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Geospatial Analysis Requires a Different Way of Thinking: The Problem of Spatial
Heterogeneity
Bin Jiang
Department of Technology and Built Environment, Division of Geomatics
University of Gävle, SE-801 76 Gävle, Sweden
Email: bin.jiang@hig.se
(Draft: May 2013, Revision: July 2013, Janurary, February 2014)
Abstract
Geospatial analysis is very much dominated by a Gaussian way of thinking, which assumes that things
in the world can be characterized by a well-defined mean, i.e., things are more or less similar in size.
However, this assumption is not always valid. In fact, many things in the world lack a well-defined
mean, and therefore there are far more small things than large ones. This paper attempts to argue that
geospatial analysis requires a different way of thinking - a Paretian way of thinking that underlies
skewed distribution such as power laws, Pareto and lognormal distributions. I review two properties of
spatial dependence and spatial heterogeneity, and point out that the notion of spatial heterogeneity in
current spatial statistics is only used to characterize local variance of spatial dependence. I
subsequently argue for a broad perspective on spatial heterogeneity, and suggest it be formulated as a
scaling law. I further discuss the implications of Paretian thinking and the scaling law for better
understanding of geographic forms and processes, in particular while facing massive amounts of social
media data. In the spirit of Paretian thinking, geospatial analysis should seek to simulate geographic
events and phenomena from the bottom up rather than correlations as guided by Gaussian thinking.
Keywords: Big data, scaling of geographic space, head/tail breaks, power laws, heavy-tailed
distributions
1. Introduction
Geospatial analysis, or spatial statistics in particular, has been dominated by a Gaussian way of
thinking, which assumes that things are more or less similar in size, and can be characterized by a
well-behaved mean. Based on this assumption, extremes are rare; if extremes do exist, they can be
mathematically transformed into normal things (e.g., by taking logarithms or square roots). This
Gaussian thinking is widespread, and has dominated the sciences for a very long time. However,
Gaussian thinking has been challenged and been accused of misrepresenting our world (Mandelbrot
and Hudson 2004; Taleb 2007). Indeed, many things in the world are not well behaved or lack of a
well-behaved mean. This can seen from the extreme events such as the September 11 attacks. The
extent of devastation of such events was enormous and beyond any predictions and estimations. This
is the same for many geographic features, which exhibit a pretty skewed or heavy-tailed distribution
such as power laws and lognormal distributions. The heavy-tailed distributions imply that there are far
more small geographic features than large ones, namely scaling of geographic space.
A power law distribution is often referred to as scale free, literally meaning a lack of average for
characterizing the sizes of things (Barabási and Albert 1999). The power law distribution has been
given different formats for it was discovered by different scientists in different disciplines over the
past 100 years. Among several alternatives, Zipf’s law (Zipf 1949) and the Pareto distribution (Pareto
1897) are the two formats most frequently referred to in the literature. Zipf’s law, with respect to city
sizes, implies that there are far more small cities than large ones, while the Pareto distribution
indicates that there are far more poor people than rich people, or equivalently far more ordinary people
than extraordinary people. The Pareto distribution has been popularized as the 80/20 principle (Koch
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1999) or the long tail theory (Anderson 2006) in the popular science and business literature. The
heavy-tailed distribution, including power laws, lognormal and others similar, is what underlies the
new way of thinking I want to advocate in this paper. The central argument is that geospatial analysis
requires a new way of thinking radically different from Gaussian thinking, and spatial heterogeneity
should be formulated as a scaling law of geography.
This is an unprecedented time when we face increasing rich geographic data sources based not only on
the legacy of traditional cartography and remote sensing imagery, but also emerging from various
social media such as Flickr, Twitter, and OpenStreetMap, collectively known as volunteered
geographic information (Goodchild 2007). Today, one can amass gigabytes of an entire country’s data
for geospatial analysis and computing, for both data volumes and computing capacity have increased
dramatically. However, our mindsets, subsequently our analysis methods, have been relatively slower
to adapt the rapid changes (Mayer-Schonberger and Cukier 2013). For example, we tend to sample data
rather than take all data for geospatial analysis; we tend to transform skewed data into “normal” by
taking logarithms for example. The sampling and logarithm transformation have distorted the
underlying property of the data before the data can yield insights. The old way of thinking, Gaussian
thinking, that relies on a well-defined mean to characterize geographic features, is a major barrier to
achieving deep and new insights into geographic forms and processes.
Many analytical techniques have been developed, in both standard and spatial statistics, to address
outliers, to measure skewness and autocorrelation, and to test significance. However, what I want to
argue in this paper is not these techniques per se, but something radical in the way of thinking.
Gaussian thinking, based on the assumption of independent things in a simple, static, and equilibrium
world, is essentially a typical linear thinking, which implies that small cause small effect, large cause
large effect, and the whole is equl to the sum of its parts. This linear thinking is a simple way of
thinking guided by the reductionism philosophy, and for understanding a simple world in essence (see
Section 2.2 for more details). The reader may argue that spatial statistics differs from standard
statistics in spatial dependence or spatial autocorrelation. It is indeed true, but the notion of spatial
dependence or autocorrelation does not help us to go beyond Gaussian thinking assumed by standard
statistics, for we tend to characterize things by a well-defined mean with a limited variance. It is well
recognized that geographic forms are fractal rather than Euclidean, and geographic processes are
nonlinear rather than linear (Batty and Longley 1994, Chen 2009). In other words, a geographic
system is a complex nonlinear world, in which there is the butterfly effect, and the whole is greater
than the sum of its parts. In this paper, I attempt to argue that the Paretian way of thinking, founded on
the assumption of interdependent things in a complex, dynamic, and nonequilibrium world, is more
appropriate for geospatial analysis, and for better understanding geographic forms and processes.
Geospatial analysis, while facing increasing amounts of social media data, should seek to uncover the
underlying mechanisms through simulations from the bottom up rather than simple causality or
correlations.
The remainder of this paper is organized as follows. Section 2 introduces, in a pedagogic manner, two
distinct statistic distributions, namely Gaussian- and Paretian-like distributions, with a particular focus
on the underlying ways of thinking. Section 3 reviews two unique properties of spatial dependence
and spatial heterogeneity, and points out that the notion of spatial heterogeneity in current spatial
statistics is only used to characterize local variance of spatial dependence. I therefore argue, in Section
4, that spatial heterogeneity should be formulated as a scaling law, and suggest some effective ways of
detecting and revealing the scaling law and pattern for geographic features. I further discuss, in
Section 5, some deep implications of Paretian thinking and the scaling law before draw a summary in
Section 6.
2. Two distint distributions and the underlying ways of thinking
In this section, I first illustrate statistical differences between a homogenous Gaussian-like distribution
and a heterogeneous Paretian-like distribution (Note the ‘homogenous’ is relative to the
‘heterogeneous’; see Section 2.1 for more details), using temperature and population of major US
cities, and based respectively on histograms and rank-size plots. The temperature is the annual average
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maximum during 1981-2010, taken from the site:
http://www.prism.oregonstate.edu/products/matrix.phtml, while the population is according to the
2010 US census. I then elaborate on the underlying ways of thinking or world views associated with
the two categories of distributions.
2.1 Gaussian- versus Paretian-like distributions
If we carefully examine two variables – temperature and population – of 720 major U.S. cities with
population greater than 50,000 people, we can see that the two variables are very distinct. Although
not a normal distribution, the temperature can be well characterized by its mean 20.6 (Figure 1a). One
can estimate a city’s temperature fairly accurate and precise based on the mean value, since the highest
is 31.6, and the lowest is 9.3. In other words, the mean 20.6 is a typical temperature for US cities. The
distribution that can be characterized by a well-defined mean is referred to as a Gaussian-like
distribution including for example the binomial and Poisson distributions. This temperature
distribution can be further assessed from the detailed statistics as shown in Table 1 (the temperature
column). The range between the highest (31.6) and the lowest (9.3) is not very big (22.3), and the ratio
of the highest to the lowest is as little as 3.4. The two measures of central tendency – mean and median
– are the same. The standard deviation is 4.9, about one quarter of the range. This statistical picture of
the temperature is very distint from that of the city size or population.
The histogram of the population is extremely right skewed (Figure 1b). This extreme skewness is
reflected in several parameters: a wide range (8,273,676), a huge ratio (166), and a large standard
deviation (393,004); see the population column of Table 1. In such a significantly skewed distribution,
the mean of 157,467 make little sense for characterizing the population. In other words, the mean of
157,467 does not represent a typical size of the U.S. cities, since the largest city is as big as 8 millions,
while the smallest city is as small as 50 thousands. The right skewed histogram indicates that there are
far more small cities than large ones in the U.S. No wonder that the two mesures of central tendency –
mean and median - differ from each other significantly; refer to Table 1 (the population column) for
more details. The standard statistics, or the histogram in particular, is little effective for describing
data with a heavy-tailed distribution such as city sizes. Instead, power law based statistics, or rank-size
plots in particular, should be adopted for characterizing this kind of data.
Figure 1: (Color online) Histograms of (a) the temperature, and (b) the population of U.S. cities
(Note: the two distint distributions indicate respectively Gaussian-like and Paretian-like distributions.)
Table 1: Statistics about temperature and population of U.S. cities
Statistics Temperature Population
Minimum 9.3 50,056
Maximum 31.6 8,323,732
Range 22.3 8,273,676
Ratio 3.4 166
Mean 20.6 157,467
Median 20.6 82,115
Mode 14.9 62,820
St. Dev. 4.9 393,004
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Instead of plotting temperature and population on the x-axis (as in the histograms), they are plotted on
the y-axis, while the x-axis is the ranking order. This way of plotting is called rank-size plot, or rank-
size distribution (Zipf 1949). The largest city (in terms of population) ranks number one, followed by
the second largest, and so on. The same arrangement is made for the temperature; the highest
temperature city ranks number one, followed by the second highest, and so on. The two distribution
curves look very different; the temperature curve drops gradually, and then reaches quickly the
minimum, while the population curve drops quickly and then gradually reaches the minimum (Figure
2). Note that the red parts in the figure are those above the averages, called the head, while those
below the averages, called the tail, are shown in blue. More specifically, 362 cities (approximately 50
percent) are above the average temperature 20.6, while only 146 cities (approximately 20 percent) are
above the average city size 157,467. Clearly, a heavy or long tail (80 percent in the tail) exists for the
population distribution, but a short tail (50 percent) for the temperature distribution. Generally, a
heavy-tailed distribution possesses an inbuilt imbalance between the head and the tail (e.g., a 70/30 or
80/20 relationship). This imbalance indicates a nonlinear relationship between the head and the tail.
Such an inbuilt imbalance, or nonlinearity, is clearly missing in a Gaussian-like distribution with a
well-balanced relationship between the head and the tail (e.g., 50/50).
Figure 2: (Color online) Rank-size plots of (a) the temperature, and (b) the population of the U.S.
cities
(Note: Values above and below the averages are respectively in red and blue; clearly, there is a short
head and a long tail for the population, forming an unbalanced contrast, while the values above and
below the averages are more or less the same for the temperature.)
2.2 The underlying ways of thinking
The differences between the two distributions lie fundamentally in different ways of thinking, or
different ways of viewing the world, rather than different techniques associated with each distribution.
Technically, data with a Paretian-like distribution can be easily transformed into a Gaussian-like
distribution, e.g., by taking logarithms. Gaussian thinking implies more or less similar things in a
simple, static, and equilibrium world, while Paretian thinking believes in far more small things than
large ones in a dynamic, complex, and nonequilibrium world (McKelvey and Andriani 2005; see Table
2). Standard statistics teaches us that if the probability of an event is small, then the event occurs
rarely. The event can be considered an outlier that is literally distant from the rest of the data.
However, in Paretian thinking, an event of small probability, or the highly improbable, has a
significant impact (e.g., the September 11 attacks) and thus be ranked highly.
Table 2: Comparison between the two ways of thinking
Gaussian thinking Paretian thinking
With a mean (or scale) Without a mean (or scale-free)
Static Dynamic
Simple Complex
Equilibrium
N
on-equilibrium
Linear
N
onlinea
r
Predictable Unpredictable
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In Gaussian thinking, the world does not change much, and all changes occur around a stable and
well-defined mean. Thus, the presumed Gaussian world is static, simple, linear, and predictable. The
Newtonian physics is sufficient to understand and deal with the Gaussian world. Why does such a
predictable world exist? Such a world reflects a lack of interaction and competition among individual
agents; every agent acts independently without influences upon or affects from others. This
assumption is fundamental to standard statistics, and of course appropriate for many events in the
world like human heights. Spatial statisics differentiate it from standard statistics in spatial
dependence, but it does not change fundamently the underlying way of thinking - Gaussain thinking
with a well-defined mean for characterizing things. On the other hand, in Paretian thinking, the world
is full of surprises, and changes are often dramatic and unexpected. Thus, there is no stable and well-
defined mean for characterizing the surprises and changes. The presumed Paretian world is essentially
dynamic, complex, nonlinear, and unpredictable. This unpredictable world is founded on the
assumption that everything is related to, or interdependent with, everything else. This interdependence
assumption implies that cooperation and competition would eventually lead to unbalanced results
characterized by a long-tail distribution (c.f. Section 3 for more discussions).
Nature is awash with phenomena such as trees, rivers, mountains, clouds, coastlines, and earthquakes
that exhibit power laws or heavy-tailed distributions in general (Mandelbrot 1982, Schroeder 1991,
Bak 1996). Accordingly, power law has been formulated as a fundamental law in various disciplines
such as physics, biology, economics, computer science, and linguistics. People’s daily activities are
also governed by power laws (Barabási 2010), indicating bursty behaviors of human mobility or
activities in general. Power laws are a signature of complex systems that are evolved in nonlinear
manners, i.e., small causes often have disproptional large effects. For instance, the top 10 percent of
the most connected streets account for 90 percent of traffic flows (Jiang 2009). In a 21-block area of
Philadelphia, 70 percent of the marriages occurred between people who lived no more than 30 percent
of that distance apart (Zipf 1949).
The examination of the two ways of thinking suggests that Paretian-like distribution, or Paretian
thinking in general, appears more appropriate for understanding geographic forms and processes, for
dependence is a key property of spatial statistics (c.f., Section 3 for more details). In spite of spatial
dependence being its key property, spatial statistics is still unfortunately very much dominated by
Gaussian thinking. The very notion of spatial heterogeneity refers to local variance of spatial
dependence, but from global to local, or from one single correlation coefficient to multiple coefficients
(c.f., Section 3 for more details). In the remainder of this paper, I review two spatial properties of
dependence and heterogeneity, and argue that spatial heterogeneity is ubiquitous, and it should be
formulated as a scaling law. And I further discuss some deep implications of the scaling law and
Paretian thinking for better understanding of geographic forms and processes in the era of big data.
3. Spatial properties of dependence and heterogeneity
It is well known that in contrast to the independence assumption of standard statistics, geographic
phenomena or events are not random or independent. Geographic events are more likely to occur in
some locations than others (spatial heterogeneity), and nearby events are more similar than distant
events (spatial dependence). Both spatial heterogeneity and spatial dependence are referred to as
spatial properties, indicating respectively that geographic events are related to their locations and to
their neighboring events. Spatial dependence is widely known or formulated as the first law of
geography: “Everything is related to everything else, but near things are more related than distant
things” (Tobler 1970). For example, your housing price is likely to be similar (positive correlation) to
those of your neighbors. Similarly, the elevations of two locations 10 meters apart are likely to be
more similar than the elevations of two locations of 100 meters apart. Note that “likely” indicates a
statistical rather than a deterministic property; one can always find exceptions in statistical trends.
Spatial heterogeneity refers to no average location that can characterize the Earth’s surface (Anselin
1989, Goodchild 2004). This is indeed true, while for example referring to the diversity of landscapes
and species (animals and plants) on the Earth’s surface (Wu and Li 2006, Bonner 2006). This diversity
or heterogeneity indicates uneven geographic and statistical distributions involving both landscapes
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and species – that is, a mix of concentrations of multiple species (biological), terrain formations
(geological), environmental characteristics (such as rainfall, temperature, and wind) on the one hand,
and various concentrations of various types of species on the other. A variety of habitats such as
different topographies, soil types, and climates can accommodate a greater number of species. These
are the natural environments of the Earth’s surface. Spatial heterogeneity in geography also concerns
human-made built environments created by human activities such as industrialization and
urbanization, and in particular, for example, the diversity of human settlements or cities in particular.
Given the diversity or spatial heterogeneity of the Earth’s surface, homogeneous Gaussian-like
distribution is unlikely to be the right means to characterize complex geographic features.
Spatial dependence and spatial heterogeneity are unique properties to spatial data and geospatial
analysis (Anselin 1989, Griffith 2003), and probably the two most important principles of geographic
information science (GIScience). Goodchild (2004) has been a key advocator for formulating general
principles for GIScience. On several occasions, he has made insightful remarks on spatial
heterogeneity or spatial properties in general. His definition of spatial heterogeneity as “no average
location” is in effect the notion of scale-free used to characterize things that exhibit a power law or
heavy-tailed distribution (Barabási and Albert 1999). On the other hand, he stated, with respect to
spatial heterogeneity, that all locations are unique, due to which geography might be better considered
as an idiographic science, studying the unique properties of places (Goodchild 2004). However, I
argue, in contrast to Goodchild, that spatial heterogeneity makes geography a nomothetic science. This
is because spatial heterogeneity itself is a law - the scaling law, implying that there are far more small
geographic features than large ones. Spatial heterogeneity is a kind of hidden order, which appears
disordered on the surface, but possesses a deep order beneath. This kind of hidden order can be
characterized by a power law or a heavy-tailed distribution in general.
Current spatial statistics suffers from what I call ‘spatial heterogeneity paradox’. Spatial heterogeneity
is defined as no average location, but we tend to use a well-defined mean or average to characterize
locations. This paradox implies that our mindsets are still constrained by Gaussian thinking. The
current notion of spatial heterogeneity refers to local variance of spatial dependence. This can be seen
from the development of local spatial statistics and local statistical models that initially brought spatial
heterogeneity into spatial statistics (Anselin 1989). Local spatial statistics concern local variants of
spatial autocorrelation, a measure to spatial dependence, including, for example, the local statistical
models (Getis and Ord 1992), the LISA techniques (Anselin 1995), and geographically-weighted
regression (Fotheringham et al. 2002). The shifting perspective of spatial autocorrelation from global
to local brings new insights into spatial dependence, or the heterogeneity of spatial dependence.
However, all these techniques and models are essentially based on Gaussian statistics, using a well-
defined mean with a limited variance. To paraphrase Mandelbrot (Mandelbrot and Hudson 2004),
spatial heterogeneity refers to ‘wild’ variances, but Gaussian-like distribution can only characterize
‘mild’ variances.
Human activities are the major forces behind spatial heterogeneity in the built environments. While
carrying out activities, human beings (and their interventions) must respect the spatial heterogeneity of
nature – that is, harmonize with rather than damage the natural environments. Geographic information
concerning urban and human geography captures essentially spatial variations of the built
environments, which demonstrate ‘wild’ heterogeneity as well. For example, Zipf’s law on city sizes
(Zipf 1949) mainly concerns such a spatial variation. Thus, I argue, in contrast to the conventional
view, that dependence, or more precisely interdependence, is a first-order effect, while heterogeneity is
a second-order effect. Let us do a thought experiment. Imagine that once upon a time, there were no
cities, only scattered villages. Over time, large cities gradually emerge through the interactions of
villages, so do mega cities through the interactions of cities. The interactions (competition and
cooperation) of villages and cities are actually those of people acting individually and/or collectively.
These interactions are what I mean by dependence and interdependence. Eventually, there are far more
small cities than large ones through for example the mechanism of “the rich get richer.” This
observation is the same for the wealth distribution among individuals in a country; far more poor
people than rich people, or far more ordinary people tha extraordinary people (Epstein and Axtell
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1996). The interactions among people and cities reflect the interdependence effect in the formation and
evolution of cities and city systems, and the built environments in general.
4. Spatial heterogeneity as a scaling law
The subtitle of this paper ‘The Problem of Spatial Heterogeneity’ is an homage to the classic work
‘The Problem of Spatial Autocorrelation’ (Cliff and Ord 1969), which popularized the concept of
spatial dependence. Similarly, spatial heterogeneity under Gaussian thinking is indeed a problem
because it cannot be characterized by a well-defined mean. However, in the Paretian way of thinking,
spatial heterogeneity is not a problem, but the norm. Spatial heterogeneity should be formulated as a
scaling law in geography.
4.1 Ubiquity of the scaling law in geography
Geographic features are unevenly or abnormally distributed, so the scaling pattern of far more small
things than large ones is widespread in geography (Pumain 2006). The scaling pattern has another
name called fractal (Mandelbrot 1983). Fractal-related research in geography has concentrated too
much on concepts such as fractal dimension and self-similarity. In fact, the scaling law is fundamental
to all of these concepts. In this regard, Salingaros and West (1999) formulated a universal rule for city
artifacts; there are far more small city artifacts than large ones, due to which the image of the city can
be formed in human minds (Jiang 2013b). With the increasing availability of geographic information,
the scaling law has been observed and examined in a wide range of geographic phenomena including,
for example, street lengths and connectivity (Carvalho and Penn 2004, Jiang 2009), building heights
(Batty et al. 2008), street blocks (Lämmer et al. 2006, Jiang and Liu 2012), population densities
(Schaefer and Mahoney 2003, Kyriakidou et al. 2011), and airport sizes and connectivity (Guimerà et
al. 2005). Interestingly, the scaling of geographic space has had an enormous effect on human
activities; human activities and interactions in geographic space exhibit power law distributions as
well (Brockmann et al. 2006, Gonzalez et al. 2008, Jiang et al. 2009). Table 3 provides a synoptic view
of the ubiquity of power laws in geography, noting that the references listed are non-exhaustive, but
for example only.
Table 3: Power laws in geographic features or phenomena
Geographic phenomena References (for example)
City sizes Zipf 1949, Krugman 1996, Jiang and Jia 2011
Fractals in cities or geographic space Goodchild and Mark 1987, Batty and Longley 1994
Coast lines and mountains Mandelbrot 1967, Bak 1999
Hydrological networks Hack 1957, Horton 1945, Maritan et al. 1996, Pelletier 1999
Urban and architectural space Salingaros and West 1999
Street lengths and connectivity Carvalho and Penn 2004, Jiang 2009
Building heights Batty et al. 2008
Street blocks Lämmer 2006, Jiang and Liu 2012
Population density Schaefer and Mahoney 2003, Kyriakidou et al. 2011
Airport sizes and connectivity Guimerà et al. 2005
Human mobility Brockmann et al. 2006, Gonzalez et al. 2008, Jiang et al. 2009
Despite its ubiquity, ironically the scaling law, or the Paretian way of thinking in general, has not been
well received in geospatial analysis as elaborated earlier in the text. Current geospatial analysis adopts
a well-defined mean or average to characterize spatial heterogeneity. The two closely related concepts
of scale and scaling must be comprehended together, i.e., many different scales, ranging from the
smallest to the largest, form a scaling hierarchy. This comprehension should be added, as a fourth one,
into the three meanings of scale in geography: cartographic, analysis, and phenomenon (Montello
2001). The essence of power laws is the scaling pattern, in which there are far more small scales than
large ones. This scaling pattern reflects the true picture of spatial heterogeneity.
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4.2 Detecting the scaling law
What were claimed to be power laws in the literature could be actually lognormal, exponential, or
other similar distributions, because the detection of power laws can be very tricky. Given a power law
relationship y=x^a, it can be transformed into the logarithm scales, i.e., ln∙, indicating
that the logarithms of the two varaibles x and y have a linear relationship. Conventionally, an ordinary
least squres (OLS) based method was widely used for the detection. In the fractal literature, the box-
counting method is usually used to compute the fractal dimension, which is the de fact power law
exponent, and its computation is also based on OLS. There are at least two issues surrounding the
power law detection. The first is that the OLS based method is found to be less reliable for detecting a
power law, so a maximum likelihood method has been developed (Clauset et al. 2009). It was found
that many claims on power laws in the literature are likely to be lognormal or other degenerated
formats such as a power law with an exponential cutoff. For the sake of readability, this paper does not
cover mathematical details on heavy-tailed distributions and their detection; interested readers can
refer to Clauset et al. (2009) and the references therein. The second is that even with the OLS-based
method, the definition of fractal dimension is so strict that many geographic features are excluded
from being fractal (Jiang and Yin 2014). Given the circumstance, the authors have recently provided a
rather relaxed definition of fractals, i.e., a geographic feature is fractal if and only if the scaling pattern
of far more small things than large ones recurs multiple times. The number of times plus one is
referred to as ht-index (Jiang and Yin 2014), an alternative index of fractal dimension, for
characterizing complexity of fractals or geographic features in particular.
The idea behind the relaxed definition of fractals, or the ht-index, is pretty simple and straightforward.
It is based on the head/tail breaks (Jiang 2013a), a new classification scheme for data with a heavy-
tailed distribution. Given a variable whose distribution is right skewed, compute its arithmetic mean,
and subsequently split its values into two unbalanced parts: those above the mean in the head, and
those below the mean in the tail. The values above the mean are a minority, while the values below are
a majority. The ranking and breaking process continues for the head part progressively and iteratively
until the values in the head no longer meet the condition of far more small things than large ones. This
way both the number of classes and the class intervals are naturally and automatically derived based
on the inherent hierarchy of data. Eventually, the number of classes, or equivalently the ht-index,
indicates hierarchical levels of the values. One can simply rely on an Excel sheet for the computation
of the ht-index. As an example, Figure 3 illustrates the scaling pattern of the US cities, discussed
earlier in Section 2.1, and it has the ht-index of 7.
Figure 3: (Color online) Scaling pattern of US cities with ht-index equal 7
(Note: The four insets from the left to the right provide the enlarged view respectively for San
Francisco, Los Angeles, Chicago and New York regions)
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4.3 Revealing the scaling pattern
The head/tail breaks can effectively reveal or visualize the scaling pattern if the data itself exhibits a
heavy-tailed distribution. This is because the head/tail breaks was developed initially for revealing the
inherent scaling hierarchy or the scaling pattern. In this regard, conventional classification methods,
mainly guided by Gaussian thinking, failed to reveal the scaling pattern. For example, the most widely
used classification natural breaks (Jenks 1967), which is set as a default in ArcGIS, is based on the
principle of minimizing within-classes variance, and maximizing between-classes variance. It sounds
very natural. In some case like the US cities, the classification result of the natural breaks may look
very similar to the one as shown in Figure 3. However, this is just by chance. Essentially, the natural
breaks is motivated by Gaussian thinking; each class is characterized by a well-defined mean with a
limited or minimized variance. In a contrast, the head/tail breaks is motivated by Paretian thinking,
and for data with a Paretian-like or heavy-tailed distribution. The iteratively or recursively defined
averages are used as meaningful cutoffs for differentiating hierarchical levels.
The reader probably has got used to the US terrain surface (Figure 4a), which is based on the natural
breaks. It is commonly seen in geography and cartography textbooks and atlases. However, I want to
challenge this conventional wisdom, arguing that the natural breaks based visualization is little natural.
I contend that the head/tail breaks derived visualization is more natural, since it reflects the underlying
scaling pattern of far more small things than large ones (Figure 4b). The things here are referred to
individual locations, or more specifically, far more low locations than high locations. The left
visualization, which distorted the scaling pattern, appears having far more high locations than the
visualization to the right, or equivalently far more high locations than what it actually has. The
visualization to the right reflects well the underling scaling pattern. This can be further seen from the
corresponding histograms of the individual classes of the two classifications (Figure 4c and 4d). What
is illustrated by the left histogram is “more low locations than high ones” which is a linear
relationship, rather than “far more low locations than high ones”, which is a nonlinear relationship.
For the left histrogram, each pair of the adjacent bars from left to right does not constitute an
unbalanced contrast of majority versus minority. For example, the first pair of bars of the left
histogram shows a well-balanced contrast of 7 to 6; in a contrast, the first pair of the right histogram is
unbalanced, 14 to 5. Therefore, the right histogram indicates clearly “far more low locations than high
ones”. Interestingly, the scaling pattern remains unchanged with respect to different scales of digital
elevation models (Lin 2013).
Figure 4: (Color online) The scaling pattern of US terrain surface is distorted by the natural breaks, but
revealed by the head/tail breaks
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5. Implications of Paretian thinking and the scaling law
Current geospatial analysis concentrates more on geographic forms, but less on why the forms. The
forms illustrated are mostly limited to whether they are random, or to what extent they are auto-
correlated. As to why the forms, it is usually ended up with simple regressions and causalities. This
way of geospatial analysis is much like short-term weather forecasting. Despite its usefulness, the
short-term weather forecast adds little to understanding the complex behavior of weather - the long-
term weather beyond two or three weeks. In essence, the long-term weather, , or climate change in
general, is unpredictable, just like earthquakes and many other events in nature and society (Bak
1996). If the real world is unpredictable, what can we do as scientists? We can simulate interactions of
things from the bottom up in order to understand the underlying mechanisms, which would help
improve predictions. In this regard, the emerging social media, in particular location-based social
media, provide valuable data for validating the simulation results (Jiang and Miao 2014). The data,
unlike traditional statistical or census data that are mainly aggregated, are not only big in size, but are
collected at individual levels. The data are not only at individual levels, but linked in time and among
individuals. The data can help track the trajectories of individuals and their associations in space and
over time. For this kind of social media data, the scaling law and fractals should be the norm.
Current spatial statistics constrained by Gaussian thinking show critical limitations for analyzing or
getting insight into big data (Mayer-Schonberger and Cukier 2013). What are illustrated by spatial
statistics, either patterns or associations, can be compared to the mental images of the elephant in the
minds of the blind men. These images reflect local truths, and are indeed correct partially, but they did
not reflect the whole of the elephant. Geospatial analysis should go beyond illustrating spatial
autocorrelation, either globally or locally, but towards uncovering the underlying scaling or fractal
patterns. Geographic features are essentially and ultimately scaling or fractal. Therefore, any patterns
deviating from the scaling pattern or that can be characterized by a well-defined mean are either
wrong or biased.
Geographic forms (or phenomena) are not the outcomes of simple processes but the results of complex
processes with positive feedbacks. In the built environments, human interventions (interdependence
and interactions) of various kinds are the major effects of spatial heterogeneity. As famously stated by
Winston Churchill (1874-1965), we shape our buildings, and thereafter they shape us. This statement
should be comprehended in a progressive and recursive manner. This comprehension, which underlies
Paretian thinking, is essentially a complex system perspective for exploring the underlying processes
related to geographic forms (e.g., Benguigui and Czamanski 2004, Blumenfeld-Lieberthal and
Portugali 2010). In this regard, complexity science has developed a range of tools such as discrete
models, complex networks, scaling hierarchy, fractal geometry, self-organized criticality, and chaos
theory (Newman 2011). All these modeling tools attempt to reveal the underlying mechanisms, linking
surface complex forms (or complexity) to the underlying mechanisms (or deep simplicity) through
simulations from the bottom up, rather than simple descriptions of forms or of geographic forms in
particular.
Paretian thinking represents a paradigm shift. Shifting from the sands to the avalanches (Bak 1996),
and from the street segments to the natural streets (Jiang 2009), enable us to see something interesting
and exciting, i.e., from the things of limited sizes to the things of all sizes. The things of all sizes imply
a scaling pattern across all scales. Recognition of the scaling pattern helps us to better understand the
underlying universal form of geographic features. This scaling pattern can further be linked to the
underlying geographic processes that are dynamic, nonlinear, and bottom-up in nature. This view
would position geography in the family of science, since we geographers are interested in not only
what things look like (the forms) but also why things look that way (the processes). Spatial
heterogeneity is thus not a problem but an underlying scaling law of geography.
6. Concluding summary
This paper argues that geospatial analysis requires a different way of thinking, or world view in
general, that underlies the Paretian-like distribution of geographic features. I put the two distinct views
11
in comparison: more or less similar things in a simple, static, and equilibrium world on the one hand,
and far more small things than large ones in a complex, dynamic, and non-equilibrium world on the
other. Geospatial analysis has been dominated by Gaussian statistics with a well-defined mean for
characterizing spatial variation (‘mild’ variance so to speak). Despite its ubiquity in geography, the
Paretian-like heavy-tailed distribution, or the underlying way of thinking in general, has not been well
received in geospatial analysis. The current geospatial analysis mainly focuses on how spatial
variation deviates from a random pattern, and measuring spatial auto-correlation from global to local
(the current spatial heterogeneity), but leaves the underlying processes unexplored. This way of
geospatial analysis is inadequate for understanding geographic forms and processes, in particular
while facing increasing amounts of social media data.
No average location exists on the Earth’s surface. Instead of more or less similar things, there are far
more small things than large ones in geographic space; small things are a majority while large things
are a minority. Importantly, the pattern of far more small things than large ones recurs multiple times
(Jiang and Yin 2014). This recurring scaling pattern reflects the true image of spatial heterogeneity
that lacks a well-defined mean (‘wild’ variance so to speak). Spatial heterogeneity is indeed a problem
in Gaussian thinking, but it is a law or scaling law in Paretian thinking. In the spirit of Paretian
thinking and the scaling law, geospatial analysis should seek to simulate individuals and individual
interactions from the bottom up rather than simple correlations and causalities. In this connection,
complexity tools such as complex networks, agent-based modeling, and fractal/scaling provide
effective means for geospatial analysis of complex geographic phenomena.
Acknowledgement
The author would like to thank the anonymous referees and the editor Daniel Z. Sui for their valuable
comments. However, any shortcoming remains the responsibility of the author.
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