Principal components analysis graphical representation

Principal components analysis is used to obtain a lower dimensional graphical representation which describes a majority of the variation in a data set. With PCA, a new set of axes arc defined in which to plot the samples. They are constructed so that a maximum amount of variation is described with a minimum number of axes. Because it reduces the dimensions required to visualize the data, PCA is a powerftil method for studying multidimensional data sets. 

Because of their fixed length, descriptors are valuable representations of molecules for use in further statistical calculations. The most important methods used to compare chemical descriptors are linear and nonlinear regression, correlation methods, and correlation matrices. Since patterns in data can be hard to find in data of high dimension, where graphical representation is not available, principal component analysis (PCA) is a powerful tool for analyzing data. PCA can be used to identify patterns in data and to express the data in such a way as to highlight their similarities and differences. Similarities or diversities in data sets and their properties data can be identified with the aid of these techniques. 

Fig. 3. Typical gas chromatographic profiles obtained after on-fibre derivatization of carbonyl compounds with pentafluorophenylhydrazine (A) from In vitro sampling of Sansivieria trifasciata flowers and (B) from a standard aqueous solution (1,58 mM of each aliphatic aldehyde, C3-C11). (C) Graphical representation of Sansevieria trifasciata flower scent composition change during the day. Principal component analysis of the compositional data permitted to discern a coordinate system with 87% of the information.

The most important method for exploratory analysis of multivariate data is reduction of the dimensionality and graphical representation of the data. The mainly applied technique is the projection of the data points onto a suitable plane, spanned by the first two principal component vectors. This type of projection preserves (in mathematical terms) a maximum of information on the data structure. This method, which is essentially a rotation of the coordinate system, is also referred to as eigenvector-projection or Karhunen-Loeve- projection (ref. 8). 

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