SPATIAL RELATIONSHIPS

When studying a spatial pattern, we may want to compare that pattern with a theoretical pattern to determine if the theory holds in the case of the pattern being studied. Alternatively, we may want to classify spatial patterns into an existing categorical structure of known patterns. If the spatial patterns we study correspond to a particular theoretical pattern, we will be able to apply properties of the theory to interpret the spatial pattern we study. Or, if the spatial pattern can be closely related to a known pattern, we will be able to borrow from experience with and knowledge of the pattern for further study. In either case, it is necessary for us to establish a structure of categories of the spatial pattern.

A spatial pattern can be clustered, dispersed, or random. In Figure 5.1, these three possibilities are shown in hypothetical patterns in the seven counties of northeastern Ohio (Cuyahoga, Summit, Portage, Lake, Geauga, Trumbull, and Ashtabula). In Case 1, the darker shade representing certain characteristics as-sociated with the counties appear to have clustered on the western side of the seven-county area. On the eastern side, the lighter shade (the other type of char-acteristics) prevails among the remaining counties. Perhaps the darker shade indi-cates the growth of the urban population in recent years, whereas the lighter shade indicates the loss of the urban population during the same period.

In Case 2, counties that are shaded darker appear to be spaced evenly and away from each other. This is often referred to as a dispersed pattern. It suggests that the geographic phenomenon displayed may be uniformly distributed across this seven-county area. For example, it may be that the political preferences between the two parties in these counties happen to change between neighboring counties.

For an extremely uniform or dispersed pattern, the checkerboard in which black and white cells distribute systematically and change between every pair of neigh-boring cells is a typical example of a dispersed spatial pattern. A repulsive spatial relationship among neighbors is implied.

The spatial pattern in Case 3 of Figure 5.1 does not appear to be either clus-tered or dispersed. It may be close to what we typically call the random pattern.

If a spatial pattern is random, it suggests that there may not be any particular

sys-Figure 5.1 Types of patterns: clustered, dispersed, and random.

tematic structure or any particular mechanism controlling the way these polygons form the pattern.

In most geographic problems, there is no clear indication that the pattern is clustered, dispersed, or random. Rather, most real-world patterns are somewhere between a random pattern and a dispersed pattern or between a random pattern and a clustered pattern. Very rarely would we find a spatial pattern that is ex-tremely clustered, exex-tremely dispersed, or purely random.

Since real-world patterns can rarely be clearly classified random, clustered, or dispersed, the question is, how close is a given spatial pattern to any of the three patterns? In the next section, we will discuss how we measure a spatial pattern

to determine its closeness to a random pattern so that the pattern can be further studied.

5.2 SPATIAL AUTOCORRELATION

In classifying spatial patterns as either clustered, dispersed, or random, we can focus on how various polygons are arranged. We can measure the similarity or dissimilarity of any pair of neighboring polygons. When these similarities and dissimilarities are summarized for the spatial pattern, we have spatial autocorre-lation (Odland, 1988).

Spatial autocorrelation means that the attribute values being studied are self-correlated and the correlation is attributable to the geographic ordering of the objects.

Many situations exhibit some degree of spatial autocorrelation. When compar-ing agricultural production levels among all the farms in a region, we would not find that all farms are producing at the same rate. Even though the local climate may be the same for all of these farms, the soil conditions or water supplies within the region may vary. Still, neighboring farms within the region share similar soil and moisture conditions, so these farms probably have similar levels of produc-tion.

Among population densities measured in each county, we would see that those counties located in or close to big central cities tend to have higher population densities than those located far away. This is because the outward influence of the central city often makes the surrounding counties its hinterlands. Once the under-lying reason is identified, it would be logical to investigate why urban population densities cluster around big central cities or why the production levels of farms cluster or disperse along known soil structures or water supplies.

The basic property of spatially autocorrelated data is that the values are not random in space. Instead, the data values are spatially related to each other, even though they may be related in different ways. Referring back to the three cases in Figure 5.1, Case 1 has a positive spatial autocorrelation, with adjacent or nearby polygons having similar shades (values). In Case 2, the dispersed pattern has a negative spatial autocorrelation, with changes in shade often occurring between adjacent polygons. As for Case 3, it appears to be close to a random pattern in which little or no spatial autocorrelation exists.

In addition to its type or nature, spatial autocorrelation can be measured for its strength. Strong spatial autocorrelation means that the attribute values of ad-jacent geographic objects are strongly related (either positively or negatively). If the attribute values of adjacent geographic objects appear to have no clear order or relationship, the distribution is said to have a weak spatial autocorrelation or a random pattern. Figure 5.2 shows five different structures with darker and lighter shades within the seven counties in northeastern Ohio. Opposite to the numeric line, the patterns toward the left end are examples of positive spatial autocorre-lation, while the patterns toward the right end are examples of negative spatial

Figure 5.2 Spatial autocorrelation.

autocorrelation. In the middle is a pattern of weak spatial autocorrelation or close to a random pattern.

Spatial autocorrelation can be a valuable tool to study how spatial patterns change over time. Results of this type of analysis often lead to further understand-ing of how spatial patterns change from the past to the present, or to estimations of how spatial patterns will change from the present to the future. Therefore, useful conclusions can be drawn to advance our understanding of the underlying factors that drive the changes in spatial patterns.

In addition, the study of spatial autocorrelation has significant implications for the use of statistical techniques in analyzing spatial data. For many classi-cal statistics including various regression models, a fundamental assumption is that observations are randomly selected and independent of each other. But when spatial data are analyzed, this assumption of independence is often violated be-cause most spatial data have certain degrees of spatial autocorrelation (Anselin and Griffith, 1988). If spatial autocorrelation in the data is significant but disre-garded, the results will be incorrect. For example, the regression parameters in a regression model that is constructed using classical statistics on geographic data may not be biased, but the significant testing of the parameters could yield wrong conclusions. Consequently, measuring and testing the significance of spatial au-tocorrelation is essential before any statistical analysis is conducted.

To measure spatial autocorrelation, there are statistics that allow us to work with points or polygons. Since the methods for measuring spatial autocorrelation in point patterns have been discussed in Chapter 3, we will focus on the methods suitable for polygons. These methods may be used to measure spatial autocor-relation of nominal and interval/ratio data. Specifically, joint count statistics can be used to measure spatial autocorrelation among polygons with binary nominal data. For interval or ratio data, we will discuss Moran’s I Index, the Geary Ratio, Local Indicators for Spatial Association, and the G-statistics. In addition, a graph-ical tool, the Moran scatterplot, used to present spatial autocorrelation visually, is included.