Comparison of point pattern analysis methods for classifying the spatial distributions of spruce-fir stands in the north-east USA

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1 Comparison of point pattern analysis methods for classifying the spatial distributions of spruce-fir stands in the north-east USA FASHENG LI 1 and LIANJUN ZHANG * 1 Pfizer, Inc. Eastern Point Road, Groton, CT 06340, USA Faculty of Forest and Natural Resources Management, State University of New York, College of Environmental Science and Forestry, One Forestry Drive, Syracuse, NY 1310, USA *Corresponding author. lizhang@esf.edu Summary The spatial distributions of tree locations in spruce-fir forest stands in the north-east USA were explored by various methods of spatial point pattern analysis. The results indicated that the 13 nearest neighbour statistics were not reliable because of the assumption violations for independent distance measures and large sample size requirements. Three other methods (i.e. refined nearest neighbour functions, Ripley s K-function and pair correlation function) seemed to capture the different aspects of the spatial patterns of these spruce-fir stands. The edge effect correction was very important to obtain unbiased results for the spatial point pattern analysis. The toroidal edge correction proved to be simple and satisfactory in this study compared with other edge correction methods. The results indicated that 4 plots (48 per cent) were classified as complete spatial random (CSR) point pattern, 17 (34 per cent) regular point pattern and 9 (18 per cent) clustered point pattern among the 50 plots. It was evident that the clustered plots were younger in age with much higher density and much smaller tree sizes than the CSR or regular plots. This classification scheme can be used as the basis for other spatial studies such as spatial point process modelling. Introduction Spatial pattern has important effects on a variety of physical and ecological processes in a forest stand including competition, size variability and distribution, crown structure, growth and mortality (e.g. Kuuluvainen and Pukkala, 1987 ; Kenkel, 1988 ; Kenkel et al., 1989 ; Miller and Weiner, 1989 ; Moeur, 1993 ; Rouvinen and Kuuluvainen, 1997, Dovciak et al., 001 ). In general, the variations in micro-environment create patches of similar tree sizes, whereas inter-tree competition tends to stimulate the variability of tree sizes ( Garcia, 199, 1994 ; Magnussen, 1994 ; Fox et al., 001 ). Furthermore, treescale spatial pattern has a strong influence on system-level and community-level properties of eco system function ( Pacala and Deutschman, Institute of Chartered Foresters, 007. All rights reserved. Forestry, Vol. 80, No. 3, 007. doi: /forestry/cpm010 For Permissions, please journals.permissions@oxfordjournals.org Advance Access publication date 1 May 007

2 338 FORESTRY 1995 ; Friedman et al., 001 ). Thus, the spatial structure of a forest stand at any given time has a decisive impact on future stand development and dynamics ( Pretzsch, 1997 ). A spatial point pattern can be defined as a set of locations, irregularly distributed within a region of interest, which have been generated by random mechanisms ( Diggle, 1983 ). Commonly, spatial point pattern analysis is used to measure how individuals are located with respect to each other over the horizontal space within a region of interest. There are three fundamental spatial point patterns: complete spatial randomness (CSR), regularity and clustering. This classification may be oversimplified, but can be useful at an early stage of spatial analysis ( Diggle, 1983 ). It is also helpful to formulate a model for the underlying spatial point process. Researchers have developed many methods of spatial point pattern analysis. These methods can be classified into different categories based on various criteria. For example, Cressie (1993) classifies the methods into five broad categories: quadrat methods, distance methods, kernel estimators of intensity, nearest neighbour distribution functions and K -function analyses. Each method has its advantages and problems, and works well for particular situations or datasets. Detailed descriptions of these methods and the associated test statistics can be found in the texts such as Ripley (1981), Cliff and Ord (1981), Diggle (1983) and Cressie (1993). In analysing spatial point patterns, sample points near the boundaries of the study region may have their neighbours located outside the region. Thus, the distance to these neighbours and any statistics involving these neighbours may be unobservable. This causes the so-called edge effects. If such edge effects are not taken into account, considerable bias can be introduced into the computation of spatial statistics for the sample points as well as the region ( Moeur, 1993 ; Haase, 1995 ). There are four main ways to deal with edge effects ( Cressie, 1993 ) (1) Buff zone method ( Sterner et al., 1986 ): A buffer (guard) zone is established inside the perimeter of the study region. The sample points within the buffer (guard) zone are not used for the spatial pattern analysis, but are used as the nearest neighbours of the selected points in the study area. () Border method ( Diggle, 1979 ; Ripley, 1988 ; Gignoux et al., 1999 ): This method corrects edge effects by eliminating all sample points that are closer to the border than to their nearest neighbours inside the study region. (3) Toroidal edge correction method ( Ripley, 1981 ): It treats a rectangular study region as a torus, where sample points near two opposite edges are considered close. This is equivalent to making eight identical copies of the study region and placing these copies around the original one and (4) Weighting method ( Ripley, 1976 ; Getis and Franklin, 1987 ; Andersen, 199). The weight, say w ( s i, s j ), is equal to the proportion of the circumference of a circle centred at s i (the location of sample point i ) passing through s j (the location of sample point j ), and that is inside the study region. This correction is based on the assumption that the region surrounding the study area has a similar point density and spatial pattern to the study area. Spruce-fir stands in the north-east USA are traditionally an important timber-producing resource. Various management tools have been developed for these stands including stocking charts and stand density management diagrams (e.g. Solomon and Zhang, 00 ). However, these traditional forest growth and yield models do not provide any information on the spatial structures of the stands. It is important for forest managers and researchers to know the details on the spatial relationships among trees within stands and to consider issues such as how the spruce-fir trees are distributed over space in the north-east region, whether there is a particular spatial pattern for a particular stand condition, whether a certain spatial pattern favours tree growth and, if not, what can be done to improve the growth condition for the trees. Information on spatial characteristics of these spruce-fir stands would help forest managers to design management strategies ( Chokkalingam and White, 001 ). The general goal of this research is to explore the spatial distributions of the sprucefir forests in the north-east, USA, as the first step towards modelling the spatial point process for these stands. Data and methods Data Data from 50 plots were collected from the evenaged spruce-fir forests in north-western Maine,

3 COMPARISON OF POINT PATTERN ANALYSIS METHODS 339 USA, located in the region between 69 W and 71 W in longitude, between 45 N and 46.5 N in latitude and between 750 and 100 m in elevation ( Kleinschmidt and Baskerville, 1980 ). The plot size ranged from to 0.0 ha. In each plot, the position of each tree over 1.37 m in height was mapped in the polar coordinate system (the azimuth was measured clockwise due north to the nearest degree, and distance from the centre of each plot was measured to the nearest 0.01 m). The polar coordinates of each tree were converted to the coordinates in the Cartesian coordinate system, whose origin was (0 m, 0 m) at the south-west corner of a plot. The plot areas were calculated by plotting the coordinates of each tree and finding the minimum plot size to include every tree. The shapes of the plots were mostly square with a few rectangular. In these plots, balsam fir ( Abies balsamea (L.) Mill.) and red spruce ( Picea rubens Sarg.) accounted for ~ 95 per cent of total number of trees and 94 per cent of total volume. Other minor species included black cherry ( Prunus serotina Ehrh.), eastern white pine ( Pinus strobus L.), white spruce ( Picea glauca (Moench) Voss), black spruce ( Picea mariana (Mill.) B.S.P) and other hardwoods. In each plot, tree diameter at breast height (d.b.h.) or 1.37 m, tree total height, crown length (top to the lowest live branch) and average crown width were recorded for each tree using a personal data recorder. The numbers of trees in each plot ranged from 1 to 109. Mean tree diameters were from. to 19. cm and mean tree total heights were from.7 to 17.1 m ( Table 1 ). Methods Four methods, namely nearest neighbour statistics, refined nearest neighbour analysis, Ripley s K -function and pair correlation function analysis, were used to perform the spatial point pattern analysis in this study. Several different edge correction methods were also applied to each of the point pattern analysis methods. We briefly describe the four spatial pattern analysis methods below. Nearest neighbour statistics Nearest neighbour statistics belongs to the category of distance methods. Thirteen of the statistics were used in this study to test the CSR for the 50 plots. Detailed formula and theoretical distributions of the 13 statistics can be found in Cressie (1993, Table 8.6, p. 604) or elsewhere in the literature. The 13 nearest neighbour statistics were (1) Pielou s (1959) index of non-randomness, () Clark and Evans s (1954) index, (3) Hopkins s (1954) aggregation index, (4) Skellam s (195) test statistic, (5) Pollard s (1971) test statistic, (6) Holgate s (1965) test statistic I, (7) Holgate s (1965) test statistic II, (8) Byth and Ripley s (1980) test statistic, (9) Diggle s (1977) test statistic, (10) Diggle, Besag and Gleaves s test statistic ( Diggle et al., 1976 ), (11) Besag and Gleaves s (1973) test statistic I, (1) Besag and Gleaves s (1973) test statistic II and (13) Cormack s (1979) test statistic. In application, at least 10 trees and at least 10 points (10 per cent if the total number of trees was more than 100) for each plot were randomly selected. One hundred runs for each plot were performed and each of the 13 statistics was calculated for each run. Then, the average of each statistic was used to calculate the P -value of the test. If the P -value was less than 0.05, it was classified as a non-csr pattern. Otherwise, it was a CSR pattern. Refined nearest neighbour analysis For a Poisson process, the event of interest is the case where a circle centred on tree i has a radius equal to the distance ( d ) to the nearest neighbour tree. The distance d has point density function Table 1: Summary statistics of the stand variables across the 50 plots Stand variables Mean SD Min Max Number of trees per plot Stand density (trees m ) Stand basal area (m ha 1 ) Stand mean d.b.h. (cm) Stand mean height (m) Stand mean crown length (m) Stand mean crown width (m)

4 fd1 fd fd Fd fd3 ˆ( fd6 fd4 ˆ 340 FORESTRY ( ) = λπdexp( λπd), (1) where λ is the intensity of events (trees) estimated by n / A ( n is the total number of trees on the plot and A is the area of the plot). The cumulative distribution of d represents the probability of the distance ( d i ) from an arbitrary tree i to its nearest neighbour trees at most d ( d i < d ) and has the form of ( ) = Pd ( < d) = 1 exp( λπ d ). () i F ( d ) can be estimated by the empirical cumulative distribution from data as the proportion of trees in the sample of size n having distances to their nearest neighbours d i ( i =1,,, k ) less than d, Fd ) number of trees with nearest neighbour distances d = n = 1 n δ i( d), (3) n i = 1 1, if di d, where d i ( d) = 0, if di > d. Two edge correction schemes, the toroidal edge and border edge corrections, were performed for the refined nearest neighbour analysis. A Monte Carlo procedure was used in this study to check the goodness-of-fit of the theoretical distribution to the observed data as follows: suppose that there were n trees in a plot (area = A ). Calculate Fd ˆ 1 ( ) for d from 0 to d 0. If ( k 1) independent realizations of the homogeneous Poisson process (each with size n ) were simulated on the plot, then ( k 1) values of Fd ˆ 1 ( )( i = 1,,, k ) can be calculated at d. A 95 per cent confidence interval (CI) at different distances d under the null hypothesis of CSR can be constructed. Assume that a significance level of α was specified. If the observed Fd ˆ 1 ( ) was among the top α per cent or the bottom α per cent of the ordered Fd ˆ 1 ( ) ( i = 1,,, k ), a non-csr spatial pattern was concluded at d. The direction of the departure indicates the pattern: regular (below the lower boundary of the interval) or clustered (above the upper boundary of the interval). Ripley s K-function Ripley s (1976, 1977) K ( d ) function is another popular tool to analyse mapped spatial point pattern. It is defined as K ( d ) = λ 1 E ( number of other events within distance d of a randomly chosen event ), where λ is the density of events (trees per unit area). Suppose there are n trees in a plot with area A and let d ij denote the distance from tree i to tree j. The empirical function Kd ˆ ( ) without edge correction is defined as n n I( dij d) Kd ( ) =. n λ 1 i = 1 j = 1 i j (4) If the above function is corrected by the toroidal edge correction, the edge-corrected function Kd ˆ ( ) is defined as n n Id ( d) Kd ˆ ij ( ) =, n λ 1 i = 1 j = 1 i j (5) where n is the events in plot A and in the eight duplicated surrounding areas, and I ( d ij d ) is an indication function with values either 0 if the condition does not hold or 1 if the condition holds. Ripley (1976, 1977) proposes a weighted edge correction. The weighted edge-corrected function Kd ˆ ( ) is defined as n n 1 w ( d) I( d d) Kd ˆ ij ij ( ) = λ 1, n i = 1 j = 1 i j (6) where the weight w ij is the proportion of the circumference of a circle centred at tree i passing through tree j and that is inside plot A, which is defined as w ( ij d ) 1, if dij dib, 1 dib 1 = 1 cos, if d dib1 + dib, d π 1 dib dib + π cos cos, if d > dib1 + d ib, d d π where d ib is the distance from tree i to the nearest boundary, d ib 1 and d ib are the distances from tree i to the nearest two boundaries. The first case is when the circle is within the plot, the second case

5 fd7 ˆ( COMPARISON OF POINT PATTERN ANALYSIS METHODS 341 is when the circle intersects with only one border and the last case is when the circle intersects two borders in a corner. Note that w ij could be unbounded as d increases in practice. Following the recommendation by Ripley (1977), w ij could be restricted to be less than or equal to 4 for tree i having distance to j greater than the distance from tree i to the nearest boundary. For a homogeneous Poisson process (or equivalently a CSR pattern), the expected value of the K ( d ) function is K ( d ) = π d. Therefore, Ripley s K ( d ) function can be used to test if a spatial point pattern departs from a Poisson process. The statistic ˆ( Kd ˆ ( ) Ld) = d can be used to replace Kd π ˆ ( ) as proposed by Besag (in discussion with Ripley, 1977 ). Ld ˆ( ) has the advantage of linearizing Kd ˆ ( ) and stabilizing its variance and has an expected value of zero under CSR. A Monte Carlo procedure can be used to evaluate the departure of a pattern from CSR. The Monte Carlo procedure is performed to get a 95 per cent CI for L ( d ) under CSR. If the observed Ld ˆ( ) at d is below the lower boundary of the 95 per cent CI, the spatial point pattern is considered regular rather than CSR; if the observed Ld ˆ( ) at d is above the upper boundary of the 95 per cent CI, the spatial point pattern is considered clustered instead of CSR. Pair correlation function analysis Ripley s K -function is a cumulative second-order statistic. The corresponding second function naturally similar to a density function is called the pair correlation function ( Stoyan and Penttinen, 000 ). This function g ( u ) has a relationship with Ripley s K -function: d Kd ( ) = gu ( ) πdd u, d> 0. 0 The pair correlation function g ( d ) is a function of inter-event distance d ( Penttinen et al., 199 ). Considering two infinitesimally small discs with areas d x and d y, which separate from each other by a distance d, the probability that each disc contains an event of the point process is P ( d ) = λ g ( d ) d x d y. For a homogeneous Poisson process, g ( d ) is equal to 1. Thus, K ( d ) = π d. If g ( d ) >1, the inter-event distances around d are relatively more frequent than under CSR indicating clustering. If g ( d ) <1, the inter-event distances around d are relatively less frequent than under CSR indicating regularity. As d increases g ( d ) approaches to 1. The estimation of the pair correlation function g ( d ) can be obtained by determining all pairs of trees having inter-tree distance in some small interval and counting their numbers. Since g ( d ) is a density function, a more elegant method can be employed. Following the recommendation of Penttinen et al. (199), a kernel estimator is used for g ( d ). The chosen kernel function is the Epanechnikov kernel: 3 x 1, if δ< x < δ, wx ( ) = 4δ δ 0, otherwise. The kernel δ is very important because it determines the degree of smoothness of the function. Based on Penttinen et al. (199), set δ = 015. / λ in this study. Then, the pair correlation function can be estimated as N N gd) = wd ( ij d)/[ λ πdsd ( )], d> 0, (7) i = 1 j = 1 i j where w( ) is the kernel function defined above, λ = n / A is the estimated density (trees per unit area), d ij is the distance between trees i and j and s ( d ) is the edge correction factor. For rectangular or square plots, the Ohser Stoyan edge correction factor (Stoyan et al., 1987) can be adopted: sd ( ) = bb d( b + b d)/ π, for 0< d< b b, where b 1 and b are the side lengths of a plot and s ( d ) < A. If no edge correction is applied, the pair correlation function g ( d ) can be estimated by replacing s ( d ) with A in equation (7). If the toroidal edge correction is applied, the pair correlation function g ( d ) can be estimated by equation (7) and using all the points in the toroidal expanded plots. On the other hand, for the Ohser Stoyan edge correction, the pair correlation function g ( d ) is estimated to compensate those points near the boundary of the plot by dividing a smaller number s ( d ) in equation (7). The Ohser Stoyan edgecorrected estimator for K ( d ) is

6 34 FORESTRY N N Id ( ) ˆ ij d Kd ( ) = sd ( ) λ i = 1 j = 1 i j ij (8) and the versions without edge correction and with the toroidal edge correction are the same as equations (4) and (5). The estimated pair correlation function gd ˆ( ) was plotted against d to check the pattern of gd ˆ( ). ˆ Kd ˆ ( ) Kd ( ) was transformed into Ld ˆ( ) = d and a π Monte Carlo procedure was used to evaluate the departure of the pattern from CSR. The Monte Carlo procedure was performed to get a 95 per cent CI for L ( d ) under CSR. If the observed Ld ˆ( ) at d was below the lower boundary of the 95 per cent CI, the spatial point pattern was considered to be regular rather than CSR; if the observed Ld ˆ( ) at d was above the upper boundary of the 95 per cent CI, the spatial point pattern was considered to be clustered instead of CSR. Results Nearest neighbour statistics According to Diggle et al. (1976), the requirement of the sampling intensity for satisfying the null distribution of CSR is at least 10 trees and at least 10 per cent of the total number of trees. However, since most of our plots had fewer than 50 trees, these two conditions could not be met simultaneously. We chose to meet the first condition in order to have a relatively large sample size. Table shows that only a few plots among the 50 plots were classified as non-csr by Pielou s, Clark and Evans s, Hopkin s, Byth and Ripley s and Besag and Gleaves s I and II statistics. No plots were classified as non-csr by the remaining seven methods. After the toroidal edge correction, even fewer plots were classified as non-csr by these 13 nearest neighbour statistics. Intensive sampling When the sample proportion is large compared with the population size, the independence of the samples may not be justified. In this study, the sampling intensities for most of the plots were high ( ~ 30 per cent on average) if sampling at least 10 trees. The violation of independence may cause the invalidation of the statistical distributions corresponding to the nearest neighbour statistics. Therefore, the possible effect of intensive sampling on the statistical tests was evaluated as follows: Suppose that there were n trees in a plot (area = A ). The n points were randomly selected within the plot region and all n trees in the plot were used to calculate the nearest neighbour statistics with the toroidal edge correction. A 95 per cent CI for each of the nearest neighbour test indices was calculated based on the corresponding theoretical distributions if the independence of distance measures was assumed. If 00 homogeneous Poisson processes were realized with intensity n / A for the plot, the 00 toroidal edge-corrected values for each of the nearest neighbour statistics were calculated. Then, if the distribution held well for each statistic, 10 values out of the 00 realizations should fall outside the 95 per cent CI ( Cressie, 1993 ). Table 3 shows the result of the intensive sampling for plot 05 as an example. Comparing the results of intensive sampling against one-sample sampling (10 trees) for the example plot 05, clearly more plots were rejected as CSR under the intensive sampling than that under one-sample sampling. Table 4 shows the classifications of the 50 plots into non-csr pattern under the intensive sampling. Comparing the results in Table 4 against those in Table, it was evident that there were many more plots under the intensive sampling classifying as a non-csr pattern than those under one-sample sampling. Thus, under the intensive sampling the null hypothesis of CSR would be rejected more often if the independence of the distance measures is assumed for the theoretical distributions. Since the assumption of independence of nearest neighbour distances is not valid under the intensive sampling, other methods not depending on this assumption should be explored. Monte Carlo tests As shown earlier, the assumption of independence of distance measures is not valid in the case of the intensive sampling, which would cause the 95 per cent CI to be too narrow, and, hence, the null hypothesis of CSR would be rejected more

7 COMPARISON OF POINT PATTERN ANALYSIS METHODS 343 Table : Summary of CSR test using the 13 nearest neighbour statistics Toroidal edge correction No edge correction Statistic Number of CSR Non-CSR Non-CSR No. Plots Number of CSR No. Plots Pielou , 49 Clark Evans , 34, 35, , 18, 1,, 34, 35, 36, 40, 44, 50 Hopkins 48 5, , 4, 49 Skellam Pollard Holgate I Holgate II Byth Ripley , 34, 35, , 1, 34, 35, 36 Diggle Diggle et al Besag Gleaves I , 35, 36 Besag Gleaves II 48 5, , 39, 49 Cormack Total 13 6 often. Furthermore, the theoretical distributions of different statistics may not hold for a relatively small sample size ( ~ 10). On the other hand, the Monte Carlo procedure offers a way to test the spatial point patterns while avoiding these issues ( Besag and Diggle, 1977 ; Diggle and Gratton, 1984 ). The Monte Carlo test for CSR was conducted as follows: suppose that there were n trees in a plot (area = A ). Based on the coordinates of the n sampled trees (or of the n sampled random points), a nearest neighbour statistic, say T, can be calculated. If ( k 1) independent realizations of the homogeneous Poisson process (each with size n ) were simulated on the plot, then ( k 1) values ( T k, i = 1,,, k ; assume the observed T is T 1 ) of the same test statistic were calculated. Assume that a significance level of α was specified. If the observed test statistic T was among the top α per cent or the bottom α per cent of the ordered T k ( i = 1,,, k ), a non-csr spatial pattern was concluded. For comparison, set k = 1000 and the summarized results of Monte Carlo test for CSR with and without the edge correction were listed in Table 5. The results indicated that the toroidal edge-corrected test using Monte Carlo procedure classified the 50 plots very differently from the no edge correction test. This may be caused by the relatively small sample size ( ~ 10 trees) in each run of the Monte Carlo test. Refined nearest neighbour analysis The refined nearest neighbour analysis was used to classify the 50 plots by the Monte Carlo test ( K = 100) with three edge corrections: (a) no edge correction, (b) the border edge correction and (c) the toroidal edge correction ( Table 6 ). The results indicated that the classifications of point patterns for these plots were different between the no edge correction and the two edge corrections. However, the two edge correction methods yielded similar results. It seemed that the refined nearest neighbour analysis with no edge correction classified too many plots as regular spatial pattern (i.e. more than twice as classified by the two edge correction methods). Since most of the 50 plots had ~ 30 trees, the toroidal edge correction was recommended since it used all the available data points compared with the border edge correction which discarded those points relatively close to the borders.

8 344 FORESTRY Table 3: Results of intensive sampling vs non-intensive sampling for the example plot 05 Intensive sampling One sample (10 trees) Method Index 95%CI + Total Pattern Index P-value Pattern Pielou 1.41 (0.77, 1.3) 30 3 N-CSR N-CSR Clark Evans 0.87 (0.88, 1.1) N-CSR CSR Hopkins.00 (0.7, 1.39) N-CSR N-CSR Skellam (11.67, ) N-CSR CSR Pollard (49.59, 96.19) N-CSR CSR Holgate I 0.54 (0.43, 0.57) CSR CSR Holgate II 0.64 (0.4, 0.58) N-CSR CSR Byth Ripley 0.53 (0.43, 0.57) CSR CSR Diggle (49.59, 96.19) N-CSR CSR Diggle et al (0.43, 0.57) CSR CSR Besag Gleaves I 0.55 (0.43, 0.57) CSR CSR Besag Gleaves II 1.99 (0.7, 1.39) N-CSR N-CSR Cormack (11.67, ) CSR CSR Note:, + and Total are the numbers of realizations that fall below, above and total of below and above the 95% CI, respectively; N-CSR indicates a non-csr pattern if an index is outside the 95% CI for intensive sampling or if a P-value is <0.05 for sample 10 (trees and/or points). Table 4: Summary of the CSR tests of the 13 nearest neighbour statistics under the intensive sampling for the 50 plots Method CSR plots No. Non-CSR (toroidal edge corrected) Pielou , 05, 1, 13, 18, 3, 4, 34, 4, 49 Clark Evans , 05, 1, 14, 19, 0, 1, 36, 49 Hopkins , 03, 05, 1, 13, 18, 19, 0, 1, 3, 4, 34, 36, 4, 49, 54 Skellam , 05, 1, 19, 36, 49 Pollard 48 05, 4 Holgate I , 34, 49, 54 Holgate II , 1, 13, 0, 34, 4, 43, 49, 54 Byth Ripley , 03, 11, 1, 14, 18, 0, 1, 3, 4, 31, 34, 36, 4, 44, 49, 54 Diggle , 03, 05, 1, 13, 18, 19, 0, 1, 5, 31, 36, 43 Diggle et al , 4 Besag Gleaves I , 04, 06, 1, 0, 3, 34, 35, 36, 4, 45, 49 Besag Gleaves II , 06, 1, 34, 36, 39, 4, 45, 49 Cormack , 18, 0, 3, 35, 4, 45, 49, 54 Plots Ripley s K-function Ripley s K -function analysis was used to classify the 50 plots by the Monte Carlo test ( K = 100) with three edge correction methods: (a) no edge correction, (b) the border edge correction and (c) the toroidal edge correction ( Table 7 ). The results indicated that the three edge correction methods had similar classifications for the 50 plots. Ripley s K -function analysis with the

9 COMPARISON OF POINT PATTERN ANALYSIS METHODS 345 Table 5: Summary of Monte Carlo CSR tests of the 13 nearest neighbour statistics for the 50 plots Toroidal edge correction No edge correction Statistic Number of CSR Non-CSR Non-CSR No. Plots Number of CSR No. Plots Pielou , 09, 11, 1, 18, , 05, 06, 13, 14, 3, 35, 36, 38 Clark Evans , 14, 18, 0, 1, 35, 4 8 1, 19,, 35, 36, 43, 56 Hopkins , 09, 11, 1, 15, 18, 0, 35, 36, 44 Skellam , 1, 14, 18, 0, 1,, 35, 36, 41, 43, 56 toroidal edge correction classified a few more CSR plots and a couple of less clustered plots than did the other two edge correction methods. Pair correlation function analysis Pair correlation function analysis was used to classify the 50 plots by the Monte Carlo test ( K = 100) with three edge corrections: (a) no edge correction, (b) the Ohser Stoyan edge correction and (c) the toroidal edge correction ( Table 8 ). The results showed that the three edge correction methods gave almost identical classifica - tions for the 50 plots. 36, 40, 4, , 05, 09, 11, 1, 18, 35, 36, , 09, 1, 19,, 35, 39, 4, 44, 50 Pollard , 04, 1, 1, 4, 37, 38, 47, 48, , 06, 1, 6, 53 Holgate I , 3, 3, 40, , 34, 35, 46, 51 Holgate II , 3, , 3, 34, 35, 46 Byth Ripley , 1, 15, 18, 0, 35, 36, 44, , 1, 13, 35, 36 Diggle , 11, 15, 18, , 08, 0, 1, 3, 31, 36, 37 Diggle et al , 3, 4, 6, 37, , 35, 38, 39, 49 Besag Gleaves I , 08, 11, , 34, 35, 36, 39 Besag Gleaves II , 06, 09, , 08, 1, 35, 37, 39, 45, 46 Cormack , 35, 39 Synthesized classification for the 50 plots The nearest neighbour statistics are based on statistical distributions with the assumptions of large sample sizes and the independence of distance measurements. However, most of our plots had less than 50 trees (median 3.5, ranging from 1 to 109 trees). In addition, a minimum of 10 trees from each plot were sampled. The sampling intensities were very high, ranging from 10 per cent (only one plot) to 84 per cent with small sample sizes (10 or at most 11 trees). Therefore, the two assumptions for the statistical test were not satisfied in this study. The results from the intensive sampling also showed that the tendency of rejecting the null

10 346 FORESTRY Table 6: Summary of the refined nearest neighbour analyses for the 50 plots No edge correction Border edge correction Toroidal edge correction Pattern No. Plots No. Plots No. Plots CSR 13 05, 11, 15, 17, 3, 33, 37, 39, 43, 45, 47, 48, 55 hypothesis of CSR occurred too often because the acceptance region obtained from the theoretical distributions was too narrow ( Table 4 ). The results of Monte Carlo indicated that different methods produced very different results as shown in Table 5. It was hard to evaluate which method would be more reasonable. Therefore, the results from these nearest neighbour statistics were not 7 04, 08, 09, 11, 17, 18, 19, 4, 5, 6, 31, 3, 34, 39, 40, 41, 43, 44, 45, 46, 47, 5, 53, 54, 55, 56, 57 Cluster 4 06, 38, 4, , 06, 15, 33, 38, 4, 48, 49, 51 Regular 33 0, 03, 04, 08, 09, 14 0, 03, 1, 13, 14, 1, 13, 14, 16, 18, 16, 0, 1,, 3, 19, 0, 1,, 3, 35, 36, 37, 50 4, 5, 6, 31, 34, 35, 36, 40, 41, 44, 46, 50, 51, 5, 53, 54, 56, 57 Table 7: Summary of the Ripley s K-function analyses for the 50 plots Pattern 3 04, 08, 09, 11, 15, 16, 17, 18,, 3, 5, 6, 31, 3, 34, 37, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 5, 54, 55, 56, , 06, 33, 38, 4, 49, , 03, 1, 13, 14, 19, 0, 1, 4, 35, 36 No edge correction Weighted edge correction Toroidal edge correction No. Plots No. Plots No. Plots CSR 16 04, 08, 17, 5, 31, 3, 34, 41, 44, 45, 47, 48, 50, 5, 55, 56 Cluster 9 05, 06, 38, 39, 4, 43, 46, 49, 51 Regular 5 0, 03, 09, 11, 1, 13, 14, 15, 16, 18, 19, 0, 1,, 3, 4, 6, 33, 35, 36, 37, 40, 53, 54, , 09, 15, 17, 19, 5, 3, 40, 41, 44, 48, 50, 5, 55, , 06, 38, 39, 4, 43, 45, 46, 49, , 03, 04, 11, 1, 13, 14, 16, 18, 0, 1,, 3, 4, 6, 31, 33, 34, 35, 36, 37, 47, 53, 54, 57 04, 06, 08, 09, 16, 17,, 5, 6, 3, 34, 35, 40, 41, 44, 48, 50, 51, 5, 53, 54, , 38, 39, 4, 45, 46, , 03, 11, 1, 13, 14, 15, 18, 19, 0, 1, 3, 4, 31, 33, 36, 37, 43, 47, 56, 57 reliable for classifying the spatial patterns of the 50 plots. The final classifications were based on the results from the refined nearest neighbour analysis, the Ripley s K -function and the pair correlation function analysis with the toroidal edge corrections. The general rule was that if the majority of the three methods agreed with a pattern, the plot

11 COMPARISON OF POINT PATTERN ANALYSIS METHODS 347 Table 8: Summary of the pair correlation function analysis for the 50 plots No edge correction Ohser Stoyan edge correction Toroidal edge correction Pattern No. Plots No. Plots No. Plots CSR 17 04, 08, 17, 19, 5, 31, 3, 41, 44, 45, 47, 48, 50, 5, 55, 56, 57 Cluster 9 05, 06, 38, 39, 4, 43, 46, 49, 51 Regular 4 0, 03, 09, 11, 1, 13, 14, 15, 16, 18, 0, 1,, 3, 4, 6, 33, 34, 35, 36, 37, 40, 53, 54 Table 9: Final classification of the spatial point pattern of the 50 plots Spatial point pattern No. Final classification Plots CSR 4 04, 08, 09, 11, 15, 16, 17,, 5, 6, 3, 34, 37, 40, 41, 44, 48, 50, 51, 5, 53, 54, 55, 56 Cluster 9 05, 06, 38, 39, 4, 43, 45, 46, 49 Regular 17 0, 03, 1, 13, 14, 18, 19, 0, 1, 3, 4, 31, 33, 35, 36, 47, 57 would be classified as that pattern. Thus, the 50 plots were classified into three spatial patterns: (1) 4 plots (48 per cent) as CSR, () 17 plots (34 per cent) as regular and (3) 9 (18 per cent) as clustered ( Table 9 ). Table 10 shows the average tree and stand variables across the plots for the three spatial patterns. It was evident that the clustered plots were younger in age with much higher density and much smaller tree sizes than were the CSR and regular plots. Although the regular plots generally had bigger and older trees, the average tree and stand characteristics between the CSR 18 04, 08, 09, 17, 5, 31, 3, 34, 41, 44, 45, 47, 48, 50, 5, 55, 56, , 06, 38, 4, 43, 46, 49, , 03, 11, 1, 13, 14, 15, 16, 18, 19, 0, 1,, 3, 4, 6, 33, 35, 36, 37, 39, 40, 53, , 08, 09, 16, 17,, 5, 6, 3, 40, 41, 44, 45, 48, 50, 51, , 06, 38, 39, 4, 43, 46, , 03, 11, 1, 13, 14, 15, 18, 19, 0, 1, 3, 4, 31, 33, 34, 35, 36, 37, 47, 53, 54, 55, 56, 57 and regular plots were not significantly different ( P -values > α = 0.05). Conclusion Among the four methods of spatial point pattern analysis, the nearest neighbour statistics was found to be unreliable in the recognition of spatial patterns. The refined nearest neighbour functions, the Ripley s K -function and the pair correlation function were related and seemed to capture different aspects of the spatial patterns. The Ripley s K -function analysis can serve as a standard tool in discriminating a spatial point pattern and as a statistic to evaluate model goodness-of-fit. Edge effect is very important in spatial point pattern analysis. In this study, the toroidal edge correction was shown to be simple and satisfactory compared with other edge correction methods. For the small plot sizes, the toroidal edge correction is recommended because it fully utilizes all the data in the plots compared with the buffer zone method (discard the trees within some distance to the borders), and would not over-correct the edge effect due to the relatively large proportions of edge trees compared with the weight methods. It should be pointed out that the marks (e.g. d.b.h., height and crown measurements of each tree) were not examined in this study. A further

12 348 FORESTRY Table 10: Average stand characteristics of the three spatial point patterns Pattern Density (trees m ) Basal area (m ha 1 ) Mean d.b.h. (cm) Mean height (m) Mean crown length (m) Mean crown width (m) Mean age (years) CSR Regular Clustering Analysis of variance: P-value Overall CSR vs regular CSR vs clustering Regular vs clustering study may be performed based on marked point pattern analyses. Disclaimer Information contained in this document is not connected to Pfizer and should not be interpreted as an opinion of Pfizer, Inc. Conflict of Interest Statement None declared. References Anderson, M. 199 Spatial analysis of two-species interactions. Oecologia 91, Besag, J.E. and Diggle, P.J Simple Monte Carlo tests for spatial pattern. Appl. Stat. 6, Besag, J.E. and Gleaves, J.T On the detection of spatial pattern in plant communities. Bull. Int. Stat. Inst. 45, Byth, K. and Ripley, B.D On sampling spatial patterns by distance methods. Biometrics 36, Chokkalingam, U. and White, A. 001 Structure and spatial pattern of trees in old growth northern hardwood and mixed forests of northern Maine. Plant Ecol. 156, Clark, P.J. and Evans, F.C Distance to nearest neighbour as a measure of spatial relationships in populations. Ecology 35, Cliff, A.D. and Ord, J.K Spatial Processes: Models and Applications. Pion, London. Cormack, R.M Spatial aspects of competition between individuals. In Spatial and Temporal Analysis in Ecology. R.M. Cormack and J.K. Ord (eds). International Co-operative Publishing House, Fairland, MD, pp Cressie, N.A.C Statistics for Spatial Data: Revised Edition. Wiley, New York. 900 pp. Diggle, P.J The detection of random heterogeneity in plant populations. Biometrics 33, Diggle, P.J On parameter estimation and goodness-of-fit testing for spatial point patterns. Biometrics 35, Diggle, P.J Statistical Analysis of Spatial Point Patterns. Academic Press, New York. 148 pp. Diggle, P.J., Besag, J. and Gleaves, J.T Statistical analysis of spatial point patterns by means of distance methods. Biometrics 3, Diggle, P.J. and Gratton, R.J Monte Carlo methods of inference for implicit statistical models (with discussion). J.R. Stat. Soc. B 46, Dovciak, M., Frelich, L.E. and Reich, P.B. 001 Discordance in spatial patterns of white pine ( Pinus strobus ) size-classes in a patchy near-boreal forest. J. Ecol. 89, Fox, J.C., Ades, P.K. and Bi, H. 001 Stochastic structure and individual-tree growth models. For. Ecol. Manage. 154,

13 COMPARISON OF POINT PATTERN ANALYSIS METHODS 349 Friedman, S.K., Reich, P.B. and Frelich, L.E. 001 Multiple scale composition and spatial distribution patterns of the north-eastern Minnesota presettlement forest. J. Ecol. 89, Garcia, O. 199 What is a diameter distribution? In Proceedings of the Symposium on Integrated Forest Management Information Systems. M. Minowa and S. Tsuyuki (eds). Japan Society of Forest Planning Press, Tokyo, Japan, pp Garcia, O The state-space approach in growth modeling. Can. J. For. Res. 4, Getis, A. and Franklin, J Second-order neighbourhood analysis of mapped point patterns. Ecology. 68, Gignoux, J., Duby, C. and Barot, S Comparing the performance of Diggle s tests of spatial randomness for small samples with and without edge-correction: application to ecological data. Biometrics 55, Haase, P Spatial pattern analysis in ecology based on Ripley s K-function: Introduction and methods of edge correction. J. Veg. Sci. 6, Holgate, P Some new tests of randomness. J. Ecol. 53, Hopkins, B A new method for determining the type of distribution of plant individuals. Ann. Bot. 18, Kenkel, N.C Pattern of self-thinning in jack pine: testing the random mortality hypothesis. Ecology. 69, Kenkel, N.C., Hoskins, J.A. and Hoskins, W.D Local competition in a naturally established jack pine stand. Can. J. Bot. 67, Kleinschmidt, S. and Baskerville, G.L Foliage weight distribution in the upper crown of balsam fir. USDA Forest Service Research Paper. NE-455. Northeastern Forest Experiment Station, Broomall, PA. 8 pp. Kuuluvainen, T. and Pukkala, T Effect of crown shape and tree distribution on the spatial distribution of shade. Agric. For. Meteorol. 40, Magnussen, S A method to adjust simultaneously for spatial microsite and competition effects. Can. J. For. Res. 4, Miller, T.E. and Weiner, J Local density variation may mimic effects of asymmetric competition on plant size variability. Ecology 70, Moeur, M Characterizing spatial patterns of trees using stem-mapped data. For. Sci. 39, Pacala, S.W. and Deutschman, D.H Details that matter: the spatial distribution of individual trees maintains forest ecosystem function. OIKOS 74, Penttinen, A., Stoyan, D. and Henttonen, H.M. 199 Marked point processes in forest statistics. For. Sci. 38, Pielou, E.C The use of point-to-plant distances in the study of the pattern of plant populations. J. Ecol. 47, Pollard, J.H On distance estimators of density in randomly distributed forests. Biometrics 7, Pretzsch, H Analysis and modeling of spatial stand structure: Methodological considerations based on mixed beech-larch stands in Lower Saxony. For. Ecol. Manage. 97, Ripley, B.D The second-order analysis of stationary point processes. J. Appl. Probab. 13, Ripley, B.D Modeling spatial patterns (with discussions). J. R. Stat. Soc. B. 39, Ripley, B.D Spatial Statistics. Wiley, New York, 5 pp. Ripley, B.D Statistical inference for spatial processes. Cambridge University Press, Cambridge, UK. Rouvinen, S. and Kuuluvainen, T Structure and asymmetry of tree crowns in relation to local competition in a natural mature Scots pine forest. Can. J. For. Res. 7, Skellam, J.G. 195 Studies in statistical ecology I. spatial patterns. Biometrika. 39, Solomon, D.S. and Zhang, L. 00 Maximum sizedensity relationships for mixed softwoods in the northeastern USA. For. Ecol. Manage. 155, Sterner, R.W., Ribic, C.A. and Schatz, G.E Testing for life historical changes in spatial patterns of four tropical tree species. J. Ecol. 74, Stoyan, D. and Penttinen, A. 000 Recent applications of point process methods in forestry statistics. Stat. Sci. 15, Stoyan, D., Kendall, W.S. and Mecke, J Stochastic Geometry and Its Applications. Wiley, New York. Received 16 October 006