Instead of more or less similar things, there are far more small things than large ones surrounding us. Given the ubiquity of the scaling pattern, head/tail breaks is found to be of use to statistical mapping, map generalization, cognitive mapping and even perception of beauty . It helps visualize big data, since big data are likely to show the scaling property of far more small things than large ones. Essentially geographic phenomena can be scaleful or scale-free. Scaleful phenomena can be explained by conventional mathematical or geographical operations, but scale-free phenomena can not. Head/tail breaks can be used to characterize the scale-free phenomena, which are in the majority. The visualization strategy is to recursively drop out the tail parts until the head parts are clear or visible enough. In addition, it helps delineate cities or natural cities to be more precise from various geographic information such as street networks, social media geolocation data, and nighttime images.
Characterizing the imbalance As the head/tail breaks method can be used iteratively to obtain head parts of a data set, this method actually captures the underlying hierarchy of the data set. For example, if we divide the array (19, 8, 7, 6, 2, 1, 1, 1, 0) with the head/tail breaks method, we can get two head parts, i.e., the first head part (19, 8, 7, 6) and the second head part (19). These two head parts as well as the original array form a three-level hierarchy: • the 1st level (19), • the 2nd level (19, 8, 7, 6), and • the 3rd level (19, 8, 7, 6, 2, 1, 1, 1, 0). The number of levels of the above-mentioned hierarchy is actually a characterization of the imbalance of the example array, and this number of levels has been termed as the ht-index. Head/tail breaks can be used to do just that with a concept called natural cities. The term 'natural cities' refers to the human settlements or human activities in general on Earth's surface that are naturally or objectively defined and delineated from massive geographic information based on head/tail division rule, a non-recursive form of head/tail breaks. Such geographic information could be from various sources, such as massive street junctions Distinctive from conventional cities, the adjective 'natural' could be explained not only by the sources of natural cities, but also by the approach to derive them Natural cities are derived from a meaningful cutoff averaged from a massive number of units extracted from geographic information. A
[https://www.arcgis.com/home/item.html?id=47b1d6fdd1984a6fae916af389cdc57d natural cities model has been created using ArcGIS model builder, it follows the same process of deriving natural cities from location-based social media,
Scaling law can also at the same time correctly be identified and the administrative borders can be created to respect this by the delineation of the natural cities. This type methodology can help urban geographers and planners by correctly identifying the effective urban territorial scope of the areas they work in. Natural cities can vary depending on the scale on which the natural cities are delineated, which is why optimally they have to be based on data from the whole world. Due to that being computationally impossible, a country or county scale is suggested as alternative. Due to the scale-free nature of natural cities and the data they are based on there are also possibilities to use the natural cities method for further measurements. One of the main advantages of natural cities is that it is derived
bottom-up instead of
top-down. That means that the borders are determined by the data of something physical rather than determined by an administrative government or administration. For example by calculating the natural cities of a natural city recursively the dense areas within a natural city are identified. These can be seen as city centers for example. By using the natural cities method in this way further border delineations can be made dependent on the scale the natural cities were generated from. Natural cities derived from smaller regional areas will provide less accurate but still usable results in certain analysis, like for example determining urban expansion over time. As mentioned before though, optimally natural cities should be based on a massive amount of for example street intersections for an entire country or even the world. This is because natural cities are based on
the wisdom of crowds thinking, which needs the biggest set of available data for the best results. Also note that the structure of natural cities can be considered to be
fractal in nature. It is important when head/tail breaks are being used to generate natural cities, that the data is not aggregated afterwards. For example, the amount of generated natural cities can only be known after they are generated. It is not possible to use a pre-defined number of cities for an area or country and aggregate the results of the natural cities to administratively determined city borders. Naturally natural cities should follow
Zipf's law, if they do not, the area is most likely too small, or data has probably been processed wrongly. An example of this is seen in a research where head/tail breaks were used to extract natural cities, but they were aggregated to administrative borders, which following that concluded that the cities do not follow
Zipf's law. This happens more often in science, where papers actually produce results which are actually false.
Color rendering DEM Current color renderings for DEM or density map are essentially based on conventional classifications such as natural breaks or equal intervals, so they disproportionately exaggerate high elevations or high densities. As a matter of fact, there are not so many high elevations or high-density locations. It was found that coloring based head/tail breaks is more favorable than those by other classifications.
Mapping scaling hierarchy The pattern of far more small things than large ones frequently recurs in geographical data. A spiral layout inspired by the
golden ratio or Fibonacci sequence can help visualize this recursive notion of scaling hierarchy and the different levels of scale. In other words, from the smallest to the largest scale, a map can be seen as a map of a map of a map, and so on.
Further applications Other applications of Head/tail breaks: • Serving as a method for efficiently estimating the absolute
Boltzmann's entropy of numerical raster data • Quantifying the multiscale representation of a polyline • Increasing computational efficiency in
data-flow analysis by emphasizing the head part of the flow dataset • Temporal analysis of urban expansion related to the
thermal environment • Image analysis where
anisotropy is measured in point patterns extracted with a digital pulse transform with the use of head/tail breaks • Visualizing and analyzing spatial patterns in
bilateral trade • To identify urban function graphs, note that this paper applies head/tail breaks on a
Gaussian kernel density estimation which reduces the accuracy of the head/tail breaks method. Essentially a natural cities approach is taken but the initial data chosen to perform head/tail breaks on has been reduced beforehand. For a better representation of urban function graphs head/tail breaks may be applied as the first step in delineating these areas. • Analyzing structures or hotspots naturally occurring within data to highlight areas of interest (Based on natural cities). •
(Over)Tourism analysis based on short term rentals (like
AirBnB) by creating hotspots out of the distribution of rented out apartments. • Measuring
tourism intensification based on the fractal dimension delineated using natural cities • Identifying urban hotspots based on taxi stops, where people are most likely to get out at major landmarks or public transport transfer areas. Head/tail breaks are applied to separate the less dense stops where few people exit, from the major stops where the most people exit. • Determining traffic hotspots or congestion zones, which can be used to in turn determine road pricing. Natural cities is an effective approach when finding these areas. • Using natural cities to identify the polycentric pattern of Chinese cities, i.e. identifying the multiple dense centers of activity found in cities. • Determining how city growth affects the
thermal environment in cities using natural cities as a measurement tool. • Identifying
polycentric cities with night time imagery, which can be used to evaluate the urban development levels. • Quantifying urban expansion by using POI data as indicators of built up areas. • Detecting hierarchical crowd data with different clustering algorithms. • Using twitter data obtained during the COVID-19 pandemic to analyze spatial hotspots with natural cities. • Reducing carbon emissions by dividing urban spaces using head/tail breaks. • Using remote sensing to identify core city expansion. • Predicting urban growth with fractal dimension logistic curve modeling and head/tail breaks. • Head/tail breaks can serve as a main indicator that phenomena are distributed long tailed and that
Paretian thinking should favor
Gaussian thinking in geographic spaces. For example within
biodiversity and
pedodiversity studies where there seem to be fractal relationships such as
taxa-area relationships. Complementary to this the polygons of soil and vegetation maps also show scaling within their structures. This can be identified and highlighted by using head/tail breaks. • In
image feature and texture extraction, certain algorithms like the discrete pulse transform, where
LULU smoothing is used to extract the features, can be sped up by using head/tail breaks in the algorithm by separating large features and smaller features more effectively. • By analysing hierarchies in urban patterns (i.e. Streets, building outlines), visual salience can be determined because it follows a similar principle, namely a scaling law, or long tailed distribution. Head/tail breaks are an aid in determining the hierarchies present because of the scaling nature of urban morphology and could be of further use when studying urban street network applications. This is especially the case for accessibility analysis, combined with
space syntax head/tail breaks allow for an in depth understanding of street network structure. • Urban structures, like street networks have been proven to be fractal in nature. An important point to note is that this structure is not consisting of only one defined fractal, it is characterized by a multifractal complex network. This means that on different scales, the defined fractal can change. Head/tail breaks can be used to determine the structure of the complex network over different scales, as it adjusts based on the data with each new hierarchy. • Head/tail breaks as a classification method can be used to visualize growth or spread patterns in for example a global pandemic, like the Covid-19 one. By using head/tail breaks, main spread events can be effectively mapped and visualized where locations with a high infection rate are highlighted specifically due to them being in the highest class. The risk measurement model based on the head/tail breaks approach can describe the spatial and temporal evolution characteristics of the risk of COVID-19, and can better predict the risk trend of future epidemics in each city and identify the risk of future epidemics even during low incidence periods. • Rock fracture networks are properties of rocks which are very important in rock engineering with applications in mining, shale gas development or slope stability. Because of the
self-similarity characteristics of these fractures combined with the
fractal nature they inhibit, head/tail breaks provide accurate measurements and analysis into these rock fracture networks. • Classifying tourist attractions into most visited, least visited and something in between for further research into the optimal route of sightseeing busses. • Measuring the heterogeneity of crime distribution quantitatively while simultaneously considering the statistical and geometrical characteristics of crime distribution. • Examining the urban sustainability of socioeconomic and environmental dynamics. The natural cities serve as the basic urban form measures to objectively capture the spatial patterns of the sustainability change. == Software implementation ==