Category Archives: Research

More on population-weighted density

Some time ago I did two posts on population-weighted density, which is the mean of the densities of small areas such as census tracts weighted by their populations. In the first post, About population-weighted density, I described this alternative to traditional, conventional density and discussed some of the issues surrounding the use of this measure. The second post was Population-weighted density and urban sprawl, which argued that both conventional density and population-weighted density were appropriate measures of the extent of urban sprawl, relevant to different consequences associated with sprawl.

Doing these posts got me more interested in population-weighted density and led to my writing a full paper exploring this alternative density measure. In addition to expanding on the topics addressed in those two earlier posts, the paper provides much new information. Two of the highlights are the demonstration of the relationship of population-weighted density to conventional density and the comparison of conventional densities and population-weighted (actually housing-unit-weighted) densities across the 59 large urban areas in my urban patterns research.

In the first post I said that to the extent that people are more heavily concentrated in some tracts with higher densities than in others, those higher density tracts will be given more weight in calculating the population-weighted average. This will cause the population-weighted density to be greater than the simple or conventional density. In the paper, I move beyond this qualitative statement to deriving the mathematical relationship between population-weighted density and conventional density. It is actually quite simple: population-weighted density is equal to conventional density plus the variance in density across the subareas divided by conventional density. To the best of my knowledge, this is the first time this relationship has been demonstrated in this manner.

Looking at conventional and housing-unit-weighted densities for the large urban areas, the distribution of the weighted densities is more highly skewed towards higher values, with New York being an extreme outlier on weighted density. The housing-unit-weighted densities were more strongly related to the size of the urban areas, including size in earlier years, suggesting that the presence of areas of concentrated high densities was established in some urban areas decades ago.

Much more information on can be found in my paper, “On Population-Weighted Density” which can be downloaded here.


Measuring urban patterns

For my research looking at 59 large urban areas from 1950 to 2010, I developed 4 measures of the urban pattern: housing unit density, the index of dissiilarity for variation in density across census tracts, the centralization ratio, and Moran’s I for clustering. (See this earlier post for more details.) These were used as measures of the urban pattern for each census year.

Over the 60-year period from 1950 to 2010, the mean values of the first 3 measures–density, variation, and centralization–dropped steadily. But as important was what was not revealed by the averages: the urban areas changed in very different ways, with some seeing large gains in the measures while others experienced major losses.

The average values for the measures varied by region of the country. The urban areas in the Northeast mostly had the highest means for both 1950 and 2010, while areas in the South were consistently lowest. But a big shift occurred with the mean densities for the urban areas in the West. In 1950, the average density was just above the South, which was the lowest. But by 2010, the density in the urban areas in the West was the highest among all regions, averaging 40 percent higher than the densities in the Northeast and the Midwest.

I used cluster analysis to develop a typology of the urban areas for 2010 based on their values for these measures, dividing the areas into 6 groups. One group consisted of the very largest urban areas, with the highest densities and the highest mean values on the other three measures as well. These were obviously the most complex urban areas. At the other extreme were the smaller urban areas, mainly in the South, with the lowest densities and among the lowest values for variation, centralization, and clustering.

More information on findings using these urban pattern measures can be found in my paper, “Measures of Urban Patterns in Large Urban Areas in the U.S., 1950–2010” which can be downloaded here.

Developing urban pattern measures

No single measure such as density can capture the complexity of urban patterns, including the distribution of housing units. For my research looking at 59 large urban areas from 1950 to 2010, I wanted to develop multiple measures of urban patterns to better characterize these areas.

Since around the turn of the century a significant amount of work has been undertaken to identify many variables to quantify urban patterns, mostly to assess levels of urban sprawl. I have raised questions about these multiple dimensions of sprawl efforts before on this blog (here and here).  I am not claiming now to be measuring sprawl, but these efforts provided many possible measures to consider. Too many, as some studies included literally dozens of variables with meanings and differences difficult to discern.

My objective was to identify a small set concepts that captured the most important aspects of urban patterns. I then selected a single variable for each concept that I felt was both among the best measures and, to the extent possible, was easy to understand and interpret.

The overall density of housing units in the urban area is the first, obvious measure.

The extent to which densities varied across census tracts came next. This is measured using the index of dissimilarity. This is a measure of the proportion of housing units that would have to be moved to other census tracts to produce a uniform distribution with equal densities in all tracts.

The centralization of housing units in the urban area, the extent to which more housing units were closer to the center, is an important aspect of the urban pattern. For this, I developed a measure I am calling the centralization ratio, which looks at the mean distance housing units are located from the center and is the proportional reduction of distance compared with the mean distance to the center if housing units were uniformly distributed.

Finally, while centralization is one form of clustering, multiple clusters of higher density housing units can exist at various locations in an urban area. This clustering is measured using Moran’s I, a measure of spatial autocorrelation. This is essentially a correlation coefficient between tract densities and the densities of the adjacent tracts.

More information on these measures, their rationale, and an empirical assessment can be found in my paper, “Developing Multiple Measures of Urban Patterns,” which can be downloaded here.

60 years of exurban growth

In an earlier post I described the tremendous variation in the sizes of exurban areas surrounding large urban areas in 2010, looking at both their land areas and numbers of housing units. In a subsequent post I looked at variation across census regions and factors associated with this. I have extended the analysis, looking at exurban areas from 1950 to 2010. The variation in size among the exurban areas was very large at every census year. For example, in 1950, the smallest area in terms of land area had only 5 square miles; the largest had near 2,000 square miles. The minimum number of housing units was under 700, the maximum was over 400,000.

The exurban areas exploded in size over this period. The mean land area across the 59 areas grew from 233 square miles in 1950 to over 1,700 square miles in 2010. Mean housing units jumped from just under 40,000 to nearly 240,000. The sizes of the exurban areas were closely related to the sizes of their urban areas throughout the period. But the exurban areas grew more rapidly in size than the urban areas.

I then looked at the sizes of the exurban areas relative to their urban areas, the ratios of exurban to urban land areas and housing units. In 1950, the mean ratios were far higher for exurban areas in the Northeast compared with the other three census regions. But by 2010, the relative sizes of exurban areas in the South had grown rapidly and the mean ratios had virtually caught up with the Northeast. The relative size of exurban areas in the West remained very low. Exurban areas in the Midwest ended up in the middle. As I found for 2010, the relative sizes of exurban areas having arid climates were significantly smaller than for other exurban areas throughout the period.

More information on the the growth and evolution of exurban areas from 1950 to 2010 can be found in my paper, “Exurban Areas Around Large Urban Areas in the U.S., 1950–2010,” which can be downloaded here.

Exurban growth and urban growth

When looking at exurban areas in 2010, I also looked at the growth (or decline) of those areas from 2000 to 2010. Just as the size of the exurban areas varied tremendously (see this post), so did the changes over the decade. Change in land area ranged from a decline of 430 square miles to an increase of 1,270 square miles. Numbers of exurban housing units dropped by over 300,000 units in one area and increased by over 150,000 in another.

One question considered was whether the change in the size of an exurban area over the decade was related to the change in the size of the urban area. Two opposing hypotheses were considered. The first and more obvious (at least to me) was that there would be a positive relationship. Metropolitan areas growing more rapidly, with greater growth of their urban areas, might likewise expect to see more growth in their exurban areas. But faster urban growth could have the opposite effect. As an urban area grows, land surrounding the urban area that was previously exurban is converted to urban, reducing the amount of exurban land. If this happens more quickly than land at the periphery is added to the exurban area, exurban growth could be inversely related to urban growth.

I compared the change in exurban land area and housing units over the decade to the change in those same measures for the urban areas, calculating the correlation coefficients. The relationships were not statistically significant. Exurban growth was not related to urban growth. At least during the decade from 2000 to 2010, the two hypothesized effects apparently canceled each other out.

More information on the exurban areas in 2010 and changes from 2000 to 2010 can be found in my paper, “Exurban Areas Around Large Urban Areas in the U.S. in 2010,” which can be downloaded here.

More on the sizes of exurban areas

The post before this one described the tremendous variation in the sizes of exurban areas surrounding large urban areas in the United States. And since exurban area land area and housing units are strongly related to those quantities for the urban area, looking at the ratios of exurban to urban size proved useful for considering the relative sizes of exurban areas.

The relative sizes of the exurban areas as measured by these ratios varied greatly across the census regions. The mean ratios in the Northeast were the largest for both land area and housing units, and exurban areas in the South were nearly as high. Exurban areas in the West were on average the smallest in relative terms, with ratios around one-third of those for the Northeast. The Midwest exurban areas fell in the middle.

This raises the question as to why exurban areas were so much smaller in the West. Ironically, the clue comes from the area having the smallest exurban area in relative terms, Miami-Fort Lauderdale-West Palm Beach. Expansion of the urban and exurban areas is constrained by the Atlantic Ocean to the east and by the Everglades to the west. The urban area has expanded to include most of the available land, drastically limiting the size of the exurban area.

So barriers to urban and exurban expansion can limit the size of exurban areas (note that I am not the first to raise this possibility). Many areas in the West have various barriers to expansion. These include mountains and federal lands not available for development. Also, the arid climates and the dependence on water often brought in from a distance by centralized agencies can mean that water utilities are only expanded at the edge of the existing urban area to serve new urban development.

I subjectively identified those urban and exurban areas for which expansion was significantly constrained by mountains or federal lands. For arid climate, an objective measure could be used, areas that received average annual rainfall of less than 20 inches. Note that the areas facing these barriers were all in the West region with the exception of El Paso. While that area is in Texas, in the South, it is literally as far west in the South region as possible. Comparing the mean exurban size ratios for areas with and without these barriers, the differences were large and highly significant. But the areas with mountains, federal lands, and arid climates overlap to a very high degree, so it is impossible to distinguish their effects.

Looking at the areas with arid climates, the one objective measure, the average ratio of exurban to urban land area for the arid areas was about 40 percent of the ratio for the non-arid areas. For housing units, the arid ratio was about one-third the non-arid ratio. Using simple regression models to predict exurban land areas and housing units yielded these differences: Areas with arid climates had exurban areas that were on average 934 square miles smaller and had 120,000 fewer housing units.

More information on the exurban areas in 2010 can be found in my paper, “Exurban Areas Around Large Urban Areas in the U.S. in 2010,” which can be downloaded here.

Houston’s urban growth and Hurricane Harvey

Hurricane Harvey has dumped massive amounts of water on the Houston area, causing widespread, calamitous flooding. Some context on the tremendous growth of Houston can show how many more people are at risk from hurricanes now than from hurricanes that have periodically struck that general area in the past.

Reports from Houston have described Houston as the fourth largest city in the United States with a population of 2,300,000 (estimated for 2016). As urbanists understand, this will always be just a portion of urban area population. Some reports have mentioned the size of the metropolitan area (MSA), which is much larger. But metropolitan areas include substantial exurban and rural territory outside of the built-up urban area. Certainly the populations included in both areas are at risk from hurricanes. But the built-up urban area is of special significance because urban development exacerbates the risks of flooding due to increased impervious surface and man-made structures that retard runoff. And flooding in urban areas poses different types of risk (not necessarily more severe) than flooding in rural areas.

My focus here is on what I have defined as the Houston urban area in my urban patterns research. This is an effort to delineate the more-or-less built-up portions of the larger metropolitan area for each census year from 1950 to 2010. These urban areas are similar to the census Urbanized Areas that delineate the area and population officially considered to be “urban,” but have the advantage of being defined in a consistent manner over time (along with other research advantages given that they are defined using consistent areal units).These urban areas consist of the contiguous census tracts having at least the urban density of 3 acres per housing unit or 213.33 housing units per square mile. This is generally equivalent to the census threshold of 500 persons per square mile currently used to delineate Urbanized Areas. (For more detail, see this earlier blog post.)

First, I’ll show maps of the extent of the Houston urban area in 1950, 1980, and 2010, showing the tremendous expansion:

Houston urban area 1950

Houston urban area 1950

Houston urban area 1980

Houston urban area 1980

Houston urban area 2010

Houston urban area 2010

Now for some basic statistics describing the urban areas at these three points in time. This table shows the land area of the urban area, the number of housing units, and the housing unit density for each census year:

Year      Area (sq mi) Housing Units         Density
1950 167 157,687 945
1980 726 874,881 1,206
2010 1,765 1,882,352 1,066

The Houston area grew 10 times larger in terms of land area, becoming the ninth largest urban area in 2010. Surprisingly it had the fourteenth largest land area in 1950 despite its much lower ranking in terms of housing units because many other urban areas were far denser at that time. This demonstrates the vast increase in the land area that could be vulnerable to urban flooding now as opposed to the middle of the last century. And the much greater impervious surface area and amount of structures that would contribute to greater flooding.

Housing units increased by 12 times. Houston was tenth among the large urban areas in 2010 in terms of housing units while it was only twentieth in 1950. So again, a huge jump in the number of dwellings, households, and people at risk in urban Houston. And from 1980 to 2010, the population of the Houston urban area increased from 2,100,000 to almost 4,900,000. (Due to the nature of the data sources used, I do not have population figures for 1950.)

The housing unit density did not change a great deal over the sixty-year period, increasing somewhat from 1950 to 1980 and then dropping back a bit to 1,066 housing units per square mile in 2010. This density was almost exactly the mean housing unit density for the 59 large urban areas in my study in 2010, which was 1,080 (and was even closer to the median value of 1,055). This is in contrast to Houston’s density in 1950, which was forty-second out of 59 with a value of 945 units per square mile compared to the mean for all areas of 1,268. Many other areas saw decreases in density over this period; Houston, with a modest increase, did not. (See this post and this paper for more on changes in density for large urban areas over this period.)