Data Analysis and Statistics for Geography, Environmental Science, and Engineering

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8

Spatial Point Patterns

8.1 TYPES OF SPATIALLY EXPLICIT DATA

In this book, we will look at two main types of spatial data: one type corresponds to point patterns

and the other type to lattice arrangements. The difference is whether we consider points or regions

placed over a spatial domain. We will focus on two-dimensional (2D) domains because they

encompass many important geographical and environmental situations.

Point patterns are typically a collection of points placed over a spatial domain. They may be

regular as in a grid or irregular. We may have values of variables associated at each point. Lattice

data correspond to regions or polygons. Regions are regularly arranged when they occur in a regular

pattern, say a rectangular grid. For example, a remote sensed image composed of pixels. Regions

are irregularly arranged when they do not follow a regular spatial pattern, as it happens for example

when we have political divisions, such as counties.

We will cover three types of spatial analysis: analysis of point patterns, that is, to examine the

spatial distribution of points, for example, check whether points are clustered or uniformly distrib-

uted; geostatistics, to predict values of variables in nonsampled points using a collection of sampled

points; spatial regression, to predict values of variables in regions from values at the neighbor-

ing regions. In this book, we will cover the fundamentals of these methods. Nowadays many of

these methods are also implemented in Geographical Information Systems (GIS) software. In this

chapter, we will cover spatial point patterns. We will wait to cover the other two methods until we

learn about matrices and linear algebra.

8.2 TYPES OF SPATIAL POINT PATTERNS

A spatial point pattern is a collection of points located within a bounded region or domain. Points in a

two-dimensional domain have coordinates x, y within the domain where x is displayed horizontally

and y vertically. Coordinate x increases from left to right and coordinate y from bottom to top. The

coordinate system can be geographical; for example x increasing towards the east and y increasing

towards the north.

Figure 8.1 illustrates four examples. At the top left, points are regularly located in the domain as

a grid; all three other patterns are irregular in distribution. At the top right, points spread uniformly,

but both patterns at the bottom are nonuniform. The one on the left follows a gradient with x, and

the one on the right shows aggregation. In this chapter, we will consider distribution in detail.

Applications are many and varied. For example, spatial distribution of plants and trees on the

landscape, sampling sites in a waste eld, and sampling sites in a lake, locations of intense events

such as quakes or tornadoes. The nature of the data could be location only (just the pair of coor-

dinates for each point), or marked point (pair of coordinates plus data values associated with each

location). A spatial point pattern is a starting point for kriging analysis as discussed later in the book.

8.3 SPATIAL DISTRIBUTION

We now consider more details of the distribution of points over space. As illustrated by Figure 8.1,

major types of distributions are uniform or homogeneous and nonuniform or nonhomogeneous.

These terms refer to the uniformity or homogeneity of the density of points, that is, spatial vari-

ability of the number of points per unit area.