Eigenvector projection

Eigenvector projections are those in which the projection vectors u and v are eigenvectors (or singular vectors) of the data matrix. They play an important role in multivariate data analysis, especially in the search for meaningful structures in patterns in low-dimensional space, as will be explained further in Chapters 31 and 32 on the analysis of measurement tables and general contingency tables. 

Forina M, Armanino C (1982) Eigenvector projection and simplified non-linear mapping of fatty acid content of Italian olive oils. Ann Chim [Rome] 72 127... 

Figure 5. Eigenvector Projection (Principal Vector Plot). A plane is least squares fitted to all the data. This plane constitutes a two-dimensional window into the multi—dimensional measurement space. The projections of the object points down to the plane are visualized in this plot.

Eigenvector projection represents the multivariate evolution of the variable-byvariable plots. This method must be considered as the fundamental method of displaying multivariate chemical information at the beginning of or during data analysis. 

Fig. 2. Eigenvector projection of French wines, with 2 categories (Bordeaux and Bourgogne) and 20 variables (elements, organic acids, etc.). (Adapted from Ref. >)...

Fig. 3. Eigenvector projection of whiskies, with 3 categories (non-Chivas, standard Chivas, commercial Chivas) and 17 variables (chromatographic peaks). (Adapted from Ref. )...

Fig. 4. Eigenvector projection of Italian red wines. 78 % of the total variance retained. 3 categories (Barolo, Grignolino, Barbera) 8 variables (1 total alcohol, 2 total pol3fphenols, 3 flavanoids, 4 color intensity, 5 tonality, 6 O.D. 280/315 nm diluted wine, 7 O.D. 280/315 jiin flavanoids, 8 proline). (Adapted from Ref.

Fig. 9. Eigenvector projection of Italian red wines. 5 variables (1 flavanoids 2 total alcohol 3 tonality 4 Mg S proline)...

In this case, the multicollinearity problem can be avoided by feature selection (e.g., by a decorrelation procedure, subtraction or eigenvector projection) followed by BA in the space of selected features. 

Although in this kind of contorted distribution it is possible to specify some subcategories by eigenvector projection and then to use BA or SIMCA with subcategory spaces, it seems desirable to have methods suitable for handling unsplit categories. 

Eigenvector projection based on the six best variables is shown in Fig. 37, and it can be compared with the projection of Fig. 3 (obtained with 17 variables). 

S. R. Lowry, G. L. Ritter, H. B. Woodruff, and T. L. Isenhour, Selecting liquid phases for multiple column gas chromatography from their eigenvector projections, J. Chromatogr. Sci., 14 126 (1976). 

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). 

At present, data projection is performed mainly by methods called PC A, FA, singular value decomposition (SVD), eigenvector projection, or rank annihilation. The different methods are linked to different science areas. They also differ mathematically in the way the projection is computed, that is, which dispersion matrix is the basis for data decomposition, which assumptions are valid, and whether the method is based on eigenvector analysis, SVD, or other iterative schemes. 

Owing to space and spin symmetry, the eigenvectors projected into the model space are necessarily orthogonal... 

M. Forina and C. Armanino, Ann. Chim. (Rome, 72, 127 (1982). Eigenvector Projection and Simplified Non-Linear Mapping of Fatty Acid Content of Italian Olive Oils. 

S. S. Williams, R. B. Lam, and T. L. Isenhour, Anal. Chem., 55,1117 (1983). Search System for Infrared and Mass Spectra by Faaor Analysis and Eigenvector Projection. 

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