Spatial point statistics

Note that evaluating the correlation functions at r = 0 yields the corresponding one-point statistics. For example, Rap(0, t) is equal to the scalar covariance Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend onlyonr = r, i.e., R,p(r, t). The scalar integral scale and the scalar Taylor microscale... 

Illian, Penttinen, Stoyan and Stoyan - Statistical Analysis and Modelling of Spatial Point Patterns... 

Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London. Gcorgatos, S. D. (1994). Towards an understanding of nuclear morphogenesis. J. Cell. Biochem. 55, 69-76. 

Illian, )., Penttinen, A., Stoyan, H., and Stoyan, D. (2008) Statistical Analysis and Modelling of Spatial Point Patterns, John Wiley Sons, Ltd., Chichester. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

Instead of nodal lines in closed systems we are interested in the statistics of NPs for open chaotic billiards since they form vortex centers and thereby shape the entire flow pattern (K.-F. Berggren et.al., 1999). Thus we will focus on nodal points and their spatial distributions and try to characterize chaos in terms of such distributions. The question we wish to ask is simply if one can find a distinct difference between the distributions for nominally regular and irregular billiards. The answer to this question is clearly positive as it is seen from fig. 3. As shown qualitatively NPs and saddles are both spaced less regularly in chaotic billiard in comparison to the integrable billiard. The mean density of NPs for a complex RGF (9) equals to k2/A-k (M.V. Berry et.al., 1986). This formula is satisfied with good accuracy in both chaotic and integrable billiards. 

In the context of spatial statistics (Cressie 1991), the estimation algorithm in PDF codes is data dense. In other words, the number of statistically independent data points is much larger than the number of parameters that must be estimated. 

These well-known results of the physics of phase transitions permit us to stress useful analogy of the critical phenomena and the kinetics of bimolec-ular reactions under study. Indeed, even the simplest linear approximation (Chapter 4) reveals the correlation length 0 - see (4.1.45) and (4.1.47), or 0 = /d for the diffusion-controlled processes. At t = 0 reactants are randomly distributed and thus there is no spatial correlation between them. These correlations arise in a course of the reaction, the correlation length 0 increases monotonously in time but 0 — 00 at t —> 00 only. Consequently, a formal difference from statistical physics is that an approach to the critical point is one-side, t0 —> 00, and the ordered phase is absent here. There is also evident correspondence between the parameter t in the theory of equilibrium phase transitions and time t in the kinetics of the bimolecular... 

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