Tag Archives: centralization

The negative exponential density gradient and decentralization

Many researchers have used the density gradient from the negative exponential model to study the decentralization of population and housing units in urban areas. The density gradient is the rate of decline of density with distance from the center of the city. A decrease or flattening of the density gradient has been considered to be evidence of the decentralization of population or housing. And the density gradient has been used as a measure of the amount of centralization in an urban area that could be used to compare levels of centralization with other urban areas.

I have estimated the density gradients for 43 large urban areas for each of the census years from 1950 to 2010. And I have developed a separate, “pure” measure of centralization of housing units which I described in the previous post. I am calling this measure the centralization ratio. So this gave me the means of actually looking at the extent to which the density gradient was a good measure of centralization and decentralization.

First, I looked at changes in the density gradient over time and compared it to changes in the centralization ratio. The relationship was reasonably strong. It is appropriate to use the change in the density gradient as a measure of decentralization.

Then I looked at the relationship between the magnitudes of the density gradient and the centralization ratio at single points in time. This time, virtually no relationship. The density gradient does not work as a measure of the level of centralization in an urban area that could be used to make comparisons with other urban areas.

What gives? Why such different findings? The key lay in the fact that the density gradient is strongly inversely related to the size of an urban area. Using the density gradient to predict the centralization ratio resulted in no relationship. But add number of housing units in the urban area to the model, controlling for the size of the area, and a strong relationship emerged. And this is why the change in the density gradient works as a measure of change in centralization over time. The size of the urban area is being subtracted out when you look at the change (with the exception of any change in size over the period).

Someone committed to the idea that the density gradient is a good measure of centralization might object that I have only shown that the centralization ratio and the density gradient are different, not that one is a better measure of centralization. I think I make a good case for the use of the centralization ratio. Also, in developing the measure, I calculated other measures of centralization for a sample of a dozen areas and they were all highly correlated. And an anecdotal point: The three urban areas in my study with the highest centralization ratios were New York, Chicago, and Philadelphia. And all three had density gradients that were below the mean for the 43 large urban areas I looked at.

Centralization in large urban areas

Many have examined the decentralization of population and housing units over time. A common approach has been to use the density gradient from the exponential model as a measure of centralization. I have estimated the parameters for the model for large urban areas since 1950. I wanted to consider how well the density gradient actually performed as a centralization measure (which will be the subject of the next post). But to do so, I needed a separate, good measure of the centralization of housing units.

I reviewed a variety of centralization measures in the literature and was not satisfied with any of them, so I developed my own. I wanted a measure that made maximum use of the data on the distribution of housing units by census tract. And I wanted the measure to be interpretable, to have meaning beyond a larger value indicating housing is more centralized. The measure involves calculating two values: One is the mean distance housing units in the urban area are from the center. The other is the mean distance they would be from the center if housing units were uniformly distributed in the area, densities everywhere the same, no centralization. The ratio of the actual to the uniform distance would, of course, be 1 if housing were uniformly distributed and would decline with decreasing mean distance to the center and greater centralization. The minimum value would be 0 if all housing were located at the center. I wanted a measure of centralization that would increase with greater centralization, so this ratio is subtracted from 1. This measure, which I am calling the centralization ratio, is the proportional reduction in mean distance housing units are located to the center compared with a uniform distribution. So a centralization ratio of 0.25, for example, would mean that the mean distance to the center is a quarter less than for an even distribution.

I calculated the centralization ratio for 59 large urban areas for each census year from 1950 to 2010. The widely expected decentralization did occur, on average, with the mean value dropping from 0.25 to 0.18 over this period. But decentralization was far from universal; 14 areas saw increases.

Levels of centralization varied greatly across the urban areas. The highest and lowest values in 2010, for example, were 0.46 and 0.08. New York, Chicago, and Philadelphia were the areas with the highest levels of centralization, not surprisingly. Tampa-St. Petersburg, El Paso, and Jacksonville were the lowest. Urban areas in the Northeast had the highest mean centralization in 2010, followed by those in the Midwest. Urban areas in the South had the lowest levels of centralization (and would have been even lower if Washington-Baltimore, more like other large urban areas in the Northeast Corridor, had been excluded). The very largest urban areas also tended to have higher levels of centralization.

More detail on this analysis using the centralization ratio is in the paper “The Degree of Centralization in Large Urban Areas in the U.S., 1950–2010,” which can be downloaded here.

The failure of two “dimensions” of sprawl

The preceding post described problems with the efforts to develop multidimensional measures of sprawl. I concluded by posing the criteria that anything offered as a measure of a dimension of sprawl meet these simple tests: If observers generally agree that one area has a greater level of sprawl than another area, then the measure should indicate more sprawl. If a change is made in one area that most observers agree results in less sprawl, then the measure should indicate less sprawl.

This post will address these criteria with respect to two measures used in the multidimensional measurement of sprawl: Variation in population or housing unit density and the degree of centralization of population or housing units within urban areas. Both the Galster, et al. and the Ewing, Pendall, and Chen sprawl indices included such measures.

For the example, consider an older large urban area in the Northeast or Midwest with a high density core and density decreasing to very low levels in the sprawling suburbs. Variation in density would be fairly high, as would be the degree of centralization. (This would be in comparison to all large urban areas in the United States.)

Now let’s imagine that reducing sprawl and promoting smart growth catches on in this urban area. In response, a major new development is built near the periphery of the urban area that embodies smart growth principles and has moderately high-density housing, far greater than the density in the surrounding area. Perhaps it is a transit-oriented development around a transit stop.

I believe that virtually everyone would agree that this development is the antithesis of urban sprawl and that building this development has resulted in the reduction of the level of urban sprawl in the area. Certainly there is less sprawl than if the area had been developed as very low-density single-family housing.

Now look at the two measures of sprawl, first variation in density. The density of the new development is much higher than the low densities in other areas at the periphery. But it is very doubtful that even what would be considered a reasonably high-density development in that setting would have a density as high as that near the center of the urban area. Instead, it would fall somewhere in the middle, closer to the mean density of the urban area. Result: The variation in density for the urban area would be reduced. But variation in density when used as a measure of sprawl says that a reduction in variation implies more sprawl. So variation in density fails as a measure of sprawl.

Now for centralization. The new development adds significant numbers of people and housing units at a location that is very distant from the center. Any measure of centralization, whether based on the estimated parameters of the negative exponential model or some other measure based on distances to the center will show a decline in centralization. And again, when used as a measure of sprawl, less centralization means more sprawl. So centralization likewise fails as a measure of sprawl.

Multidimensional measures of sprawl

Around the beginning of this century, increased attention was being devoted to urban sprawl and its measurement. Numbers of those efforts conceived of sprawl as being a complex, multidimensional phenomenon and sought to develop sprawl indices combining multiple measures. The two most-cited studies doing this were by Galster and his colleagues, “Wrestling Sprawl to the Ground” (Housing Policy Debate, 2001) and by Ewing, Pendall, and Chen, Measuring Sprawl and Its Impact (from Smart Growth America).

As I am going to be making some criticisms of this approach to sprawl measurement, let me begin by making full disclosure: I was involved in research on multidimensional measurement of sprawl at this time. So the criticisms apply to me as well as others. Mea culpa.

The inspiration for using multiple measures for different dimensions of sprawl comes from at least two sources. First, the literature on sprawl provides widely varying definitions. So it is may not be unreasonable to conclude that sprawl might be a complex phenomenon that has multiple dimensions. And second, sprawl is one type of urban pattern. Urban patterns are obviously complex and vary over many different dimensions. The conclusion is that therefore sprawl has multiple dimensions.

Neither of these arguments is logically very tight. The existence of multiple definitions of sprawl does not imply that the various definitions are even consistent, much less that they represent aspects of a more general concept of urban sprawl. To say that urban patterns are complex and multidimensional and that sprawl is an aspect of the urban pattern does not necessarily imply that urban sprawl (whatever it means) is complex and multidimensional. It could be, but needn’t be.

The multidimensional sprawl efforts did not address these issues. Rather, they begin by asserting that sprawl is a phenomenon to be measured along multiple dimensions. Then comes the identification of the dimensions of sprawl. At least for some of the measures, the approach appeared to be identifying measures of the general urban pattern and adopting these as measures of sprawl. The overall population or housing unit density of the urban area is a measure of the urban pattern, so this can then be considered a measure of sprawl. (Given generally accepted ideas about sprawl, most would not argue with that.) Some other measures of the urban pattern identified were the extent of the variation in density across the urban area and the degree of centralization of the population or housing units within the urban area. So these could be considered to be measures of sprawl as well.

When you identify other measures, it becomes necessary to specify which direction is associated with greater amounts of sprawl and which with less sprawl. This was not explicitly addressed. Instead, the direction was simply asserted, apparently as being obvious. Less variation in density was associated with more sprawl and lower levels of centralization were associated with more sprawl. I believe that these “obvious” choices arose from looking at the characteristics of urban areas thought to be more or less sprawling. The areas seen to be more sprawling had less variation in density and less centralization than areas seen to have less sprawl.

Multiple problems exist with this process. First, there is no logical or empirical connection established between the various measures and any fundamental conception of sprawl. This is, of course, difficult without a clear conception of what sprawl is. And an empirical relationship cannot be established without measures of sprawl, which these projects are attempting to provide. The logical problems go away, however, if the authors are simply asserting that they are providing a definition of sprawl with these measures. They can certainly proceed in this manner. But their work can then be judged on the basis of whether their definition is consistent with other notions of sprawl.

Establishing the direction associated with more sprawl for the various measures is likewise problemmatic. First, to the extent that it is based on the direction of the variation between more-sprawling and less-sprawling urban areas, it is grounded in the subjective judgement about degrees of sprawl in those areas. And second, the judgement is based on differences among urban areas in the United States in the early twenty-first century. Conditions in other eras or in other parts of the world may present a different relationships between sprawl and the directions of variation in different dimensions of the urban pattern.

Consider just the simple measure of the variation in density. Less variation is associated with more sprawl because urban areas exist in the United States with overall low densities (and hence high levels of sprawl) that have very limited variation in density. But what about an urban area with uniformly high densities? I don’t think you would want to consider that area as having high levels of sprawl, higher than an area with some high densities and some low densities. Some will argue that this is not a realistic possibility. But it is a logical possibility, so where should one come down on this?

But let’s ignore this. I think most would agree to two closely related criteria for judging whether a measure can be considered to be a reasonable measure of urban sprawl. If you have two urban areas and most observers would agree (on whatever basis they are using), that one area has more sprawl than the other, the measure should indicate more sprawl for that area. And if a change is made in an urban area that results in what most observers would agree creates less sprawl, then the measure should indicate less sprawl.

The next post will provide a simple example demonstrating how the measures of variation in density and centralization fail to meet these criteria.