112 林业科学 49卷 aged spruce-fir forests in northwestern Maine,USA, trees.The forest plot A is a sampling window within a located in the region between 69 W and 71 W in much larger forest region.Therefore,the events longitude,between 45 N and 46.5 N in latitude,and (trees)are the points of a partial realization of a between 750 m and I 200 m in elevation planar point process within A.For the patterns of N (Kleinschmidt et al.,1980).The plot size ranged from trees in a bounded region A,the pairwise interaction 0.002 5 to 0.02 hm2.The shapes of the plots were processes (i.e.,pair-potential processes)are suitable mostly square with a few rectangular.In each plot,the stochastic models.The joint density for these processes position of each tree over 1.37 m in height was mapped generally takes the form of Li et al,2007).In these plots balsam fir (Abies )=A”。 balsamea)and red spruce (Picea rubens)accounted e即{-A(Ix-xID}, for about 95%of total number of trees and 94%of (1) total volume.Other minor species included black where is the Euclidean distance between trees i cherry (Prunus serotina),eastern white pine (Pinus and j,B is a parameter determining the intensity of the strobus),white spruce Picea glauca),black spruce process,()is a pair potential function,and C is a Picea mariana),and other hardwoods.Tree diameter normalizing constant to make f(x)a density depending at breast height (DBH),total height,crown length on B and中(·). (top to the base of crown),and average crown width The homogeneous Gibbs point process is a special were recorded for each tree (Solomon et al,2002). case of pairwise interaction processes.It is The numbers of trees in each plot ranged from 12 to characterized by the potential functionΦ(·）and 109.Mean tree diameters were from 2.2 to 19.2 cm another parameter a =-log(B).Although the explicit and mean tree total heights were from 2.7 to 17.1 form of the distribution is hard to obtain,the following m (Tab.1). relationship holds for a homogeneous Gibbs point process Diggle et al.,1994) Tab.1 Summary statistics of the stand AE[Z(X)]= variables across the 50 plots Stand variables Mean Std.Min.Max. Number of trees per plot 34.4 17.7 12 109 Efz(X)exp[-a-((2) Stand density/(tree.m2) 0.61 0.54 0.18 2.52 where A is the density of events,Eo is the expectation Stand basal area/(m2.hm-2)50.2 14.3 10.0 79.6 Stand mean DBH/cm 11.53.9 2.2 19.2 with respect to the Palm distribution of the process,E Stand mean height/m 11.63.7 2.7 17.1 is the expectation,and Z(X)is any random function Stand mean crown length/m 4.41.5 1.6 8.4 of the process where X includes all events of the Stand mean crown width/m 1.I 0.3 0.5 1.9 process in the whole plane.For a strict definition of Using the methods of refined nearest neighbor Palm distribution,see Stoyan et al.(1987),Tomppo statistic,Ripley's K-function (Ripely,1976)analysis (1986)and Cressie (1993). and pair correlation function analysis,24 plots were However,when it is conditional on the number of classified as completely spatial random (CSR)point points N,Gibbs distribution can be expressed as pattern,17 plots as regular point pattern,and 9 plots as clustered point pattern out of the 50 plots (Li et al., )=2ep{-会Ax.-XD}.3) where Z is the normalizing constant in the form of 2007).The classification was used as the basis for modeling the spatial patterns in this study z=ep-合点X,-XD],d 2.2 Methods The pair potential function()can be interpreted as 2.2.1 Theoretical background of Gibbs model followings:(d)>0 indicates that the probability pairwise interaction process model)Suppose the density for inter-event distance d is lower than that spatial point pattern in a mapped forest plot A has N under Poisson process and,thus,there is repulsion at events (trees).Denote=X =(u:,v)EA:i=1, distance d;(d)<0 indicates that the probability 2,..,N as the set of coordinates (u:,v)of the density for inter-event distance d is higher than that 万方数据112 林业科学 49卷 aged spruce—fir forests in northwestern Maine，USA， located in the region between 69。W and 7 1。W in longitude，between 45。N and 46．5。N in latitude，and between 750 m and l 200 m in elevation (Kleinschmidt et a1．，1980)．The plot size ranged from 0．002 5 to 0．02 hm2．The shapes of the plots were mostly square with a few rectangular．In each plot，the position of each tree over 1．37 m in height was mapped (Li et a1．，2007)．In these plots batsam fir(Abies balsamea)and red spruce(Picea rubens)accounted for about 95％of total number of trees and 94％of total volume．Other minor species included black cherry(Prunus serotina)，eastern white pine(Pinus strobus)，white spruce(Picea glauca)，black spruce (Picea mariana)，and other hardwoods．Tree diameter at breast height(DBH)，total height，crown length (top to the base of crown)，and average crown width were recorded for each tree(Solomon et a1．，2002)． The numbers of trees in each plot ranged from 1 2 to 109．Mean tree diameters were from 2．2 to 19．2 am and mean tree total heights were from 2．7 to 17．1 m(Tab．1)． Tab．1 Summary statistics of the stand variables across the 50 plots Stand variables Mcan Std． Min． Max． Number oftrees per plot 34．4 17．7 12 109 Stand density／(tree·m一2)0．61 0．54 0．18 2．52 Stand basal area／(m2．hm一2) 50．2 14．3 10．0 79．6 Stand mean DBH／cm 11．5 3．9 2．2 19．2 Stand mean height／m 11．6 3．7 2．7 17．1 Stand mean crown length／m 4．4 1．5 1．6 8．4 Stand mean crown width／m 1．1 0．3 0．5 1．9 Using the methods of refined nearest neighbor statistic．Ripley’s K-function(Ripely，1 976)analysis and pair correlation function analysis，24 plots were classified as completely spatial random(CSR)point pattern，1 7 plots as regular point pattern，and 9 plots as clustered point pattern out of the 50 plots(Li et a1．， 2007)．The classification was used as the basis for modeling the spatial patterns in this study． 2．2 Methods 2．2．1 Theoretical background of Gibbs model (pairwise interaction process model) Suppose the spatial point pattern in a mapped forest plot A has N events(trees)．Denote X={X。=(Mi，口。)∈A：i=1， 2，…，Ⅳ}as the set of coordinates(Mi，u。)of the trees．The forest plot A is a sampling window within a much larger forest region．Therefore，the events (trees)are the points of a partial realization of a planar point process within A．For the patterns of N trees in a bounded region A，the pairwise interaction processes(i．e．，pair—potential processes)are suitable stochastic models．The joint density for these processes generally takes the form of 州=篇唧{-乏。三驯l xi．Xj∽)， (1) where㈩I is the Euclidean distance between trees i and j，8 is a parameter determining the intensity of the process，多(·)is a pair potential function，and C is a normalizing constant to make八z)a density depending on口and中(·)． The homogeneous Gibbs point process is a special case of pairwise interaction processes． It is characterized by the potential function中(‘)and another parameter a=一log(f1)．Although the explicit form of the distribution is hard to obtain，the following relationship holds for a homogeneous Gibbs point process(Diggle et a1．，1994) AE“Z(X)]= E{z(x)exp[一a一萎中(…I)])’(2) where A is the density of events，Eo is the expectation with respect to the Palm distribution of the process，E is the expectation，and Z(X)is any random function of the process where X includes a11 events of the process in the whole plane．For a strict definition of Palm distribution，see Stoyan et a1．(1987)，Tomppo (1986)and Cressie(1993)． However．when it is conditional on the number of points N，Gibbs distribution can be expressed as 以并)=Z-1exp{-∑∑西(1IXi一划)}，(3) where Z is the normalizing constant in the form of z=exp[一乏。三中(IIX；一圳h，…，d‰· The pair potential function多(·)can be interpreted as followings：中(d)>0 indicates that the probability density for inter．event distance d is lower than that under Poisson process and，thus，there is repulsion at distance d；中(d)<0 indicates that the probability density for inter．event distance d is higher than that 万方数据