Principal component analysis mapping

Figure 17.17. Principal component analysis map of sample (left) and color-coded spectra (right) from a sample of marine suspended particulate matter. The lower three spectra are characteristic of low organic mineral phases, while the upper three organic phases have distinctively different C-NEXAFS spectra. Background regions are shown in black (J. Brandes, unpublished data 2007). See color insert.

Input/Output sequence encoding methods (see Table 9.1) PCA - Principal Component Analysis Map -Kohonen map (with dimensions) G = Global (member/non-member) M = Motif (member/non-member). 

According to Andersen [12] early applications of LLM are attributed to the Danish sociologist Rasch in 1963 and to Andersen himself. Later on, the approach has been described under many different names, such as spectral map analysis [13,14] in studies of drug specificity, as logarithmic analysis in the French statistical literature [15] and as the saturated RC association model [16]. The term log-bilinear model has been used by Escoufier and Junca [ 17]. In Chapter 31 on the analysis of measurement tables we have described the method under the name of log double-centred principal components analysis. 

FIG. 7 Principal component analysis similarity map defined by the principal components 1 and 2 (Al, A2) for Laurdan fluorescence excitation spectral data recorded during the BLG5 milk coagulation. The digits correspond to the elapsed time. Each label corresponds to a spectrum. 

To reduce intensity effects, the data were normalized by reducing the area under each spectrum to a value of 1 [42]. Principal component analysis (PCA) was applied to the normalized data. This method is well suited to optimize the description of the fluorescence data sets by extracting the most useful data and rejecting the redundant ones [43]. From a data set, PCA assesses principal components and their corresponding spectral pattern. The principal components are used to draw maps that describe the physical and chemical variations observed between the samples. Software for PCA has been written by D. Bertrand (INRA Nantes) and is described elsewhere [44]. 

In general, the evaluation of interlaboratory studies can be carried out in various ways (Danzer et al. [1991]). Apart from z-scores, multivariate data analysis (nonlinear mapping, principal component analysis) and information theory (see Sect. 9.2) have been applied. 

Chapter 3 starts with the first and probably most important multivariate statistical method, with principal component analysis (PC A). PC A is mainly used for mapping or summarizing the data information. Many ideas presented in this chapter, like the selection of the number of principal components (PCs), or the robustification of PCA, apply in a similar way to other methods. Section 3.8 discusses briefly related methods for summarizing and mapping multivariate data. The interested reader may consult extended literature for a more detailed description of these methods. 

How is dimension reduction of chemical spaces achieved There are a number of different concepts and mathematical procedures to reduce the dimensionality of descriptor spaces with respect to a molecular dataset under investigation. These techniques include, for example, linear mapping, multidimensional scaling, factor analysis, or principal component analysis (PCA), as reviewed in ref. 8. Essentially, these techniques either try to identify those descriptors among the initially chosen ones that are most important to capture the chemical information encoded in a molecular dataset or, alternatively, attempt to construct new variables from original descriptor contributions. A representative example will be discussed below in more detail. 

When molecules are represented by low-dimensional descriptors, then the descriptors can be used to define the axes of a chemistry space. Typical descriptors are a small number of physicochemical properties or the principal components generated by the application of principal components analysis to high-dimensional descriptors. Each descriptor then defines one axis and is divided into a series of bins. The combination of all bins over all descriptors defines a set of cells over a chemistry space. Molecules can be mapped onto the cells according to their physicochemical properties. A diverse library is one that occupies a large number of cells in the space, whereas a focused library is one where the molecules occupy a small localized region of the space. 

The virtual library was then characterized using the Cerius2 default topological descriptors and physicochemical properties (35). The 50 default descriptors were reduced to three principal components using principal components analysis, and this defined a 3D chemistry space into which the virtual library could be plotted. The chemistry space consisted of 1134 cells and, when the virtual library was mapped into the space, it was found to occupy 364 of the cells thus, this represents the maximum cell coverage that is achievable. 

Here E is the solute excess molar refractivity, S is the solute dipolarity/ polarizability A and B are the overall or summation hydrogen-bond acidity and basicity, respectively and V is the McGowan characteristic volume lower-case letters stand for respective coefficients which are characteristic of the solvent, c is the constant. By help of sfafisfical methods like the principal component analysis and nonlinear mapping, the authors determined the mathematical distance (i.e., measure of dissimilarify) from an IL fo seven conventional solvents immiscible with water. It appears that the closest to the IL conventional solvent is 1-octanol. Even more close to IL is an aqueous biphasic system based on PEG-200 and ammonium sulfate (and even closer are ethylene glycol and trifluoroethanol, as calculated for hypofhefical water-solvenf sysfems involving fhese solvenfs). 

Figure 8. Kriged image map showing the distribution of Factor I scores from the principal components analysis (see the caption for Figure 5 for the results of the principal components analysts).

A principal component analysis was also performed [20]. It was found that relative positions among different brands of beer (for example, with respect to K2 as an origin) are similar to those in Figure 14. This assures the conventional taste expressions such as "sharp touch" and "rich taste" in the taste map. Simultaneous consideration of output patterns with various methods will make it possible to describe these obscure human taste expressions using the five basic taste qualities. 

Figure 17.19. Carbon (1 s) NEXAFS spectra of clusters obtained from principal component analysis. Cluster maps are shown in Figure 17.18. Further reduction in the number of clusters will reduce redundancy, but can mask minor features such as those in cluster 19 (J. Lehmann, unpublished data 2006, measured as described in Lehmann et al., 2007).

Unsupervised learning methods - cluster analysis - display methods - nonlinear mapping (NLM) - minimal spanning tree (MST) - principal components analysis (PCA) Finding structures/similarities (groups, classes) in the data... 

Unsupervised multivariate statistical methods [CA, principal components analysis, Kohonen s self-organizing maps (SOMs), nonlinear mapping, etc.], which perform spontaneous data analysis without the need for special training (learning), levels of knowledge, or preliminary conditions. 

Astel, A., S. Tsakovski, P. Barbieri, and V. Simeonov. 2007. Comparison of self-organizing maps classification approach with cluster and principal components analysis for large environmental data sets. Water Res. 41 4566-4578. 

---