Multivariate data projection

Yin, H. (2002). ViSOM - A novel method for multivariate data projection and structure visualization. IEEE Trans. Neural Networks, 13, 237-243. 

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. 

Partial least squares (PLS) projections to latent structures [40] is a multivariate data analysis tool that has gained much attention during past decade, especially after introduction of the 3D-QSAR method CoMFA [41]. PLS is a projection technique that uses latent variables (linear combinations of the original variables) to construct multidimensional projections while focusing on explaining as much as possible of the information in the dependent variable (in this case intestinal absorption) and not among the descriptors used to describe the compounds under investigation (the independent variables). PLS differs from MLR in a number of ways (apart from point 1 in Section 16.5.1) ... 

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). 

Preference mapping can be accomplished with projection techniques such as multidimensional scaling and cluster analysis, but the following discussion focuses on principal components analysis (PCA) [69] because of the interpretability of the results. A PCA represents a multivariate data table, e.g., N rows ( molecules ) and K columns ( properties ), as a projection onto a low-dimensional table so that the original information is condensed into usually 2-5 dimensions. The principal components scores are calculated by forming linear combinations of the original variables (i.e., properties ). These are the coordinates of the objects ( molecules ) in the new low-dimensional model plane (or hyperplane) and reveal groups of similar... 

PLS is a method by which blocks of multivariate data sets (tables) can be quantitatively related to each other. PLS is an acronym Partial Least Squares correlation in latent variables, or Projections to Latent Structures. The PLS method is described in detail in Chapter 17. 

In the first mentioned type of application, electrophoretic data are subjected to exploratory analysis techniques, such as principal component analysis (PCA) (5-8), robust PCA (rPCA) (9-13), projection pursuit (PP) (6,14-18), or cluster analysis (8, 19, 20). They all result in a simple low-dimensional visualization of the multivariate data. As a consequence, it will be easier for the analyst to get insight in the data in order to see whether there is a given... 

An insight is provided here into the current state-of-the-art for the compositional analysis of molecules in food utilising high-resolution NMR spectroscopy in conjunction with multivariate data analysis techniques. The recent Human Metabolome Project (http //www.metabolomics.ca) has identified 2,500 metabolites, 1,200 drugs, and 3,500... 

Visualization of Latent Projections. Visualization is of prime importance for interpreting multivariate data. Many plots are available. The most important are score and loading plots and their combination into biplots. 

PCA is a method based on the Karhunen-Loeve transformation (KL transformation) of the data points in the feature space. In KL transformation, the data points in the feature space are rotated such that the new coordinates of the sample points become the linear combination of the original coordinates. And the first principal component is chosen to be the direction with largest variation of the distribution of sample points. After the KL transformation and the neglect of the components with minor variation of coordinates of sample points, we can make dimension reduction without significant loss of the information about the distribution of sample points in the feature space. Up to now PCA is probably the most widespread multivariate statistical technique used in chemometrics. Within the chemical community the first major application of PCA was reported in 1970s, and form the foundation of many modem chemometric methods. Conventional approaches are univariate in which only one independent variable is used per sample, but this misses much information for the multivariate problem of SAR, in which many descriptors are available on a number of candidate compounds. PCA is one of several multivariate methods that allow us to explore patterns in multivariate data, answering questions about similarity and classification of samples on the basis of projection based on principal components. 

Principal component analysis (PCA) is frequently the method of choice to compress and visualize the structure of multivariate data [13]. The original experimental data are compressed by representing the total data variance using only a few new variables, called principal components (PCs). These PCs, which are orthogonal to each other, are ranked in a descending order of the variance they model. This means that with PCA, samples are projected onto an optimal direction in the multivariate data space explaining the largest possible variance. As mentioned earlier, the variance of a projection is not robust and the presence of outliers in the data will affect the construction of PCs. A direct way to obtain robust principal components (RPCs) is to replace the classic variance estimator with its robust counterpart. 

As mentioned above, there are several well-known methods of multivariate data analysis (Juts et al, 2(XX)), but peihaps the most popular is PCA (JoUife, 1986). This feature extraction method consists in projecting the A-dimensional data set (in this case N is the number of sensors) onto a new base of the same dimension N, but now defined by the eigenvectors of the covariance or the correlation matrix of the data set. The components (projections) of the original data vectors onto this new base are the so-called Principal Components,... 

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