Corrected pair correlation functions for environments with obstacles

Environments with immobile obstacles or void regions that inhibit and alter the motion of individuals within that environment are ubiquitous. Correlation in the location of individuals within such environments arises as a combination of the mechanisms governing individual behavior and the heterogeneous structure of the environment. Measures of spatial structure and correlation have been successfully implemented to elucidate the roles of the mechanisms underpinning the behavior of individuals. In particular, the pair correlation function has been used across biology, ecology, and physics to obtain quantitative insight into a variety of processes. However, naively applying standard pair correlation functions in the presence of obstacles may fail to detect correlation, or suggest false correlations, due to a reliance on a distance metric that does not account for obstacles. To overcome this problem, here we present an analytic expression for calculating a corrected pair correlation function for lattice-based domains containing obstacles. We demonstrate that this obstacle pair correlation function is necessary for isolating the correlation associated with the behavior of individuals, rather than the structure of the environment. Using simulations that mimic cell migration and proliferation we demonstrate that the obstacle pair correlation function recovers the short-range correlation known to be present in this process, independent of the heterogeneous structure of the environment. Further, we show that the analytic calculation of the obstacle pair correlation function derived here is significantly faster to implement than the corresponding numerical approach.

Figure 1

(a) Experimental image of the nervous system within the mouse colon, containing neurons (magenta), glial cells (green), and glial processes (white). Glial cells within clusters (known as ganglia) are highlighted with orange squares. Yellow lines indicate inaccessible regions. An example of path distance and Cartesian distance between cells are highlighted in cyan (dashed and solid, respectively). (b) Experimental image of pedestrian locations (red squares) in the presence of obstacles (yellow lines). Image is obtained from the freely available data set provided by the authors of Ref. [1].

Figure 2

Distance between the center lattice site (highlighted in red) and other lattice sites under the (a) rectilinear $x$, (b) rectilinear $y$, (c) taxicab, and (d) square uniform distance metrics. Note that blue sites correspond to a distance of zero.

Figure 3

Distance between the center lattice site (highlighted in red) and other lattice sites under (a), (e) rectilinear $x$, (b), (f) rectilinear $y$, (c), (g) taxicab, and (d), (h) square uniform distance metrics in the presence of (a)–(d) one or (e)–(h) nine inaccessible sites (cross-hatched). Note that blue sites correspond to a distance of zero.

Figure 4

Example domains with a single inaccessible site (cross-hatched). The color of individual sites corresponds to the distance between that site and the inaccessible site. Accessible-inaccessible pairs are highlighted in (a)–(c) green or (d)–(f) cyan, depending on whether the accessible site is within the same row or column as the inaccessible site.

Figure 5

(a) Example domain with a single inaccessible site (cross-hatched) and sites contributing to shifted pairs highlighted in pink. Shifted pairs consist of a pair of pink sites, where there is one site on either side of the inaccessible site. (b)–(e) Distance between an example lattice site (highlighted in red) and other lattice sites for (b)–(c) the row of pink sites and (d)–(e) the column of pink sites if the inaccessible site (b), (d) does not have to be avoided (green) or (c), (e) must be avoided. (c), (e) Note the equivalence between a subdomain with a single inaccessible site that must be avoided and a subdomain with three inaccessible sites that do not need to be avoided, with two additional sites in the subdomain.

Figure 6

(a) Example domain with a cluster of inaccessible sites (cross-hatched) and sites contributing to shifted pairs highlighted in pink. (b) Subdomain that contains all lost pairs for the row of pink sites in panel (a); that is, all pairs in this subdomain are contained within $A\left(m\right)$ but must be removed to incorporate the influence of the inaccessible site. (c) Subdomain that contains all gained pairs for the row of pink sites in panel (a); that is, all pairs in this subdomain are not contained within $A\left(m\right)$ but need to be included to incorporate the influence of the inaccessible site. (d) Subdomain, and associated transformation to multiple subdomains, which contain all lost pairs for the three columns of pink sites in panel (a). (e) Subdomain, and associated transformation to multiple subdomains, which contain all gained pairs for the three columns of pink sites in panel (a).

Figure 7

(a) Example domain with a cluster of inaccessible sites (cross-hatched) and sites contributing to shifted pairs highlighted in pink. (b)–(c) Subdomain, and associated transformation to multiple subdomains, which contain all lost and gained pairs, respectively, for the two rows of pink sites in panel (a). (d)–(e) Subdomain, and associated transformation to multiple subdomains, which contain all lost and gained pairs, respectively, for the three columns of pink sites in panel (a).

Figure 8

(a) Example domain with multiple inaccessible sites (cross-hatched) and sites contributing to shifted pairs highlighted in pink. (b) Subdomain, and associated transformation to multiple subdomains, for the leftmost column of pink sites. (c) Subdomain, and associated transformation to multiple subdomains, for the bottommost row of pink sites. Note that each transformed subdomain has both lost and gained pairs as described previously and that the gained pair subdomains are not shown.

Figure 9

(a), (c), (e), (g) Domains containing various configurations of inaccessible sites (black) and (b), (d), (f), (h) the corresponding count of pair distances obtained from the analytic expression (cyan) or numerical approach (black, dashed). The domain in panel (a) has zero inaccessible sites. The domain in panel (c) has one cluster of inaccessible sites, with 676 sites per cluster. The domain in panel (e) has 25 clusters of inaccessible sites, with 16 sites per cluster. The domain in panel (g) has 576 clusters of inaccessible sites, with one site per cluster.

Figure 10

(a), (d), (g), (j) Domains containing various configurations of inaccessible sites (black) with agents (red) randomly placed on accessible sites with the corresponding (b), (e), (h), (k) standard PCF, that is, the pair correlation calculated ignoring inaccessible sites, and (c), (f), (i), (l) oPCF. The dashed black line corresponds to no correlation. All PCFs are the average of 100 identically prepared domains.

Figure 11

(a), (d), (g), (j) Domains containing various configurations of inaccessible sites (black) with agents (red) located at accessible sites after undergoing a birth-movement random walk with the corresponding (b), (e), (h), (k) standard PCF, that is, the pair correlation calculated ignoring inaccessible sites, and (c), (f), (i), (l) oPCF. The dashed black line corresponds to no correlation. All PCFs are the average of 100 identically prepared domains and the subsequent realization of the random walk process.

Figure 12

(a)–(c) Pair distance count corresponding to the domains presented in Figs. 11, 11, and 11, respectively. Blue corresponds to pair distances obtained solely from $D_{\text{NO}}\left(m\right)$ (i.e., uncorrected), green corresponds to pair distances from $D_{\text{NO}}\left(m\right)$ corrected by $I\left(m\right)$ and $A\left(m\right)$, and orange represents the correction associated with $S\left(m\right)$. The corrected count of pair distances and the approximation are superimposed by the dashed black and red lines. (d)–(f) The approximate oPCF (blue) and exact oPCF (dashed black) for randomly located agents on the domains presented in Figs. 11, 11, and 11, respectively.

Figure 13

(a) Approximate oPCF for 50 randomly generated domains containing 100 (cyan), 200 (red), 300 (green), 400 (purple), or 500 (orange) inaccessible sites for agents placed at random on accessible sites. The exact oPCF is shown via the dashed black line. (b) Error (mean $\pm$ one standard deviation) in the approximate oPCF, compared to the exact result, as a function of the number of inaccessible sites.

Figure 14

(a), (d), (g), (j) Domains containing various configurations of inaccessible sites (black) with agents (red) randomly placed on accessible sites with the corresponding (b), (e), (h), (k) oPCF for $1\le m\le 80$ and (c), (f), (i), (l) oPCF for $1\le m\le L_x+L_y-2$. The dashed black line corresponds to no correlation. All PCFs are the average of 1000 identically prepared domains.