Aims of PCA

There are two principal needs in chemistry. In the case of the example of case study 1, we would like to extract information from the two way chromatogram. 

Below we will look in more detail how this information is obtained. However, the ultimate information has a physical meaning to chemists. 

Plot of scores of PC2 versus PCI after standardisation for case study 2 

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The aim of PCA is to create a set of latent variables which is smaller than the set of original variables but still explains all the variance in the matrix X of the original variables. 

Principal component analysis (PCA) can be considered as the mother of all methods in multivariate data analysis. The aim of PCA is dimension reduction and PCA is the most frequently applied method for computing linear latent variables (components). PCA can be seen as a method to compute a new coordinate system formed by the latent variables, which is orthogonal, and where only the most informative dimensions are used. Latent variables from PCA optimally represent the distances between the objects in the high-dimensional variable space—remember, the distance of objects is considered as an inverse similarity of the objects. PCA considers all variables and accommodates the total data structure it is a method for exploratory data analysis (unsupervised learning) and can be applied to practical any A-matrix no y-data (properties) are considered and therefore not necessary. 

FIGURE 3.2 Matrix scheme for PCA. Since the aim of PCA is dimension reduction and the... 

In many chemical studies, the measured properties of the system can be regarded as the linear sum of the fundamental effects or factors in that system. The most common example is multivariate calibration. In environmental studies, this approach, frequently called receptor modeling, was first applied in air quality studies. The aim of PCA with multiple linear regression analysis (PCA-MLRA), as of all bilinear models, is to solve the factor analysis problem stated below ... 

There are a number of important features of scores and loadings. It is important to recognise that the aim of PCA involves finding mathematical functions which contain certain properties which can then be related to chemical factors, and in themselves PCs are simply abstract mathematical entities. 

In PCA as with many other multivariate statistical techniques, sample-to-sample or object-to-object variation can be represented by a series of principal components (PCs), which preserve the structure of the underlying variance between two or more variables. The general aim of PCA is the reduction of the dimensionality of a dataset by the computation of a small number of these components (typically much less than the number of variables) that are parameterized by so-called scores cuid loadings. Each component derived from PCA contcuns within it some proportion of the overall variance (generally expressed as a percentage of the total), with the first principal component (PCI) being... 

PLS is related to principal components analysis (PCA) (20), This is a method used to project the matrix of the X-block, with the aim of obtaining a general survey of the distribution of the objects in the molecular space. PCA is recommended as an initial step to other multivariate analyses techniques, to help identify outliers and delineate classes. The data are randomly divided into a training set and a test set. Once the principal components model has been calculated on the training set, the test set may be applied to check the validity of the model. PCA differs most obviously from PLS in that it is optimized with respect to the variance of the descriptors. 

One aim of chemometrics is to obtain these predictions after first treating the chromatogram as a multivariate data matrix, and then performing PCA. Each compound in the mixture is a chemical factor with its associated spectra and elution profile, which can be related to principal components, or abstract factors, by a mathematical transformation. 

By plotting the magnitudes of each successive components (or errors in modelling the V block), it is also possible to determine prediction errors for the V block. However, the main aim of calibration is to predict concentrations rather than spectra, so this information, although useful, is less frequently employed in calibration. More details have been discussed in the context of PCA in Chapter 4, Section 4.3.3, and also Section 5.4.2 the ideas for PLS are similar. 

Another example of a descriptor pre-processing is a multivariate analysis of HlV-1 protease inhibitors. The aim of the study is to develop a method to predict on the basis of simple descriptors whether compounds are likely to trigger resistance or are effective against mutant HIV strains. PCA is used to reduce the 12 original descriptors to 4 uncorrelated descriptors. These four are then used to build a LDA model. 

Detailed survey on thermal destruction of a wide range of polymers and PA, specifically, is presented in monograph of Kovarskaya [34]. There are a few works in literature devoted to investigation of the kinetics of oxidation at relatively low temperatures. Kinetics of PCA thermooxidation has been studied in the work [35] having the aim to determine the possibility of forecasting of shelf lives and usage of PA-materials and products on their basis, proceeding from the temperature dependence of their induction periods of oxidation . 

These various chemometrics methods are used in those works, according to the aim of the studies. Generally speaking, the chemometrics methods can be divided into two types unsupervised and supervised methods(Mariey et al., 2001). The objective of unsupervised methods is to extrapolate the odor fingerprinting data without a prior knowledge about the bacteria studied. Principal component analysis (PCA) and Hierarchical cluster analysis (HCA) are major examples of unsupervised methods. Supervised methods, on the other hand, require prior knowledge of the sample identity. With a set of well-characterized samples, a model can be trained so that it can predict the identity of unknown samples. Discriminant analysis (DA) and artificial neural network (ANN) analysis are major examples of supervised methods. 

The mathematical background of PCA consists in the transformation of the initial coordinate system into a new one in order to display the variance of the experimental data much more clearly. To this aim the mathematical algorithms provide that... 

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