Continuum Percolation and Spatial Point Pattern in Application to Urban Morphology

Abstract

One of the important problems in studying urban morphology is to be able to quantify urban objects using mathematical descriptions or measures. Doing so allows one to make a comparison amongst different types of geometry/morphology in urban systems, and effectively classify them. A number of techniques have indeed been developed in urban/geographical analysis. Here in this work, we propose a method to quantify spatial patterns of a set of points based on the idea of percolation, that is rooted in mathematical physics and has been applied to various fields. Employing the idea of percolation enables us to characterise the relative spatial arrangement of points in a set. In particular, the method could be applied to analyse the distribution of different point datasets in urban context and illustrate how it could characterise and classify their spatial patterns. We show that different spatial distributions of points can generally be classified into four groups with distinctive features: clustered, dispersed or regularly distributed at single or multiple length-scales. In applying to urban morphology, the results could enable quantitative discussion on the existence of two different forms of urban system: well-planned and organically grown.

1 Introduction

For a long time, the pattern of human settlement has always been a subject of much interest across different fields of Science, from Geography, Archaeology to Statistics (Small et al. 2005; Helbing et al. 1997; Anschuetz et al. 2001; Mellars 2006; Schumacher 1976; Zucchetto 1983). Over time, urban systems have become ever more important as a form of human settlement on the globe. As such, the study of urban morphology, the study of the spatial structure of people’s physical living space in which their daily activities take place, forms a major component of settlement or urban research as a whole. A good understanding of the morphology of an urban system provides us with the comprehension of its current status of development or even the living condition of people inside it. On the other hand, the knowledge of spatial organisation of an urban system could unveil the dynamical processes behind its growth and development, and allows us to gain insights into the way the system operates and evolves.

Traditionally, as mentioned, urban study has been performed by geographers, archaeologists or statisticians. However, urban systems, or ‘cities’ in modern terms, are typical examples of highly complex systems in which overwhelmingly many agents interact in non-trivial and nonlinear manners over a wide spectrum of spatial and temporal scales (West 2017; Bettencourt et al. 2007; Batty 2017). The results of such tangled interactions are the emergence of unexpected global patterns that cannot be derived from solely knowing the local behaviours of individual agents. It is this emergence property that has attracted researchers in theoretical fields like Mathematics or Physics to apply their tools to provide different and unconventional perspectives on urban systems.

One of the toolboxes which proves very useful in tackling complexity problems is that from Statistical Physics. The framework of Statistical Physics is particularly suitable for dealing with complex systems which have multiple agents, interacting locally with each other in simple but nonlinear manners (Viswanathan et al. 2014; Bertin 2016). The result of such interactions is a complex pattern at the level of the entire system, usually with long-range correlation that normally coexists with a critical state, signified by a phase transition and characterised by the notion of universality class. One such tool is the well-studied percolation phenomena with long-established literature in both Applied Mathematics and Theoretical Physics. As will be seen in this chapter, we will apply the techniques in percolation to analysing the cluster structure of sets of points, which could represent either transport points, be it road junctions or transport stations, or various sites in the urban context.

Spatial point pattern analysis is a well-studied field with various applications in biology (e.g. spatial distribution of cells obtained from bio-imaging (Burguet and Andrey 2014)), ecology (e.g. location and distribution of species (Knegt et al. 2010)) or geography (e.g. spatial patterns of people’s location or movement and physical entities, both natural like terrain and artificial like infrastructure (Fotheringham et al. 2000)). There have been many different methods developed either specific to each problem or aimed to be a general approach applicable to a wide range of problems (Illian et al. 2008). Most of them have been mainly concerned with identifying whether or not a point pattern distribution is complete spatial randomness (CSR), which implies that the mechanism underlying the process of generating the points is likely to be a homogeneous Poisson with little inter-event correlation. It appears that most of the analysis methods utilise the nearest-neighbour information based on some metric. Well-known methods like Ripley’s K-function (Ripley 1976, 1977) or pair correlation function (Stoyan and Penttinen 2000) also require proper identification of observation window which can become very complicated in the case of locations of urban entities because the domain of such is normally not a simple square or rectangle due to the geography and topography of the system. Furthermore, these methods are also known to be sensitive with respect to the boundary conditions for the domain (Li and Zhang 2007). Here we propose a method that is simply based on the relative position among points in a domain without requiring the knowledge of its boundary. The method is inspired by the ideas in ordinary critical phenomena (e.g. Uzunov 2010), percolation in particular (Stauffer et al. 1994). In percolation or ordinary critical phenomena in general, there is a control parameter that can be tuned to drive the system of interest to different desired states possessing distinct properties. The transition from one state to another takes place at some special value of the control parameter which is called a critical value. A percolation system is a collection of many sites each of which can be either percolating or non-percolating with a certain probability. This probability of a site being percolating (or non-percolating) is the control parameter in the percolation system. When the percolating probability is low, most of the sites are non-percolating, blocking the path to go from one end of the system to the other. When the percolating probability is high, most of the sites are percolating, allowing such path to exist. The length of the shortest such path is typically the (linear) size of the largest contiguous cluster connecting the neighbouring percolating sites. It has been shown that there exists a critical value of the percolating probability—the critical point—at which the system transits from being not percolating to being so. The behaviours of the system approaching the critical point can then be used to classify the system, i.e. identifying its universality class (Stanley 1999). These behaviours include the critical exponents (and scaling function (Hinrichsen 2000)) characterising the scaling of the size of the percolating cluster, among many other quantities.

Bringing this idea of percolation to studying the spatial point pattern, we introduce a control parameter that changes the state of a spatial point. To serve this purpose, the state of a point should reflect its connectivity to the surrounding points. A cluster should then be a group of neighbouring sites that share the same state. As we vary the control parameter and, hence, change the state of the points, the clusters would change. Analysing the properties of these clusters would allow us to classify the system of points of interest.

In what follows, we first present the method we employ to study the pattern of a spatial point distribution, including the quantities we define to analyse its structure. The analysis yields the characteristic distances of a distribution of points and how they can be used to classify different distributions. Finally, we discuss several technical aspects of the proposed method including the comparison with other known methods in the literature, and the applicability and interpretation when applying to study urban morphology.

2 Percolation in Mathematical Physics

2.1 Description

The idea of percolation is indeed very simple. On a lattice (in discretised space), which can be thought of as a porous media, every site¹ can be either occupied, meaning a liquid is passing through it, with a certain probability p or otherwise empty. At a small value of p, it is very difficult for the liquid to pass through the sites and only small clusters of (adjacent) occupied sites exist (see Fig. 1a–c). But at large value of p, a large cluster is formed spanning one edge to the opposite (Fig. 1d–f). That means the liquid can now easily percolate through the media. The graph of percolation probability against the occupation probability would show a sharp pick up near p equal to 0.6, which is known as the percolation threshold ($p_{c} \approx 0.5927$) (Newman and Ziff 2000) on a two-dimensional square lattice, when there ‘suddenly’ exists a giant cluster that spans one side to the other of the lattice.

Fig. 1

Realisations of sites’ configuration on a square lattice with different occupation probabilities p. On average, for uniform site-percolation model, the density (or fraction) of occupied sites equals the probability p of every site being occupied

Simply put, at small value of p, most sites are empty and no large clusters exist. At the other end of large value of p, most sites are occupied and a so-called giant cluster exists. So, what happens in between? By the argument of continuity, we know the transition from small clusters to a giant cluster must take place somewhere. It has been studied and found that there exists a critical value p_c of the occupation probability p. When p approaches p_c both from above and below, all sorts of scaling behaviours come in to play, for example the divergence of correlation length in the system that is characterised by a power law (Stauffer et al. 1994).

2.2 Phase Transition

It is clear that the higher the percolating probability is, the easier the sites are connected to one another other, and vice versa. The percolating probability is then viewed as a control parameter in the system. The transition occurs when the control parameter is adjusted to a critical value (called the critical point) at which the system transits from one state (or phase) to another with distinct properties, namely the non-percolating and percolating phases respectively at low and high percolating probabilities. The behaviour of the system approaching the critical point can then be used to classify the system, i.e. identifying its universality class (Stanley 1999).

When the system is at the critical point, many physical quantities diverge in a manner that can be described by a power law

$$ X\left( p \right) \propto \left| {p - p_{c} } \right|^{{ - \beta_{X} }} , \quad \left| {p - p_{c} } \right| \ll 1, $$

(1)

whose behaviour is solely characterised by the exponent $\beta_{X} > 0$. X could quantify some relevant properties of the system, for example, correlation length (distance beyond which the states of two points, on average, become uncorrelated) or average cluster size, etc. A small value of the exponent $\beta_{X}$ indicates a slow divergence of X, whilst a larger one means that X diverges fast. In physics literature, the set of exponents $\beta_{X}$ together with other mathematical descriptions are used to characterise the universality class of a physical system (Hinrichsen 2000).

Applying that idea of transition in standard percolation, we find the clusters of points in our data by defining a parameter distance $\rho$ that sets the maximum nearest neighbour distance between a pair of points in the same cluster. Once a cluster is identified, its size is defined as the number of points in the cluster, and area as the union area of circles of radius $\rho$ centred at all the points. Intuitively, we know that for small parameter distance $\rho$, most points belong to distinct clusters while at large parameter distance, most points belong to a single cluster. A transition is expected to take place in between.

3 Continuum Percolation and Spatial Arrangement of Points

3.1 General Ideas

We propose a procedure to characterise the spatial pattern of a set of points, that is based on the idea of continuum percolation (Meester et al. 1996) (see also random disk models or Gilbert disk (Boolean) model (Balister et al. 2008)). The procedure involves identifying clusters of points, whose pairwise distance does not exceed a parameter value $\rho$, and quantifying the growth of the clusters as the parameter $\rho$ changes its value. This distance parameter (also known as buffer radius in spatial analysis) $\rho$ sets out the neighbourhood of a point, which could be related to its connectivity. A larger value of $\rho$ means an extensive vicinity for the point, allowing access or connection to more other points to form a common cluster.

Mathematically, we consider a domain ${\mathfrak{D}}$ in ${\mathbb{R}}^{2}$, in which there are N points, each of which is located at coordinates $\left( {x_{i} ,y_{i} } \right)$. We then construct the clusters based on the buffer radius $\rho$. Any point j, whose Euclidean distance

$$ d_{ij} = \sqrt {\left( {x_{i} - x_{j} } \right)^{2} + \left( {y_{i} - y_{j} } \right)^{2} } $$

(2)

from point i is less than or equal to $\rho$, belongs to the same cluster as i (Fig. 2).² We denote $\eta \left( \rho \right)$ as the number of clusters given the buffer radius $\rho$. For each cluster, we define its size $\xi \left( \rho \right)$ as the number of points in the cluster whereas its area $A\left( \rho \right)$ the union of area of circles of uniform radius $\rho$ centred at the points in the cluster. To make different domains comparable, we normalise the clusters’ size $\xi \left( \rho \right)$ by the number of points N in the domain (i.e. as fraction of points), while the clusters’ area $A\left( \rho \right)$ by the union of area $\Xi \left( \rho \right)$ of circles of radius $\rho$ centred at all N points in the domain ${\mathfrak{D}}$. In this analysis, we are interested in the cluster with either largest size $\xi_{max} \left( \rho \right)$ or largest area $A_{max} \left( \rho \right)$, which may not be necessarily the same one. For simplicity of discussions, and unless stated otherwise explicitly, the descriptions of cluster size $\xi$ below also hold for cluster area A.

Fig. 2

Illustration of the cluster identification based on buffer radius $\rho$. Points (indicated as ${ \star }$, with circle of radius $\rho$ centred at it) belonging to the same cluster are marked with same colour. The cluster with largest size is coloured red, and it grows as $\rho$ increases. A cluster of size 1 is coloured grey with dashed circle around it. (Tip to read clusters: a point (${ \star }$) must be of the same colour as the circle (${ \odot }$) enclosing it.)

In applying the idea of the percolation cluster to analyse the spatial pattern of a set of points, the buffer radius $\rho$ plays the role of the control parameter. For different values of $\rho$, the system could be argued to be in different phase, namely, a segregate phase for small values of $\rho$ and an aggregate phase for large values of $\rho$. The transition from one phase to the other takes place in the intermediate regime of $\rho$. When $\rho$ is small, there are many small clusters as most points are not connected due to limited vicinity, resulting in a large number of clusters $\eta \left( \rho \right)$. The size of largest cluster $\xi_{max} \left( \rho \right)$, therefore, stays small and slowly increases due to this disjoint structure. As $\rho$ enters an intermediate regime, $\xi_{max} \left( \rho \right)$ increases faster than it does in the small-$\rho$ regime. This is when the larger clusters merge together making a significant expansion in size of the largest cluster. At the same time, $\eta \left( \rho \right)$ starts to decrease when small clusters merge to form larger ones. By this time, the largest cluster would have already encompassed most of the points in the domain, leading to very mild change in its size $\xi_{max} \left( \rho \right)$ as $\rho$ increases further into the regime of large $\rho$.³ The picture above illustrates that as $\rho$ is tuned, the system undergoes a change in its state, from a segregate to an aggregate phase, via a region of ‘phase transition’ corresponding to the intermediate regime of $\rho$, which has been well studied in physics (Domb et al. 2001), with a prime example of percolation (Stauffer et al. 1994; Meester et al. 1996).

The idea of percolation, in fact, has been employed in a number of urban-related studies. These include modelling the growth of cities (Makse et al. 1998) as a variant of percolation (i.e. correlated percolation), analysis of road network in Britain using percolation cluster to determine the hierarchical and organisational structure of cities and regions (Arcaute et al. 2016), analysis of the global urban land cover pattern to show a transition from separated clusters to a gigantic component at the country scale (Fluschnik et al. 2016), or application to modelling of urban retail locations (Piovani et al. 2017). However, while the cluster size appears in most of the studies, the cluster area has not previously been considered. In this chapter, we shall see that the combination of analysis of profiles of both the largest system size $\xi_{max} \left( \rho \right)$ and area $A_{max} \left( \rho \right)$ transit through the intermediate region would allow us to characterise the spatial pattern of a set of points.

3.2 Characterisation of Spatial Distribution of Points

The growth pattern of the clusters of points as the buffer radius parameter $\rho$ varies could be used to quantify the spatial pattern of the set of points. When $\rho$ increases, more points at farther distance can now join to belong to the same cluster as previously disjoint points, making the clusters merge to increase their size. Since the largest cluster size $\xi_{max} \left( \rho \right)$ increases monotonically with $\rho$, there exists a representative characteristic distance $\rho_{\xi }^{{ \star }}$ at which $\xi_{max} \left( \rho \right)$ effectively exhibits the most significant increase. As $\rho$ approaches $\rho_{\xi }^{{ \star }}$, the profile $\xi_{max} \left( \rho \right)$ could exhibit different behaviours, showing either a sharp narrow or a gradual wide-range increase. To account for that, it is also meaningful to introduce a quantity $\sigma_{\xi }$, called spread of transition, to measure the overall width of the increases in the profile of $\xi_{max}$.

3.2.1 Peak Analysis

In order to calculate the characteristic distance $\rho_{\xi }^{{ \star }}$ and its spread $\sigma_{\xi }$, we make use of the profile of the first derivative $\xi_{max}^{'} \left( \rho \right) = \frac{{{\text{d}}\xi_{max} \left( \rho \right)}}{{{\text{d}}\rho }}$ of the largest cluster size (similarly, we use $A_{max}^{'} \left( \rho \right) = \frac{{{\text{d}}A_{max} \left( \rho \right)}}{{{\text{d}}\rho }}$ for the cluster area), which measures its growth. If the profile of $\xi_{max} \left( \rho \right)$ exhibits a sharp increase, its first derivative produces a single dominant peak. On the other hand, if $\xi_{max} \left( \rho \right)$ increases gradually over a wide range of $\rho$, multiple scattering peaks are present in the profile of $\xi_{max}^{'} \left( \rho \right)$. To identify these peaks, we consider the profile of $\xi_{max}^{'} \left( \rho \right)$ (and $A_{max}^{'} \left( \rho \right)$) at every value of the buffer radius $\rho$ ranging from $\rho_{min} = \rho_{1}$ to $\rho_{max} = \rho_{M}$ in the step of $\delta \rho = \rho_{i + 1} - \rho_{i}$, $\forall i$. Since the values of the buffer radius are discrete, a point $\rho_{i} ,\xi_{,max}^{'} \left( {\rho_{i} } \right))$ is a peak if and only if

$$ \left\{ {\begin{array}{*{20}c} {\xi_{max}^{'} \left( {\rho_{i} } \right) > \xi_{max}^{'} \left( {\rho_{i - 1} } \right)} \\ {\xi_{max}^{'} \left( {\rho_{i} } \right) > \xi_{max}^{'} \left( {\rho_{i + 1} } \right)} \\ \end{array} } \right. $$

(3)

This discrete nature also produces a lot of small noisy peaks. For practical purpose, these noisy peaks could be filtered out by offsetting the entire profile of $\xi_{max}^{'} \left( \rho \right)$ by a sufficiently small amount (usually less than 0.5% of the maximum possible peak, which is 1, after normalisation) and considering only the positive remaining peaks.

3.2.2 Characteristic Distances Among Points

Once the peaks have been properly identified, the characteristic distance $\rho_{\xi }^{{ \star }}$ could be calculated to be the mean of $\rho$ of all peaks, as the representative value of $\rho$ the transition in largest cluster size takes place, since all peaks in the derivative $\xi_{max}^{'} \left( \rho \right)$ contribute to the growth of the cluster $\xi_{max} \left( \rho \right)$. However, since not every peak is contributing equally to the growth, a measure of the characteristic distance $\rho_{\xi }^{{ \star }}$ must take into account the effects of their strength. It follows that a high peak indicates a more significant increase in cluster size (a major merger) than those indicated by a lower one. Hence, the average of all values of $\rho_{\xi ,i}^{\dag }$ at which a peak i occurs, weighted by the height $\xi_{max}^{'} \left( {\rho_{\xi ,i}^{\dag } } \right) = \left. {\frac{{d\xi_{max} \left( \rho \right)}}{d\rho }} \right|_{{\rho = \rho_{\xi ,i}^{\dag } }}$ of the peaks, is an appropriate measure of this characteristic distance, i.e.

$$ \rho_{\xi }^{{ \star }} = \frac{{\mathop \sum \nolimits_{i} \xi_{max}^{'} \left( {\rho_{\xi ,i}^{\dag } } \right)\rho_{\xi ,i}^{\dag } }}{{\mathop \sum \nolimits_{i} \xi_{max}^{'} \left( {\rho_{\xi ,i}^{\dag } } \right)}} $$

(4)

Similarly, we have for the cluster area

$$ \rho_{A}^{{ \star }} = \frac{{\mathop \sum \nolimits_{i} A_{max}^{'} \left( {\rho_{A,i}^{\dag } } \right)\rho_{A,i}^{\dag } }}{{\mathop \sum \nolimits_{i} A_{max}^{'} \left( {\rho_{A,i}^{\dag } } \right)}} $$

(5)

It should be noted that every peak in $\xi_{max}^{'} \left( \rho \right)$ signifies the existence of one or a few clusters of points located at a farther distance than those in the current largest cluster. This thus provides us with information on the length scales of distribution of points within the set. If there are many peaks, the points are distributed in clusters separated by different distances. On the other hand, the existence of a few peaks implies an almost uniform distribution of points that are (approximately) equidistant from one another. In either case, it is without a doubt that there exists a characteristic distance in the spatial distribution of points. This characteristic distance should tell us the length scale above which the points are (largely) connected and below which they are disconnected.

It is noteworthy that $\rho_{\xi }^{{ \star }}$ and $\rho_{A}^{{ \star }}$ are different from the average of pairwise distance among all points in the set because they encode the connectivity of the points in terms of spatial distribution. In other words, the two characteristic distances are the measure of typical distance between points in the set in the perspective of global connectivity of all points. In the context of points in an urban system, they translate to the distance one has to traverse to get from one point to another in order to explore the entire system. It then follows that a large value of characteristic distance implies a sparsely distributed set of points, i.e. a low density of points per unit area (Huynh et al. 2018).

The analyses of peaks in size profile $\xi_{max}^{'} \left( \rho \right)$ and area profile $A_{max}^{'} \left( \rho \right)$ provide different perspectives on the spatial distribution of points. The size quantifies the number of points with respect to the distance while the area further takes into account the relative position of the points. There is no redundancy in the consideration of the two measures, but rather, one is complementary to the other. This comes to light in the next Sect. 3.2.3 when the combination of the two allows us to classify distinct types of distribution of points.

3.2.3 Quantification and Classification of the Spatial Distributions of Points

The characteristic distances introduced above inform us on the points of transition of cluster size and area but they do not tell us how the size and area of the cluster transit from small to large value, i.e. how the cluster grows. This, however, can be easily characterised by further exploiting the analysis of peaks in $\xi_{max}^{'} \left( \rho \right)$ (and $A_{max}^{'} \left( \rho \right)$) discussed in Sec. 3.2.1. As discussed above, if the cluster grows rapidly through the transition, there are very few peaks in $\xi_{max}^{'} \left( \rho \right)$, and all of which are sharp and localised. On the other hand, the peaks are scattered over a wide range of $\rho$ should the cluster grow gradually. The standard deviation of the location $\rho_{\xi ,i}^{\dag }$ of the peaks, or the spread of transition $\sigma_{\xi }$ for cluster size, is a good measure of such scattering. Again, however, a low peak that is distant from a group of localised high peaks should not significantly enlarge the spread. Therefore, the standard deviation of $\rho_{\xi ,i}^{\dag }$ needs to be weighted by the height $\xi_{max}^{'} \left( {\rho_{\xi ,i}^{\dag } } \right)$ of the peaks, i.e.

$$ \sigma_{\xi } = \sqrt {\frac{{\mathop \sum \nolimits_{i} \xi_{max}^{'} \left( {\rho_{\xi ,i}^{\dag } } \right)\left( {\rho_{\xi ,i}^{\dag } - \rho_{\xi }^{{ \star }} } \right)^{2} }}{{\mathop \sum \nolimits_{i} \xi_{max}^{'} \left( {\rho_{\xi ,i}^{\dag } } \right)}}} $$

(6)

Similarly, we have the spread of transition for cluster area

$$ \sigma_{A} = \sqrt {\frac{{\mathop \sum \nolimits_{i} A_{max}^{'} \left( {\rho_{A,i}^{\dag } } \right)\left( {\rho_{A,i}^{\dag } - \rho_{A}^{{ \star }} } \right)^{2} }}{{\mathop \sum \nolimits_{i} A_{max}^{'} \left( {\rho_{A,i}^{\dag } } \right)}}} $$

(7)

An illustration of the characteristic distance $\rho_{\xi }^{{ \star }}$ (and $\rho_{A}^{{ \star }}$) and the spread $\sigma_{\xi }$ (and $\rho_{A}$) is depicted in Fig. 3.

Fig. 3

Illustration of the quantities yielded from analysis of peaks in the size and area profile of the largest cluster

Once we have quantified the measures of the largest cluster size and largest cluster area, we shall argue that the combination of these quantities allows us to characterise the pattern of relative position of points in a set. In particular, the combination of the two spreads of transition defined in Eqs. (6) and (7) enables us to construct the $\left( {\sigma_{\xi } ,\sigma_{A} } \right)$ diagram (see Fig. 4) to discuss different regions corresponding to different types of spatial point distribution. In general, there are four types of distribution that can be identified from the spreads of transition in size $\sigma_{\xi }$ and area $\sigma_{A}$. The first one is the region of small $\sigma_{\xi } \approx \sigma_{A}$ in which both $\xi_{max} \left( \rho \right)$ and $A_{max} \left( \rho \right)$ exhibit a sharp rise. This is the case of a system of regularly distributed points in which the points are (approximately) equally distant from each other, e.g. grid points. Outside this region, going along the diagonal line of $\sigma_{\xi } \approx \sigma_{A}$, the second type could be identified in which both $\xi_{max} \left( \rho \right)$ and $A_{max} \left( \rho \right)$ exhibit a gradual increase and almost every peak in $\xi_{max}^{'} \left( \rho \right)$ has a respective peak in $A_{max}^{'} \left( \rho \right)$. When the $\sigma_{\xi }$ and $\sigma_{A}$ differ significantly, we have two other types of pattern, one below and one above the diagonal line. To discuss these two types of pattern, we recall that the size of a cluster is the number of points in that cluster, whereas its area the union area of circles of radius $\rho$ centred at the points in the cluster. Above the diagonal, in the region of $\sigma_{\xi } \gg \sigma_{A}$, the peaks in $\xi_{max}^{'} \left( \rho \right)$ tend to spread over a wider range of $\rho$ than those in $A_{max}^{'} \left( \rho \right)$, implying clustered points and compact coverage area. In such distributions, there are jumps in the size of the largest cluster size that do not give rise to a jump in its area. This happens when the points of an acquired cluster are compact, contributing very little increase in the area of the largest cluster. If the acquired cluster is not compact, i.e. its points span a larger area, there might be a significant increase in the area of the largest cluster. On the other side, below the diagonal, in the region of $\sigma_{\xi } \ll \sigma_{A}$, the peaks in $\xi_{max}^{'} \left( \rho \right)$ tend to be more localised than those in $A_{max}^{'} \left( \rho \right)$, implying dispersed points and broad coverage area. In such distributions, there are jumps in the area of the largest cluster area that do not give rise to a jump in its size. This happens when the points of an acquired cluster are stretched apart (but still within the buffer radius so that they belong to the same cluster). This way, the increase in the area of the largest cluster is more significant than that in its size.

Fig. 4

Interpretation of different patterns of spatial point distributions given different values of the pair $\left( {\sigma_{\xi } ,\sigma_{\text{A}} } \right)$

3.3 Comparison with Other Methods

There are a few well-established methods in the literature to analyse the spatial pattern of a set of points. Among them, the most popular method in the geographical analysis is probably the Ripley’s K-function (Ripley 1976, 1977). The method of pair correlation function (Stoyan and Penttinen 2000) is also widely used in the field of forestry to understand the pattern of trees’ location in an area. These methods are mainly concerned with the null hypothesis whether the pattern of the points in the set is complete spatial randomness (CSR). If not CSR, the pattern can further be determined to be either regular or clustered.

The Ripley’s K-function method centralises about the function K, which is defined as the expected number of neighbours within distance d from a randomly chosen point rescaled by the density $\lambda$ of points in the domain of interest. The function K(d) of the set of points is then compared against that of a homogeneous Poisson process⁴ (or CSR pattern), which is known to be $K_{Poisson} \left( d \right) = \pi d^{2}$. When the value K(d) is larger than that of $K_{Poisson} \left( d \right)$, the points are said to be clustered at distance d because more neighbours are expected within distance d of a point than in a Poisson process. Conversely, when the value K(d) is smaller than that of $K_{Poisson} \left( d \right)$, the points are said to be regular at distance d. On the other hand, the pair correlation function method utilises the pair correlation function g between two points at distance u (Penttinen et al. 1992) which is related to the Ripley’s K-function via

$$ K\left( d \right) = \mathop \smallint \limits_{0}^{d} g\left( u \right)2\pi d{\text{d}}u $$

In both cases, the function K (or g), however, involves the density $\lambda$ of points in the domain and, hence, requires a proper identification of the area of the domain. An overestimate or underestimate of the density can significantly alter the behaviour of the function K. Visually, an immense domain seeing the points clustering in a small area would yield very small $\lambda$ and, hence, significantly boost up K(d), making it incorrectly larger than $K_{Poisson} \left( d \right)$. Contrarily, a tight domain may artificially make K(d) smaller than $K_{Poisson} \left( d \right)$. In the study of forestry, the domain is usually taken to be the range of the coordinates of the trees’ location, which is a rectangular bounding box, due to the nature of the observation. However, in the case of urban entities like road junctions, the bounding box might not be a proper identification of the domain of the points because of their irregular distribution pattern due to unknown underlying urban growth processes. These methods could be difficult to apply in analysing the spatial patterns of points in an urban context like in the distribution of road junctions.

4 Interpretation of the Patterns in Urban Morphology

In the coming section, we discuss how the analysis method described in Sect. 3.2 could be applied to quantify different patterns in urban morphology. For this discussion, we refer to the points in an urban system as spatial locations of important entities that could be used as a representative set of points in an urban system. Such a set could nominally be, for example the locations of road junctions. In a recently published study, the method has been applied to a case study of spatial distribution of locations of public transport points in different cities in the U.S. (Huynh et al. 2018).

4.1 Single-scale Regular Pattern

The bottom-left corner of the $\left( {\sigma_{\xi } ,\sigma_{A} } \right)$ plot (Fig. 4) represents the set of points whose values of the spreads $\sigma_{\xi }$ and $\sigma_{A}$ are very small. That means the profiles of $\xi_{max} \left( \rho \right)$ and $A_{max} \left( \rho \right)$ exhibit sharp transition with localised peaks in both $\xi_{max}^{'} \left( \rho \right)$ and $A_{max}^{'} \left( \rho \right)$. This signifies a characteristic length scale at which most of the points are (approximately) equally spaced from each other, e.g. grid points (see Fig. 5). There is a sharp transition at the value of the characteristic distances $\rho_{\xi }^{{ \star }} \approx \rho_{A}^{{ \star }}$: the largest cluster transits from occupying a modest fraction to almost the entire of the points in the set. One could take the boroughs of Bronx, Brooklyn and Manhattan of New York city as typical examples of such kind of distribution. The pattern of spatial points in these cities appears very regular. In fact, by inspecting their street patterns, one can easily tell the pattern of parallel roads in one direction cutting those in the other, dividing the land into well-organised polygons with almost perfect square and rectangular shapes. Apparently, this feature was a result of top-down planning and design that happened before the infrastructure was being built in the city (Barthelemy et al. 2013).

Fig. 5

An example of single-scale regular point pattern. The pattern is generated by adding small random displacement to points of a regular lattice, which comprises 2025 (45 × 45) points spanning an area of 10 km × 10 km. The values of the characteristic distances are $\rho_{\upxi}^{{ \star }} = 240\,{\text{m}}$ and $\rho_{\text{A}}^{{ \star }} = 240\,{\text{m}}$. The measures of spread of transition are $\sigma_{\xi } = 0\,{\text{m}}$ and $\sigma_{\text{A}} = 0\,{\text{m}}$ (there is only a single peak, resulting in vanishing standard deviation as obtained from Eqs. 6 and 7; in general, $\sigma_{\xi }$ and $\sigma_{\text{A}}$ should stay small)

4.2 Multi-scale Regular Pattern

The set of points in a city can also be distributed in a regular manner but at different length scales (Chen and Wang 2013; Nie et al. 2015). For example, the entire set of points can be divided into several subsets and within each subset, the points are (quasi-)equally distant from each other (see Fig. 6). At larger length scale, i.e. $\rho$ increases further, these subsets of points are again (quasi-)equally distant from each other, i.e. hierarchical structure. The buffer radius $\rho$ can thus be thought to play the role of a zooming parameter. In this multi-scale regular pattern, the profile of the largest cluster size $\xi_{max} \left( \rho \right)$ and area $A_{max} \left( \rho \right)$ experience a significant jump every time $\rho$ changes its zooming level. At the lowest level are individual transport point. When $\rho$ zooms out to the second level, the points that are closest to each other start to form their respective clusters. Moving to the next level, the nearby clusters start joining to form larger cluster but there will be many of these ‘larger clusters’, i.e. the largest cluster is of comparable size or area to several other clusters. The most important feature of this spatial pattern is that the jumps in the profile of $\xi_{max} \left( \rho \right)$ correspond well to those in $A_{max} \left( \rho \right)$, even though the locations of the jumps are spread apart. That leads to the (approximate) equality of the spread of transitions $\sigma_{\xi }$ and $\sigma_{A}$ despite their not being small.

Fig. 6

An example of multiple-scale regular point pattern. The pattern is generated by superimposing different sets of single-scale pattern (of different densities) like in Sect. 4.1. There are in total 2041 points spanning an area of 10 km × 10 km. The values of the characteristic distances are $\rho_{\xi }^{{ \star }} = 315.080\,{\text{m}}$ and $\rho_{\text{A}}^{{ \star }} = 314.357\,{\text{m}}$. The measures of spread of transition are $\sigma_{\xi } = 23.185\,{\text{m}}$ and a similar value for $\sigma_{\text{A}} = 25.172\,{\text{m}}$

4.3 Clustered Pattern

There are cases in which the jumps in the profile of largest cluster size $\xi_{max} \left( \rho \right)$ do not correspond to those in the area $A_{max} \left( \rho \right)$ and vice versa. In such cases, the spatial distribution of the transport points deviates from regular patterns. We first consider the scenarios in which $\sigma_{\xi } \gg \sigma_{A}$. For such distributions, the points are clustered and tend to minimise the coverage area (see Fig. 7). When $\sigma_{\xi } \gg \sigma_{A}$, there are jumps in the size of the largest cluster size that do not give rise to a jump in its area. This happens when the points of an acquired cluster are compact, contributing very little increase in the area of the largest cluster. If the acquired cluster is not compact, i.e. its points span a larger area, there might be significant increase in the area of the largest cluster and, hence, a peak would be reflected by its contribution to $\sigma_{A}$. However, the size measure is not affected as it only tells the number of points that are included in the cluster but not their relative location with respect to each other.

Fig. 7

An example of clustered point pattern. The pattern is generated by first defining a fixed number of centres (3 in this case) and then randomly adding points surrounding them with density decaying away from the centres. There are in total 2000 points spanning an area of 10 km × 10 km. The values of the characteristic distances are $\rho_{\xi }^{{ \star }} = 258.747\,{\text{m}}$ and $\rho_{\text{A}}^{{ \star }} = 277.752\,{\text{m}}$. The measures of spread of transition are $\sigma_{\xi } = 65.474\,{\text{m}}$ and a smaller value for $\sigma_{\text{A}} = 17.603\,{\text{m}}$

4.4 Dispersed Pattern

At the other end, we have the scenarios of $\sigma_{\xi } \ll \sigma_{A}$, in which the points are dispersed and tend to maximise the coverage area. When $\sigma_{\xi } \ll \sigma_{A}$, there are jumps in the area of the largest cluster that do not give rise to a jump in its size. This happens when the points of an acquired cluster are dispersed (but still within the buffer radius so that they belong to the same cluster, see Fig. 8). This way, the increase in the area of the largest cluster is more significant than that in its size, resulting $\sigma_{\xi } \ll \sigma_{A}$.

Fig. 8

An example of dispersed point pattern. The pattern is generated by scattering points with uneven density. There are in total 2000 points spanning an area of 10 km × 10 km. The values of the characteristic distances are $\rho_{\xi }^{{ \star }} = 124.322\,{\text{m}}$ and $\rho_{\text{A}}^{{ \star }} = 191.146\,{\text{m}}$. The measures of spread of transition are $\sigma_{\xi } = 30.875\,{\text{m}}$ and a larger value for $\sigma_{\text{A}} = 67.361\,{\text{m}}$

If the feature of single-scale regular spatial pattern (when both $\sigma_{\xi }$ and $\sigma_{A}$ are small) is a result of well-designed and top-down planning in an urban system, the other spatial patterns (either $\sigma_{\xi }$ or $\sigma_{A}$ is not small) can be interpreted as a consequence of urban system development under local constraints. In the former case, the urban system appears to be of organised type while in the latter, it can be said to be of organic type because its spatial features develop in an ad hoc manner as the city grows. Thus, our analysis on the spatial patterns in urban systems provides more nuances of their morphology and insights into different possible growth processes as the systems progress through their development.

The proposed method in this chapter provides quantitative measures of spatial pattern of a set of points, that can be applied to, for example, the road junctions in the context of urban systems. The public road junctions play an important role in forming the backbone of any modern urban system. For example, street network serves the essential role of enabling flow or exchange of various processes in the city, whose overall patterns could be captured by the intersections, which in turn define the pattern of land parcels.

The results from the analysis in this work unveil different types spatial patterns in urban systems. The patterns are shown to be either of organised type, in which the entities are well spaced as if they were built top-down, or of organic type, in which the entities are spaced with multiple length scales as if they grew spontaneously (Cheng et al. 2003; Makse et al. 1998; Longley et al. 1992). It should be noted that while these two types of patterns have been discussed in other literature like architecture or urban geography, the emphasis of this study is the quantification of the perceived spatial patterns found in different urban systems, via $\sigma_{\xi }$ and $\sigma_{A}$, that allows comparison among them.