The Autocovariance Function for Marked Point Processes: A Comparison between Two Different Approaches

Capobianco, Rosa; Renshaw, Eric

doi:10.1002/(sici)1521-4036(199808)40:4<431::aid-bimj431>3.3.co;2-m

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…Lancaster et al 2003, Lancaster and Downes 2004)]. There are several possible test functions (Capobianco and Renshaw 1998, Stoyan and Penttinen 2000, Schlather 2001), which may all result in similar ecological interpretations (Parrott and Lange 2004), and I have used κ mm (t), the scaled counterpart of the k mm ‐function (Penttinen et al 1992) which is easy to interpret and apply in ecological applications: where: μ is the mean number of marks, averaged over all points, and the relationship between marks is described by the density function k mm (t). Empirically, k mm (t) is the mean product of all pairs of marks, i.e.…”

Section: Numerical Analysis Of Mppamentioning

confidence: 99%

Using neutral landscapes to identify patterns of aggregation across resource points

Lancaster

2006

Ecography

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Many organisms are aggregated within resource patches and aggregated spatially across landscapes with multiple resources. Such patchy distributions underpin models of population regulation and species coexistence, so ecologists require methods to analyse spatially‐explicit data of resource distribution and use. I describe a method for analysing maps of resources and testing hypotheses about how resource distribution influences the distribution of organisms, where resource patches can be described as points in a landscape and the number of organisms on each resource point is known. Using a mark correlation function and the linearised form of Ripley's K‐function, this version of marked point pattern analysis can characterise and test hypotheses about the spatial distribution of organisms (marks) on resource patches (points). The method extends a version of point pattern analysis that has wide ecological applicability, it can describe patterns over a range of scales, and can detect mixed patterns. Statistically, Monte Carlo permutations are used to estimate the difference between the observed and expected values of the mark correlation function. Hypothesis testing employs a flexible neutral landscape approach in which spatial characteristics of point patterns are preserved to some extent, and marks are randomised across points. I describe the steps required to identify the appropriate neutral landscape and apply the analysis. Simulated data sets illustrate how the choice of neutral landscape can influence ecological interpretations, and how this spatially‐explicit method and traditional dispersion indices can yield different interpretations. Interpretations may be general or context‐sensitive, depending on information available about the underlying point pattern and the neutral landscape. An empirical example of caterpillars exploiting food plants illustrates how this technique might be used to test hypotheses about adult oviposition and larval dispersal. This approach can increase the value of survey data, by making it possible to quantify the distribution of resource points in the landscape and the pattern of resource use by species.

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Section: Numerical Analysis Of Mppamentioning

confidence: 99%

Using neutral landscapes to identify patterns of aggregation across resource points

Lancaster

2006

Ecography

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“…Another function, the mark variogram γ mm ( r ), expresses the conditional expectation of the mark difference given that there are points at both locations, where

γ_{m m} false(r false) = double-struckE ([], \frac{1}{2} {false(m false(\circ false) - m false(r false) false)}^{2}), r \geq 0

. Finally, the mean product of marks U ( r ) for points separated by the distance r is defined by

U false(r false) = λ^{2} sans-serifg false(r false) k_{m m} false(r false) d {bolds}_{1} d {bolds}_{2}

, where

sans-serifg

( r ) is the pair correlation function, k mm ( r ) is the mark correlation function, and d s 1 and d s 2 are two infinitesimal small areas separated by the distance r (see Capobianco & Renshaw, ). Extending this function to the bivariate case, two functions are of interest, namely, the “auto”‐type and the “cross”‐type mean product of marks defined by

U_{i i} false(\cdot false) = λ_{i}^{2} {sans-serifg}_{i} false(\cdot false) k_{m m}^{i} false(\cdot false) d {bolds}_{i} d {bolds}_{i}^{'}

and

U_{i j} false(\cdot false) = λ_{i} λ_{j} {sans-serifg}_{i j} false(\cdot false) k_{m m}^{i j} false(\cdot false) d {bolds}_{i} d {bolds}_{j}

.…”

Section: Classical Analysis Of Qualitatively and Quantitatively Markementioning

confidence: 99%

Partial characteristics for marked spatial point processes

Eckardt

Mateu

2019

Environmetrics

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This paper introduces a new class of partial summary characteristics for two types of marked planar point processes. We consider purely qualitatively marked spatial point processes and multitype spatial point processes with quantitative marks. After a survey of classical functional summary characteristics for qualitatively and quantitatively marked point processes, the class of partial characteristics is presented. These characteristics are defined through the periodogram, which makes the computations efficient. We demonstrate the application of our method in the analysis of a multispecies tree data recorded in the City of Melbourne.

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The Autocovariance Function for Marked Point Processes: A Comparison between Two Different Approaches

Cited by 2 publications

References 6 publications

Using neutral landscapes to identify patterns of aggregation across resource points

Using neutral landscapes to identify patterns of aggregation across resource points

Partial characteristics for marked spatial point processes

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