Statistic function | No departure from the null model | Positive departure | Negative departure | Examples of application | |||
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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 O_{12}(r) (Wiegand et al. 1999; Wiegand and Moloney 2004) | O_{12}(r) = λ_{2} | Patterns 1 and 2 are independent | O_{12}(r) > λ_{2} | Patterns 1 and 2 are attracted | O_{12}(r) < λ_{2} | 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 g_{12}(r) (Stoyan and Stoyan 1994) | g_{12}(r) = 1 | Patterns 1 and 2 are independent | g_{12}(r) > 1 | Patterns 1 and 2 are attracted | g_{12}(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 g_{21}(r) (Stoyan and Stoyan 1994) | g_{21}(r) = 1 | Patterns 1 and 2 are independent | g_{21}(r) > 1 | Patterns 1 and 2 are attracted | g_{21}(r) < 1 | Patterns 1 and 2 are segregated | Nguyen et al. 2016; Li et al. 2020a |
Ripley’s L-function L_{12}(r) (Lotwick and Silverman 1982) | L_{12}(r) = 1 | Patterns 1 and 2 are independent | L_{12}(r) > 1 | Patterns 1 and 2 are attracted | L_{12}(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 k_{m1m1}(r) (Stoyan and Stoyan 1994) | k_{m1m1}(r) = 1 | The marks of points are similar to the mean marks (of the study plot) | k_{m1m1}(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 | k_{m1m1}(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 k_{m1.} (r) (Illian et al. 2008) | k_{m1.} (r) = 1 | The marks of neighbouring points did not show any spatial correlation | k_{m1.} (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 | k_{m1.} (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 I_{mm}(r) (Schlather et al. 2004) | I_{mm}(r) = 1 | Absence of correlation between point marks | I_{mm}(r) > 1 | High correlation between point marks | I_{mm}(r) < 1 | Low correlation between point marks | Fedriani et al. 2015; Jácome-Flores et al. 2016 |
Mark variogram γ_{m}(r) (Illian et al. 2008) | γ_{m}(r) = 1 | No correlation between point marks | γ_{m}(r) < 1 | The pairs of points tend to have similar marks (positive correlation) | γ_{m}(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 k_{m1m2}(r) (Stoyan and Stoyan 1994) | k_{m1m2}(r) = 1 | The marks of two pattern points are not spatially correlated | k_{m1m2}(r) > 1 | The marks of the two pattern points tend to have larger marks than the mean mark (positive correlation) | k_{m1m2}(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 k_{m.} (r) (Illian et al. 2008) | k_{m.} (r) = 1 | There is no effect of a pattern 2 point on the mark of the pattern 1 point. | k_{m.} (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) | k_{m.} (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 γ_{m1m2}(r) (Illian et al. 2008) | γ_{m1m2}(r) = 1 | The distribution of point patterns 1 and 2 is independent from their marks | γ_{m1m2}(r) < 1 | The points of patterns 1 and 2 tend to have similar marks (positive correlation) | γ_{m1m2}(r) > 1 | The points of patterns 1 and 2 tend to have dissimilar marks (negative correlation) | Erfanifard and Stereńczak 2017 |
Qualitatively univariate marked analysis | |||||||
g_{11}(r) (Stoyan and Stoyan 1994) | g_{11}(r) = 1 | Points of pattern 1 are randomly distributed | g_{11}(r) > 1 | Points of pattern 1 are aggregated | g_{11}(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 |
g_{22}(r) (Stoyan and Stoyan 1994) | g_{22}(r) = 1 | Points of pattern 2 are randomly distributed | g_{22}(r) > 1 | Points of pattern 2 are aggregated | g_{22}(r) < 1 | Points of pattern 2 are dispersed | Velázquez et al. 2014; Miao et al. 2018 |
Mark connection function p_{11}(r) (Gavrikov and Stoyan 1995; Illian et al. 2008) | p_{11}(r) = p_{1}p_{1} | Points of pattern 1 are randomly distributed | p_{11}(r) > p_{1}p_{1} | Points of pattern 1 are clustered, i.e. two points taken randomly have a higher probability of being both of pattern 1 | p_{11}(r) < p_{1}p_{1} | 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 p_{22}(r) (Gavrikov and Stoyan 1995; Illian et al. 2008) | P_{22}(r) = p_{2}p_{2} | Points of pattern 2 are randomly distributed | P_{22}(r) > p_{2}p_{2} | Points of pattern 2 are aggregated, i.e. two points taken randomly have a higher probability of being both of pattern 2 | p_{22}(r) < p_{2}p_{2} | 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 | |||||||
g_{12}(r) (Stoyan and Stoyan 1994) | g_{12}(r) = 1 | Patterns 1 and 2 are independent | g_{12}(r) > 1 | Patterns 1 and 2 are attracted | g_{12}(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 |
g_{21}(r) (Stoyan and Stoyan 1994) | g_{21}(r) = 1 | Patterns 1 and 2 are independent | g_{21}(r) > 1 | Patterns 1 and 2 are attracted | g_{21}(r) < 1 | Patterns 1 and 2 are segregated | Yuan et al. 2018 |
g_{1,1+2} - g_{2,1+2} (Raventós et al. 2010) | g_{1,1+2} − g_{2,1+2} = 0 or g_{1,1+2} = g_{2,1+2} | Density of patterns 1 and 2 around pattern 1 is similar to that around pattern 2, i.e. absence of density-dependent effect | g_{1,1+2} − g_{2,1+2} > 0 | Pattern 1 occurs preferably in areas with high density of patterns 1 and 2, i.e. negative density-dependence (density-dependent mortality) | g_{1,1+2} − g_{2,1+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 |
g_{12}(r) – g_{11}(r) (Getzin et al. 2006) | g_{12}(r) − g_{11}(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 | g_{12}(r) − g_{11}(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 | g_{12}(r) − g_{11}(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 |
g_{21}(r) – g_{22}(r) (Getzin et al. 2006) | g_{21}(r) − g_{22}(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 | g_{21}(r) − g_{22}(r) > 0 | Pattern 1 are relatively more frequent around pattern 2 than pattern 2 around pattern 2, i.e. pattern 1 is negatively correlated | g_{21}(r) − g_{22}(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 |
g_{11}(r) − g_{22}(r) (Wiegand and Moloney 2014) | g_{11}(r) − g_{22}(r) = 0 | Patterns 1 and 2 are similar | g_{11}(r) − g_{22}(r) > 0 | Pattern 1 are more clustered than pattern 2 | g_{11}(r) − g_{22}(r) < 0 | Pattern 2 are more clustered than pattern 1 | Omelko et al. 2018; Li et al. 2020b |
Mark connection function p_{12}(r) (Gavrikov and Stoyan 1995; Illian et al. 2008) | p_{12}(r) = p_{1} p_{2} | No association of pattern 1 to pattern 2 | p_{12}(r) > p_{1} p_{2} | Patterns 1 and 2 are attracted, i.e. pattern 1 points tend to appear in pairs with pattern 2 | p_{12}(r) < p_{1} p_{2} | 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 |