第5期 金星姬等：树木位置空间模式建模与预测 113 under Poisson process.Thus,there is attraction at length of parameter vector (a,) distance d;and (d)=0 indicates a Poisson process. There is no agreed rule for choosing the function 2.2.2 Pair potential functions The pair potential Z.However,if the test function Z:is chosen to be function ()can be parameterized in various forms. equal to N()Diggle et al,1994),which is the In this study,three parametric models were adopted. number of events i with:-X‖≤rk,where X,(U= The first one is called Very-Soft-Core (VSC)pair 1,2,...,L)is the jth randomly chosen testing point, potential function (Ogata et al,1984): the left side of equation (2)can be estimated as Φ(r)=-lgi1-exp[-(r/6)2]},(4) i.(a,0)=A2K(r), (8) where 6 is a scaling parameter.For the VSC pair where K(r)is Ripley's K-function (Ripley,1976; potential function,the range of interaction is infinite 1977).The right side of equation (2)can be for all distance r.The second one is called Diggle's estimated as pair potential function that has the following form Diggle et al.,1994): 元(e,9)=它N() ()=-lg(1-[1-(/8)2]2),fr≤； N (9) 0, if rx6. exp-&-∑Φ(Ix-X,I)] =1 (5) The choices of L,m and r.are arbitrary although where the scaling parameter 6 defines the range of different values can be tried and compared.For the interaction.The third one is called Ogata and data used in this study,we set the constants as Tanemura's pair potential function (Ogata et al., follows:1)L=2N because we wanted to ensure that 1985): there were enough random points surrounding a tree, (r)=-lg(1+[r(r/8)-1] and 2)m =30 and r=0.Ik which made the exp[-(r/8)2]),r≥0,8>0， (6) maximum value 3.0 m)because we wanted there to be where the scaling parameter 6 defines the range of enough but not too many trees within the plot which are interaction.Notice that equation(4)is a special case of considered interacting with each other. equation(6)when the parameter 7=0.For equation The optimization of equation(7)was performed (5),(r)=0 when r=6 representing the Poisson using a direct search polytope algorithm one available process.For equation(6),(r)=0 when T =5/r, subroutine is the UMPOL in the IMSL Math Library of which represents the Poisson process. Fortran PowerStation 4.0 C 1994-95 Microsoft 2.2.3 Parameter estimation method One of the Corporation).Details of the polytope algorithm can be methods for estimating the parameters of the pair found in the literature (Nelder et al.,1965;Gill et al., 1981). potential functions is the Takacs-Fiksel method (Diggle et al.,1994).It is based on the fundamental 2.2.4 Edge corrections Typically,the study relationship in equation (2).Suppose is the regions would be the sampled plots within a forest stand.Each of these plots covers only a part of the parameter vector of a pair potential function0=8 for entire spatial pattern.Interactions can only be the potential functions(4)and(5)or =(7,)for the potential function(6)).If a series of test functions Z observed among the trees within a plot,while interactions are unobserved between the trees within (k=1,2,...,m)are chosen to estimate both sides of the plot and the trees outside the plots.It is known as equation(2)(denoting the left side as L(a,0)and the edge or boundary effect.If no correction is taken the right side as R(a,0)),the sum of squared for the boundary effect in the point pattern analysis, differences of both sides can be minimized to estimate the estimation of the interactions is biased.In the the model parameters (a,0)as, previous study Li et al.,2007),the toroidal edge S(a,6)=∑[i(a,0)-(a,0)]2,(7) correction method was proven to be simple and satisfactory compared with other available edge where m is some arbitrary number no less than the correction methods such as buffer zone,border 万方数据第5期 金星姬等：树木位置空间模式建模与预测 113 —————————————————————————————————————————————————一一 under Poisson process．Thus，there is attraction at distance d；and中(d)=0 indicates a Potsson Drocess． 2．2．2 Pair potential functions The pair potential function中(·)can be parameterized in various forms． In this study，three parametric models were adopted． The first one is called Very．Soft．Core(VSC)pair potential function(Ogata et a1．，1 984)： 中(r)=一lg{1一exp[一(r俗)2] where 6 is a scaling parameter．For the VSC Dair potential function，the range of interaction is innnite for all distance，．The second one is called Diggle’s pair potential function that has the foliowing form (Diggle et a1．，1994)： 西(r)：r 19(1～[1一(r／a)2]2)，if r≤6； 【 o， if r>占． (5) where the scaling parameter 6 defines the range of interaction．The third one is called Ogata and Tanemura’8 pair potential function(Ogata et a1．． 1985)： 西(r)=一lg(1+[丁(r／8)一1] exp[一(r／8)2])，r≥0，6>0， (6) where the scaling parameter 6 defines the range of interaction．Notice that equation(4)is a special case of equation(6)when the parameter 7．=0．For equation (5)，中(r)=-0 when r=6 representing the Poisson process．For equation(6)，西(r)-=0 when丁=6／r． which represents the Poisson process． 2．2．3 Parameter estimation method 0ne of the methods for estimating the parameters of the pair potential functions is the Takacs—Fiksel method(Diggle et a1．，1994)． It is based on the fundamental relationship in equation(2)．Suppose 0 is the parameter vector of a pair potential function[0=6 for the potential functions(4)and(5)or 0=(下，∞for the potential function(6)]．If a series of test functions Z‘ (k=1，2，…，m)are chosen to estimate both sides of equation(2)(denoting the left side as L^(ot，0)and the right side as R^(ot，0))，the sum of squared differences of both sides can be minimized to estimate the model parameters(ot，0)as， 5(a，口)=∑哦(a，日)一壶。(a，∽]2，(7) t：1 where m is some arbitrary number no less than the length of parameter vector(a，0)． There is no agreed rule for choosing the function ZI．However， if the test function Z‘is chosen to be equal to％(r女)(Diggle et a1．，1994)，which is the number of events i with lIX。一x，II≤k，where x，(，= 1，2，…，L)is thejth randomly chosen testing point． the left side of equation(2)can be estimated as L^(o／，0) where K(r^)is Ripley’s 1977)．The right side estimated as =A 2K(0)， (8) K—function(Ripley，1 976； of equation(2)can be 交如∽=÷萎L_㈠) exp[一a一萎4(IIx。一训】． (9) The choices of￡，m and^are arbitrary although different values can be tried and cornpared．For the data used in this study， we set the constants as follows：1)L=2N because we wanted to ensure that there were enough random points surrounding a tree， and 2)m=30 and o=0．1七(which made the maximum value 3．0 m)because we wanted there to be enough but not too many trees within the plot which are considered interacting with each other． The optimization of equation(7)was performed using a direct search polytope algorithm(one available subroutine is the UMPOL in the IMSL Math Library of Fortran PowerStation 4．0 c|C 1 994—95 Microsoft Corporation)．Details of the polytope algorithm can be found in the literature(Nelder et a1．，1 965；Gill et of．． 1981)． 2．2．4 Edge corrections Typically，the study regions would be the sampled plots within a forest stand．Each of these plots covers only a part of the entire spatial pattern． Interactions can only be observed among the trees within a plot， while interactions are unobserved between the trees within the plot and the trees outside the plots．It is known as the edge or boundary effect．If no correction is taken for the boundary effect in the point pattern analysis． the estimation of the interactions is biased．In the previous study(Li et a1．，2007)，the toroidal edge correction method was proven to be simple and satisfactory compared with other available edge correction methods such as bufier zone． borde。 万方数据