Principal component regression, results

As we saw in the last chapter, by discarding the noise eigenvectors, we are able to remove a portion of the noise from our data. We have called the data that results after the noise removal the regenerated data. When we perform principal component regression, there is not really a separate, explicit data regeneration step. By operating with the new coordinate system, we are automatically regenerating the data without the noise. 

Because of peak overlappings in the first- and second-derivative spectra, conventional spectrophotometry cannot be applied satisfactorily for quantitative analysis, and the interpretation cannot be resolved by the zero-crossing technique. A chemometric approach improves precision and predictability, e.g., by the application of classical least sqnares (CLS), principal component regression (PCR), partial least squares (PLS), and iterative target transformation factor analysis (ITTFA), appropriate interpretations were found from the direct and first- and second-derivative absorption spectra. When five colorant combinations of sixteen mixtures of colorants from commercial food products were evaluated, the results were compared by the application of different chemometric approaches. The ITTFA analysis offered better precision than CLS, PCR, and PLS, and calibrations based on first-derivative data provided some advantages for all four methods. ... 

Principal component regression (PCR) is an extension of PCA with the purpose of creating a predictive model of the Y-data using the X or measurement data. For example, if X is composed of temperatures and pressures, Y may be the set of compositions that results from thermodynamic considerations. Piovoso and Kosanovich (1994) used PCR and a priori process knowledge to correlate routine pressure and temperature measurements with laboratory composition measurements to develop a predictive model of the volatile bottoms composition on a vacuum tower. 

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 main goal of this chapter is to present the theoretical background of some basic chemometric methods as a tool for the assessment of surface water quality described by numerous chemical and physicochemical parameters. As a case study, long-term monitoring results from the watershed of the Struma River, Bulgaria, are used to illustrate the options offered by multivariate statistical methods such as CA, principal components analysis, principal components regression (models of source apportionment), and Kohonen s SOMs. 

Perform principal components regression as follows. Take each compound in turn, and regress the concentrations on to the scores of the two PCs you should centre the concentration matrices first. You should obtain two coefficients of the form c = t r + /2r2 for each compound. Verify that the results of PCR provide a good estimate of the concentration for each compound. Note that you will have to add the mean concentration back to the results. 

There is an approach in QSRR in which principal components extracted from analysis of large tables of structural descriptors of analytes are regressed against the retention data in a multiple regression, i.e., principal component regression (PCR). Also, the partial least square (PLS) approach with cross-validation 29 finds application in QSRR. Recommendations for reporting the results of PC A have been published 130). 

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. 

Equally important is to check the descriptor set for multicolinearity. Correlation between descriptor values results in unreliable MLR with overestimate goodness-of-fit parameters and poor predictive capability. Crosscorrelation matrices provide information on descriptor multicolinearity. It is worth mentioning that when two descriptors, X and Z, are statistically correlated, it does not necessarily mean that physicochemically they are also redundant. Principal component regression or partial least squares regression can be used to address multicolinearity. Alternatively, the impact of descriptors X and Z on the QSRR should be inspected separately. It should be possible to select from the solute set those solutes with varying X values and constant Z values and vice versa. 

TABLE 16.8. Partial least squares regression (PLS) and principal components regression (PCR) results for the prediction of the ripening times of ewe s milk cheeses and cheeses made from cow s and ewe s milk... 

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). 

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]. 

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