Scatter plots, structural analysis

Spectral features and their corresponding molecular descriptors are then applied to mathematical techniques of multivariate data analysis, such as principal component analysis (PCA) for exploratory data analysis or multivariate classification for the development of spectral classifiers [84-87]. Principal component analysis results in a scatter plot that exhibits spectra-structure relationships by clustering similarities in spectral and/or structural features [88, 89]. 

The wa in equation (6) are the PLS loading weights. They are explained in the theory in references 53 - 62. Equation (7) shows how X is decomposed bilinearly (as in principal component analysis) with its own residual Epls A. T is the matrix with the score vectors as columns, P is the matrix having the PLS loadings as columns. Also the vectors of P and wa can be used to construct scatter plots. These can reveal the data structure of the variable space and relations between variables or groups of variables. Since PLS mainly looks for sources of variance, it is a very good dirty data technique. Random noise will not be decomposed into scores and loadings, and will be stored in the residual matrices (E and F), which contain only non-explained variance . 

Especially scatter plots should be interpreted very carefully. Principal component analysis produces latent variables and at the same time orthogonal scores and orthonormal loadings. The latent variable and the Euclidean interpretation of scatter plots both come from the same model. For three-way models, the latent variables do not allow a direct Euclidean interpretation of loading plots. A recalculation can give this Euclidean interpretation, but then the original latent variable structure gets lost. 

Figure 9.15 Scatter plots of the first three principal components of all structural parameters and variance components. Each index patient was singularly removed from the data set and the model in Eq. (9.14) was refit using FOCE-I. The resulting matrix of structural parameters and variance components was then analyzed using principal components analysis. Influential observations are noted in the figures. Patient 100, who had a BSA of 2.52m2 and a BMI of 31.2kg/m2, is denoted as a solid square.

PCA is by far the most important method in multivariate data analysis and has two main applications (a) visualization of multivariate data by scatter plots as described above (b) data reduction and transformation, especially if features are highly correlating or noise has to be removed. For this purpose instead of the original p variables X a subset of uncorrelated principal component scores U can be used. The number of principal components considered is often determined by applying a threshold for the score variance. For instance, only principal components with a variance greater than 1% of the total variance may be selected, while the others are considered as noise. The number of principal components with a non-negligible variance is a measure for the intrinsic dimensionality of the data. As an example consider a data set with three features. If all object points are situated exactly on a plane, then the intrinsic dimensionality is two. The third principal component in this example has a variance of zero. Therefore two variables (the scores of PCI and PC2) are sufficient for a complete description of the data structure. 

One of the most useful techniques for the study of the structure and thermodynamics of dilute polymer solutions is light scattering. The principles of light scattering from an ensemble of solute molecules will be presented and illustrated. The analysis of a Zimm plot will also be explained and discussed. 

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