An overriding concern of a number of scholars over the years has been their attempts at differentiating one pattern from another, by deriving or describing various measures of shape, form, density, intensity, clustering, centrality, and dispersion (see the recent review by Wentz, 2000). The literature tends to emphasise a variety of subjects; among them are the patterns of towns, the clustering of diseases, the form of physical features, and the shape of economic regions. There are those, however, who have developed universal, systematic ways to measure two-dimensional patterns without reference to such subject areas; these include: Mandelbrot (1983) and fractals; Ahuja and Schachter (1983) and pattern models; Okabe, Boots, and Sugihara (1992) and tesselations; Getis and Boots (1978) and points, lines, and areas; Wentz (2000) and shape; Haggett, Cliff, and Frey (1977) and scale; Tobler (1979) and map projections; and Neft (1966) and measures of centrality. Our work follows in this tradition.

In this paper, we first introduce the nature of the reference area and its statistical distribution (Section 2) and then present a series of descriptive devices: measures of concentration or dispersion (Section 3), eccentricity (Section 4), randomness (Section 5), geophenograms (Section 6), and clustering (Section 7). We consider a spatial pattern as a multidimensional concept where each dimension requires a specific measuring rod. Our goal is to first present the theory for spatial patterns, in general, giving examples along the way, and then, in a subsequent paper, develop a focused approach in order to demonstrate how spatial pattern may be described within subsets of large data sets. The paper will close with a simple, more comprehensive example (Section 8) and some observations on the practical usefulness of the pattern measures (Section 9).

The choice of the particular reference area is that its principle, adding elementary squares, can be generalized to irregular shapes for the study of observed areas (see e.g. Kuiper, 1997). In addition, in the present context, the reference area offers the property of having a central square, c, which allows us to emphasise our particular spatial point of view; it has an odd number of elementary squares, and could, in later investigations, be complemented by an «even» type.

The following assumptions are introduced:

A₁: spatial contiguity is defined as the rook’s case, which implies that the distance separating elementary squares having a side in common is one;

A₂: n elementary units (or objects) can be freely dispersed among the n elementary squares, in the sense that all patterns to be described further down are equiprobable and independent.

In order to fix ideas, it will be helpful to envisage a number of units of one kind (firms, residences, trees, crimes, infected persons, etc.) placed, accordingly to A₂, in the v squares (districts, residential blocks, pixels, etc.). The model is simple, but it is able to generate a large number of different spatial patterns.

A partition is described by the number of units in each of the squares. For each partition, there are one or more different arrangements of units in squares. Each different arrangement is called a pattern. The following shows the different types of patterns when n = 5 and v = 5; there are 27 different types in seven different partitions. We shall first use the convention of representing the contents of the cells, a, b, c, d, e, of Figure 1 as:

The total number of spatial patterns is a function of the number of different ways one can divide n objects into n squares for each partition keeping in mind that the number of patterns for each partition is a function of the distance each square is from the central square. For example, the central square can contain objects (X) or it can be empty (0). The four squares one distance unit away from the central square (peripheral squares when r = 1) are related either by being diagonally next to each other or separated by the central square. Thus, when r = 1 there are four general types of spatial patterns; these are shown as:

Thus, in order to identify the total number of patterns for each partition, one permutes the n objects in the arrangements shown. If a value of the same kind is placed in both a squares, spatially they cannot be differentiated from each other; that is, direction does not matter. For example, in the case discussed above, if the value 5 is placed in the central square, the resulting pattern counts as one pattern; if, however, the value 5 is placed in a peripheral square, that also constitutes one pattern. Thus, there are two possible patterns when all objects are placed in one cell. Again, if each object is different, in this case, there will remain just two possible patterns.

We contrast this approach with the combinatorial problem of determining the number of arrangements for each partition. Table 1 gives the number of spatial and non-spatial patterns for each partition. The formula for finding the number of arrangements (non-spatial) for each partition is given as:

where v₁ is the number of cells having the largest partition, v₂ the second largest, and so on. Thus, for the case n=5, v=5, and the partition 3,1,1, we have:

Table 1
Statistical distribution of patterns n = 5, v = 5

If one considers the presence of 3 objects or more in a cell as representing a spatial concentration, then the number of realizations is 16 (Types D, E, F, G) out of the 27 possible patterns or 59%.

where P is the set of all possible patterns, p, and the Δ_jc’s are the absolute differences between the cell values within one unit of distance from the central square (index c); the indicator has values in the closed interval [0,1]: indeed, for complete dispersion, it takes on a value of zero, and for total concentration in one cell the value n is divided by itself. Thus, the range, 0 to 1, represents a useful index of concentration or dispersion.

If distance were not a factor in this system, then location over the cells is not relevant to the calculation; in this case, for a given n and v, all pattern types will have the same value of c_p. Table 2 gives the value of c_p for each pattern type given above.

Table 2
Values of c_p

It can be proved that c_p equals the Gini concentration coefficient; indeed, the Gini coefficient can be written as:

where the sum is taken over v-1 cumulative terms of the Gini concentration graph.

which is defined over the closed interval [0,1]; p and P have been defined above. The values μ _i are the number of «outliers» in a partition f of a given type within a pattern p; table 3 gives the values of e_pf for each of the spatial partitions identified above in Section 2.

Table 3
Values of e_p

From this it is evident that there can be a wide range of eccentricity values for each pattern type; this is to be expected given that the measure is a function of the relative position of the more or less concentrated units.

where n_p is the required number of non-zero parameters, and v is defined by equation (1).

Table 4
Randomness equations

The last two examples show that the relative positioning of the units strongly influences the degree of randomness of the pattern.

It is true that for larger v, one has to solve large systems of linear equations, but with present computational facilities this would not be a problem.

One could imagine representing numerically such a pattern; a possible measure could be the surface of the typical triangles divided by the maximum value which is 1.3 for unit radius. For sides of length a, b, and c, using Heron’s formula, one has:

where s = (a+b+c)/2, and, therefore, for the pattern 02120, the measure would be .393.

Another possibility is the length of the sides of those triangles starting from the center, divided by its maximum, i.e., three. For the 02120 pattern, this value is .633.

An easy measure to compute is the average score (which is already normalized to one) and the coefficient of variation, of which the maximum value is √3; this can be proved as follows.

The coefficient of variation is defined as:

the square of which can be expanded, using definitions of ρ and μ, and n_i being the number of indicators considered, to:

and, given non-negativity in the case of the indicators discussed, this quantity reaches a maximum for Σ_ijx_ix_j = 0, implying that only one indicator is non-zero. For the 02120 pattern this value is 1.039.

Differences between patterns can be expressed as normalized Hamming distances, defined as the sum of absolute differences between indicators divided by their number. Again, for the 02120 pattern, with respect to the type A pattern, this value is .617.

For all of these synthetic indicators, it should be noted that the same value may correspond to different underlying joint patterns, as is well known from the Gini coefficient.

When all cell values are found within the v squares, the v squares constitute a cluster of one. The measure resides in the closed interval [0,1]; clearly, as d approaches r, the probability is high that the measure will approach one, thus, the measure is only useful when d is considerably less than r (see Getis and Ord, 1992, for a test of significance on the hypothesis of no clustering). In Table 5, we show the measure for distance 0 from the central square, i.e., only the value in the central square enters the numerator; clearly, those partitions with the highest values of k_p0 are those with the highest values in the central squares.

Table 5
Values of k_pd

There are 30 different partition types for 10 cities in 5 states. The total number of spatial patterns is 174 (the non-spatial total is 1001). For the partition 4 4 1 1 0, there are 6 spatial patterns (04411, 14401, 44101, 44110, 41104, 41014). Of the 174 spatial patterns, those having at least 5 cities in one state is 106 or 61%. Thus, it is not unusual for one state to have 4 cities, but only 12 of 174 patterns (6.9%) have exactly two states each having 4 cities in them.

The index of concentration, c_p , is 0.25, indicating a relatively high level of dispersion.

Eccentricity for the observed pattern of cities is 0.60, a moderate value indicating that there is a tendency for the areas around Colorado (especially Kansas) to draw the intensity of the pattern away from Colorado.

As far as clustering is concerned, having four cities in Colorado, of the ten in the region under study, indicates a tendency for clustering but, as mentioned above, such a pattern is not unexpected.

As a second example, the regional products (10⁶ Dfl, at market prices) in the 12 Dutch provinces will be submitted to an analysis; Table 6 presents the data (source: Dutch Central Statistical Office, Statline).

Table 6
Regional product of the Dutch provinces (1997)

The index of concentration, c_p, is calculated to be 0.4657. The computed value of the eccentricity indicator, e_pf, is 0.4456; in order to derive this value, the central «elementary square» was located at the province Utrecht, which has the lowest total distance to the other provinces, those distances having been measured «as the crow flies» between provincial capital cities (a better distance measure, e.g. a Hausdorff distance, would take into account more locations internal to the provinces). The clustering indicator, k_pd, is computed to be 0.810 for 7 out of 12 provinces lying within half the maximal observed distance from the central unit, but this value is not statistically significant (Getis and Ord’s (1992) test reveals Z=+1.88). Finally, the randomness indicator, r_pf, has the value 1, the crude data as well as in a stylised version, where a reference area with radius 2 (the central square not being used) is introduced together with five classes of GRP. The provinces were allocated to squares according to a rough nearest neighbour assignment (for demonstration purposes).

This example shows that even outside the straitjacket of a stylised reference area, analytical descriptive results can be obtained from various statistical sources at the spatial level.

An obvious next step in this work is to identify problems that might arise when r —and therefore v—is a large number, and to treat multidimensional observations. Applications in various endeavors in regional science would contribute to a better understanding of the nature of spatial patterns.