Characterisation of spatial patterns

Spatial point process methods are useful tools to analyse point distributions in various areas of study (e.g. biology, ecology, epidemiology, seismic, and geologic).

Point pattern analysis can be tested by measuring second-order properties to determine spatial dependence, based on measuring the distances between each data point (Bailey and Gatrell 1995). Such second-order techniques include: (1) the nearest-neighbour distance method (Clark and Evans 1954); (2) Ripley’s K function (Ripley 1977; Cressie 1993); (3) Besag’s L function (Besag 1977); and (4) lacunarity (Plotnick et al. 1996). Spatial point process analysis tests whether the architectural elements of channelised fluvial sandbodies are distributed randomly (every point is equally likely to occur at any location and the position of each point is not effected by the position of any other point), clustered (point distributions are concentrated close together and are surrounded by large areas that contain very few data points), or regularly (each data point is positioned at equal distances apart from every other point) (Diggle 1983; Hajek et al. 2010). The statistical compensation index, which measures the degree in which fluvial or marine stratigraphic architecture fills basins randomly or by compensational stacking,has also been used to compare stacking patterns of channelised fluvial sandbodies (Straub et al. 2009; Wang et al. 2011;

Hajek and Wolinsky 2012). Lacunarity and Besag’s L function are useful methods to apply to this study because they can summarise spatial distributions over a wider range of scales (Plotnick et al. 1996), are independent of the shape of the study area (Cressie 1993), assume isotropy (Dixon 2002), can be easily applied to geologic datasets (e.g. Hajek et al. 2010; Zhao et al. 2011; Flood and Hampson 2015), and generate clear and easily interpretable graphic outputs (Plotnick 1999; Ripley 1977).

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Isotropy displays the same spatial pattern in a variety of different directions whereas anisotropy displays variable spatial patterns in a variety of directions (Fig. 2.19).

Figure 2.19 Schematic diagrams to illustrate isotropic and anisotropic point pattern distributions. A) Points are arranged randomly with respect to direction within each cluster and represent an isotropic distribution. B) Points are aligned in a NW to SE direction within a cluster and represent an anisotropic distribution (after Rosenberg and Anderson 2011

2.5.1. Lacunarity

Lacunarity is a scale-dependent measure of spatial dispersion (Elliot 1977), and characterises the distribution of gap sizes, as a function of scale (Gefen et al. 1983;

Mandelbrot 1983; Allain and Cloitre 1991; Plotnick et al. 1996). Although the potential uses of lacunarity are still relatively unclear, this technique has been recently used in ecology (e.g. Plotnick et al., 1993) and for some geological studies (e.g. Plotnick et al., 1996). Algorithms for calculating lacunarity (e.g. gliding-box algorithm, standard box-counting algorithm, and differential box-counting algorithm) have been developed by numerous authors (Mandlebrot 1983; Gefen et al. 1984; Lin and Yang 1986; Allain and Cloitre 1991). A square of a given length (𝐿𝐿) is placed at

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the top left origin of a binary black and white image, and the box mass is subsequently scanned and counted (Allain and Cloitre 1991; Plotnick et al. 1993).

The box is then moved one column along to the right and the process is repeated over all rows and columns until the entire area of the image is scanned and counted.

The mean value of lacunarity is defined as:

(𝜆𝜆) = 𝐿𝐿(𝐹𝐹) =^�

𝛴𝛴 [𝐹𝐹Ʌ]�

𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺

(F) represents the total number of pixels in the scanned part of the image per box count. ‘Grids’ refers to the total number of times a complete scan was run over the

Figure 2.20 Examples of different spatial distribution patterns that consist of a similar number of points. A) Clustered, B) random, and C) regular distributions are illustrated. The value of lacunarity increases from regular to clustered spatial patterns (after Hsui and Wang 2013).

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entire image. (Λ) represents the average value of lacunarity over all grid sizes (Karperien 2007). The values of lacunarity are dependent on the geometry of the pattern in which the objects are distributed (Plotnick et al., 1993). A low value of lacunarity (minimum = 0) is suggestive of a homogeneous and translationally invariant spatial pattern consisting of gap sizes of similar size (low range of gap sizes) (Fig. 2.20). In comparison, a high value of lacunarity (maximum = 1) indicates a heterogeneous pattern which displays a varied range of gap sizes (Plotnick et al.

1993; 1996) (Fig. 2.20).

2.5.2. Ripley’s K function and Besag’s L function

Analysis of second-order spatial point-process patterns commonly involves the use of Ripley’s K function (Ripley 1977), which measures the extent of clustering and spatial dispersion over different length scales. The K function compares the predicted number of points within a distance (h) of each event in the study area to the average rate of the point process (𝜆𝜆) as outlined in the following equation:

𝐾𝐾(ℎ)=𝜆𝜆⁻¹𝐸𝐸 �𝑁𝑁(ℎ)� 𝑓𝑓𝑓𝑓𝑓𝑓 ℎ > 0

where 𝜆𝜆 is the number of centroid points in the study area (N) divided by the total area of the study region, and E(N(h)) is the expected number of points in the same region (Cressie 1991; Hajek et al. 2010). Besag’s L function (Besag 1977) is often easier to interpret because it allows an appropriate comparison of an observed distribution against a benchmark of zero (𝐿𝐿�(ℎ) = 0) (Besag 1977; Rosenberg and Anderson 2011). Besag’s L function (Besag 1977), is a linearized version of Ripley’s K function and defines the spatial scales at which points tend to be more or less

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clustered than expected by chance (Besag and Diggle 1977; Ripley 1977; Cressie 1991). Statistical isotropy (i.e. that a unit of distance in one direction has the same result as a unit of distance in an alternative direction; Dixon 2002) allows every centroid in a two-dimensional panel to display a constant mean and a constant variance (Masihi et al. 2006). The normalized L function is defined as:

𝐿𝐿�(ℎ) = ℎ − �^{𝐾𝐾�(ℎ)}_𝜋𝜋

To test whether 𝐿𝐿(ℎ) varies significantly different from zero (𝐿𝐿�(ℎ) = 0), the Monte Carlo technique is applied (Diggle 1983). If 𝐿𝐿�(ℎ) < 0, points are arranged in a clustered distribution, the L function results plot negatively below the CSR envelope (Fig. 2.21A; Besag 1977). If the expected value of 𝐿𝐿(ℎ) found at a certain scale is equal to the number of points estimated, taking into account the intensity of the point process, then 𝐿𝐿�(ℎ) = 0 and the distribution pattern represents complete spatial randomness (CSR; Besag 1977; Dixon 2002) (Fig. 2.21B). In contrast, if 𝐿𝐿�(ℎ) > 0, then points are regularly dispersed, and the L function plots positively above the CSR envelope (Fig. 2.21C; Besag 1977). The confidence intervals that define the minimum and maximum limits of complete spatial randomness are determined by running a number of Monte Carlo test simulations (Besag 1977; Diggle 1983). A 95%

confidence interval limit is determined by running 99 individual simulations using a set of randomly distributed ‘newly’ constructed data points which are placed within the confines of each panel area (Lancaster and Downes 2004; Hajek 2010). The

‘new’ points are analysed, and the upper and lower 95% confidence interval limits are plotted in combination with the values obtained from analysis of the original coordinates (Monte Carlo significance level, α = 0.05).

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Figure 2.21 Graphical results for the L function for: A) clustered, B) random, and C) regularly distributed points, respectively. CSR = complete spatial randomness (after Hajek et al. 2010).

The ‘edge effect’ can create statistical errors if centroid points are situated within close proximity to the boundary area of the panel (Haase 1995). Various methods have been produced to correct for the edge effect (e.g. Besag 1977; Ripley 1977;

Diggle 1983). Ripley’s weighted method (Ripley 1988) centres a circle, of radius (r) around every centroid within the region which is situated inside of a smaller panel area (Fig. 2.22). The smaller panel area is 75% of the total panel areas and is centred within the confines of the whole panel area (Rosenberg and Anderson 2011).

Ripley’s method weights each count based on how much of a circles proportion, falls within the confines of the study area (Ripley 1977). All centroids are counted if they

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fall within a circle, irrespective of whether they are situated inside or outside of the smaller panel area (Rosenberg and Anderson 2011).

Figure 2.22 schematic diagram illustrating the weighted edge correction technique.

Circles of a given radius h are centred on centroids situated inside of the red rectangle (75% smaller than the full plot). All centroids are counted if they fall within a circle, irrespective of whether they are situated inside or outside of the red rectangle (after Rosenberg and Anderson 2011).