To Go or Not to Go --> (Urban Planning and the Distance Decay Function)

The fine art of problem articulation 

The important thing about mathematical urban models is not the mathematics itself but its application to simulate urban phenomena that we are trying to understand. Therefore, even a failed attempt at creating a model may help an urban planner understand and articulate a phenomenon with greater clarity. Trying to explain an urban phenomenon in the form of an equation compels us to cut through the fog of vagueness and confusion in our minds and seek clarity.

Try to imagine what we expressed by the equation --> A=f(B, C, 1/D) in the previous blog and then try to explain it in words instead of the equation.

We would have to say something like - "When we consider shopping or any such pattern in a city...it depends on how many people are shopping....where they are shopping...it depends also on which shopping areas are large or attractive...also we must consider which are far away or close by...its a pretty complex process....but also very basic etc etc"

While the equation used just 5 letters and 1 digit, the verbal explanation used about 250 letters (and counting). 

This is not to say that the equation is better than the description, but that one should attempt to formulate an equation from the description - if only to check if that is even possible. It is an iterative process where a description helps us to form an equation and the equation in turn helps us to give a clearer description and so on.

Distance decay and the Gravity function

Distance is a crucial topic in urban planning. As we have seen in the previous blog, "distance", in this case, is not just a physical distance, but the measure of difficulty (or ease) involved in reaching where we wish to reach -- it could be kilometers of roads; delays due to traffic jams; the high cost of petrol; the physical and emotional stress of spending hours in travel etc.

The funny thing with transportation is that it is the only land-use that does not exist for itself, but for the primary purpose of facilitating interaction among other land-uses. We are generally not on a road because we want to "go" to the road. We are on it because it links where we are (our point of origin) to where we wish to go (our destination).

The other funny thing is that distance...decays.

What this essentially means is that when distances increase between us and a particular destination, our desire to go to that destination also decreases (or decays). This is easy to imagine. Let's say that there are two cafes which are equally attractive to you. Would you rather go to the one that is 500 meters away from you or to the one which is 12 kilometers away ?  

This is what we expressed in our equation as 1/D, i.e. as D increases the A (number of trips) decreases. But by how much ?

Way back, in the 1930s, the American economist William J. Reilly discovered from his empirical studies on the flow of retail trade, that the volume of trade between cities increased in direct proportion to the population of the cities and in inverse proportion to the distance between the cities. The trade not only decreased  but it decreased in proportion to the square of the distance between cities. 

Based on these observations, Reilly formulated a law which states:

"A city will attract retail trade from a town in its surrounding territory, in direct proportion to the population size of the city and in inverse proportion to the square of the distance from the city."

Mathematically it is expressed as -

R_i = P_i / d²_ki

Where R_i is the attraction of city i felt by city k; P_i is the population of city i; and d_ki is the distance between city i and city k. It has an uncanny similarity with Newton's law of gravitation where the attraction between two bodies also decreases in inverse proportion to the square of the distance between them. 

Reilly tested his model extensively by studying the breaking point between cities in the United States.  

If we accept Reilly's findings for now, then we have already developed our original equation further and we now have this -

A_ij = f(B_i, C_j, 1/D²_ij)

Nothing Super-Natural about it

It is clear from the way we constructed our equation, that there is nothing super-natural or god-given or mysterious about any of it.

We are just trying to analyse and describe how a certain kind of spatial interaction takes place.

This implies that there is nothing holy about the fact that the distance is raised to the power 2. Depending on the local context, it may be something else. In fact, empirical work over the years has shown that it tends to vary between 1.5 and 3 depending on a range of contextual factors.

However, many planning text books in India (including the venerable book on transportation planning by L.R. Kadiyali which every planning student is familiar with) continue to use distance to the power 2 without explaining that while it was derived out of extensive empirical research, that research corresponded to large cities in the United States, separated by an average distance of about 100 miles and was conducted in the 1930s.

(The fact that the equation still holds was precisely due to Reilly's empirical rigour - something that our scholars and researchers tend to shy away from most of the time.)

In reality you are free to play around with the power of D in order to check which number makes the equation simulate an observed reality best. Consider the following graph which shows how the distance decay curve would vary if we assume distance to decay if we raise the D parameter in the equation to a power of 1 or 2 or 3 -

This graph shows how quickly the likelihood of traveling a certain distance will decline if we increase the power of D in the model.

As the power increases from 1 (the green line) to 3 (red line) the tendency to travel farther declines. The red line corresponds to a reality where the tendency to travel decreases precipitously when distance increases from 0 to 3 kilometers and becomes almost nil when distance increases beyond 5 kilometers. 

Understanding the rate at which the tendency to travel declines based on increase in the distance can help us understand how appropriately or inappropriately important facilities are located with respect to each other in a city. It can also help us to concretely evaluate planning decisions aimed at locating facilities in a certain way. 

In the next blog we will play around with this equation a bit more by applying it to situations familiar to us.