Methods of spatial point pattern analysis applied in forest ecology

relation 
 structure in the marks, conditional on spatial pattern of the trees carrying the marks. The 
 analysis of quantitative marks addresses questions concerning the numerical difference among 
 the marks that is dependent on the distances of the corresponding points, for example, why 
 neighboring points tend to have smaller (larger) marks than the mean mark (Getzin et al. 2008; 
 Getzin et al. 2011). 
 2
 The mark correlation function kmm(r) is defined as kmm(r) = cmm(r)/ for r> 0, where cmm(r) 
 is the conditional mean of the product of the marks of a pair of points with distance r; is the 
 mean mark (Illian et al. 2008). This normalization allows comparing the strength of mark 
 correlation between different processes. If the empirical mark correlation functions are not 
 constantly equal 1, there is reason to assume that marks are not independent. Applied to forest 
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 ecology, kmm(r) < 1 is assumed to indicate inhibition, individual trees compete against each other 
 and thus have smaller than average marks if they are distance r apart. Inversely, kmm(r) > 1 
 indicates that points at distance r have average marks larger than the mean mark. 
 Fig. 5: Typical forms of the mark correlation function 
 II. DISCUSSION 
 Even though applications of point process methods have been developed and widely 
 implemented in various scientific fields, these tools are bounded in practical uses. There are 
 three main reasons: (1) requirement of mapped data, (2) pairwise-based second-order 
 characteristics, and (3) snapshot analysis of pure spatial patterns (Comas & Mateu 2007). 
 Basically, in point process models and spatial statistical tools, pair-wise analysis is a major part 
 of second-order characteristics and analysis tools for multi-specific interaction do not analyze 
 more than two variables. However, spatial correlation provides a sensitive indicator of 
 ecological interactions structuring spatial patterns of plant species in communities (Wiegand et 
 al. 2007). Moreover, snapshot observations combined with time series analyses have specific 
 advantages (e.g., less time consuming and cheaper) and are appropriate approaches for dynamic 
 assessments of long-lived tree species (Wiegand et al. 2000; Halpern et al. 2010). 
 In spatial point patterns analysis, an observed spatial pattern from the K-statistics is 
 compared to a hypothetical model and evaluated via confidence envelopes, which are 
 constructed by the maximum and minimum results computed across the simulated patterns. 
 However, results from this approach are problematic because of violation of Monte Carlo 
 methods and incorrect type I error performance rate (Loosmore & Ford 2006). A proposed 
 solution is goodness-of-fit test as implemented in the software Programita 
 ( However, other authors stated that Monte Carlo method is appropriate 
 and can be used to assess whether the spatial pattern is significantly different from random 
 (Dale et al. 2002). 
 In computing point-pattern statistics, edge correction is required since events near the edge 
 of the study region have fewer neighbors than centered events, leading to incorrectly calculated-
 intensities.Therefore, circle or ring samples will produce a biased estimation of the point pattern 
 if used without edge correction (Wiegand & Moloney 2004). Three approaches are proposed for 
 dealing with edge effect: Ripley‘s weighted correction, a toroidal correction and a guard area 
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 correction (Haase 1995; Yamada & Rogerson 2003). The major finding of Yamada & 
 Rogerson(2003) is that the K-function method adjusted by either the Ripley or toroidal edge is 
 more powerful than the guard area method. Among two alternatives to estimate the bivariate K-
 function (numeric and analytical methods), numeric approaches are integrated and use an 
 underlying grid of cells. Therefore, analyses using Programita software do not require edge 
 correction (Wiegand & Moloney 2004). 
 Generally, plant community dynamics are driven by spatially dependent birth, death and 
 growth processes and closely embedded in a heterogeneous landscape (Law et al. 2009). From 
 multispecies spatial patterns analyzed, ecologists may use spatio-temporal information to tackle 
 basic questions. Moreover, plants are obviously not points, marks characterized for individual 
 plants (e.g., species, biomass, height, so on) and environmental data need to be considered as 
 they are highly relevant (Illian & Burslem 2007). Therefore, theoretical and empirical issues are 
 closely connected with their estimations and infer to the real dynamic processes generating 
 patterns of plant communities. Spatial point pattern analysis is stimulated by large and technical 
 literature in mathematics and plant ecology, moreover by strongly developed applications in 
 computer science (Law et al. 2009). 
 III. CONCLUSION 
 Analysis of spatial point pattern has a long history in plant ecology and there are a number 
 of tests available to characterise and explore such data. However, these tests do not all perform 
 equally and all have their weaknesses and strengths. As a result, it is suggested that a suite of 
 statistics is used to characterise spatial point patterns, otherwise there is a risk that the 
 description of the pattern will be partially determined by the test chosen. We have to note that 
 point-pattern analysis is a descriptive analysis. Even if a particular null model describes our 
 pattern well, it is not appropriate to conclude that the mechanism behind the null model is the 
 mechanism responsible for our pattern. Other mechanisms may lead to exactly the same pattern. 
 However, point-pattern analysis helps to characterize our pattern and to put forward hypotheses 
 on the underlying mechanisms that should be tested in subsequent steps in the field. 
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 CÁC PHƢƠNG PHÁP PHÂN TÍCH MÔ HÌNH ĐIỂM KHÔNG GIAN 
 ỨNG DỤNG TRONG SINH THÁI RỪNG 
 Nguyễn Hồng Hải 
 Đại học Lâm nghiệp Việt Nam 
 TÓM TẮT 
 Rất nhiều các phương pháp phân tích về mô hình điểm đã được phát triển trong nhiều lĩnh 
 vực khoa học. Thống kê bậc nhất mô tả sự biến động của mật độ các điểm trên phạm vi lớn của 
 vùng nghiên cứu, trong khi tính chất của bậc hai là tổng hợp thống kê của khoảng cách giữa các 
 điểm và cung cấp khả năng nhận dạng các loại mô hình ở các phạm vi khác nhau. Thống kê bậc 
 hai dựa vào hàm Ripley‘s K được sử dụng rộng rãi trong sinh thái để mô tả mô hình không gian 
 và để phát triển giả thuyết về quá trình đang diễn ra. Mục tiêu của bài báo này là thống kê lại 
 các phương pháp phân tích mô hình điểm không gian có thể ứng dụng trong sinh thái rừng. 
 Chúng tôi đã tổng hợp các vấn đề liên quan trong phần thảo luận và kết luận. Chúng tôi cũng 
 giới thiệu phần mềm Programita để ứng dụng tất cả các phương pháp được trình bày trong bài 
 báo. 
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