Spatial autocorrelation

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]. 

There must be some spatial autocorrelation in the image, that means that the local frequency have to be low or medium according to Fig. 2.6b,c (in contrast, in cases of zero- and high local frequencies, see Fig. 2.6a,d, there is no autocorrelation). 

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). 

Figure 4.13 Concentration fluctuations and the spatial autocorrelation function.

Spatial autocorrelation analysis was used to quantify patterns in two-dimensional microscale distribution of chlorophyll a concentrations (pgChl-a l-1) and relative seawater excess viscosity rj (%). The objective is to link these parameters to a common patch size to quantify their spatial linkage. 

Generally speaking, positive spatial autocorrelation indicates that abundances in adjacent localities are similar (aggregation) while negative-autocorrelation signifies that adjacent localities alternate between high and low values, i.e. peaks and valleys, and hot spots and cold spots. The theory and use of spatial autocorrelation procedures to analyse spatial associations has been extensively discussed (Sokal and Oden 1978a, b), and the reader is referred to these papers for more-detailed descriptions. 

Moran s I and Geary s c spatial autocorrelation statistics (Moran 1950 Geary 1954) are defined as ... 

The results of spatial autocorrelation analysis showed that none of the investigated patterns were uniform nor random (Table 4), indicating the existence of structural complexity in 2D microscale patterns of chlorophyll a concentration and seawater viscosity. Except on June 20, consistent spatial patterns were found for chlorophyll and excess... 

Spatial autocorrelation and correlograms were used to describe the microscale spatial patterns of chlorophyll a concentration and seawater excess... 

Table 4 Results of spatial autocorrelation analyses of chlorophyll a and seawater excess viscosity two-dimensional microscale distributions...

Sokal R, Oden N (1978a) Spatial autocorrelation in biology. I Methodology Biol J Linn Soc London 10 199-228 Sokal R, Oden N (1978b) Spatial autocorrelation in biology. II. Some biological implications and four applications of evolutionary and ecological interest. Biol J Linn Soc London 10 229-249... 

Sadowski, J., Wagener, M. and Gasteiger, J. (1995) Assessing Similarity and Diversity of Combinatorial Libraries by Spatial Autocorrelation Functions and Neural Networks. Angew. Chem. Int. Ed. Engl., 1995, 34,2674-2677. 

Assessing Similarity and Diversity of Combinatorial Libraries by Spatial Autocorrelation Functions and Neural Networks. 

J. Sadowski, M. Wagener and J. Gasteiger, Assessing similarity and diversity of combinatorial libraries by spatial autocorrelation functions and neural networks, Angew. Chem. Inti. Ed. Engl, 1995, 34, 2674-2677. 

To obtain spatial autocorrelation molecular descriptors, function /(x,) is any physico-chemical property calculated for each atom of the molecule, such as atomic mass, polarizability, etc., and - local vertex invariants such as - vertex degree. Therefore, the molecule atoms represent the set of discrete points in space and the atomic property the function evaluated at those points. 

For spatial autocorrelation molecular descriptors calculated on a - molecular graph, lag I is defined as the - topological distance d. 

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

Average spatial autocorrelation descriptors are obtained by dividing each term by the corresponding number of contributions, thus excluding any dependence on molecular size ... 

General index of spatial autocorrelation that, if applied to a molecular graph, can be defined as i a a... 

The properties p of the atoms i and j are considered for a particular topological distance d. Sjj is a Kronecker delta that represents additional constraints or conditions. The topological distance may also be replaced by the Euclidean distance, thus accounting for two- or three-dimensional arrangement of atoms. Three-dimensional spatial autocorrelation of physicochemical properties has been used to model the biological activity of compound classes [24]. In this case, a set of randomly distributed points is selected on the molecular surface, and all distances between the surface points are calculated and sorted into preset intervals. These points are used to calculate the spatial autocorrelation coefficient for particular molecular properties, such as the molecular electrostatic potential (MEP). The resulting descriptor is a condensed representation of the distribution of the property on the molecular surface. 

The most common spatial autocorrelation molecular descriptors are obtained by taking the molecule atoms as the set of discrete points in space and an atomic property as the function evaluated at those points. 

This is the most known spatial autocorrelation defined on a molecular graph Q as [Moreau and Broto, 1980a, 1980b Broto, Moreau et al, 1984a]... 

PEST Autocorrelation Descriptors (or PAD descriptors) are spatial autocorrelation descriptors defined on the basis of TAE and PEST descriptors [Breneman, Bundling et al., 2003]. For each ray in PEST, the length of the ray and the product of the property values at starting and ending points are computed. The distribution is birmed into 20 bins along the ray length and the autocorrelation values for each bin calculated. For 10 TA E properties, this yields a total of 200 PEST autocorrelation descriptors. 

Employing Local Indicators of Spatial Autocorrelation to Anthrax Data... 

Figure 16 provides67 a comparison between the classical [Eq. (4.6)], and quantum [Eq. (4.1)] spatial correlation functions for four chaotic stadium eigenfunctions. The classical and quantum correlation functions are seen to agree very well for all distances other than those comparable to the size of the stadium s linear dimension. Thus, the rapid decay of the quantum spatial autocorrelation is a measure relating directly to classical ergodic behavior. 

Rubin and Gdme -Hemdndez (1990) and Indelman and Dagan, (1993) have developed analytical expressions fotyupscaling both the mean and the variance of spatially autocorrelated lnOfc fdatcL Since macroscale dispersion can be related to the variance of ln( sat), it may be possible to apply their expressions for predicting variance as a function of block size to the problem of upscaling estimates of K determined in the laboratory to the field scale. Unfortunately, numerical simulations based upon this approach are likely to be computationally expensive. 

Purposive spatially distributed sampling design accounting explicitly for spatial autocorrelation of indicator variables... 

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