Spatial autocorrelation vectors

Let us start with a classic example. We had a dataset of 31 steroids. The spatial autocorrelation vector (more about autocorrelation vectors can be found in Chapter 8) stood as the set of molecular descriptors. The task was to model the Corticosteroid Ringing Globulin (CBG) affinity of the steroids. A feed-forward multilayer neural network trained with the back-propagation learning rule was employed as the learning method. The dataset itself was available in electronic form. More details can be found in Ref. [2]. 

Each virtual compound was represented by a 12-dimensional spatial autocorrelation vector computed according to equation (23) ... 

In the calculation of a 3D autocorrelation vector the spatial distance is used as given by Eq. (20). 

In contrast to the topological autocorrelation vectors in the 3D autocorrelation vector, the spatial distance between atoms is used for calculation. Flence, using 3D autocorrelation vectors, it is possible to distinguish between different conformations of a molecule. The calculation of autocorrelation vectors of surface properties is similar to equation (10.2) ... 

Spatial autocorrelation coefficients can be used to produce a 3D-QSAR that does not require alignment of the structures. The user must choose the conformation to be compared, however. The physical basis of the autocorrelation vector is the observation that properties at one point in space are often correlated with those at another point in space for example, adding a methyl group to a carbon atom typically changes the steric energy at several CoMEA lattice points and/or the distances between several surface points. The autocorrelation vector for enantiomers will be identical, because the autocorrelations are based on distances between points, and enantiomers have the same distances between atoms and properties based on them. Hence, the user must decide which is the bioactive enantiomer after the analysis. 

For the case of spatial motion, Figs. 47b and 47c, it is convenient to start from the following expression for the autocorrelator of the F-projection rE(t) of the radius vector r(t) [147] ... 

Here B is an optical constant, or is the total polarizability of the particle, and n is the number of components in each particle. The indexes i and j refer to components of the same particle. If the assumption of independent particles was not made, then the indexes could refer to components of any two particles, and the autocorrelation expression could not be written as a simple sum of contributions from individual particles. The spatial vector r(r) refers to the center of mass of the particle. R(r). In the case of a nonspherical particle (arbitrary shape), Eq. (I0) would describe the coupled motion of the center of mass and the relative arrangement of the components of the particle. For spherical particles, translational and rotational motion arc uncoupled and we have a simplified expression for the electric field time correlation function ... 

The spatial distribution of matter in a porous medium can be represented by the phase function, Z(x), which is equal to 1 if x belongs to the pore space, or 0 if x belongs to solid (x is the position vector fiom an arbitrary origin). A reliable 3D representatian of a porous medium should possess the same statistic properties as those determined in a single two-dimensional section, properly reflected by the various moments of Z(x) [6] and in most cases, the first two moments are considered sufficient. The one-point correlation function. Si, is the probability that any point lies in the pore space and is defined as 5i= (< > indicates spatial average, hence, Si is equal to the porosity, e). On the other hand, the two-point correlation function, -S2(u), is the probability that two points a specified distance u apart, both lie in the pore space and is defined as S 2(u)=. SiCa). is directly related to the autocorrelation function, RJyi) (ii (u)=[iS2(u)-fi ]/(E-0). For an isotropic m ium, iyu) is only a function of m=1u( [1] and thus degenerates to the one-dimensional autocorrelation function, Rz(u), of Eq. (1). 

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