In my explorations of road density versus population density (see yesterday’s post on the topic for an introduction to the idea), I have been wondering whether the scale of observation would influence the relationship between the two measures.

It should seem intuitively obvious that scale of observation will affect either the measured road density or population density, particularly in higher density areas. As you zoom out to include surrounding lower density areas, this will obviously dilute the average density, hence measured density is dependent on scale of measurement, in some ways similar to the way measurements of coastlines depend on the length of the ruler. Ergo there is a suggested fractal aspect to road density and population density.

Anyhow, what I wondered was whether these scale effect would influence any inferred relationship(s) between the two measures. the following images illustrate the outcome of a preliminary investigation. First, here is a graph showing a comparison of actual road density and population density for roughly 1500 census blocks – census tracts (CTs) in Quebec, and census subdivisions (CSDs) in Ontario. The area of these blocks ranges from .04 to 400,000 sq. km, or a span of 8 orders of magnitude.

This image seems to suggest a relatively consistent relationship across these 8 orders of magnitude, with the two measures being related by a power law with an exponent of 2. Note the “best fit” as determined by a least squares minimization has an exponent closer to 1.9, but the square fit is essentially as good, and use of a second significant digit with a fit of this nature seems unmerited.

When we plot these same data (less the Ontario data for clarity) on a natural scale, we see a fairly nice lower bound trend with what appears to be a tight grouping of most data, with what looks to be a transition to near chaos (i.e. much greater evident scatter) for road density > 5 km/sq. km:

This graph made me wonder if there is a real transition from fairly “nice” (well controlled) behaviour to chaos at some critical point (i.e. ~ 5 km/sq. km), so I zoomed in closer to look at the lower density data:

This graph seems to show a fairly marked increase in variability above 5 km/sq. km, with the highest population density values being roughly 4 times greater than the inferred best fit.

When we zoom in further, we see that at < 2 km/sq. km, where our lowest densities lie, and generally largest census blocks, we still have population densities significantly higher than the best fit, at about 3 times greater.

This doesn’t seem to differ much between the Quebec CTs and Ontario CSDs.

So, the data seem to me to be suggesting that, within the limits imposed by the variability in the data, the relationships we can infer between road density and population density are fairly consistent across a very wide range of scales of measurement. In other words, the relationships seem to be consistent across a wide range of scales of measurement.

We’re working with data from a Caribbean island and I’m anxious to see how the two data sets compare. I suspect that given the scatter in the data for Canada, their data will likely overlap substantially so that the same general relationships can be applied. But at the moment that’s only a suspicion, not yet borne out by evidence. More to follow… 🙂