MTA TAKI, H-1022 Budapest Herman Ottó út 15, Hungary
L. V. Tóth
Dept. of Astr., Eötvös Univ., H-1083 Budapest Ludovika tér 2, Hungary
Consider ; where . Here T is the index set, is the spatial process, is a realization of the process. In the present paper we give a brief overview on the most important spatial statistical models,
to illustrate the range of problems that can be addressed and the wide applicability of spatial statistical models in astronomy.
A usual spatial point process is defined as (i.e., the index set is the points of ) or [the number of points within A]; (i.e., the index set is the units of ), where both and T are random. First- and second-order properties of a spatial point process are the intensity function: ; and the second-order intensity function: . Spatial point processes are the mathematical models producing point patterns as their realization.
A number of processes are available for modeling the patterns that arise in nature:
Examples of applicability in astronomy include: (1) revealing regularity in the spatial distribution of point-like objects, (2) identification of important scales in the spatial distribution of point-like objects, (3) stellar statistics (deriving distributions, testing of predicted distribution functions, identification of clusters and associations of stars, search for wide binaries and multiple systems), and (4) cosmological problems (testing of predicted distribution functions, identification of galaxy clusters, voids, etc.).
The spatial index t varies continuously throughout a fixed subset T of a d-dimensional Euclidean space. Term ``regionalized'' was introduced in order to emphasize the continuous spatial nature of the index set T. The prefix ``geo'' reflects the fact that the theory's roots are in geographical and geological applications. Random processes are usually characterized by their moment measures. In geostatistics, ``semivariogram'' plays a crucial role. If for ; is called semivariogram. If for and exist, is intrinsically stationary. Semivariogram is conditional negative-definite. If is second-order stationary . Linear, spherical, and exponential models are simple isotropic (semi)variogram.
The most important application of the (semi)variogram is ``kriging,'' a stochastic spatial interpolation method which depends on the second-order properties of the process. The principal aim of kriging is to provide accurate spatial predictions from observed data. Kriging techniques are all related and refined versions of the weighted moving average originally used by Krige (1951) and based on the simple linear model: , where . Kriging provides optimum prediction in a sense of minimizing mean-squared prediction error, and also
provides the estimation.
A useful decomposition is , where is the large-scale variation, is the smooth small-scale variation, is the micro-scale variation, is the measurement error. These models are widely applied in geosciences.
A number of astronomical applications of the method come to mind: (1) the creation of contour and/or surface maps in the case of incompletely sampled maps in extended radio surveys, (2) testing for completeness in sampling (whether the expected structure is revealed as spiral or filamentary), (3) testing whether resolution is achieved (in the cores of galaxies), (4) the creation of maps with resolution higher than the physical resolution of the observation (interpolations arising from the co-addition of separate sky coverage by IRAS or ISO), and (5) interpolations to reach a higher virtual resolution for comparisons (e.g., IRAS 12 and 100micron images).
Examples of applicability to astronomy include: (1) 2-D classification of objects by their shape on images (e.g., star, galaxy identification on CCD or photographic images), (2) cloud identification from coordinate-velocity ``data cubes'' (e.g., radio spectroscopic observations), and (3) any advanced image processing technique, like maximum entropy or deconvolution (e.g., maximum correlation method in ``HIRES'' IRAS data processing at IPAC).
This research was partially supported by the Hungarian State Research Found (Grant No. OTKA-F 4239). L. Pásztor is grateful to ADASS and the Hungarian State Research Found for the travel grants.
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