Autocorrelation of a topological structure

Spatial autocorrelation is a quantitative measure of the probability of finding objects of defined properties within a distance of interest [9, 10]. The concept of autocorrelation is mainly applied in fields such as geography, economics, ecology or meteorology to describe the spatial distribution of features. The idea of a molecular descriptor based on the autocorrelation concept was first introduced into the field of cheminformatics by Moreau and Broto in 1980 [11] with the ATS (autocorrelation of a topological structure) descriptor. For this approach, the atoms of a molecule were represented by properties such as atomic mass or partial charge. The distance between atoms was measured as the number of bonds between the respective atoms (topological distance). 

Moreau-Broto autocorrelation (. Autocorrelation of a Topological Structure, ATS) This is a spatial autocorrelation defined on a molecular graph C as ... 

Moreau, G. and Broto, P. (1980b). The Autocorrelation of a Topological Structure A New Molecular Descriptor. Nouv.J.Chim., 4,359-360. 

Autocorrelation Vectors (topological autocorrelation vectors, autocorrelation of a topological Structure, ATS) are molecular descriptors for a property distribution along the topological structure. 

Autocorrelation of a Topological Structure = Moreau-Broto autocorrelation autocorrelation descriptors... 

Broto-Moreau autocorrelation of a topological structure- lag7/ weighted by atomic masses... 

Bauknecht et al. [48] and Sadowski et al. [49] have also developed molecular descriptors that are based on the physicochemical properties of the atoms in a molecule. They calculate several different electronic properties for each atom in a molecule and then use autocorrelation to generate a fixed-length vector that is independent of the size of the molecule. Autocorrelation was first applied to the topology of a molecular structure by Moreau and Broto [50] using the following function ... 

Gasteiger et al. [20,21] have used autocorrelation vectors as molecular descriptors in diversity studies and also in library comparisons. Autocorrelation is a method for converting information of variable length into fixed-length information. Moreau and Broto [22] were the first to apply an autocorrelation function to the topology of a molecular structure ... 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

Autocorrelation of Topological Structure Moreau and Broto [24,25] have suggested the autocorrelation vector of a molecular graph as the source for molecular descriptors. This method assumes that each atom i in the graph is uniquely associated with a numeric quantity, qit such as the atomic number, atomic mass, (v)(, (ds), Sv, or electronegativity. The intrinsic atom values of the electrotopological state [26] and the atomic Rd and log Kow parameters [27,28] are other potential atomic descriptors suitable to construct autocorrelation vectors. Generally, the fcth element of the autocorrelation vector is defined as... 

For this reason, molecular interaction potentials based on three-dimensional structures are calculated for the compounds of the test dataset. The potentials include all possibibhes of interaction between the small molecule and the en-2ymes, as well as the shape of the molecule. Twenty autocorrelation coefficients are derived from these potentials for each molecule and used as a descriptor set. The new descriptors are mapped in the same way as the topological descriptors by means of a self-organizing map. 

Topological autocorrelation vectors are calculated from the two-dimensional structure of a molecule that can be expressed as molecular graph. One of the original... 

In order to transform the information fi om the structural diagram into a representation with a fixed number of components, an autocorrelation function can be used [8], In Eq. (19) a(d) is the component of the autocorrelation vector for the topological distance d. The number of atoms in the molecule is given by N. 

Atom-type topological indices are used to describe a molecule by information related to different atom types in the molecule. An atom-type index is usually derived from some properties of all the atoms of the same type and their structural environment. Atom-type Estate indices of Kier and Hall, perturbation connectivity indices, atom-type path counts, and atom-type autocorrelation descriptors are examples of these molecular descriptors. 

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