第5期 金星姬等：树木位置空间模式建模与预测 111 Furthermore,the spatial patterns at tree-scale have One such model is the Gibbs point process model large influences on system-level and community-level or pairwise interaction process model )which properties of ecosystem functions (Pacala et al.,1995; assumes the interaction between two events (e.g., Friedman et al.,2001).In general,positive spatial trees)depends on the relative location of them.It dependence among neighboring trees is mainly due to originates from statistical physics Wood,1968; the effects of the micro-site,whereas inter-tree Landau et al,1980)and has been intensively used in competition tends to create negative spatial dependence spatial statistics (Stoyan et al.,1987;Ogata et al, among neighboring trees Fox et al.,2001). 1981;1984;1985;Diggle et al.,1994).The Gibbs Spatial point pattern analysis is commonly used to point process model with pair potential functions is a study the distributions of the horizontal spacing among class of models for point patterns exhibiting interactions trees within a region of interest.The objective of between the events.These models have been applied spatial point pattern analysis is to measure how in forestry and ecological studies due to their great individual trees are located with respect to each other flexibility and realism in modeling different spatial by using quantitative characteristics and to describe the point patterns (Tomppo,1986;Penttinen et al.,1992; "laws"regulating the locations with mathematical Stoyan et al.,1998;Sarkka et al.,1998;Stoyan et al., models Diggle,1983;Tomppo,1986;Pretzsch, 2000). 1997).There are three fundamental spatial point Different methods have been developed to estimate patterns:complete spatial randomness (CSR ) the parameters of the pair potential functions.Diggle et regularity,and clustering.This classification is over- al.(1994)summarizes three general methods,namely simplified,but useful at an early stage of spatial approximations of maximum likelihood,maximum analysis.It is also helpful to formulate a model for the pseudo-likelihood,and Takacs-Fiksel method.These underlying spatial point process Diggle,1983).A methods can be routinely used in applications and there spatial point pattern is the realization of a spatial point are no restrictions on the parametric form of the pair process (Cressie,1993).Once the spatial point potential functions Tomppo,1986;Stoyan et al., patterns are recognized,it is desirable to fit appropriate 1987;Jensen et al.,1991). point process models to the spatial patterns in order to Spruce-fir stands in the Northeast,USA are understand the mechanisms generating the patterns and traditionally important resources for timber production. to perform analyses of goodness-of-fit for the models. Forest managers and researchers have developed a Detailed review on spatial point process models can be variety of tools for managing these stands.These tools found in the literature (Cressie,1993;Tomppo, include stocking charts and stand density management 1986;Stoyan et al.,2000). diagrams for the pure and mixed softwood stands in the The spatial point process generating CSR is called region (Solomon et al.,2002).However,very little a homogeneous Poisson process,such that the number spatial information is available for these stands to date. of events realized by this process 1)follow a Poisson The objectives of this research were 1)to explore the distribution with mean equal to AA,where A is the spatial structures of the spruce-fir softwood stands in density or intensity of events (e.g.,trees)and A is the Northeast,USA,2)to fit the Gibbs point process the area of the region A;and 2)are independent model with three pair potential functions to the spatial random samples from the uniform distribution on A patterns of trees within the stands,and 3)to develop (Cressie,1993;Diggle,1983).Intuitively,CSR empirical regression models for predicting the means that events are equally likely to occur anywhere parameters of the Gibbs point process model using within A and that events do not interact with each other available stand variables. (Cressie,1993).The homogeneous Poisson point 2 Data and methods process usually serves as a null model Stoyan et al., 2000).If the null model is rejected,more complicated 2.1 Data models are necessary to be tested (Tomppo,1986). Fifty (50)plots were collected from the even- 万方数据第5期 金星姬等：树木位置空间模式建模与预测 111 Furthermore，the spatial patterns at tree-scale have large influences on system·level and community．1evel properties of ecosystem functions(Pacala et a1．，1 995； Friedman et a1．，200 1)．In general，positive spatial dependence among neighboring trees is mainly due to the effects of the micro—site，whereas inter．tree competition tends to create negative spatial dependence among neighboring trees(Fox et a1．，2001)． Spatial point pattern analysis is commonly used to study the distributions of the horizontal spacing among trees within a region of interest．The objective of spatial point pattern analysis is to measure how individual trees are located with respect to each other by using quantitative characteristics and to describe the “laws”regulating the locations with mathematical models(Diggle，1983；Tomppo，1986；Pretzsch， 1 997)．There are three fundamental spatial point patterns： complete spatial randomness(CSR)， regularity，and clustering．This classification is over． simplified，but useful at an early stage of spatial analysis．It is also helpful to formulate a model for the underlying spatial point process(Diggle，1983)．A spatial point pattern is the realization of a spatial point process(Cressie，1 993)．Once the spatial point patterns are recognized，it is desirable to fit appropriate point process models to the spatial patterns in order to understand the mechanisms generating the patterns and to perform analyses of goodness—of—fit for the models． Detailed review on spatial point process models can be found in the literature(Cressie，1 993；Tomppo， 1986；Stoyan et a1．，2000)． The spatial point process generating CSR is called a homogeneous Poisson process，such that the number of events realized by this process 1)follow a Poisson distribution with mean equal to A A l，where A is the density or intensity of events(e．g．，trees)and A|is the area of the region A；and 2)are independent random samples from the uniform distribution on A (Cressie，1993；Diggle，1983)．Intuitively，CSR means that events are equally likely to occur anywhere within A and that events do not interact with each other (Cressie，1 993)．The homogeneous Poisson point process usually serves as a null model(Stoyan et a1．， 2000)．If the null model is models are necessary to be rejected，more complicated tested(Tomppo，1986)． One such model is the Gibbs point process model (or pairwise interaction process model)，which assumes the interaction between two events(e．g．， trees)depends on the relative location of them．It originates from statistical physics(Wood，1 968； Landau et a1．，1 980)and has been intensively used in spatial statistics(Stoyan et a1．，1987；Ogata et a1．， 1981；1984；1985；Diggle et a1．，1994)．The Gibbs point process model with pair potential functions is a class of models for point patterns exhibiting interactions between the events．These models have been applied in forestry and ecological studies due to their great flexibility and realism in modeling different spatial point patterns(Tomppo，1 986；Penttinen et a1．，1 992； Stoyan et a1．，1998；Sarkka et a1．，1998；Stoyan et a1．， 2000)． Different methods have been developed to estimate the parameters of the pair potential functions．Diggle et a1．(1 994)summarizes three general methods，namely approximations of maximum likelihood， maximum pseudo—likelihood，and Takacs-Fiksel method．These methods can be routinely used in applications and there are no restrictions on the parametric form of the pair potential functions(Tomppo，1986；Stoyan et a1．， 1987；Jensen et a1．，1991)． Spruce-fir stands in the Northeast，USA are traditionally important resources for timber production． Forest managers and researchers have developed a variety of tools for managing these stands．These tools include stocking charts and stand density management diagrams for the pure and mixed softwood stands in the region(Solomon et a1．，2002)．However，very little spatial information is available for these stands to date． The objectives of this research were 1)to explore the spatial structures of the spruce—fir softwood stands in the Northeast，USA，2)to fit the Gibbs point process model with three pair potential functions to the spatial patterns of trees within the stands，and 3)to develop empirical regression models for predicting the parameters of the Gibbs point process model using avai】ab】e stand variables． 2 Data and methods 2．1 Data Fifty(50)plots were collected from the even- 万方数据