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Comparison of relative density of two random geometric digraph families in testing spatial clustering

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Abstract

We compare the performance of relative densities of two parameterized random geometric digraph families called proximity catch digraphs (PCDs) in testing bivariate spatial patterns. These PCD families are proportional edge (PE) and central similarity (CS) PCDs and are defined with proximity regions based on relative positions of data points from two classes. The relative densities of these PCDs were previously used as statistics for testing segregation and association patterns against complete spatial randomness. The relative density of a digraph, D, with n vertices (i.e., with order n) represents the ratio of the number of arcs in D to the number of arcs in the complete symmetric digraph of the same order. When scaled properly, the relative density of a PCD is a U-statistic; hence, it has asymptotic normality by the standard central limit theory of U-statistics. The PE- and CS-PCDs are defined with an expansion parameter that determines the size or measure of the associated proximity regions. In this article, we extend the distribution of the relative density of CS-PCDs for expansion parameter being larger than one, and compare finite sample performance of the tests by Monte Carlo simulations and asymptotic performance by Pitman asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each alternative in finite samples and in the limit as the sample size tending to infinity. As a result of our comparisons, we demonstrate that in terms of empirical power (i.e., for finite samples) relative density of CS-PCD has better performance (which occurs for expansion parameter values larger than one) for the segregation alternative, while relative density of PE-PCD has better performance for the association alternative. The methods are also illustrated in a real-life data set from plant ecology.

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Abbreviations

PCD:

proximity catch digraph

CCCD:

class cover catch digraph

CS-PCD:

central similarity PCD

PE-PCDs:

proportional edge PCD

PAE:

Pitman asymptotic efficiency

CSR:

complete spatial randomness

NNCT:

nearest neighbor contingency table

NN:

nearest neighbor

D(V,A):

Vertex random PCD with vertex set V and arc set A. See Sect. 2, paragraph 1.

N(⋅) and N(x):

Proximity map and the proximity region associated with a point x. See Sect. 2, paragraph 5.

\(\mathcal {C}_{H}(\mathcal {Y}_{m})\) :

Convex hull of \(\mathcal {Y}_{m}\). See Sect. 2, paragraph 7.

\(T(\mathcal {Y}_{3})=T(\mathsf {y}_{1},\mathsf {y}_{2},\mathsf {y}_{3})\) :

The triangle with vertices y 1,y 2,y 3. See Sect. 2, paragraph 7.

\(\mathcal {S}(F)\) :

Support of the distribution F and \(\mathcal {U}(T (\mathcal {Y}_{3} ))\): Uniform distribution on \(T (\mathcal {Y}_{3} )\). See Sect. 2, paragraph 7.

N PE(x,r):

The proportional edge proximity map with expansion parameter r. See Sect. 2, paragraph 8.

R V (y 1), R V (y 2), and R V (y 3):

The vertex regions for vertices y 1,y 2,y 3. See Sect. 2, paragraph 8.

N CS(x,τ):

The central similarity proximity map with expansion parameter τ. See Sect. 2, paragraph 9.

R E (e 1), R E (e 2), R E (e 3):

The edge regions for edges e 1,e 2,e 3 opposite to the vertices y 1,y 2,y 3 See Sect. 2, paragraph 9.

ρ(D):

The relative density of the digraph D. See Sect. 3, paragraph 1.

ρ PE(n,r) and ρ CS(n,τ):

The relative densities of PE-PCDs and CS-PCDs, respectively. See Theorem 3.1.

μ PE(r) and ν PE(r):

The arc probability (or the asymptotic mean) and asymptotic variance of relative density of PE-PCDs. See Sect. 3.1, paragraph 4.

μ CS(r) and \(\nu_{_{Cs}}(r)\) :

The arc probability (or the asymptotic mean) and asymptotic variance of relative density of CS-PCDs. See Sect. 3.1, paragraph 4.

\(\widetilde{\rho}_{\mathrm{PE}}(n,m,r)\) and \(\widetilde{\rho}_{\mathrm{CS}}(n,m,\tau)\) :

The relative density for the PE-PCD and CS-PCD in the multiple triangle case. See Sect. 3.2, paragraph 2.

\(\widetilde{\mu}_{\mathrm{PE}}(m,r)\) and \(\widetilde{\nu}_{\mathrm{PE}}(m,r)\) :

The asymptotic mean and asymptotic variance of relative density of PE-PCDs in the multiple triangle case. See Corollary 3.4.

R PE(r):

The standardized test statistic based on the relative density of PE-PCD in the one-triangle case. See (13).

\(\widetilde{R}_{\mathrm{PE}}(r)\) :

The standardized test statistic based on the relative density of PE-PCD in the multi-triangle case. See Sect. 4.1, paragraph 3.

\(\operatorname {PAE}_{\mathrm{PE}}(r)\) and \(\operatorname {PAE}_{\mathrm{CS}}(\tau)\) :

Pitman asymptotic efficiency score for relative density of PE-PCD and CS-PCD, respectively. See Sect. 7, paragraph 2.

\(\pi_{\operatorname {out}}\) and \(\widehat{\pi}_{\operatorname {out}}\) :

Proportion of class 1 points outside the convex hull of class 2 points and its estimate. See Sect. 8, paragraphs 1 & 2, respectively.

C ch :

The correction coefficient for the class 1 points outside the convex hull of class 2 points. See (14).

\(\widetilde {R}^{\mathrm{ch}}_{\mathrm{PE}}(r)\) and \(\widetilde {R}^{\mathrm{ch}}_{\mathrm{CS}}(\tau)\) :

The convex hull corrected versions of the standardized test statistics based on the relative density of PE- and CS-PCDs. See (15).

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Acknowledgements

I would like to thank an anonymous associate editor and referees whose constructive comments and suggestions greatly improved the presentation and flow of the paper. Most of the Monte Carlo simulations presented in this article were executed at Koç University High Performance Computing Laboratory. This research was supported by the European Commission under the Marie Curie International Outgoing Fellowship Programme via Project # 329370 titled PRinHDD.

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  1. Department of Mathematics, College of Sciences, Koç University, 34450 Sarıyer, Istanbul, Turkey

    Elvan Ceyhan

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Ceyhan, E. Comparison of relative density of two random geometric digraph families in testing spatial clustering. TEST 23, 100–134 (2014). https://doi.org/10.1007/s11749-013-0344-4

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