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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*

Statistic function | No departure from the null model | Positive departure | Negative departure | Examples of application | |||
---|---|---|---|---|---|---|---|

Value | Interpretation | Value | Interpretation | Value | Interpretation | ||

Unmarked univariate analysis | |||||||

â€ƒ |
| Points of the pattern are randomly distributed |
| Points of the pattern are aggregated |
| 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 |

â€ƒPair-correlation function |
| Points of the pattern are randomly distributed |
| Points of the pattern are aggregated |
| Points of the pattern are segregated | PÃ©lissier 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; MartÃnez et al. 2010; Lan et al. 2012; Liu et al. 2014; Petritan et al. 2014; VelÃ¡zquez et al. 2014; Petritan et al. 2015; Wang et al. 2015 ; JÃ¡come-Flores et al. 2016; Nguyen et al. 2016; JanÃk et al. 2016; Fibich et al. 2016; Gradel et al. 2017; Wang et al. 2017; Erfanifard and StereÅ„czak 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;Â Muvengwi et al. 2018; Yang et al. 2018; Szmyt and Tarasiuk 2018; Erfanifard et al. 2018; Nguyen et al. 2018b; YÄ±lmaz 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 |

| Points of the pattern are randomly distributed |
| Points of the pattern are aggregated |
| Points of the pattern are segregated | Kenkel 1988; Ward et al., 1996; Haase et al. 1996; Haase et al. 1997; PÃ©lissier 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; IszkuÅ‚o et al. 2012; Zhang et al. 2013; Navarro-Cerrillo et al. 2013; Ebert et al. 2015; Wehenkel et al. 2015; JÃ¡come-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 | |

â€ƒ |
| Points of the pattern are random |
| Points of the pattern are segregated |
| Points of the pattern are aggregated | Omelko et al. 2018 |

Unmarked bivariate analysis | |||||||

â€ƒ |
| Patterns 1 and 2 are independent |
| Patterns 1 and 2 are attracted |
| 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 |

â€ƒPair correlation function |
| Patterns 1 and 2 are independent |
| Patterns 1 and 2 are attracted |
| Patterns 1 and 2 are segregated | PÃ©lissier 1998; Wiegand et al. 2007a; LeMay et al. 2009; Comas et al. 2009; Wang et al. 2010a; Zhang et al. 2010; MartÃnez 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 StereÅ„czak 2017; GarcÃa-CervigÃ³n 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; YÄ±lmaz et al. 2019; Erfanifard et al. 2019; Li et al. 2020a; Ben-Said et al. 2020; Ribeiro et al. 2021 |

â€ƒPair correlation function |
| Patterns 1 and 2 are independent |
| Patterns 1 and 2 are attracted |
| Patterns 1 and 2 are segregated | |

â€ƒRipleyâ€™s |
| Patterns 1 and 2 are independent |
| Patterns 1 and 2 are attracted |
| Patterns 1 and 2 are segregated | Kenkel 1988; Ward et al. 1996; Haase et al. 1996; Haase et al. 1997; PÃ©lissier 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; IszkuÅ‚o 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 | |||||||

â€ƒMark correlation function |
| The marks of points are similar to the mean marks (of the study plot) |
| The marks of point that had another point nearby tend to be larger than the mean marks, i.e. positive correlation or mutual stimulation |
| 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 StereÅ„czak 2017; Ziegler et al. 2017; Das Gupta and Pinno 2018; Muvengwi et al. 2018; Erfanifard et al. 2018; YÄ±lmaz et al. 2019; Erfanifard et al. 2019; Ben-Said et al. 2020; Li et al. 2020b |

â€ƒ |
| The marks of neighbouring points did not show any spatial correlation |
| 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 |
| 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 | |

â€ƒ |
| The marks of points did not show any spatial correlation |
| The marks of points are larger than the mean if they are nearby to a focal point |
| The marks of points are smaller than the mean if they are nearby to a focal point | |

â€ƒSchlatherâ€™s |
| Absence of correlation between point marks |
| High correlation between point marks |
| Low correlation between point marks | |

â€ƒMark variogram |
| No correlation between point marks |
| The pairs of points tend to have similar marks (positive correlation) |
| 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 StereÅ„czak 2017; Kuehne et al. 2018; Erfanifard et al. 2018; Li et al. 2020b |

Quantitatively bivariate marked analysis | |||||||

â€ƒMark correlation function |
| The marks of two pattern points are not spatially correlated |
| The marks of the two pattern points tend to have larger marks than the mean mark (positive correlation) |
| The marks of the two pattern points tend to have smaller marks than the mean mark (negative correlation) | Das Gupta and Pinno 2018; RaventÃ³s et al. 2011; Erfanifard and StereÅ„czak 2017; Erfanifard et al. 2019 |

â€ƒ |
| Marks of points do not show a spatial pattern |
| 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) |
| 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) | RaventÃ³s et al. 2011; JÃ¡come-Flores et al. 2016; Ziegler et al. 2017 |

â€ƒ |
| There is no effect of a pattern 2 point on the mark of the pattern 1 point. |
| The mean mark of focal points of pattern 1 that have a pattern 2 neighbour is larger than the plot mean mark (positive correlation) |
| 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 |

â€ƒMark variogram |
| The distribution of point patterns 1 and 2 is independent from their marks |
| The points of patterns 1 and 2 tend to have similar marks (positive correlation) |
| The points of patterns 1 and 2 tend to have dissimilar marks (negative correlation) | Erfanifard and StereÅ„czak 2017 |

Qualitatively univariate marked analysis | |||||||

â€ƒ |
| Points of pattern 1 are randomly distributed |
| Points of pattern 1 are aggregated |
| Points of pattern 1 are dispersed | RaventÃ³s et al. 2010, 2011; VelÃ¡zquez et al. 2014; Petritan et al. 2015; Szmyt and Tarasiuk 2018; Abellanas and PÃ©rez-Moreno 2018; Miao et al. 2018; Szmyt and Tarasiuk 2018 |

â€ƒ |
| Points of pattern 2 are randomly distributed |
| Points of pattern 2 are aggregated |
| Points of pattern 2 are dispersed | |

â€ƒMark connection function |
| Points of pattern 1 are randomly distributed |
| Points of pattern 1 are clustered, i.e. two points taken randomly have a higher probability of being both of pattern 1 |
| Points of pattern 1 are segregated, i.e. two points taken randomly have a lower probability of being both of pattern 1 | |

â€ƒMark connection function |
| Points of pattern 2 are randomly distributed |
| Points of pattern 2 are aggregated, i.e. two points taken randomly have a higher probability of being both of pattern 2 |
| Points of pattern 2 are dispersed, i.e. two points taken randomly have a lower probability of being both of pattern 2 | RaventÃ³s et al. 2011 |

Qualitatively bivariate marked analysis | |||||||

â€ƒ |
| Patterns 1 and 2 are independent |
| Patterns 1 and 2 are attracted |
| Patterns 1 and 2 are segregated | RaventÃ³s et al. 2010, 2011; Petritan et al. 2014, 2015; VelÃ¡zquez et al. 2014; Szmyt and Tarasiuk 2018; Abellanas and PÃ©rez-Moreno 2018; Yuan et al. 2018; Miao et al. 2018; Szmyt and Tarasiuk 2018 |

â€ƒ |
| Patterns 1 and 2 are independent |
| Patterns 1 and 2 are attracted |
| Patterns 1 and 2 are segregated | Yuan et al. 2018 |

â€ƒ |
| Density of patterns 1 and 2 around pattern 1 is similar to that around pattern 2, i.e. absence of density-dependent effect |
| Pattern 1 occurs preferably in areas with high density of patterns 1 and 2, i.e. negative density-dependence (density-dependent mortality) |
| Pattern 1 occurs preferably in areas with low density of patterns 1 and 2, i.e. positive density dependence (density-dependent survival) | RaventÃ³s et al. 2010, 2011; VelÃ¡zquez et al. 2014; JÃ¡come-Flores et al. 2016; Szmyt and Tarasiuk 2018; Miao et al. 2018 |

â€ƒ |
| 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 |
| 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 |
| 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; VelÃ¡zquez et al. 2014; Das Gupta and Pinno 2018; Yuan et al. 2018 |

â€ƒ |
| 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 |
| Pattern 1 are relatively more frequent around pattern 2 than pattern 2 around pattern 2, i.e. pattern 1 is negatively correlated |
| 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; VelÃ¡zquez et al. 2014; Das Gupta and Pinno 2018; Yuan et al. 2018 |

â€ƒ |
| Patterns 1 and 2 are similar |
| Pattern 1 are more clustered than pattern 2 |
| Pattern 2 are more clustered than pattern 1 | |

â€ƒMark connection function |
| No association of pattern 1 to pattern 2 |
| Patterns 1 and 2 are attracted, i.e. pattern 1 points tend to appear in pairs with pattern 2 |
| Patterns 1 and 2 are segregated, i.e. pattern 1 points tend to appear segregated from pattern 2 points | Getzin et al. 2008b; RaventÃ³s et al. 2011; JÃ¡come-Flores et al. 2016; YÄ±lmaz et al. 2019; Ben-Said et al. 2020 |