Eigenvalue Based Methods

There are two fundamental groups of approaches. The first involves performing PCA on an expanding window which we will refer to as EFA. There are several variants on this theme, but a popular one is indicated below. 

Perform uncentred PCA on the first few datapoints of the series, e.g. points 1-4. In the case of Table 6.1, this will involve starting with a 4 x 12 matrix. 

Record the first few eigenvalues of this matrix, which should be more than the number of components expected in the mixture and cannot be more than the smallest dimension of the starting matrix. We will keep four eigenvalues. 

Extend the matrix by an extra point in time to a matrix of points 1 -5 in this example and repeat PCA, keeping the same number of eigenvalues as in step 2. 

Continue until the entire data matrix is employed, so the final step involves performing PCA on a data matrix of dimensions 25 x 12 and keeping four eigenvalues. 

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Alternating least squares (ALS) methods are both slower, due to their numeric intensity, and more flexible than eigenvalue-eigenvector problem-based methods for solving Equation 12.1a and Equation 12.1b. The basic PARAFAC model of Equation... 

In any Newton-based optimization - which, as discussed in Section 10.8, implicitly or explicitly requires the inversion of the Hessian matrix - the inclusion of redundant parameters is not only unnecessary but also undesirable since, at stationary points, these parameters make the electronic Hessian singular. The singularity of the Hessian follows from (10.2.8), which shows that the rows and columns corresponding to redundant rotations vanish at stationary points. Away from the stationary points, however, the Hessian (10.1.30) is nonsingular since the gradient elements that couple the redundant and nonredundant operators in (10.2.5) do not vanish. Still, as the optimization approaches a stationary point, the smallest eigenvalues of the Hessian will tend to zero and may create convergence problems as the stationary point is approached. Therefore, for the optimization of a closed-shell state by a Newton-based method, we should consider only those rotations that mix occupied and virtual orbitals ... 

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

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