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Table 1 Second-order summary statistics commonly used in spatial point-pattern analysis. In the univariate analysis, only one pattern is involved (e.g. one species, one size or age class, one life stage, etc.), while in the bivariate version two patterns (1 and 2) are investigated (e.g. two different species, two size classes, two life stages, etc.). For the mark correlation analysis, the most studied mark is by far tree diameter. In the case of random labelling analysis, the marks usually consist of tree status (e.g. dead vs living). For all analysis types, positive, negative, or absence of departure from the null model simulation envelopes occurs at a given scale r

From: Spatial point-pattern analysis as a powerful tool in identifying pattern-process relationships in plant ecology: an updated review

Statistic function

No departure from the null model

Positive departure

Negative departure

Examples of application

Value

Interpretation

Value

Interpretation

Value

Interpretation

Unmarked univariate analysis

O-ring statistic O(r) (Wiegand et al. 1999; Wiegand and Moloney 2004)

O(r) = λ

Points of the pattern are randomly distributed

O(r) > λ

Points of the pattern are aggregated

O(r) < λ

Points of the pattern are segregated

Hao et al. 2007; De Luis et al. 2008; Navarro-Cerrillo et al. 2013; Kang et al. 2014; Cordero et al. 2016; Hu et al. 2017; Miao et al. 2018; Bassil et al. 2018; Zhang et al. 2020; Wang et al. 2020b

 Pair-correlation function g(r) (Stoyan and Stoyan 1994)

g(r) = 1

Points of the pattern are randomly distributed

g(r) > 1

Points of the pattern are aggregated

g(r) < 1

Points of the pattern are segregated

Pélissier 1998; Wiegand et al. 2007a; Suzuki et al. 2008; LeMay et al. 2009; Comas et al. 2009; Batllori et al. 2010; Wang et al. 2010a; Zhang et al. 2010; Martínez et al. 2010; Lan et al. 2012; Liu et al. 2014; Petritan et al. 2014; Velázquez et al. 2014; Petritan et al. 2015; Wang et al. 2015 ; Jácome-Flores et al. 2016; Nguyen et al. 2016; Janík et al. 2016; Fibich et al. 2016; Gradel et al. 2017; Wang et al. 2017; Erfanifard and Stereńczak 2017; Collet et al. 2017; Ghalandarayeshi et al. 2017; Ziegler et al. 2017; Du et al. 2017; Omelko et al. 2018; Kuehne et al. 2018 ; Das Gupta and Pinno 2018; Carrer et al. 2018; Yuan et al. 2018; Muvengwi et al. 2018; Yang et al. 2018; Szmyt and Tarasiuk 2018; Erfanifard et al. 2018; Nguyen et al. 2018b; Yılmaz et al. 2019; Erfanifard et al. 2019; Li et al. 2020a; Ben-Said et al. 2020; Garbarino et al. 2020; Li et al. 2020b; Wang et al. 2020a; Meyer et al. 2020

 Ripley’s L-function L(r) (Ripley 1977; Besag 1977)

L(r) = 0

Points of the pattern are randomly distributed

L(r) > 0

Points of the pattern are aggregated

L(r) < 0

Points of the pattern are segregated

Kenkel 1988; Ward et al., 1996; Haase et al. 1996; Haase et al. 1997; Pélissier 1998; Eccles et al. 1999; Chen and Bradshaw 1999; Mast and Veblen 1999; Grau 2000; He and Duncan 2000; Mori and Takeda 2004; North et al. 2004; Motta and Lingua 2005; Motta and Edouard 2005; Suzuki et al. 2005; Fajardo et al. 2006; Gray and He 2009; Szmyt 2010; Nanami et al. 2011; Iszkuło et al. 2012; Zhang et al. 2013; Navarro-Cerrillo et al. 2013; Ebert et al. 2015; Wehenkel et al. 2015; Jácome-Flores et al. 2016; Jia et al. 2016; Zhang et al. 2016; Owen et al. 2017; Gradel et al. 2017; Zheng et al. 2017; Nguyen et al. 2018b; Omelko et al. 2018; Muvengwi et al. 2018; Vandekerkhove et al. 2018; Kazempour Larsary et al. 2018; Lv et al. 2019; Baran et al. 2020; Meyer et al. 2020

K-2 function K2-(r) (Schiffers et al. 2008)

K2(r) = 0

Points of the pattern are random

K2(r) > 0

Points of the pattern are segregated

K2(r) < 0

Points of the pattern are aggregated

Omelko et al. 2018

Unmarked bivariate analysis

O-ring statistic O12(r) (Wiegand et al. 1999; Wiegand and Moloney 2004)

O12(r) = λ2

Patterns 1 and 2 are independent

O12(r) > λ2

Patterns 1 and 2 are attracted

O12(r) < λ2

Patterns 1 and 2 are segregated

Riginos et al. 2005; Hao et al. 2007; De Luis et al. 2008; Batllori et al. 2010; Navarro-Cerrillo et al. 2013, Kang et al. 2014; Cordero et al. 2016; Hu et al. 2017; Miao et al. 2018; Bassil et al. 2018; Zhang et al. 2020

 Pair correlation function g12(r) (Stoyan and Stoyan 1994)

g12(r) = 1

Patterns 1 and 2 are independent

g12(r) > 1

Patterns 1 and 2 are attracted

g12(r) < 1

Patterns 1 and 2 are segregated

Pélissier 1998; Wiegand et al. 2007a; LeMay et al. 2009; Comas et al. 2009; Wang et al. 2010a; Zhang et al. 2010; Martínez et al. 2010; Lan et al. 2012; Liu et al. 2014; Petritan et al. 2014, 2015; Wang et al. 2015; Ghalandarayeshi et al. 2017; Collet et al. 2017; Erfanifard and Stereńczak 2017; García-Cervigón et al. 2017; Ziegler et al. 2017; Ramage et al. 2017; Das Gupta and Pinno 2018; Yang et al. 2018; Szmyt and Tarasiuk 2018; Erfanifard et al. 2018; Yılmaz et al. 2019; Erfanifard et al. 2019; Li et al. 2020a; Ben-Said et al. 2020; Ribeiro et al. 2021

 Pair correlation function g21(r) (Stoyan and Stoyan 1994)

g21(r) = 1

Patterns 1 and 2 are independent

g21(r) > 1

Patterns 1 and 2 are attracted

g21(r) < 1

Patterns 1 and 2 are segregated

Nguyen et al. 2016; Li et al. 2020a

 Ripley’s L-function L12(r) (Lotwick and Silverman 1982)

L12(r) = 1

Patterns 1 and 2 are independent

L12(r) > 1

Patterns 1 and 2 are attracted

L12(r) < 1

Patterns 1 and 2 are segregated

Kenkel 1988; Ward et al. 1996; Haase et al. 1996; Haase et al. 1997; Pélissier 1998; Eccles et al. 1999; Chen and Bradshaw 1999; Mast and Veblen 1999; Grau 2000; He and Duncan 2000; North et al. 2004 ; Motta and Edouard 2005; Motta and Lingua 2005; Suzuki et al. 2005; Fajardo et al. 2006; Nanami et al. 2011; Iszkuło et al. 2012; Navarro-cerrillo et al. 2013; Wehenkel et al. 2015; Jia et al. 2016; Zhang et al. 2016; Owen et al. 2017; Zheng et al. 2017; Lv et al. 2019

Quantitatively univariate marked analysis

 Mark correlation function km1m1(r) (Stoyan and Stoyan 1994)

km1m1(r) = 1

The marks of points are similar to the mean marks (of the study plot)

km1m1(r) > 1

The marks of point that had another point nearby tend to be larger than the mean marks, i.e. positive correlation or mutual stimulation

km1m1(r) < 1

The marks of point that had another point nearby tend to be smaller than the mean marks, i.e. negative correlation or mutual inhibition

Getzin et al. 2008a; Suzuki et al. 2008; Gray and He 2009; Zhang et al. 2013; Fibich et al. 2016; Erfanifard and Stereńczak 2017; Ziegler et al. 2017; Das Gupta and Pinno 2018; Muvengwi et al. 2018; Erfanifard et al. 2018; Yılmaz et al. 2019; Erfanifard et al. 2019; Ben-Said et al. 2020; Li et al. 2020b

r-mark correlation function km1. (r) (Illian et al. 2008)

km1. (r) = 1

The marks of neighbouring points did not show any spatial correlation

km1. (r) > 1

The marks of a focal point that has another neighbour are larger than the mean mark, i.e. positive effect of nearby points on the marks

km1. (r) < 1

The marks of a focal point that has another neighbour are smaller than the mean mark, i.e. negative effect of nearby points on the marks

Raventós et al. 2011; Fedriani et al. 2015

r-mark correlation function k.m1(r) (Illian et al. 2008)

k.m1(r) = 1

The marks of points did not show any spatial correlation

k.m1(r) > 1

The marks of points are larger than the mean if they are nearby to a focal point

k.m1(r) < 1

The marks of points are smaller than the mean if they are nearby to a focal point

Raventós et al. 2011; Ziegler et al. 2017

 Schlather’s I function Imm(r) (Schlather et al. 2004)

Imm(r) = 1

Absence of correlation between point marks

Imm(r) > 1

High correlation between point marks

Imm(r) < 1

Low correlation between point marks

Fedriani et al. 2015; Jácome-Flores et al. 2016

 Mark variogram γm(r) (Illian et al. 2008)

γm(r) = 1

No correlation between point marks

γm(r) < 1

The pairs of points tend to have similar marks (positive correlation)

γm(r) > 1

The pairs of points tend to have dissimilar marks, i.e. large marks are close to small ones (negative correlation)

Suzuki et al. 2008; Fibich et al. 2016; Ghalandarayeshi et al. 2017; Erfanifard and Stereńczak 2017; Kuehne et al. 2018; Erfanifard et al. 2018; Li et al. 2020b

Quantitatively bivariate marked analysis

 Mark correlation function km1m2(r) (Stoyan and Stoyan 1994)

km1m2(r) = 1

The marks of two pattern points are not spatially correlated

km1m2(r) > 1

The marks of the two pattern points tend to have larger marks than the mean mark (positive correlation)

km1m2(r) < 1

The marks of the two pattern points tend to have smaller marks than the mean mark (negative correlation)

Das Gupta and Pinno 2018; Raventós et al. 2011; Erfanifard and Stereńczak 2017; Erfanifard et al. 2019

r-mark correlation function k.m2(r) (Illian et al. 2008)

k.m2(r) = 1

Marks of points do not show a spatial pattern

k.m2(r) > 1

The mark of a pattern 2 point is larger than the mean mark if it is nearby to a point of the focal pattern 1 (positive correlation)

k.m1(r) < 1

The mark of pattern 2 points is smaller than the mean mark if it is nearby to a point of the focal pattern 1 (negative correlation)

Raventós et al. 2011; Jácome-Flores et al. 2016; Ziegler et al. 2017

r-mark correlation function km. (r) (Illian et al. 2008)

km. (r) = 1

There is no effect of a pattern 2 point on the mark of the pattern 1 point.

km. (r) > 1

The mean mark of focal points of pattern 1 that have a pattern 2 neighbour is larger than the plot mean mark (positive correlation)

km. (r) < 1

The mean mark of focal points of pattern 1 that have a pattern 2 neighbour is smaller than the plot mean mark (negative correlation)

Ribeiro et al. 2021

 Mark variogram γm1m2(r) (Illian et al. 2008)

γm1m2(r) = 1

The distribution of point patterns 1 and 2 is independent from their marks

γm1m2(r) < 1

The points of patterns 1 and 2 tend to have similar marks (positive correlation)

γm1m2(r) > 1

The points of patterns 1 and 2 tend to have dissimilar marks (negative correlation)

Erfanifard and Stereńczak 2017

Qualitatively univariate marked analysis

g11(r) (Stoyan and Stoyan 1994)

g11(r) = 1

Points of pattern 1 are randomly distributed

g11(r) > 1

Points of pattern 1 are aggregated

g11(r) < 1

Points of pattern 1 are dispersed

Raventós et al. 2010, 2011; Velázquez et al. 2014; Petritan et al. 2015; Szmyt and Tarasiuk 2018; Abellanas and Pérez-Moreno 2018; Miao et al. 2018; Szmyt and Tarasiuk 2018

g22(r) (Stoyan and Stoyan 1994)

g22(r) = 1

Points of pattern 2 are randomly distributed

g22(r) > 1

Points of pattern 2 are aggregated

g22(r) < 1

Points of pattern 2 are dispersed

Velázquez et al. 2014; Miao et al. 2018

 Mark connection function p11(r) (Gavrikov and Stoyan 1995; Illian et al. 2008)

p11(r) = p1p1

Points of pattern 1 are randomly distributed

p11(r) > p1p1

Points of pattern 1 are clustered, i.e. two points taken randomly have a higher probability of being both of pattern 1

p11(r) < p1p1

Points of pattern 1 are segregated, i.e. two points taken randomly have a lower probability of being both of pattern 1

Raventós et al. 2011; Jácome-Flores et al. 2016

 Mark connection function p22(r) (Gavrikov and Stoyan 1995; Illian et al. 2008)

P22(r) = p2p2

Points of pattern 2 are randomly distributed

P22(r) > p2p2

Points of pattern 2 are aggregated, i.e. two points taken randomly have a higher probability of being both of pattern 2

p22(r) < p2p2

Points of pattern 2 are dispersed, i.e. two points taken randomly have a lower probability of being both of pattern 2

Raventós et al. 2011

Qualitatively bivariate marked analysis

g12(r) (Stoyan and Stoyan 1994)

g12(r) = 1

Patterns 1 and 2 are independent

g12(r) > 1

Patterns 1 and 2 are attracted

g12(r) < 1

Patterns 1 and 2 are segregated

Raventós et al. 2010, 2011; Petritan et al. 2014, 2015; Velázquez et al. 2014; Szmyt and Tarasiuk 2018; Abellanas and Pérez-Moreno 2018; Yuan et al. 2018; Miao et al. 2018; Szmyt and Tarasiuk 2018

g21(r) (Stoyan and Stoyan 1994)

g21(r) = 1

Patterns 1 and 2 are independent

g21(r) > 1

Patterns 1 and 2 are attracted

g21(r) < 1

Patterns 1 and 2 are segregated

Yuan et al. 2018

g1,1+2 - g2,1+2 (Raventós et al. 2010)

g1,1+2g2,1+2 = 0 or g1,1+2 = g2,1+2

Density of patterns 1 and 2 around pattern 1 is similar to that around pattern 2, i.e. absence of density-dependent effect

g1,1+2g2,1+2 > 0

Pattern 1 occurs preferably in areas with high density of patterns 1 and 2, i.e. negative density-dependence (density-dependent mortality)

g1,1+2g2,1+2 < 0

Pattern 1 occurs preferably in areas with low density of patterns 1 and 2, i.e. positive density dependence (density-dependent survival)

Raventós et al. 2010, 2011; Velázquez et al. 2014; Jácome-Flores et al. 2016; Szmyt and Tarasiuk 2018; Miao et al. 2018

g12(r) – g11(r) (Getzin et al. 2006)

g12(r) − g11(r) = 0

Pattern 1 is surrounded by pattern 2 in the same way as pattern 1 surrounds pattern 1, i.e. patterns 1 and 2 have similar spatial distributions

g12(r) − g11(r) > 0

Pattern 2 is more frequent around pattern 1 than pattern 1 around pattern 1, i.e. pattern 1 is negatively correlated.

Pattern 2 show additional aggregation that is independent from pattern 1

g12(r) − g11(r) < 0

Pattern 1 are more frequent around pattern 1 than pattern 2 around pattern 1, i.e. positive correlation for pattern 1

Getzin et al. 2008b; Velázquez et al. 2014; Das Gupta and Pinno 2018; Yuan et al. 2018

g21(r) – g22(r) (Getzin et al. 2006)

g21(r) − g22(r) = 0

Pattern 2 are surrounded by pattern 1 in the same way as pattern 2 surrounds pattern 2, i.e. patterns 1 and 2 have similar spatial distributions

g21(r) − g22(r) > 0

Pattern 1 are relatively more frequent around pattern 2 than pattern 2 around pattern 2, i.e. pattern 1 is negatively correlated

g21(r) − g22(r) < 0

Pattern 1 are relatively more frequent around pattern 2 than pattern 2 around pattern 2, i.e. positive correlation for pattern 1

There is additional aggregation of pattern 2 independently of pattern 1

Getzin et al. 2006, 2008b; Velázquez et al. 2014; Das Gupta and Pinno 2018; Yuan et al. 2018

g11(r) − g22(r) (Wiegand and Moloney 2014)

g11(r) − g22(r) = 0

Patterns 1 and 2 are similar

g11(r) − g22(r) > 0

Pattern 1 are more clustered than pattern 2

g11(r) − g22(r) < 0

Pattern 2 are more clustered than pattern 1

Omelko et al. 2018; Li et al. 2020b

 Mark connection function p12(r) (Gavrikov and Stoyan 1995; Illian et al. 2008)

p12(r) = p1 p2

No association of pattern 1 to pattern 2

p12(r) > p1 p2

Patterns 1 and 2 are attracted, i.e. pattern 1 points tend to appear in pairs with pattern 2

p12(r) < p1 p2

Patterns 1 and 2 are segregated, i.e. pattern 1 points tend to appear segregated from pattern 2 points

Getzin et al. 2008b; Raventós et al. 2011; Jácome-Flores et al. 2016; Yılmaz et al. 2019; Ben-Said et al. 2020

  1. λ: intensity of the pattern, m1 and m2: marks of two neighbouring points of one pattern (univariate mode) or two different patterns (bivariate mode). Note that the values of the p(r) vary between 0 and 1, it equals 1 if the first point is of pattern 1 and the second of pattern 2, and zero otherwise