(Diggle 1979). Kalso presents the advantage of being density-independent, unlike the two other tests (Ripley 1981). In their analysis of the tropical palm tree Borassus aethiopum, Barot et al. (1999) found that use of F,G, and Ktogether, as proposed by the developers of these methods (Ripley 1981, Diggle 1983), provided greater insight than did the use of individual methods in isolation. The functions for these statistics are presented by Barot et al. (1999).
During the past few years, methods based on Ripley’sK-function have under- gone a rapid development and are now widely used in plant ecology. This method is reviewed in detail by Wiegand and Moloney (2004), who cite a large number of studies that have used the technique. Recent examples of the use of Ripley’sK function to assess the spatial structure of tree populations include Aldrich et al.
(2003), Barot et al. (1999), Condit et al. (2000), He and Duncan (2000), and McDonald et al. (2003). Statistical significance is usually evaluated by comparing the observed data with Monte Carlo simulations of a null model, which is most commonly complete spatial randomness (Wiegand and Moloney 2004). The statistics can be calculated with appropriate statistical analysis software, such as the spatial statistics module of S-Plus software (produced by Insightful Corporation;
具www.insightful.com/典). Thorsten Wiegand (Department of Ecological Modelling, UFZ-Centre for Environmental Research, Leipzig, Germany) has developed a freely available software program (Programita) for analysis of point data, which can be used to calculate Ripley’sLfunction (a transformation of Ripley’sK) and the Wiegand–Moloney O-ring statistic, described by Wiegand and Moloney (2004).
TheO-ring statistic is a probability density function that is complementary to the Kstatistic and can detect aggregation or dispersion at a given distance. Wiegand and Moloney (2004) also provide a valuable step-by-step series of recommenda- tions for the use of these methods.
Morisita’s index (I␦) is another measure of dispersion that has been widely used to examine the spatial pattern of trees. This may be calculated as follows (Dale 1999):
where xiis the number of individuals of a particular kind in the ith quadrat, nis the total number, and s2is the sample variance. Typically, values of the index are calcu- lated from measures of density or presence–absence data collected from contigu- ous quadrats. Quadrats can be combined into squares of increasing size, and values of the index calculated for each. As the size of the area analysed increases, values of the index remain constant until the mean clump size is reached, and then it increases (Dale 1999). However, if there is more than one scale of pattern in the data, the method does not provide clear results (Dale 1999). On the other hand, the method does not require the positions of all trees to be mapped, a process that can be very labour intensive. Examples of use of this method to analyse the spatial
I␦n
冦
1s2xx2冧
(n1)pattern of trees are provided by Veblen (1979), Taylor and Halpern (1991) and Bunyavejchewin et al. (2003).
Increasing interest in the spatial ecology of plant populations has led to increas- ing awareness of spatial autocorrelation(Figure 3.13). This refers to the situation when the observed value of a variable at one locality is dependent on values of the variables at other localities (Johnston 1998). Importantly, spatial autocorrelation impairs the ability to perform standard statistical tests; for example, the assump- tion that different samples are independent may be invalid. Two measures are commonly used to assess spatial autocorrelation: Moran’sIstatistic and Geary’sc statistic, both of which indicate the degree of spatial autocorrelation summarized for the entire data set (Johnston 1998). The formulae are given below.
● Moran’s Iis calculated for Nobservations on a variable xat locations i,jas:
where is the mean of the xvariable, wijare the elements of the spatial weights matrix, and S0is the sum of the elements of the weights matrix:
S0冱ijwij
I
(
N/S0)
冱i冱jwij(
xi)(
xj)
/冱i(
xi)
2Spatial structure of tree populations | 123
(a)
l = –0.393 l = –1.000
l = +0.393 l = +0.857 l = 0.000 (c)
(b)
(e) (d)
Fig. 3.13 Field arrangements of cells (which could represent a variety of measured variables, or the distribution of individuals) exhibiting (a) extreme negative spatial autocorrelation, (b) a dispersed arrangement, (c) spatial independence, (d) spatial clustering, and (e) extreme positive spatial
autocorrelation. Values of Moran’s Iare given next to the figures. (From Longley et al. 2005, Geographical Information Systems and Science, 2nd ed., Copyright John Wiley and Sons Limited. Reproduced with permission.)
● Geary’s c statisticis expressed in the same notation as:
Positive spatial autocorrelation is indicated by a value of Moran’sIthat is larger than its theoretical mean of 1/(N1), or a value of Geary’scless than its mean of 1 (Johnston 1998).
Another statistical approach that can be used for analysis of spatial data is geostatistics. This provides a set of techniques that can be used for the analysis and prediction of spatially distributed phenomena. The approach was originally developed for use in geology, but is increasingly being used by ecologists. Its use is likely to become more widespread among ecologists in future with the increasing availability and power of software tools. As noted in section 2.6, some GIS software programs are now able to do geostatistical analysis, either as part of the program itself (for example IDRISI) or as additional software modules or extensions (for example Geostatistical Analyst, which is an ArcGIS extension produced by ESRI; see section 2.5.1 for URLs). Some statistical software programs (such as the spatial statistics module of S PLUS, see above) can also be used to do geostatistics.
Geostatistical methods measure the similarity or dissimilarity between variables based on the spatial dependence of measurements taken at different locations, and then use this information for interpolation, extrapolation, or simulation.
Geostatistics is based on the assumption that data values located closely together in space are likely to be more related than locations that are further apart (Johnston 1998). Spatial dependence decreases with increasing distances among sample points. Analyses are usually done by producing a semivariogram (or variogram for short), which illustrates the degree of dissimilarity between values at different intervals of distance and direction.
Once a variogram has been produced, interpolation methods can be used to esti- mate values at unsampled locations (Johnston 1998). The most commonly used geostatistical method of interpolation is kriging, which uses weighted average of sample measurements to estimate the value at non-sampled points. The weights are calculated from expected spatial dependence between estimated points and sampled points. In practice, kriging is done by selecting a mathematical function from a variety of alternatives, and fitting the function to the observed data points to obtain the best possible fit. The fitted variogram is then used to estimate values at locations of interest (Longley et al. 2005). A number of different options are available regarding the choice of mathematical function for the variogram; an appropriate option has to be determined by the analyst (Longley et al. 2005).
Introductions to kriging are provided by DeMers (2005), Longley et al. (2005) and Johnston (1998).
Textbooks describing geostatistical techniques include those by Goovaerts (1997), Isaaks and Srivastava (1989), Wackernagel (2003), and Webster and Oliver (2000). Use of geostatistics in ecology is described by Johnston (1998),
c(N1)/2S0
[ 兺
i兺
jwij(xixj)2/兺
i(xi)2]
Robertson (1987), and Wagner (2003). Examples of the use of geostatistics in forest ecology include:
● Köhl and Gertner (1997), who used these methods in forest damage surveys
● Bebber et al. (2002), who used geostatistics to determine the spatial relation- ships between canopy openness and seedling performance in secondary low- land forest in Borneo
● Hohn et al. (1993), who used three-dimensional kriging in space and time to predict defoliation caused by gypsy moth in Massachusetts
● Nanos et al. (2004), who used geostatistics to analyse the stand characteristics of a pine forest in Spain
● Schumeet al. (2003), who carried out a spatiotemporal analysis of soil water content in a mixed Norway spruce–European beech stand in Austria.