necessary to be a spectroscopic expert to use the methods for chemical analyses. Presently, the methodologies are easy and fast and take advantage of all or most of the spectral data. In order to understand the basis for most of the current processing methods, we will address two important techniques principal component analysis (PCA) and partial least squares (PLS). When used for quantitative analysis, PCA is referred to as principal component regression (PCR). We will discuss the two general techniques of PCR and PLS separately, but we also will show the relationship between the two. 

PCR and PLR are useful when the matrix does not contain the full model representation. The first step of PCR is the decomposition of the data matrix into latent variables through PCA and the dependent variable is then regressed onto the decomposed independent variables. PLS performs, however, a simultaneous and interdependent PCA decomposition in a way that makes that PLS sometimes handles dependent variables better than does PCR. 

Simplified equations result because A A equals the identity matrix I, B B equals I, and C C equals I. The same equations are valid for PARAFAC models, but the middle cross-product is not identity. To summarize the properties of the squared Mahalanobis distances, the following can be said. Leverage can be defined for multiple linear regression as an influence measure. It is related to a specific Mahalanobis distance. The term leverage is sometimes also used for similar Mahalanobis distances in low-rank regression methods such as PCR and PLS. Then it becomes dependent on the rank of the model. The squared Mahalanobis distances can also be defined for PCA and multi-way models and can be calculated for both variables and objects. 

PCA step of PCR with the regression step. Latent variables, like PCs, are calculated to explain most of the variance in the x set while remaining orthogonal to one another. Thus, the first latent variable (LVi) will explain most of the variance in the independent set, LV2 the next largest amount of variance and so on. The important difference between PLS and PCR is that the latent variables are constructed so as to maximize their correlation with the dependent variable. Unlike PCR equations where the PCs do not enter in any particular order (see eqns 7.6 to 7.8) the latent variables will enter PLS equations in the order one, two, three, etc. The properties of latent variables are ... 

Principal component analysis (PCA) and principal component regression (PCR) were used to analyze the data [39,40]. PCR was used to construct calibration models to predict Ang II dose from spectra of the aortas. A cross-validation routine was used with NIR spectra to assess the statistical significance of the prediction of Ang II dose and collagen/elastin in mice aortas. The accuracy of the PCR method in predicting Ang II dose from NIR spectra was determined by the F test and the standard error of performance (SEP) calculated from the validation samples. 

It must be stressed that, differently than in the case of MLR and PCR, in building the PLS model for multiple responses, the correlation structure of the dependent block is explicitly taken into account and actively used while MLR (and, as a consequence PCR, which is nothing more than MLR on PCA scores) assumes that the responses are independent, so that the same results would be obtained by modelling each dependent variable individually or the Y-block as a whole, PLS relies on the identification of a common latent structure between the responses, which is supposed to be covariant with the component space of the predictor block. 

It may look weird to treat the Singular Value Decomposition SVD technique as a tool for data transformation, simply because SVD is the same as PCA. However, if we recall how PCR (Principal Component Regression) works, then we are really allowed to handle SVD in the way mentioned above. Indeed, what we do with PCR is, first of all, to transform the initial data matrix X in the way described by Eqs. (10) and (11). 

Sections 9A.2-9A.6 introduce different multivariate data analysis methods, including Multiple Linear Regression (MLR), Principal Component Analysis (PCA), Principal Component Regression (PCR) and Partial Least Squares regression (PLS). 

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