Research Notes and Comments

Transcription

1 Research Notes and Comments Frequency Distributions of Distances and Related Concepts byj. H. Kuiper and J. H. R Paelinck. FREQUENCY DISTRIBUTIONS IN SPACE Measurements on economic quantities with spatial dimensions are in principle interdependent; distance between the points in space plays an important role, mutual influence being often stronger as distances become smaller. For instance, one could take as a measure of the distance between two regions the degree of contiguity, represented by the minimum number of borders one has to cross to travel from one region to the other; a synthesis of the distances that prevail between regions in an area is given by the frequency distribution of distances in that area, which shows for each observable distance in the area the total number of times (absolute frequency) a distance actually appears. One can also imagine deriving a frequency distribution of the number of interregional contacts (or visits) classified according to the distance at which the contact is made or the visit paid. As an example a rectangular area consisting of eight regions was chosen (Fig. l), the degree of contiguity being a measure of the distance between the regions FIG.. Area with Eight Regions The results of computing its frequency distribution are shown in Table and Figures a and b; distances not mentioned in the table have frequency zero. TABLE Frequency Distributions of Distances and Contacts Frequency Distribution of Distances Contact Frequency (bv distance of visit) TOTAL 8 6. H. Kuiper is assistant professor and]. H. P. Paelinck is professor, Department of Theoretical Spatial Economics, Erasmus University Rotterdam, The Netherlands. 6-76/8/78-5$.5/ 98 Ohio State Univesity Press GEOGRAPHICAL ANALYSIS, vol. 4, no. (July 98)

2 54 I Geographical Analysis observeil numberr? 4 + distance FIG. a (left). Frequency of Distances FIG. b (right). Frequency of Distances of Visits 4 + distance Given these distributions it is easy to compute the average distance; the results are, = 8* + * + * + 6* + *4 6 =.55 and ; = * + * + * + * + 4*4 =. 56 This example can be generalized by considering differently structured spaces, for instance: - Square, rectangular, circular spaces TABLE Possible Cases for the Determination of Frequency Distribution Road Area Euclidean Manhattan Continuous Discrete Continuous Discrete

3 Research Notes and Comments I 55 - Spaces with points (in the above example regions) dispersed in a continuous or discrete way, all points lying in undifferentiated space - Metric spaces with Euclidean, Manhattan (rectangular), or other norms - Next to distance or contact frequency, it is possible to derive other frequencies, for example, road-area frequency. From each point in space a number of trips can be made over a certain distance; this distance is bridged along a road, the shape of which depends on the distance measure used. In this way an area within the given space will be covered with roads; each distance is now related to a certain road area, and considering all distances one finds the road-area frequency. Table sums all relevant cases. Here attention will be paid only to spaces in which the points are distributed continuously; the assumption will be that the points are spread homogeneously across the space considered (uniform distribution). To compute the frequency distributions one has to measure () the number of points; () the distances between points; our concern now will be with the problems involved.. MEASUREMENT PROBLEMS A. Measurement of the Distance and Point Frequency In all cases the starting point is a well-defined space of known dimensions; to illustrate the measuring of a distance frequency, one case from Table will be closely considered, the others being comparable. Let us suppose a square-shaped space with side a, within which the frequency distribution of Euclidean distances has to be calculated. Consider a point (x,y) within a square (Fig. ); fromjhis point other points can be reached at distances varying from = to Z =, Z depending on (x,y). Only distances Zl, I,, and Z are distinguished, with < Z < min {a-x, a- y} () min(a-x,a-y) <, < max{(a-x+y), (x+a-y)} Z max{(a-x+y), (x+a-y)}. (4) (5) Starting from (x, y) one has relatively more distances Z than Z, Z not occurring at all, because all points at distance Z from (x,y) lie outside the square. Now let us use as a measure of the incidence of distances Z the value, as a measure of the incidence of distance Z the value, and as a measure of the incidence of distances, a figure between and, to be calculated from the ratio between the part of the perimeter of the circle (radius ) falling within the square and the total perimeter of the circle. To derive the frequency distribution of the distances it is necessary to find out for each point what distances occur and how often each distance occurs ; summation or integration over all points then produces the frequency distribution. As a measure of the number of points within a certain Z-property the area containing the points concerned is chosen. To get an impression of the shape of the distance frequency distribution, let us consider some extreme values of. First take = ; the incidence of this distance is. The number of points for which the distance falls entirely within the defined space, is the entire space itself; their number is a; so the frequency for I = is

4 56 I Geographical Analysis FIG.. Euclidean Distances in a Square (a) a'* = a'. The largest distance one can find in the square is = aj; the incidence of this distance is (the circle with radius aj lies always outside the square) and the frequency for = aj is, therefore,. Two points of the frequency distribution looked for are thus known; Figure 4 shows what the shape of the frequency distribution would probably look like. B. Measurement of the Contact Frequencies To illustrate the measurement of the contact frequency, once more Euclidean distances on a square are considered. From the frequency distribution of distances the number of distances of length in the space is known; the number of possible visits (contacts) from (x, y) (Fig. ) at distance is equal to the number of distances that can be measured in that point, multiplied by the number of contacts that can be made at distance. As a measure of the number of contacts has been chosen the length of the perimeter of the circle (radius ) as far as it falls within the space considered (= ~~7~); the number of contacts from point (x,y) with points at distance then becomes: FIG. 4. Frequency Distribution of Distances. Number of distances on vertical axis.

5 Research Notes and Comments I 57 FIG 5. Contact Frequency Distribution. Number of contacts on vertical axis. For distances with =, the measure of the number of distances is for each point, and the number of points is a; the contact frequency is a a$mz = a m =. For distances with I = aj, the number of distances for each point is, and the number of points is also ; the accessory contact frequency is a m aj =. The contact frequency distribution can now be tentatively be sketched (Fig. 5). C. Measurement of the Road-Area Frequency From point (x,y)(fig. ) a number of contacts are possible over a distance. Because we are considering Euclidean distances, distance is bridged along the radius (I), so the area within the circle () and the space (a) can be covered with roads (Fig. 6). From the distance frequency the number of distances can be derived; this number multiplied by the relative number of roads yields the road-area frequency. As a measure of the relative number of roads is chosen the ratio between the area of circle () that can be covered with roads within the square and the total area of the circle. Let = ; the number of distances is a; a measure of the number of roads is ; the road-area frequency is a = a. Let = aj; the number of distances is ; a measure of the number of roads is [~(aj)~] = ; the road-area frequency is. Tentatively the road-area frequency can now be sketched (Fig. 7). FIG. 6. Roads from Point (x,y), Distance

6 58 I Geographical Analysis FIG. 7. Road-Area Frequency Distribution. Road area on vertical axis.. SOME RESULTS As an example some results of the calculations follow below. A. Distance Density of Manhattan Distances in a Square OSlSa a - a + 6 l aslga (a - ) 6 average distance ~() =.4 a. B. Visiting Density of Manhattan Distances in a Square OSZSa lj(az - a + 59 ) average distance ~() =.48 a. C. Road-Area Density of Manhattan Distances in a Square OSlSa aslsa a - a a + a - 9aZ - 4a4 l Clearly, the density expressions do not satisfy the same functional relationship across the whole area on which is defined; for values > a the relative frequency of starts to decrease less rapidly, the reason being that though greater distances occur relatively less frequently (Fig. 8), the measure of their occurrence declines less rapidly. 4. DISCUSSION For a complete picture of the incidence of distances in a space it is necessary to determine a distance density. In this note some distances in a few geometrical 'The detailed calculations can be found in Paelinck (98, chap., app.). 'A check on the correctness of the calculations is the continuity of the density curve for = u; in fact for the case discussed, the first derivative is also continuous, but the second derivative is not.

7 Research Notes and Comments I 59 FIG. 8. Distance Frequency in a Square (Manhattan Distances) spaces have been calculated, only the continuous case having been considered; similar densities can be derived for discontinuously spread points in space. The densities obtained in the continuous case should be equal to the densities of the discontinuous case if the number of points in the space becomes infinite. The expected distance &(I) within a space is calculated via the distance frequency f(); the formula then holds. For Manhattan distances in a square (a) space the formula becomes This value can be regarded as the expected distance between two points chosen at random in the space; it can be used in theoretical geography and spatial economics and also in spatial-operations research. LITERATURE CITED Paelinck, J. (with the assistance of J.-P. Ancot and J. H. Kuiper) (98). Formal Spatial Economic Analysis. Aldershot, England: Gower.