Matrix computations principal component analysis

Principal component analysis (PCA) can be considered as the mother of all methods in multivariate data analysis. The aim of PCA is dimension reduction and PCA is the most frequently applied method for computing linear latent variables (components). PCA can be seen as a method to compute a new coordinate system formed by the latent variables, which is orthogonal, and where only the most informative dimensions are used. Latent variables from PCA optimally represent the distances between the objects in the high-dimensional variable space—remember, the distance of objects is considered as an inverse similarity of the objects. PCA considers all variables and accommodates the total data structure it is a method for exploratory data analysis (unsupervised learning) and can be applied to practical any A-matrix no y-data (properties) are considered and therefore not necessary. 

Computing the sensitivities is time consuming. Fortunately the direct integral approximation of the sensitivity matrix and its principal component analysis can offer almost the same information whenever the direct integral method of parameter estimation applies. 

Principal component analysis is ideally suited for the analysis of bilinear data matrices produced by hyphenated chromatographic-spectroscopic techniques. The principle component models are easy to construct, even when large or complicated data sets are analyzed. The basis vectors so produced provide the fundamental starting point for subsequent computations. Additionally, PCA is well suited for determining the number of chromatographic and spectroscopically unique components in bilinear data matrices. For this task, it offers superior sensitivity because it makes use of all available data points in a data matrix. 

The method of the principal component analysis of the rate sensitivity matrix with a previous preselection of necessary species is a relatively simple and effective way for finding a subset of a large reaction mechanism that produces very similar simulation results for the important concentration profiles and reaction features. This method has an advantage over concentration sensitivity methods, in that the log-normalized rate sensitivity matrix depends algebraically on reaction rates and can be easily computed. For large mechanisms this could provide considerable time savings for the reduction process. This method has been applied for mechanism reduction to several reaction schemes [96-102]. 

Fig. 6-15 Principal component analysis of multidimensional, chemical-genetic data, (a) Eigenvalues and associated variance, and eigenvectors and associated factor scores computed from the data in Fig. 6-14(a). The matrix of eigenvectors...

The singular value decomposition (SVD) method, and the similar principal component analysis method, are powerful computational tools for parametric sensitivity analysis of the collective effects of a group of model parameters on a group of simulated properties. The SVD method is based on an elegant theorem of linear algebra. The theorem states that one can represent an w X n matrix M by a product of three matrices ... 

Dimension reduction is, as the name implies, a technique for reducing the dimensionality of a dataset, which is most often applied to the columns (variables) but may also be applied to the rows (cases or compounds) and results in a reduction from p variables to q variables or dimensions where q is often 2 or 3 (for ease of display of the resulting data matrix). A common method of dimension reduction is principal component analysis (PCA). A less-frequently used but related method is factor analysis (FA). Insufficient space exists here for a complete description of these techniques, so the reader is directed to references 26 and 27 for PCA and 28 and 29 for FA. Briefly, each computes new variables... 

If the origin ( 0 ) is chosen at the centroid of the atoms, then it can be shown that distances from this point can be computed from the interatomic distances alone. A fundamental theorem of distance geometry states that a set of distances can correspond to a three-dimensional object only if the metric matrix g is rank three, i.e., if it has toee positive and N — 3 zero eigenvalues. This is not a trivial theorem, but it may be made plausible by thinking of the eigenanalysis as a principal component analysis all of the distance properties of the molecule should be describable in terms of three components , which would be the x, y and z coordinates. If we denote the eigenvector matrix as w and the eigenvalues A., the metric matrix can be written in two ways ... 

An alternative to the use of principal components or factor analysis is the BCUT method of Pearlman [Pearlman and Smith 1998]. In this method, three square matrices are constructed for each molecule. Each matrix is of a size equal to the number of atoms in the molecule and has as its elements various atomic and interatomic parameters. One matrix is intended to represent atomic charge properties, another represents atomic polarisabilities and the third hydrogen-bonding capabilities. These quantities can be computed with semi-empirical... 

---