Conceptual framework

The literature on regional and urban economics is, in one way or another, about where people and economic activity locate and how this changes over time. In this paper we revisit this issue using a newly available, highly dis-aggregated dataset on buildup in a West African country. While this dataset has considerable breadth in terms of coverage and geographical detail, it also lacks many variables that have received attention in the literature—such as population, jobs, sectors of activity, production, and infrastructure. This forces us to adopt a reduced-form approach that emphasizes breadth instead of depth.

To begin, we regard the presence of buildings in a cell as indicating the presence of population and, hence, of economic activity nearby. The denser buildup is in a cell, the more people live in that cell, deriving a livelihood from work in the vicinity. This starting point is unproblematic given the level of generality of our analysis. Our data contain lots of cells that are only partially built. Hence, in order to make full use of our data, we need to put together a conceptual framework on partially built-up cells that is inspired from the literature while remaining sufficiently sparse to match the limited depth of our data. This conceptual framework also must include all cells – not just metropolitan areas, which often are the sole focus of spatially dis-aggregated empirical analysis [17, 18, 21, 48–50].

The lowest level of building density—e.g., a lone building surrounded by unbuilt cells—is probably a farm. Ghana is a large agricultural country with a relatively undifferentiated geography, but a strong rainfall gradient running from North West (semi-arid) to South (humid). More details are provided in Appendix C in S1 Appendix. Since humid areas can support a higher population density than dry ones, we expect a higher frequency of farms in the South. More generally, agricultural potential varies with agro-ecological features and so does agricultural population density. What is less clear is whether, in denser areas, farms cluster in villages—in which case buildup is concentrated in a few cells – or whether they disperse across the land—in which case there are many cells with a single homestead on them. Which pattern best describes Ghana’s rural population is not entirely clear—some factors militate in favor of clustering (e.g., for defense purposes during the slave trade); others in favor of dispersion (e.g., long fallows). Data on partially built-up cells provides an easy way of characterizing this distribution.

Population growth has also fostered a process of agricultural expansion whereby land previously un-farmed gets newly settled. This process is common to much of sub-Saharan Africa, and was still underway during our study period. We therefore expect that, over time, new farms spring up in previously empty cells. What is unclear is whether these new farms form new clusters, or whether they spread out more or less evenly over the landscape as time passes. The answer to this question is of obvious interest to policy markets interested in rural infrastructure and agricultural marketing: it is a lot easier to provide agricultural and public services to a population that is relatively concentrated. To cast light on this question, we examine how the distribution of partially built-up cells evolves over time: do we observe a gradual dispersion of buildings across the land, or a gradual concentration. We also investigate what is the long-term trend in terms of building (and thus population) dispersion across the country. This is done by constructing the transition matrix of built-up areas and using it to predict buildup distributions by the middle and the end of this century. This research question is related to well-studied processes in developed countries, such as rural flight and and its relative converse, urban sprawl [51].

A closely related issue is the location of rural market towns. As rural population increases, a demand for specialized services arises (e.g., retail shops, hair salon, bicycle repair) that favors the emergence of rural towns [28]. Rural settlements can also be jump-started by investments in agricultural infrastructure and production [52] or mining [53]. Sub-Saharan Africa—as in other LDCs—has experienced a rapid rise in the growth of rural non-farm enterprises, often in the form of small shops and artisans [41]. This process of rural small-scale urbanization—and the accompanying distribution of human settlement across space—was already the focus of von Thunen’s work [31], and has since attracted the attention of numerous others [26, 27, 32, 33]. The common thread running through this literature is the concept of rural hinterland: each market town serves a surrounding area, the diameter of which depends on transportation costs. For instance, if farmers will only walk two hours to sell their crops and purchase retail goods, the diameter of the hinterland of a market town is two hours walking time. In this case, we expect the landscape to be characterized by small market towns surrounded by farms scattered over a radius of a few kilometers at most. We show in Fig A1 in S1 Appendix what this pattern of settlement may look like, together with a spatial covariogram that shows, for each pairwise distance between cells, the proportion of pairs made of two built-up cells. We note that, at close distances, there is an excess of built-up pairs. This captures the strength of agglomeration forces: buildings are located in and around small local hubs. At longer distances, there is no systematic pattern because settlements are located a random distances from each other.

If population density is low, rural hubs can locate at will—e.g., in locations that have good access to arable land and drinking water. When density gets high enough, however, the hinterland of these rural hubs start to overlap. This generates competition between them at the margin, e.g., to attract farmers selling their crops and buying consumer items. Provided that rural consumers enjoy a diverse offer of goods and services in a single location, rural market towns – like shopping malls—cannot locate immediately next to each other. This results in a regular spacing of rural towns, either on a plane (in which case hinterlands form a honeycomb shape with market towns at the center—[33]) or along a road [16, 38] or railway line [45]. The hinterland concept is illustrated in Fig A2 in S1 Appendix where we show, on the left, a simulated example with an equal-spacing distribution of rural hubs. There is some limited evidence that rural towns are fairly equally spaced in Africa [30] but this theory has never been tested at the level of an entire country. Regular spacing of small market towns on a plane would show up in a spatial covariogram as a spike at regular intervals, as shown in the right-hand graph of Fig A2 in S1 Appendix. Regular spacing along roads would show the same signature when limiting the attention to cells along roads. We test these ideas in the empirical part of the paper. We do not expect this regular spacing to hold when rural population density is too low.

In our data, small rural market towns may occupy one or two cells at most. What of larger settlements, what distribution of buildup cells can we expect to observe? While there is a lot of literature on the growth of large cities (see [3] for a review), less attention has been devoted to the growth of urban settlements of small and middle sizes, especially on the African continent. In particular, little is known regarding the expansion of partially built-up cells during rapid urbanization: do towns and cities grow as densely packed entities; or do they spawn a large hinterland of peri-urban agriculture [54] or urban commuters [55]? To articulate these ideas formally, we develop a toy model of cities as concentric circles: at the middle of the city, all cells are fully built according to our data definition (i.e., cells contains buildings, roads, sidewalks, parking lots, and the like). This center is then surrounded by a sequence of donuts of different sizes characterized by a decreasing density of buildup (see Fig A3 in S1 Appendix, top left). We think of falling buildup density as resulting from a varying mix of sub-urban residences for commuters—which can be densely packed—and those of farmers who provide urban consumers with vegetables, fruits, and dairy products and need land to farm. As distance to the center grows, fewer commuters move in and more land is left for farming. This is obviously a simplification, but it is sufficient for our purpose.

How do the donuts change as the town grows? Regarding farmers, we can imagine that demand for peri-urban crops increases more or less proportionally with population and thus with the size of the town center. Similarly, we can expect the number of town workers to increase more or less proportionally with town size—and thus the number of aspiring sub-urban commuters. This implies that if the town center – made of fully built-up cells—expands by a factor of k, so does the area covered by the partially built-up donuts that surrounds it. To see this, let C denote the fully built-up area at the town center. We have C = πr² where r is the radius of the center. Let R denote the outer radius of a partially built donut that surrounds the town center. In the discussion here we imagine a single donut, but the same reasoning applies to multiple donuts of varying density.

Imagine that the area of the center grows by a factor k. By how much must R increase for the area of the donut to also increase by k? Since the area of the donut is πR² − πr², the answer is k. We want the ratio of the donut area to the center area to remain constant, i.e., we want = constant when r is set to kr. It is immediately clear that this requires that R be multiplied by the same k. This means that, in this world, the proportion of partially built-up land around towns and cities increases at the same rate as the fully built-up land. This prediction is testable with the data we have. If we combine this prediction with the rise of partially built-up cells due to agriculture and small market towns that we discussed earlier, we expect the fraction of total land that is partially built-up across the country to increase at the same rate or faster than the fraction of fully built-up cells. One specific version of this prediction is Zipf’s Law applied to the size of human settlements. The size distribution of cities is said to follow Zipf’s Law if we observe a linear relationship between the log of city size and the log of the rank of a city in terms of size [34]. Applied to buildup in cells, Zipf’s Law would imply a linear relationship between the log of buildup proportion and the log of the rank of the cell in terms of buildup. If Zipf’s Law were to hold for buildup, it would also imply that the frequency distribution of buildup across cells would remain constant as urbanization takes place—which in our toy model could only arise if city growth implies proportional growth in the size of the hinterland where partially built-up cells are located. The reverse is, however, not true: the frequency distribution of buildup could be time-invariant but not follow a power law. We test this hypothesis as well.

So far we have assumed that the area traveled by commuters does not matter. As the town grows, however, suburbs move further out, and sub-urban commuters must travel a longer distance. Presumably, as R keeps increasing, the length and duration of the commute increase in roughly the same proportion (if not more, e.g., due to increased congestion in the center). Without investment in rapid transit infrastructure for commuters, this is likely to reduce the appeal of suburban living, convincing some would-be commuters to shift to high density residential areas closer to the town center. In this case, the radius of the outer donut grows less and the radius of the center grows more for a given population increase. In the extreme case where R represents an absolute limit on commutable distance, urban growth is characterized by an increase in r while R remains unchanged. The town grows up to the point at which r = R and the entire area is fully built-up, at which point it stops: the city has reached the limit set by the available transport infrastructure and technology [19, 26, 27]. A similar prediction holds even if some rapid transit infrastructure is introduced, e.g., in large cities, but it is insufficient to allow R to increase as the same speed as r: the city still densifies, even if at a slower rate. These cases are illustrated in Fig A3 in S1 Appendix. Here the prediction is that the ratio of partially built-up urban cells over fully built-up cells falls as the proportion of fully built-up cells grows. This is because cities densify as they grow (i.e., the ratio of falls as r increases). This is illustrated in Fig A4 in S1 Appendix, which shows the simulated frequency distribution of partially and fully built-up cells when a town grows as shown in Fig A3 in S1 Appendix. Initially there is a large fraction of partially built-up cells, especially low buildup density cells. As the town grows, the proportion of these cells falls while the proportion of fully built-up cells rises. But the proportion of built-up cells of intermediate density does not change much. In contrast, if a city’s hinterland expands proportionally to the city center, the frequency distribution of partially and fully built-up cells remains unchanged—which implies that the proportion of sparsely built-up cells rises at par with fully built-up cells.

In the paper, we look at the long-term evolution of the frequency distribution of partially and fully built-up cell to throw light on whether towns and cities crowd out their hinterland as they grow, or whether their hinterland grows with them. We nonetheless recognize that a fall in the ratio of sparsely built-up cells to fully built-up cells due to crowding-out may be partially offset by growth in the number of rural market towns. Here too, however, the same logic applies for individual towns if commuting time is a constraint: as small settlements grow, they absorb their hinterland, and the fraction of partially built-up land falls there as well. Only the creation of a sufficiently large number of new low-density villages and rural market towns can compensate this evolution. Whether this took place in Ghana over our study period—and is likely to continue over the foreseeable future—is thus an empirical question.

The visual representation of the concept of hinterland used so far is spherical, which essentially assumes that travel time is the same in all directions. This may be a reasonable assumption for travel by foot or bicycle on the relatively flat rural terrain that characterizes much of Ghana. It does, however, fail to recognize the role played by roads in facilitating transport and commuting. Conceptually, roads can be thought of as reshaping space to make certain locations closer to each other (in terms of travel time or cost) than they were without it. We mentioned this idea earlier when discussing workers commuting from sub-urban areas: roads—and transit systems more generally – help shape cities [19]. Roads can also shape the locations of rural market towns and even farmers themselves, e.g., because they allow inter-city trade [56]. We take this into account in our empirical analysis by examining the long-term effect that past roads may have on the location of human settlements, while simultaneously examining agglomeration effects.

Data

Our main variable of interest y_it measures the proportion of a cell i in year t that is covered by buildings, as inferred from a high-definition satellite image of Ghana. This variable comes from the Global Human Settlement Layer (GHSL) data that have been put together by the Joint Centre of the European Commission. A Mollweide projection was used to divide the earth’s surface into square cells of approximately 250 by 250 meters, which is rather small for an analysis of built-up area. The proportion of buildings on each cell was predicted using a machine learning algorithm trained on observations of actual buildings made on the ground. Pesaresi et al. [25] describes the machine learning algorithm in detail. This algorithm predicts buildup in squares of 38x38 meters. These predictions are then aggregated using GIS into 250x250m cells, thereby generating an accurate prediction of the proportion of the cell that has buildings on it. Using code written in R, we overlay the GHSL data on a GIS map of Ghana and extract the cells that lie within the country’s borders. A little over 3.8 million cells are needed to cover the entire area of Ghana. Additional information about the data is provided in Appendix D in S1 Appendix.

The GHSL dataset provides information on built-up cells for four separate years: 1975, 1990, 2000, and 2014. Cells for each of these years have been carefully aligned using detailed GPS coordinates so as to form a panel dataset for y_it. Aligning satellite photographs with each other and over time was achieved projecting each photograph using the same Mollweide projection. Variable y_it takes values between 0 and 1, with 0 meaning no building detected in the cell, and 1 meaning the cell is entirely occupied by buildings—i.e., as in a city center. More precisely, y_it = 1 when all its area is covered by impervious surfaces, such as roofs, sidewalks, streets, and courtyards. As shown in Table 1, most cells (98.5%) in 1975 are predicted to be unbuilt, which is hardly surprising given that Ghana at the time only had a national population density of 43.9 per square km. Built-up area grows over time as population increases to 119.4 per sqkm in 2014 and Ghana becomes more urbanized (urban population share increasing from 30% in 1975 to 53% in 2014). In 2014, 93.8% of cells were unbuilt.

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Table 1. Built-up area.

https://doi.org/10.1371/journal.pone.0250847.t001

For each cell i in year t we construct two measures of buildup in the cells surrounding i. The first of these measures is denoted as and is the average built-up area in the cells immediately adjacent to i. Since the data is divided into a square grid, each cell on that grid in principle has 8 immediate neighbors, which constitute the set n₁. We thus have: (1) where by an abuse of notation n₁ represents both the set of immediate neighbors and their number. Together with y_it, this variable captures the proportion of built-up area in an approximate cell 675 meters wide by 825 meters high (approximately 2025 by 2475 feet). Cells located on the border or along the sea or large body of water do not have a full complement of 8 adjacent cells located wholly or partly within Ghana. For these cells, the immediate neighborhood is limited to those cells included in the dataset and the average is adjusted accordingly. We also construct a variable that captures the average buildup in cells immediately adjacent to . There are in principle 16 cells in set n₂, except in the vicinity of borders and large bodies of water where that number gets truncated. The formula for is that same as Eq (1) but with n₁ replaced by n₂. Together, y_it, and cover an area 1125 meters wide and 1375 meters wide, roughly 1.55 square kilometers. These are generated by using spatial lag functions in R.

We combine data on built-up area with information on agro-ecological zones to capture the wide variety of landscapes and agricultural potential that characterize Ghana. The data for agro-ecological zones are downloaded from the GAEZ-Global Agro-Ecological Zones data portal that was developed by the Land and Water Division of the Natural Resources Management and Environment Department of FAO. The resolution for GAEZ dataset is 10 km. We overlay the 250 by 250 meter grid of the GHSL dataset over the GAEZ dataset to extract soil types for each cell. We divide this information into two variables, which do not vary over time. The first of these variables characterizes the humidity of a particular region, i.e., whether moist/dry, semi-humid, or humid. In Ghana as in the rest of West Africa, the rains follows a monsoon pattern and rainfall generally decreases along a North-South gradient, with the North being drier and the South being more humid. There are exceptions to this general pattern, however—the immediate coastal area, for instance, tends to be drier than inland. In the data, humid soils are associated with the entire valley of the Volta river and with the forested region North of Takoradi.

The second agro-ecological variable we construct captures soil quality – going from poor to moderately good to good. There are two additional categories: hydromorphic soil, which is characterized by excess moisture which tends to suppress aerobic factors in soil-building; and soils unsuitable for buildings or agriculture, such as steep soils or soils covered by standing water at least part of the year—these soils require special investments (e.g., terracing, drainage) in order to become usable. Given that Ghana is relatively flat with no mountains and a relatively homogenous physical landscape, we abstract from altitude: whatever variation in altitude there may be, it translates into variation in rainfall and soil quality and is already captured.

To this data we add longitude and latitude information of cells from the GHSL dataset expressed in decimal degrees, which we normalize to have mean zero and unit variance. A higher value of longitude means more to the West; a higher value of latitude means more to the North. Population density in Ghana is in general higher in the humid South than in the drier North, largely due to the higher agricultural potential and better soils in the South. But there is a lot of local variation, something we aim to capture with our set of agro-ecological and geographical variables.

We also have information about the presence on a cell of a land border, sea border, or large body of water from the GIS shape files of Ghana. We drop from the y_it database of all the cells outside Ghana, as well as those that are entirely covered by water, since they cannot be built on. This includes lake Volta, a large artificial lake that covers 8,502 square kilometers within Ghana. Some cells are partially outside Ghana, or partially covered in water or by the sea, something we capture with dummies. We also construct a dummy equal to one if there is a large body of water on any of the eight adjacent cells n₁, and another one for the next 16 adjacent cells n₂. This is done to investigate whether proximity to a lake or lagoon attracts human settlement, e.g., if the lake is use to fish or transport goods. The same thing is done for proximity to the sea and to a land border, for similar reasons.

The last piece of information that we integrate into our analysis is road data for 1976 and 1986, at the beginning of our period of analysis. We digitized road maps from the Ghana population censuses. These maps are then used to extract information on the presence and/or absence of roads in GHSL cells. These roads are classified into four categories, from 1 to 4. There is also a class 5 which is labeled ‘unknown road’. Out of an abundance of caution, we ignored these roads in our analysis. Class 1 roads are the highest quality, and class 4 roads the poorest. Typically, a class 1 road is paved and hence usable all year round. In contrast, a class 4 is a dirt track and many of these roads are seasonal. Class 2 and 3 occupy an intermediate position. Road availability in 1976 is of particular interest for our purpose because these roads predate our period of analysis and thus cannot be construed as caused by buildings constructed after 1976. The same can be said of 1986 roads regarding new buildings constructed afterwards. Using the digitized maps, for each cell we construct a dummy r_kit equal to 1 if there is a road of class k in cell i in year t = 1976 or t = 1986 and 0 otherwise. This is done by overlaying road maps on the GHSL cells defined earlier. Next we calculate the proportion of immediately adjacent cells that contain a road of each particular class, which we denote . Mirroring what we did for built-up area, we also calculate the proportion of roads in the n₂ neighborhood of i, which we denote .

Empirical methods

Three types of analyses, described below, are used to shed lights on the evolution of buildup areas in Ghana.

Descriptive statistics

We start by showing in Tables 1 to 3 key descriptive statistics for our variables of interest. From Table 1 we see that proportion of empty cells (y_it = 0) fell 98.5% to 93.8% between 1975 and 2014. Averages are essentially identical for y_it, and , which is hardly surprising since they average essentially the same cells. The proportion of fully built-up cells (y_it = 1) among built-up cells increased a lot between 1975 and 1990—from 2% to 6.6%, but does not show any trend after that. We revisit this point more in detail below. We check whether Zipf’s Law applies to built-up area at this micro level. To do this, we sort partially built-up cells by the proportion of cell area that is built up—from largest to lowest. We then plot the log of the rank to the log of the built-up area. Here cells with a built-up proportion close to 1 represent ‘cities’ while those with a built-up proportion close to 0 represent the smallest possible town. Results are shown in Fig B1 (in levels) and B2 (in logs) in S1 Appendix. If Zipf’s law holds, then we expect a downward sloping straight line in Fig B2 in S1 Appendix. As clear from the figures, we find absolutely no evidence of a Zipf’s Law for partial buildup similar to what has been discussed for cities [34, 35].

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Table 2. Presence of roads in a cell.

https://doi.org/10.1371/journal.pone.0250847.t002

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Table 3. Time-invariant features of cells.

https://doi.org/10.1371/journal.pone.0250847.t003

In Maps 1a to 1d (Fig 1), we illustrate what the data looks like for a randomly chosen location in Ghana—the rural town of Yeji. Buildup is color coded, with white meaning no evidence of any building in the cell and black meaning high density buildup—i.e., 90% and above. Intermediate shades of blue and grey correspond to different levels of partial buildup. We see that in 1975 there is very little evidence of buildup – most of Map 1a is empty. But the small settlement of Yeji can be seen in the upper middle of the map. Over time, this initial settlement grows and expands geographically. It also spawns small satellite settlements at more or less random but fairly equally spaced intervals. New settlements also appear farther away from the initial settlement, often in the proximity of a road. But the initial settlement remains the most densely built-up area of the map throughout the 40 years period covered by the data, suggesting some form of preferential attachment/first-mover advantage. If we enter the geographical coordinates of Yeji (8° 05’56.4”N 0° 54’10.8”W) in Google Earth, it is immediately clear that the data contain a wealth of detail about buildup that comes to view under magnification. It also reveals the presence of the Volta lake nearby. Some of the roads shown in Google Earth correspond to 1975 roads. However, one road running to the west from Yeji seems no longer in use, even though its outline is still visible from the air. Similar map sequences for other parts of Ghana show a similar evolution, at least visually (Maps 2a to d (Fig 2) show the evolution of buildup in the coastal town of Takoradi). The rest of the paper is devoted to analyzing these patterns in a formal way.

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Fig 1. Map 1.

Buildup of Yeji over time.

https://doi.org/10.1371/journal.pone.0250847.g001

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Fig 2. Map 2.

Buildup of Takoradi over time.

https://doi.org/10.1371/journal.pone.0250847.g002

Road information is presented in Table 2. For 1976 and 1986, the road information we have captures road quality, divided into four categories, plus a small ‘quality unknown’ category. At that time, most of the roads included in this list were unpaved. According to these data, the proportion of cells with at least one road was 3% in 1976 and 3.4% in 1986. The 2008 data use a more stringent road definition, according to which 1.8% of cells had at least one road. The 2014 data includes two new categories: highways, which at the time only include on short stretch of motorway/freeway between Accra and a nearby town; and local roads. The meaning of the other three categories also seems to have changed between 2008 and 2014, precluding a direct comparison. In the analysis we rely heavily on the 1976 data because it captures the predetermined road infrastructure at the beginning of our study period. Averages for road availability in the 8 n₁ and 16 n₂ neighboring cells are not shown to save space but, by construction, they are identical to the cell averages reported here.

In Table 3 we summarize our main time-invariant variables. Longitude and latitude are presented in decimal degrees. They are used to measure the distance between cells in the subsequent analysis. A tiny fraction of cells are on the border with a neighboring country or on the seashore, but 2.7% are partially covered by a water body such as Lake Volta. The predominant soil quality is either poor or moderate, a situation that typifies much of Sub-Saharan Africa and reflects its tropical nature: heavy rainfall combined with high evapo-transpiration means considerable leaching of nutrients from the soil. Only 7.4% of soils are categorized as good. There is a small fraction of cells with hydroponic soils, which means they are located in river estuaries or flood plains. Some of these soils can be made fertile, but often at some cost in terms of infrastructure and equipment. Some 5% of cells have land unsuited for traditional agriculture either because they are waterlogged or too steep. Cultivating these soils requires investment in drainage or terracing. In terms of humidity, most of Ghana’s land is classified as sub-humid, with some areas being either drier or more humid. Rainfall typically follows a North-South gradient in western Africa, with more rains along the coast than in the North. There are exceptions, however: coastal areas of Ghana tend to receive less rainfall than those located more inland. The humidity of the soil also depends on its texture. In Ghana, soils in the valley of the Volta river are classified as humid as they tend to retain moisture better.

In Table 4 we break down y_it by agro-ecological zone and proximity to borders and water. We find strong initial differences in buildup density across zones with, in 1975, more buildings (and hence a higher population density) in semi-humid areas and in areas with humid or hydromorphic soils. Over the study period, however, we observe dramatic changes in relative buildup densities: cells with initially the lowest buildup densities—i.e., areas with soils that are humid, hydromorphic, or waterlogged or steep—experience the fastest growth, possibly suggesting that investments were successful in making these soils or locations more productive. Reduction in onchocercosis—also known as river blindness—may have also attracted settlement in these areas.

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Table 4. Breakdown of built-up proportion by agro-ecological zones.

https://doi.org/10.1371/journal.pone.0250847.t004

Table 5 presents a similar breakdown of built-up areas by the presence or not of a road in the cell. Not surprisingly, we find a strong difference in buildup density between cells with and without a road. This difference falls over time however: comparing cells with and without a road in 1976, the difference falls from 8.5 more buildup in 1975 to 6.3 times in 2014. We also note that the new road definitions adopted in the 2014 data match heavily built-up areas much more closely than before, suggesting that this new road nomenclature puts more emphasis on residential streets [57] and lesson dirt roads serving rural communities.

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Table 5. Breakdown of built-up proportion by presence of a road.

https://doi.org/10.1371/journal.pone.0250847.t005

In Table 6 we show buildup proportions in ‘edge cells’, that is, cells that overlap with a land border or a water body. We see that the presence of a water body is in general associated with a much lower buildup density. This may be because the largest inner body of water in Ghana is a dam lake and hence probably has a water level that varies over time depending on annual rainfall: people don’t want to live right next to it. Land borders also have a lower density than no-edge cells, while cells near the sea have a much higher buildup density, probably reflecting the presence of towns and cities along the coast, but also of fishing communities that launch their fishing boats directly from the beach. We also observe a much faster buildup over time in cells near inner water bodies, probably reflecting the fact that Lake Volta was only completed in 1965 and it took time for people to relocate along its shore.

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Table 6. Breakdown of built-up proportion by boundary status.

https://doi.org/10.1371/journal.pone.0250847.t006

Buildup frequency distribution

Next, we look in more detail at the frequency distribution of buildup for cells that are neither empty of buildings or fully occupied by them. Fig 3 shows this distribution for the four data years we have. It is immediately clear that the frequency distribution of partially built cells remains constant over time, even if the proportion of unbuilt cells falls while the proportion of fully built-up cells increases. We then break down this distribution by cells that are located near built-up areas and cells that are not, i.e, by whether or = 0. Unsurprisingly perhaps, we find that isolated built-up cells only have a small fraction of built-up area: half of them are less than 4% built up, which is equivalent to having at most a 50m² (550 square feet) structure on them; three quarters of them have structures covering at most 83m² or 915 square feet. In contrast, half of built-up cells in the vicinity of other built-up cells have structures covering 106m² or more—with a fairly even distribution of y_it across the entire spectrum of built-up proportion, a finding in line with the idea that these cells correspond to the edges of towns and cities. This pattern suggests that population growth at the aggregate level is not associated with a densification of all areas simultaneously. Rather, what we observe is a fall in the proportion of empty cells, an increase in the proportion of densely built cells, and a more or less constant distribution of partially built cells.

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Fig 3. After dropping unbuilt cells (which constitute more than 90% of cells in all years) and the fully built-up cells (which are discussed separately in the text), Fig 3 plots four histograms of the frequency distribution of partially built-up cells by their built-up proportion.

The horizontal axis shows the built-up proportion of the cell. The vertical axis shows the proportion of cells in each buildup percentage interval. There is one histogram for each year. To facilitate viewing, each of the four histograms is represented as a colored line and the four histograms are overlaid on top of each other. This is done to facilitate a visual comparison of the frequency distribution across years.

https://doi.org/10.1371/journal.pone.0250847.g003

To investigate this issue further, we plot the forecasts based on the transition matrix M in Fig 4, together with the actual built-up proportions in 1975 and 2014. The histogram confirms that the evolution of built-up proportions in Ghana is characterized by a transfer from unbuilt to fully built, with partially built-up cells serving a transitory role. Over the foreseeable future, the calculations forecast 89.5% unbuilt cells circa 2050 and 85% unbuilt cells circa 2100. At the same time, the proportion of densely built areas (90% built-up or more) is forecasted to go from 0.9% in 2014 to 3% in 2053 and 6.2% in 2092. No equivalent increase is predicted for partially built cells. In other words, the additional population will not spread itself evenly on the land. Rather, it will congregate in dense towns and cities. Assuming that the historical rate of population growth continues, this represents more than a tripling of dense urban areas in about 40 years, and a multiplication by 7 by the end of the century. Similar forecast are obtained if we construct the transition matrix using only 1976 and 1986, or 2000 and 2014: the resulting transition matrices are very similar, and the forecasts as well. Although our forecasts apply to buildings only, they imply a dramatic change in the distribution of population across the country, with an increased concentration in towns and cities without much of a densification of rural areas. To illustrate the magnitude of these changes, an equivalent increase in the population of Accra would imply an increase from 2.2 million in 2014 to 7.3 million people by 2053 and more than 15 million by the end of the century. These are not small changes.

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Fig 4. Fig 4 shows a stacked histogram of the frequency distribution of built-up proportion divided into deciles of buildup proportion in 1975.

Fully built cells are included in the top decile. Since an overwhelming proportion of cells are unbuilt in all years, they are omitted from the Figure for the sake of clarity. The 1975 and 2014 values are taken directly from the data. The 2053 and 2092 forecasts are obtained as follows. We first form the decile transition matrix using data from 1975 and 2014 (a 39-years interval). We then project the built-up frequency distribution from 2014 to 2053 and 2092 as two iterations of the transition matrix. The resulting forecasts should be understood as predictions of what the distribution of buildup will look like if the transition matrix remains unchanged. Similar forecasts are obtained if we use transition matrices constructed using different data years. The forecasts do not specify which cells will be built.

https://doi.org/10.1371/journal.pone.0250847.g004

Roads

In Fig 5a and 5b, we repeat the exercise separately for cells that were close to a road in 1976 and those that were not. We define being close to a road as having at least one of the five categories of roads documented in 1976 either in the cell itself, or in its neighborhood n₁ or n₂. With this definition, 14.24% of cells at most 1.5km from a road in 1976. The year 1976 is chosen because it is close to the starting point of our data. Hence the transition matrices that are constructed on that basis can be seen as predictions based on road conditions existing in 1976. Results show a stark contrast between cells near road or not. Cells near a 1976 road were already less likely to be unbuilt in 1975: 93.7% vs. 99.3% for cells not near a road. This gap increased to 81% vs. 96% in 2014 and is predicted to grow to 61% vs. 90% by the end of the century. Furthermore, the distribution of built-up areas also varies greatly depending on the presence of a road in 1976. In 1975, only 0.05% of cells not near a road had a value of y_it in excess of 90%, compared to 0.73% of those near a road. By 2014 the difference was 0.46% vs. 3.71%. By the end of the century, the forecasted difference is 4% vs. 19%. Put differently, by the end of the century nearly 19% of the area within 1.5km of a 1976 road will be nearly fully built up. Of course new roads will be built that can affect that, a point we revisit below. But the main message here is that the early presence of a road is predicted to have long lasting effects on urbanization patterns. Fig 3a and 3b also show that while proximity to a road affects the probability of subsequent buildup, it does not markedly change the distribution of partially built-up cells: while the levels are different between the two figures, the shape of the distribution is similar. This is true even though each transition matrix is estimated from two distinct sets of observations.

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Fig 5. The Figures show a stacked histogram of the frequency distribution of built-up proportions divided into 1975 deciles.

In Fig 5a only those cells with a road in 1976 are used in the histograms. Fig 5b does the same thing for cells without a road in 1976. In both Figures, fully built cells are included in the top decile. Since an overwhelming proportion of cells are unbuilt in all years, they are omitted from the Figures for the sake of clarity. The 1975 and 2014 values are taken directly from the data. The 2053 and 2092 forecasts are obtained as follows. We first form the decile transition matrix using data from 1975 and 2014, only using cells with a road in 1976 (for Fig 5a) or those without a road in 1976 (for Fig 5b). We then separately project the built-up frequency distribution from 2014 to 2053 and 2092 as two iterations of the transition matrix. The resulting forecasts should be understood as predictions of what the distribution of buildup will look like in 1976 if the transition matrices remain unchanged. Similar forecasts are obtained if we use transition matrices constructed using different data years. The forecast does not specify which cells will be built.

https://doi.org/10.1371/journal.pone.0250847.g005

Next, we expand on the above analysis to investigate the effect of adding or removing a nearby road. We use the data from 1976 and 1986 since they are comparable. We define four categories: cells not close to a road in either 1976 or 1986; cells close to a 1976 road that no longer existed in 1986; cells close to a road that did not exist in 1976 but was present in 1986; and cells close to a road that existed both in 1976 and 1986. Abandoning a road may signal that the road was no longer economically useful which, if true, should predict less growth and less subsequent buildup. In contrast roads that remained in existence across both years should be correlated with a pre-existing and persistent economic usefulness. Hence, we expect them to be most associated with economic activity and thus most likely to attract population and buildup. Roads that were added in 1986 should occupy an intermediate position: they were not initially deemed useful but were added later, perhaps when sufficient funds became available to add less essential roads. For the analysis, we construct four transition matrices from the 1975–2014 data on these four categories of cells. We then predict buildup in 2053 and 2092 as before. To save space, we present in Fig 6a and 6b only the results for unbuilt and fully built cells, respectively. Indeed, as noted earlier, the actual and forecasted distributions of partially build cells are fairly stable across time. We see that, as anticipated, cells near a road are predicted to receive a lot more buildup over time. The buildup forecast is particular strong for locations that had a road in 1986. Whether or not they also had a road in 1976 matters less, contrary to what we expected. Cells near an abandoned road do less well, but still better than cells with no road in either years.

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Fig 6. Fig 6a shows the percent of unbuilt cells in various years, conditional on whether the cell had a road in 1976 (r76 = 1) and a road in 1986 (r86 = 1).

Some cells with a road in 1976 no longer have one in 1986 (e.g., because the road was not maintained or was moved). Fig 6b shows the percent of fully built cells in various years, also depending on whether the cell had a road in 1976 (r76 = 1) and a road in 1986 (r86 = 1). Values for 1975 and 2014 are take from the data. Values for 2053 and 2092 are forecasts obtained in the same manner as for Fig 5a and 5b, except that we now use four different transition matrices and four 2014 buildup vectors from which to iterate.

https://doi.org/10.1371/journal.pone.0250847.g006

Agglomeration

Having discussed roads, we now turn to agglomeration effects. Table 7 reports the results from the estimation of Eq (2). The results, shown in Table 7, are very stable across data years: we note a strong contemporaneous correlation between y_it and but, conditional on , the correlation with is negative. Put differently, if we observe high buildup in n₁ and n₂, we would predict a lower buildup y_it than if we only observed high buildup in n₁ alone. This potentially suggest that buildups on either side of n₁ are substitutes: if happens in n₂, it is less likely to occur in cell i.

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Table 7. Correlation in buildup between neighboring cells.

https://doi.org/10.1371/journal.pone.0250847.t007

Next, we investigate whether cells in a built-up neighborhood subsequently experience more buildup. Results are presented in Table 8 for the six separate regressions corresponding to specification given in Eq (3). Unsurprisingly, they show a lot of persistence in buildup: the estimated coefficient β₁ is large in all regressions, indicating that past buildup is a strong predictor of future buildup. This predictive power, however, falls with the time gap s: the more distant in the past is y_it−s, the smaller the value of β₁, and the smaller is the precision of the prediction, i.e., the R². We also notice that, when the time interval s is short, the additional predictive power of and is relatively small. As the time interval increases, the coefficients—and thus the predictive power—of and increases and, unlike in Table 7, it is positive for both. This is consistent with a process by which much of the diffusion of buildup to neighboring cells occurs over a relatively short time interval of less than 10 to 15 years. Once this diffusion has begun, own past buildup takes over the predictive power of buildup in neighboring cells. This finding is suggestive of some form of hysteresis: once a cell has started to build up, it tends to remain that way, irrespective of neighboring effects.

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Table 8. Correlation in buildup between neighboring cells over time.

https://doi.org/10.1371/journal.pone.0250847.t008

This conjecture about persistence is confirmed when we estimate a nonparametric version of Eq (3) whose specification is given in Eq (4). Coefficient estimates for the buildup variables of interest are presented in graphical form in Fig 7. Results by and large confirm earlier findings from Table 8. They also provide some evidence of a stronger lag effect for higher deciles of y_it−s, suggesting more persistence for high buildup. Interestingly, we see that the lagged effect of buildup in neighboring cells tends to taper off at high buildup values, consistent with some form of substitution effect.

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Fig 7. Fig 7 summarizes the coefficients from three separate regressions.

The dependent variable in the three regressions is the proportion of a cell that is built up in 1990, 2000, and 2014 respectively. Each regression includes lagged values of buildup in the cell and in the neighboring cells n1 and n2. The lagged values are for 1975, 1990, and 2000 respectively. Each of the three lagged buildup variables appears in the regression as a series of 10 dummies, one per buildup decile. The Figure shows the estimated decile coefficients in each of the three regressions for (1) buildup in the cell itself [lines y(1975), y(1990) and y(2000)]; buildup in neighboring cells n1 [lines n1(1975), n1(1990), and n1(2000)]; and buildup in neighboring cells n2 [lines n2(1975), n2(1990), n2(2000)]. Each regression also includes the following time invariant regressors: longitude, latitude, and their product [three variables]; soil humidity and soil quality dummies [six variables]; and dummies for the presence of a water body, a land border, or the sea in the cell, the neighboring cells n1 and the neighboring cells n2, respectively [9 variables]. The number of observations in each regression is just shy of 3.8 million. R-squares are 0.799, 0.845 and 0.796, respectively. All coefficients shown here are significant at the 1% or better.

https://doi.org/10.1371/journal.pone.0250847.g007

Next, we explore whether the agglomeration and persistence patterns found above hold true for different road types. We estimate the regression specification given in Eq (5). We focus our attention on outcome years 1990 and 2000 because for each of them we have comparable road data approximately 14 years earlier, i.e., in 1976 and 1986. Road data for 2008 and 2014 are not comparable to either of these or to each other. Coefficient estimates for γ₁, γ₂ and γ₃ are presented in Fig 8, for each of the four road categories and each of the two outcome years, 1990 and 2000. Coefficient γ₁ gives the predicted % point increase in buildup if there is a road in the cell 14 years before. Coefficients γ₂ and γ₃ give the predicted % point increase in buildup if neighborhoods n₁ and n₂, respectively, go from no cell with a road to all cells with a road. We note a lot of similarity between the two regressions in terms of relative magnitude of the coefficients: the presence of a class 1 road in the cell or nearby predicts more buildup 14 years later. For the other three categories, the presence of a road in the cell predicts slightly higher buildup later on, but the presence of a road in the more distant neighborhood n₂ has a consistently negative effect on subsequent buildup predictions. A similar negative coefficient is observed in some regressions for neighborhood n₁ as well. These findings are again consistent with the existence of a substitution effect in buildup: if the road is in a neighboring cell but not in my cell, my cell is likely to be built-up a decade and a half later—an outcome we would expect if local inhabitants prefer to locate immediately near these smaller roads. When we estimate regression (5) with 2014 buildup y_it and the 2008 road data, we find that all the road coefficients but one are positive. The only exception is for the presence of a primary road in the cell itself, which by 2014 may have too much traffic to be an attractive location. The 2008 road data is completely different from the earlier data, however, with very little correlation between the two, thereby making comparisons problematic. The 2014 road data is strongly correlated with buildup in that year, but since it is contemporary to our last data year, it cannot be regarded as a useful predictor.

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Fig 8. Fig 8 summarizes the coefficients from two separate regressions.

The dependent variable in the two regressions is the proportion of a cell that is built up in 1990 and 2000, respectively. The Figure shows the coefficients of the presence of a road of a certain class (i.e., category) in the cell, the neighborhood N1 and the neighborhood N2. The road quality ordering is: class 1 (top category), class 2, class 3, and track (bottom category). In the 1990 regression, the road data refers to 1976; in the 2000 regression, the road data refers to 1986. Each of the regressions also includes all the regressors that were present in those for Fig 7. More precisely, each regression includes lagged values of buildup in the cell and in the neighboring cells n1 and n2. The lagged values are for 1975 and 1990, respectively. Each of the three lagged buildup variables appears in the regression as a series of 10 dummies, one per buildup decile. Each regression also includes the following time invariant regressors: longitude, latitude, and their product [three variables]; soil humidity and soil quality dummies [six variables]; and dummies for the presence of a water body, a land border, or the sea in the cell, the neighboring cells n1 and the neighboring cells n2, respectively [9 variables]. The number of observations in each regression is just shy of 3.8 million. R-squares are 0.796 and 0.796, respectively. All coefficients shown here are significant at the 1% or better.

https://doi.org/10.1371/journal.pone.0250847.g008

We also estimate a different version of regression (5) setting t − s = 1975 for lagged buildup and t − s = 1976 for the road data. By regressing each of the subsequent data years 1990, 2000 and 2014 on these initial buildup and road conditions, we examine the delayed predictive effect of buildup and roads over time. Fig 9 summarizes the buildup coefficients in a way similar to Fig 6, the main difference being that now all regressions use the same 1975 buildup data as regressors. The Figure shows clearly that, over time, the predictive effect of past buildup changes. For buildup in neighborhoods n₁ and n₂, we see a systematic increase in buildup predictions as time passes. This is consistent with some form or preferential attachment—the presence of buildings in the past attracts new buildings later—possibly due to agglomeration externalities. Past buildup in the cell itself varies depending on the original buildup intensity: for heavily built-up cells in 1975, the own-cell predictive effect falls over time. In contrast, cells with less buildup in 1975 see their buildup predictions increase over time. The fall in predictions for heavily built-up cells may suggest some form of congestion, unless it is compensated by buildup in neighboring cells, which makes it harder to move to a nearby cell.

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Fig 9. Fig 9 summarizes the coefficients from three separate regressions.

The dependent variable in the three regressions is the proportion of a cell that is built up in 1990, 2000, and 2014 respectively. Each regression includes the 1975 values of buildup in the cell and in the neighboring cells n1 and n2. Each of the three 1975 buildup variables appears in the regression as a series of 10 dummies, one per buildup decile. The Figure shows the estimated decile coefficients in each of the three regressions for (1) 1975 buildup in the cell [lines y(1990), y(2000) and y(2014)]; 1975 buildup in neighboring cells n1 [lines n1(1990), n1(2000), and n1(2014)]; and 975 buildup in neighboring cells n2 [lines n2(1990), n2(2000), n2(2014)]. Each regression also includes all the road presence dummies for 1976 (see Fig 10) and the following time invariant regressors: longitude, latitude, and their product [three variables]; soil humidity and soil quality dummies [six variables]; and dummies for the presence of a water body, a land border, or the sea in the cell, the neighboring cells n1 and the neighboring cells n2, respectively [9 variables]. The number of observations in each regression is just shy of 3.8 million. R-squares are 0.799, 0.711 and 0.582, respectively. All coefficients shown here are significant at the 1% or better.

https://doi.org/10.1371/journal.pone.0250847.g009

Fig 10 does the same thing for coefficient estimates for γ₁, γ₂ and γ₃. The interpretation of the coefficients is similar to that in Fig 8, except that we are now comparing the evolution of buildup predictions over time, using roads in 1976 as predictor. We immediately see that the predicted effect of class 1 roads increases massively over time both for the presence of a road in the cell, as well as for road density in neighboring cells. The other three road categories produced similar results: we observe an increasing positive effect of roads in the cell and in the immediate neighborhood n₁ on buildup predictions, while road presence in neighborhood n₂ has a negative effect that is initially stronger and falls later on. This is consistent with an initial substitution effect that starts to reverse itself as population density increases and moving to a neighboring cell to get near the road becomes harder.

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Fig 10. Fig 10 summarizes other coefficients from three separate regressions presented in Fig 9.

The dependent variable in the three regressions is the proportion of a cell that is built up in 1990, 2000, and 2014 respectively. The Figure shows the coefficients of the presence in 1976 of a road of a certain class in the cell, the neighborhood N1 and the neighborhood N2. The road quality ordering is: class 1 (top category), class 2, class 3, and track (bottom category). Each regression includes the 1975 values of buildup in the cell and in the neighboring cells n1 and n2. Each of the three 1975 buildup variables appears in the regression as a series of 10 dummies, one per buildup decile (see Fig 9). Each regression also includes the following time invariant regressors: longitude, latitude, and their product [three variables]; soil humidity and soil quality dummies [six variables]; and dummies for the presence of a water body, a land border, or the sea in the cell, the neighboring cells n1 and the neighboring cells n2, respectively [9 variables]. The number of observations in each regression is just shy of 3.8 million. R-squares are 0.799, 0.711 and 0.582, respectively. All coefficients shown here are significant at the 1% or better.

https://doi.org/10.1371/journal.pone.0250847.g010

Spatial covariogram

So far we have been looking at agglomeration effects at fairly close range—i.e., up to 1.5km at most. We found mostly evidence of agglomeration effects, except occasionally when we uncovered patterns suggestive of spatial substitution effects. We now explore how buildup areas evolved over a much broader range of distances. This is done by estimating spatial covariograms as described in the empirical strategy section (2.3.3). First, we present the results when Ghana is divided into 856 equal sized tiles resulting in maximum distance of about 26km within each tile. The results are shown in Fig 11 for all four years combined. We see that, since the proportion of built-up cells increases over time, the proportion of built-up cell pairs not surprisingly increases as well. Otherwise the spatial pattern is similar across all years: cells located close to a built-up cell are more likely to be built up as well: E[z_it z_jt] falls with distance. The correlation is particularly strong at short distances of up to 3 km, but the pattern is present across the entire distance distribution and, given how tight the confidence intervals are, differences in spatial correlation by distance are significant up to distances of at least 20km. Beyond 20 km, the number of observations in distance bins N_bm drops rapidly, confidence intervals increase in size, and the covariogram becomes unreliable, a point we revisit below.

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Fig 11. Fig 11 shows, for each distance value within small tiles, the proportion of cell pairs that are both built-up.

For instance, a value of 2% at distance 4Km means that pairs with two built-up cells make up 2% of all the cell pairs distant from each other by 4Km. Four correlograms are displayed, one for each of the years for which we have buildup data.

https://doi.org/10.1371/journal.pone.0250847.g011

We repeat the exercise separately for cell pairs both located near a road in 1976, and other cell pairs. Proximity to a road is defined as having a road in the cell itself or in neighborhoods n₁ or n₂—i.e., as having , , or for t₀ = 1976. The resulting spatial covariogram is shown in Fig 12 for all four years of data we have. The general pattern is similar to that observed for all cell pairs, except that the proportion of built-up cells is higher near roads, and thus there are more built-up cell pairs near roads. We also observe the sharply higher spatial correlation at close distances of around 2–3 km. But the spatial correlation in buildup falls less fast near roads than away from roads. This is consistent with our earlier finding that proximity to a road predicts higher buildup independently from proximity to a pre-existing agglomeration, and this effect is distinct from the fact that settlements often are near roads to start with. The spatial covariogram for cell pairs that are not close to roads is very similar to that reported in Fig 11 and is not shown here to save space.

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Fig 12. Fig 12 is constructed in the same way as Fig 11, except that Fig 12 only includes cell pairs in which both cells are near a road, either in the cell itself or in a neighboring cell n₁ or n₂.

In Fig 12 we also show the 95% confidence interval for two of the years. The bounds of the confidence interval are only visible at long distances; for other values they are too tight to be visible on the graph.

https://doi.org/10.1371/journal.pone.0250847.g012

Contrary to what is predicted by the von Thunen [31] model subsequently expanded by Chistaller [32] and Isard [33], we find no strong evidence of a ‘bump’ in the covariogram at any specific distance. If farm buildings tended to be spaced equally so as to locate in the middle of their fields, we would expect a spike in the covariogram at the relevant distance. For instance, if each farming households occupies, say, 20 Ha of cultivated land, fallows and pastures, we would expect farms to be, on average, 500 meters from each other. This would more or less correspond to one farm per cell, and thus a flat spatial covariogram between farm buildings. We also would not expect to find a buildup concentration along roads—which is the contrary of what we find. In contrast, if farmers live in villages of, say, 100 households needing each 20 Ha of combined cultivated land, fallows and pastures, each village would require an area of 2,000 Ha, which would imply a spacing of villages each 5 km. If this were the case, we would expect a spike in the spatial covariogram at 5km: the presence of village buildings at distance 0 would predict village buildings 5 km away. We do not find evidence consistent with any of these conjectures—or any equal spacing of villages up to 20 km. We do however find evidence of agglomerations covering an area of up to 3 km in radius—6 km or 4 miles in diameter. These localities may be larger than simple villages, and they may be located further apart than 20 km. Could there be, therefore, spikes in the spatial covariogram at distances above 20km—something that is possibly suggested by the appearance of a spike at distances slightly above 20km in Fig 12?

To investigate this possibility, we recalculate the spacial covariogram for distances up to 75 km using the procedure described in section (2.3.3). Results comparable to Figs 11 and 12 are shown in Figs 13 and 14, respectively. We see that the reported patterns are similar to those reported earlier. The fall in buildup correlation that was observed at short distances in Fig 11 is confirmed in Fig 13, and it is seem to continue then flatten our around about 50 km of distance. A similar pattern is shown in Fig 14 for cell pairs located near a road, except that the fall is steeper at short distances and the slope is steeper for longer. The curve also flattens our at around 50 km distance. We also observe that the bump at distances above 20 km in Fig 12 is not a robust result and is probably an artifact of the small number of tiles and cell pairs with observations at those distances. Using larger tiles solves that problem. These results confirm our earlier findings, namely that, across Ghana, human settlements (as proxied by buildings) are not evenly spaced. This is true across cells in general, but also along roads. We do, however, find strong evidence of agglomeration at short distances, as well as evidence of weak agglomeration forces operating up 50 km away—in the sense that being less than 50 km from a built-up location predict a slightly higher probability of buildup. The figures also confirm that buildings tend to concentrate along roads and near each other.

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Fig 13. Figs 13 and 14 show, for each distance value within big tiles, the proportion of cell pairs that are both built-up.

For instance, a value of 2% at distance 4Km means that pairs with two built-up cells make up 2% of all the cell pairs distant from each other by 4Km. Four correlograms are displayed, one for each of the years for which we have buildup data. The two Figures are constructed in the same way, except that Fig 14 only includes cell pairs in which both cells are near a road, either in the cell itself or in a neighboring cell n₁ or n₂. Confidence intervals are not shown because they are too tight to be visible in the graph.

https://doi.org/10.1371/journal.pone.0250847.g013

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Fig 14. Figs 13 and 14 show, for each distance value within big tiles, the proportion of cell pairs that are both built-up.

For instance, a value of 2% at distance 4Km means that pairs with two built-up cells make up 2% of all the cell pairs distant from each other by 4Km. Four correlograms are displayed, one for each of the years for which we have buildup data. The two Figures are constructed in the same way, except that Fig 14 only includes cell pairs in which both cells are near a road, either in the cell itself or in a neighboring cell n₁ or n₂. Confidence intervals are not shown because they are too tight to be visible in the graph.

https://doi.org/10.1371/journal.pone.0250847.g014

In the last figure, we perform a similar exercise using small tiles as unit of observation. Since nearly all these tiles include at least one building, here we define buildup as being in the top decile in term of buildup, averaged over the whole tile. We then form all tile pairs and conduct the same analysis as in Figs 11 and 12. Distance between tiles is calculated from the tile’s centroid which, for our purpose, is defined as the point with the average longitude and latitude of the tile. As we have much fewer observations, we need not put tile pairs into distance bins: we can work directly on the full sample of pairs. Given that nearly all tiles contain at least one road, we ignore that dimension here. Fig 15 shows the result with distance on the horizontal axis and, on the vertical axis, the share of tile pairs that are both in the top buildup decile. By construction, the horizontal axis is truncated at distances less than 18 km, which is the size of a tile itself. The four curves show, for each of the data year, the predictions from a fractional polynomial regression of the share of top decile pairs on distance. If buildup were distributed randomly across space, we should observe a flat curve. This is not what we see: as in Figs 11 to 14 we find evidence of agglomeration, which is strongest at distances up to around 60 km, which is the range of distances covered in Figs 13 and 14. Correlation in high buildup falls after that, but only gradually—suggesting that agglomeration continues to have an effect on buildup even at long distances.

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Fig 15. Fig 15 shows, for each distance value between small tiles, the proportion of tile pairs that are both in the top buildup decile.

For instance, a value of 2% at distance 50Km means that tile pairs that are both in the top build decile make up 2% of all the cell pairs distant from each other by 50Km. Four correlograms are displayed, one for each of the years for which we have buildup data. The four curves show, for each data year, the predictions from a fractional polynomial regression of the share of top decile pairs on distance between pairs.

https://doi.org/10.1371/journal.pone.0250847.g015