Exploratory Point Pattern Analysis for Modeling Biological Data

Exploratory Point Pattern Analysis for Modeling Biological Data

Stelios Zimeras (Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi, Samos, Greece)
DOI: 10.4018/ijsbbt.2013010101
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Abstract

Data in the form of sets of points, irregular distributed in a region of space could be identified in varies biological applications for examples the cell nuclei in a microscope section of tissue. These kinds of data sets are defined as spatial point patterns and the presentation of the positions in the space are defined as points. The spatial pattern generated by a biological process, can be affected by the physical scale on which the process is observed. With these spatial maps, the biologists will usually want a detailed description of the observed patterns. One way to achieve this is by forming a parametric stochastic model and fitting it to the data. The estimated values of the parameters could be used to compare similar data sets providing statistical measures for fitting models. Also a fitted model can provide an explanation of the biological processes. Model fitting especially for large data sets is difficult. For that reason, statistical methods can apply with main purpose to formulate a hypothesis for the implementation of biological process. Spatial statistics could be implemented using advance statistical techniques that explicitly analyses and simulates point structures data sets. Typically spatial point patterns are data that explain the location of point events. The author’s interest is the investigation of the significance of these patterns. In this work, an investigation of biological spatial data is analyzed, using advance statistical modeling techniques like kriging.
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Spatial Statistics Modeling

Spatial modelling is analysing various data of spatial process: 1. geostatistical data, 2. lattice data, and 3. point patterns (Figure 1).

Figure 1.

General spatial model

The notations that will be used are based on (Cressie, 1993; Diggle, 1983; 2003; Ripley, 1981). The study region (domain) is given by D. Usually D is a subset of 2-dimensional space, but it could be 1-dimensional or 3-dimensional or even beyond so , where d is the dimension we choose. The vector s denotes the data location. Locations in D are denoted by the vector s. For example, in 2-dimensional space, s will have 2 components containing the coordinates (x, y), such as latitude and longitude. At location s, we obtain some value z. So z(s) is the values for each value z that correspond location s. Finally, we will assume that Z(s) is a random variable at each location. The general spatial model could be defined as . Basic models in spatial statistic are: 1. Geostatistical data where D is a continuous fixed subset of ; Z(s) is a random vector at location . 2. Lattice data where in this case D is a fixed countable subset of such as a grid some representation with nodes; Z(s) is a random vector at location . 3. Point Patterns where D is a random subset of denoted as a point process; if Z(s) is a random vector at location then it is a marked spatial point process; if Z(s) ≡ 1 so that it is a degenerate random variable, then only D is random and it is called a spatial point process.

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