The Correlation-vector Approach

We have found [1] that orthogonalization of the correlated vectors not only takes care of the above problem very effectively but also gives vital insight into the phenomenon of memory storage in the brain. The orthogonalization process in fact acts like a decision making process wherein when a new object is compared with the stored ones, the nature of its correlation with them decides, in an efficient (or economical) manner, in what form it should be stored. The noise is also eliminated using this approach. 

An alternative to fingerprint based similarities are those based on BCUTs (Burden, CAS, University of Texas). This method uses a modified connectivity matrix (the Burden matrix) onto which are mapped atomic descriptors (such as atomic mass and polarizability) and connectivity information. The eigenvectors of this matrix represent a compressed summary of the information in the matrix and are used to describe a molecule. Typically 5-6 BCUT descriptors suffice to describe the chemical space of a set of molecules, and the space is usually partitioned into distinct bins , with each molecule assigned to the appropriate partition. In this format, similarity calculations become very simple molecules which are mapped into the same partition are similar. As an alternative, one could use larger numbers of molecular properties and a correlation vector approach. 

The idea behind this approach is simple. First, we compose the characteristic vector from all the descriptors we can compute. Then, we define the maximum length of the optimal subset, i.e., the input vector we shall actually use during modeling. As is mentioned in Section 9.7, there is always some threshold beyond which an inaease in the dimensionality of the input vector decreases the predictive power of the model. Note that the correlation coefficient will always be improved with an increase in the input vector dimensionality. 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

In this particular example, the salt and temperature information was recorded, however with the inverse modeling approach, the values are not used in the computation of the PLS model. One might be tempted to want to account for these variables by including them as additional columns of R. However, this is not necessary, because the effects of these variables are already captured by the spectra. Complimenting the R matrix with variables related to or correlated with the c vector may be helpful if that correlation is different from what is already in R. This is in contrast to a more classical approach for analyzing these same data, discussed in Section 5.2.2.2. 

It should be noted that other approaches to selecting basis vectors for PCR have been proposed [47 and references therein]. The most popular approach includes those basis vectors that are maximally correlated to y [48 and references therein]. 

Using the matrix notation approach [13 15] that was introduced to describe multicomponent (here also consider n components) flexible polymer systems, the RPA equations are reviewed here for an incompressible stiff polymer mixture. As before, the idea is to isolate a matrix component (denoted component M) from the rest of the blend (denoted R). The various correlations are described through a scalar part XjiM(Q), a vector part Xmr(Q) and a matrix part XRR(Q), and similarly for potentials W s. The RPA equations for the n-vector fluctuating densities are ... 

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