 Simulation tests of planning in centuriations |
Simulation tests of planning in centuriations |
By contrast, we must base the choice of a random line length on empirical evidence of the lengths of segments of Roman roads. It seems a reasonable estimate that they will be between 2 to 16km long. For squares of 700m this would be 3 to 23 times the grid size. We assume that they are distributed uniform randomly in the range. Clearly, some segments of Roman road are considerably longer than this, but, since such longer lengths are less likely to fit the grid, the chosen values are likely to overestimate, rather than underestimate, the fraction of road lengths which would fit by chance.
For a given line, it can be decided that it is a hit if it passes within a given distance (the tolerance) of at least two grid intersections, and does not miss intersections it ought to hit (see diagram below). Note that in this diagram the points are shown as small black circles, whose radius is (of course) the tolerance.
Search algorithm1) Select a random starting point within the unit square bounded by the y-axis to the left and the x-axis below.2) Select a random orientation between 0 and 90°. 3) Select a length between 3 and 23 times the grid size. 4) Test for all lines at intervals of one unit parallel to the y-axis which are intersected by the oblique line, and
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The likelihood that the oblique line will fit depends upon the tolerance selected for the grid points. This tolerance is the 'radius' of the point. If the line passes closer to the point than the tolerance, then it hits. If the tolerance is zero, the chance of a line hitting is zero.
The relationship between the tolerance and the percentage length of lines hitting was investigated by a Basic program which tested 10,000 lines for each of 41 tolerance values from zero to 20% of the grid size, with the following results.
It may seem surprising that the percentage of lines which have a rational fit does not tend towards 100% for large tolerances, but if we consider how the algorithm works, it is explicable. We can see in the figure on the left that in certain cases the algorithm will decide from two points that the orientation is, as in this example, 0:1. However, this is an incorrect decision, since the next point is missed and the line is ac cordingly rejected. If the tolerance had been smaller this particular line would have stood a chance of being accepted as a 1:5.
If we are studying the relationship of linear features to cadastral grids on a map, the question is what tolerance to assume. A reasonable figure to use is 0.4mm on the map. On 1:50,000 maps, this is equivalent to 20m. It gives coordinate points on the map with a diameter of 0.8mm, which is large when compared to the width of the conventional representation of a single carriageway road, which varies between about 0.5mm and 0.7mm. We can thus be confident that, with this tolerance, a fit or (more importantly) a lack of fit will be determined without excessive ambiguity. If we use a radius of 20m, this represents a tolerance of 2.8%.
From the simulation result we can see that, for this tolerance, under 4% of random lines will fit at 'specifiable' angles. This may be examined more closely by further simulation runs to produce a distribution of percentages fitting (at these angles) in 500 trials of 1000 lines with a tolerance 2.8%.
mean: 3.65
standard deviation: 1.28
Thus there is less than a 16% probability that more than 5% of any sample of lines will fit, and there is only a 0.003% (1 in 33,000) chance of more than 9% fitting. These figures represent respectively: the mean plus one standard deviation and the mean plus four standard deviations.
The conclusion that we can draw from the simulation study is that we cannot neglect the possibility that a supposed fit of an oblique feature to a cadastral grid has occurred by chance. About 1 in 25 segments of roman road will fit a grid positioned in a completely arbitrary fashion. However, if more than 10% of any randomly selected set of road segments do fit, then it is highly unlikely that this is a chance event. Thus we can be prepared to bet that the associations are not chance when this is the case for a reasonably large sample, say 25 segments.
Unfortunately this is hardly ever achieved in practice. There may not be enough segments of road, and the situation is inevitably complicated by the presence of roads which were established earlier than the cadastre. They are not expected to fit, except by chance, so if they are included in the sample they may unfairly bias the result against the hypothesis that the cadastre exists.
Nevertheless, so long as we accept these caveats, we can say that there is only about a 4% chance that a length of road, chosen at random, will fit a grid to the tolerance given here. When we see such a thing in an otherwise arbitrary grid, we are looking at what would be, on the basis of chance, a fairly rare phenomenon.
There are also some cases, in real cadastres, in which oblique segments meet at grid square corners. On the basis of pure chance this should be an even more rare event. I have carried out a simulation, using the same tolerance as before (2.8%), taking pairs of lines with length ten units, starting at a fixed point (0.999, 0.999) very close to a corner of the unit square, with a random orientation. The results (on the right) showed that for each 10,000 lines (5,000 pairs) there were about 100 cases in which both lines fitted the grid at a specifiable angle.
Given that the intersection of the two lines, defined to a tolerance of 0.028, has a chance of 0.0025 of falling at a given point in the unit square, there is a chance of about 1 in 40,000 that two lines will meet at an intersection and both fit. Thus the presence of this, as a road which has segments going through grid points and which turns at a grid point, is strong evidence for the grid's existence, and for the fact that the road is probably later than it.
(e-mail j.peterson@uea.ac.uk)
