Reduced Eigenvector Space

Not many years ago, computer memory was very precious and it was important to write code that was economic with respect to memory requirements. More recently, this aspect of computing has changed dramatically and as a consequence this sub-chapter is no longer of vital 

In the standard equation for multiwavelength spectrophotometric investigations, based on Beer-Lambert s law, the matrix Y is written as the product of the matrices C and A. According to the Singular Value Decomposition (SVD), Y can also be decomposed into the product of three matrices 

We discuss this decomposition again in great depth in Chapter 5, Model-Free Analyses. For the moment we need to identify a few essential properties of the Singular Value Decomposition and in particular of the matrices U, S, and V. 

SVD is completely automatic. It is one of the most stable algorithms available and thus can be used blindly. It is one command in Matlab [U, S, Vt]=svd (Y, 0). The matrices U and V1 contain as columns so-called eigenvectors. They are orthonormal (see Orthogonal and Orthonormal Matrices, p.25) which means that the products 

The important point is that the above manipulations do not affect the matrix C which is directly related to the model and to the non-linear parameters. Only the matrices Y and A are reduced to Yred and Ared. 

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All this is not completely new. In Reduced Eigenvector Space (p,180, we did just that the matrix US was used to represent the complete matrix Y. The matrix US we called Yred. The component spectra A can also be represented in the eigenvector axes Ated=AVt. As mentioned then, the reduction in the size of the matrices Y and A can be substantial. 

For the sake of completeness, we also include the relevant equations if data reduction, according to Reduced Eigenvector Space (p. 180), is applied ... 

This empirical statistical function, based on the residual standard deviation (RSD), reaches a minimum when the correct number of factors are chosen. It allows one to reduce the number of columns of R from L to K eigenvectors or pure components. These K independent and orthogonal eigenvectors are sufficient to reproduce the original data matrix. As they are the result of a mathematical treatment of matrices, they have no physical meaning. A transformation (i.e. a rotation of the eigenvectors space) is required to find other equivalent eigenvectors which correspond to pure components. 

The sums in Eq. (61) always run over the complete sets of eigenvectors of the corresponding Hilbert sp2ices. The sums over u, in particular, run over all exact states with N particles, p and A lable the iV + 1, and k and a the N — 1 particle states. As already mentioned in the discussion of the Hilbert space Y of Eq. (6), symmetry considerations may reduce the number of states that actually couple to the primary states. Along with a reduced Hilbert space Y, then also less terms are needed in the diagonal representation (61) of the extended operator H. 

Factor spaces are a mystery no more We now understand that eigenvectors simply provide us with an optimal way to reduce the dimensionality of our spectra without degrading them. We ve seen that, in the process, our data are unchanged except for the beneficial removal of some noise. Now, we are ready to use this technique on our realistic simulated data. PCA will serve as a pre-processing step prior to ILS. The combination of Principal Component Analysis with ILS is called Principal Component Regression, or PCR. 

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

When the first direction, v has been found, the data are reflected such that the first eigenvector is mapped onto the first basis vector. Then the data are projected onto the orthogonal complement of the first eigenvector. This is simply done by omitting the first component of each (reflected) point. Doing so, the dimension of the projected data points can be reduced by 1 and, consequently, all the computations do not need to be done in the full r-dimensional space. 

The global dimensionality of the system is always that of the original data ( v), but, since the last dimensions explain only a very small part of the information, they can be neglected and one can take into account only the first dimensions (the significant components ). The projection of the objects in this space of reduced dimensionality retains almost all the information that can now also be analyzed in a visual way, by bi- or tri-dimensional plots. These new directions, linear combinations of the original ones, are the Principal Components (PC) or Eigenvectors. 

Alternative approaches to the many-electron problem, working in real space rather than in Hilbert space and with the electron density playing the major role, are provided by Bader s atoms in molecule [11, 12], which partitions the molecular space into basins associated with each atom and density-functional methods [3,13]. These latter are based on a modified Kohn-Sham form of the one-electron effective Hamiltonian, differing from the Hartree-Fock operator for the inclusion of a correlation potential. In these methods, it is possible to mimic correlated natural orbitals, as eigenvectors of the first-order reduced density operator, directly... 

Figure 3 The original data, X, comprising n objects or samples described by m variables, is converted to a dispersion (covariance or correlation) matrix C. The eigenvalues, A, and eigenvectors, L, are extracted from C. A reduced set of eigenvectors, L, i.e., selected and the original data projected into this new, lower-dimensional pattern space Y.

The projection of the objects in this space of reduced dimensionality retains almost all the information that can now also be analyzed in a visual way, by two- or three-dimensional plots. These new directions, linear combinations of the original ones, are the principal components (PCs) or eigenvectors. 

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