Chapter 4. Pattern analysis

Transcription

1 Chapter 4. Pattern analysis by Guy BOUXI Contents Introduction... The analysis of dispersion in contiguous quadrats... 6 Dispersion indices... 6 Example with a relevé strip and presence-absence data... 5 Example with a relevé strip and frequency data... Example with a relevé strip and abundance data... 5 Example with a grid of contiguous quadrats... 6 Example of correlogram Pattern analysis with distance measurements Calculation of coordinates with distance measurements Description of the dispersion pattern... 4 GALIAO s technique... 4 The RIPLEY s K function Other informations On the appropriateness of corrections due to border effects Complementary tests General conclusions Computations References rue des Sorbiers, 33, à B. 5 Erpent mail : guy.bouxin@synet.be

2 Introduction Plant associations, or other types of communities, are described by means of their species composition or by means of environmental, physiognomic, structural or other characteristics, but always with the same objective of superposing biotic and physical characteristics of similar dispersions. Pattern analysis is a fundamental tool to achieve this. GREIG-SMITH (964) points out that one of the main contributions that can be expected from the use of quantitative methods is the most accurate detection and description of the dispersion patterns. ROCHE (995) highlights the importance of spatial analysis of ecological data and cites numerous references in this direction. We stress that, in the context of this wor, the study of the dispersion patterns, although useful in itself, is not a purpose in itself but a necessary step in the description and conceptualization of communities. The final choice of methods is closely lined to this objective. In studying the spatial patterns, it is necessary first to distinguish three wholly setups that depends on the nature of the space and the organisms (PIELOU, 977) : - case in which the organisms are confined to discrete habitable sites or units; for example the epiphytes in trees; each tree constitutes a habitable site and is a natural sampling unit; - case in which the organisms have a continuum of space that they can occupy; for example, trees in forest; there are now no natural sampling unit; if we wish to count individuals per unit, the unit has to be arbitrarily defined; - case in which, in addition to the absence of natural sampling units, there are no clearly delimited individuals that can be counted; several individuals can thus mix without being able to individualize them. We will estimate abundance-dominance, for example, in relevés. The last two cases are taen into account in this boo. We are interested here in the horizontal dispersal of the species taen one by one, that is to say the spatial arrangement of the individuals. The definitions previously given are reproduced in BOUXI (99b). The multispecies analysis is discussed in chapters on ordination techniques and multivariate analyses. We will see that it is beneficial to carry out both types of analysis in a sequential manner and that the second advantageously benefits from the results of the first one. Two characteristics of the patterns must be highlighted: intensity and grain. Intensity is the extent to which density or other property varies from place to place. In a high-density pattern, the differences in density are pronounced, that is, areas of high density alternate with areas of low density. When the intensity is low, density contrasts are also low. The notion of grain is to be related to the size of the units constituting the general pattern (aggregates, clumps, gaps of small or large size). However, only simple patterns such as those cited are completely defined with intensity and grain. Often, patterns are complex and the notion of

3 grain disappears. A complete definition of a pattern implies to specify its intensity as well as its type such as aggregate, clump, gap, density variation, gradient or any type of combination between these types. There is in fact an infinity of possible types of patterns. In this ind of study, we must avoid confusion "distribution" and "dispersion". The term "distribution" will be used in its statistical meaning. A variate has a distribution whereas organisms have a dispersion (PIELOU, 977). The analysis of dispersion attempts to answer questions such as: - are individuals, presences, abundances, biomass dispersed at random; if they are not, are they scattered aggregatively or regularly? - Are the same individuals dispersed following defined patterns? Let us first recall classical definitions. Following PIELOU (96), GOUOT (96 in CALLEJA, DAGELIE & GOUOT), GOUOT (969), and BOUXI (974), we may say that the dispersion of a species is: - random, when the probability of encountering the species at any point on the surface is a priori the same everywhere; - aggregative (or contagious), when the probability of encountering the species is greater in certain places than in others, following the formation of aggregates; authors lie PIELOU (977) use the terms "aggregated, clumped, clustered or patchy"; - regular, when the presence of the species is dispersed more regularly than in the random dispersion. The first question has been, and still is, the starting point of many wors, the distribution of Poisson serving as a reference model (GREIG-SMITH, 964, DAGELIE, 969, inter alia). But experience shows that individuals of species (or presences, abundances, etc.) are only exceptionally scattered at random. Starting from such a question is not only unnecessary but also leads to the misuse of statistical techniques. LEGEDRE & FORTI (989) affirm "In nature, living beings are distributed neither uniformly nor at random. Rather, they are aggregated in patches, or they form gradients or other inds of spatial structures". umerous references are given by BOUXI (99b) in which the analytical diagrams consist of plots arranged randomly. A major drawbac of the randomly scattered plot technique is that dispersion can only be observed on a single scale, corresponding to the size of the plot. The results are therefore strongly influenced by the size of the plot and we will not retain this technique. The second question alone therefore serves us as a starting point and we see that as we move forward, we are led to as more precise questions. Our paper deals with the description of patterns and the estimation of their intensity from data collected in grids and transects of contiguous plots or from mapped individuals. We therefore deal with presence, frequency, abundance in plots or measurements of distance between mapped individuals (e.g. trees). Historically, it has been proposed by GREIG-SMITH (95) to use a grid of contiguous plots, which maes it possible to combine neighboring plots into blocs of increasing size; that technique is called nested-bloc analysis blocs of variance and covariance. GREIG-SMITH suggested using a grid of 6 x 6 plots; blocs of,, 4, 8, 6, 3

4 3, 64 and 8 plots are thus obtained. Kershaw (957) completed the technique by adding the transect, that is, a row of contiguous plots oriented in a precise direction. The analysis of the dispersion of species in contiguous plots has been presented by BARY-LEGER (967), CALLEJA, DAGELIE & GOUOT, (96), GREIG-SMITH (964), KERSHAW (964). It allowed the description of the patterns of species (GREIG-SMITH, 95, 96, 964, KERSHAW, 957, 96, HILL, 973). Particular attention must be paid to spatial autocorrelation: it is the property of a set of mapped points each time it shows a pattern (UPTO & FIGLETO, 985). These two authors differentiate between the study of point pattern and the study of quantitative pattern in which each point has a value; if the localization of the points is not uninteresting, it is mainly the spatial variation of the values that is studied. For CLIFF & ORD (973), there is autocorrelation when there is a spatial variation of values in a given territory. We therefore consider the patterns through values recorded in given sites, rather than the patterns defined by the position of the sites. A number of statistical indices have been developed but a general model can be presented (UPTO & FIGLETO, 985): r W ij Y i j ij in which W ij is a measure of the spatial proximity of points i and j and Y ij is a measure corresponding to values observed at points i and j. Points i and j may very well correspond to contiguous plots. Several definitions of Y ij have been proposed. Correlograms are graphs or tables showing how autocorrelation changes with distance. UPTO & FIGLETO (985) recommend the permutation techniques to test the significance of coefficients such as MORA. The current development of computers allows a very large number of permutations (, or,). The autocorrelogram, the MATEL correlogram and the semi-variogram presented and applied by ROCHE (995) are methods based on the detection of repeated patterns in a continuous space. Applications of spatial autocorrelation in ecology have also been proposed by COLE & SYMS (999), DOVCIAK et al. (), FRIEDMA et al. () SOG et al. () and SOKAL & THOMSO (987); LITTLE & DALE (999) generalize autocorrelation in the search for spatio-temporal patterns. LEGEDRE & FORTI (989) show with examples how the correlograms provide a description of the spatial structure. The MORA index I was used by VA DER HOEVE et al. (99) in the study of the dispersion on a fine scale of two herbaceous species by ASI (993) in the study of the spatial structure of several herbaceous Malaysian species, resulting in a description of dispersal patterns and by ALMEIDA & LEWISOH (4) in the study of flower production and related phenological variables. CARRER & URBIATI () used variograms and autocorrelograms in the study of spatial patterns of structural, dendrochronological and dendroclimatological characteristics in mixed forests of Larix decidua and Pinus cembra in the eastern Italian Alps. Self-covariance and cross-covariance were used by PURVES & LAW () to detect spatial structure on a fine scale; the directionality of structures is also discussed. Spectral analysis (RIPLEY, 978) is a comparable technique that uses the periodogram. It has been extensively used by statisticians, economists and engineers with observational data at constant time intervals. The idea is to 4

5 express the observations as a linear combination of a set of waves. The analysis decomposes the pattern into sinusoidal and cosinusoidal waves. It is the only system that avoids the pitfall of the starting point. This analysis is easily compared to the analysis with blocs of increasing size in which the patterns are decomposed into square waves. The main tool is the periodogram Ip with p =, m/ - (m is the number of observations along a line), the corresponding bloc sizes being m/p. The null hypothesis is the randomness of the pattern. Tests are possible, which only apply if the observations have a normal distribution. As expected, the results are relatively independent of the starting point of the grid or transect, but the interrelation between the peas and troughs of the periodogram is complex and difficult to interpret. Two-dimensional spectral analysis of spatial patterns was presented by FORD & RESHAW (984) and RESHAW & FORD (984). They consider that the technique is able to detect all the pattern scales and is sensitive to the directional components. Of course, the periodogram and the R and Q spectra are very sensitive to data repetition but do not detect spatial patterns that do not involve repeatability (LEGEDRE & FORTI, 989). CARPETER & CHAEY (983) compared hierarchical variance analysis, random pairing and two-term local variance methods, and spectral analysis using simulated data. They concluded that the "random pairing" method estimated the size of the patterns more accurately than the other methods; the spectral analysis was unable to partition the grain components. Applications of spectral analysis can be found in CARPETER & TITUS (984), COUTERO (), FRAKLI et al. (985) and KEKEL (988). EWBERY et al. (986) consider that their experiment with spectral analysis should be looed at in an exploratory way only and not as a statistical test of formal hypothesis. A proposal for the unification of spatial models in spatial statistics has been proposed by VER HOEF et al. (993). Let us also note that the variogram was used by DIBLASI & BOWMA () to verify the independence in the spatial data. We do not use the techniques of the correlogram or the variogram as originally presented. However, it was possible to adapt the spatial autocorrelation index of GEARY (CHESSEL, 98) to a grid of plots. It is thus possible to combine the techniques resulting from the technique of the grid of plots and the contributions of techniques such as the correlogram to arrive at an accurate description of the patterns. ASI (993) also shows a very rich use of the MORA s index in a grid of 64 plots. The use of random sampling by ROCHE (995) considerably impoverishes the search for patterns by destructuring data at the finest scales, but such methods nevertheless have had the great merit of showing the continuous variation of spatial interactions as a function of the observation scale; they are particularly interesting in the study of landscape ecology. According to BELLEHUMEUR et al. (997), the parameters of the variogram for several bloc sizes show that the phenomena dispersed in space do not have discrete scales but a continuum of spatial structures whose perception depends on the size of the unit plots. In the this study, we do not develop all the existing techniques but we present, with examples, those which seem to us the most interesting according to the objectives fixed, that is to say the description of vegetation. 5

6 The reader interested in sampling will find further information in BOUXI (99b). Let us note that the study of patterns is often associated with that of the change of patterns, or process, generated by a set of constraints or directions caused by the components of the system (SMALLWOOD, 993). The analysis of dispersion in contiguous quadrats Dispersion indices From its origin (GREIG-SMITH, 95), the analysis of the data obtained from contiguous plots or transect grids is presented as graphs with the abscissa as the size of the blocs (one bloc consists of a single plot or the joining of two or more contiguous plots) and on the ordinate, the variance calculated for each bloc size ("mean square"). In such a graph, the peas indicate the pattern sizes. However, the GREIG-SMITH test on the relationship between the variance at a given scale and the unit plot variance, or the THOMPSO test (958), which defines confidence limits for the ratio of variance to mean ( which was later taen up by GREIG-SMITH) were strongly criticized. For the THOMPSO test, the presence of true aggregation (high variability between neighboring elementary cells) is a case of absolute non-use. The same holds true when there are many empty cells and therefore a small mean (CHESSEL, 978). Finally, the division by the mean gives non-independent statistics to the different bloc sizes. For the GREIG-SMITH method, only the validity of the analysis of variance (independence of the variances to the different bloc sizes) is concerned. The objective is no longer simply to as whether the individuals of the species are randomly, aggregatively or regularly dispersed since, in any case, the random and regular dispersions are exceptional! However, such studies are still under way, but the interpretation of data is incomplete (PATIS & STAMOU, 99). Substantial progress has been made with the adoption of non-parametric models (CHESSEL, 978). Let us first tae an example: that is to say a set of individuals of an annual plant liely to find place in another set of plots. There are a very large number of possible configurations of individuals: - the individuals in the same plot, - one individual in each of the plots, - all other possible configurations. A configuration observed in the field is thus one of a set of theoretically possible configurations. Some are not surprising, while others seem extraordinary and require interpretation. How to separate the extraordinary from that which is not? CHESSEL (978) replies in the following way: "There is a formalization procedure well developed in 6

7 such an approach: assuming a priori equiprobable each element of the configuration file, studying random variables on the probability space thus defined and using the technique of statistical tests, we define a strategy of description of a figure by confrontation with the other figures of the same type (the set of possibilities).this clarifies a important point: the hypothesis of randomness is not the formalization of a process, is not an expression of ignorance of the mechanisms intended or experienced and is not necessarily induced by the reproducibility of the experiment, but is the starting point of an analysis of the data. Parametric is therefore a collection of figures with a uniform probability distribution and a group of random variables ". Three models are used: - the presence-absence model based on the equiprobability of ( / M) figures formed by aligned points, of which M are mared + and (-M) are mared -. The symbol + characterizes the presence of a species at a point, a segment or in a plot; - the classical occupation model in which the random variables are defined on the space of the P allocations of P objects in cells, space with the equiprobability; this model is adapted to the frequency data; - the permutation model or model of the equiprobability of! permutations of numerical values; this model is adapted to abundance-dominance, biomass or any other measure of abundance. Appropriate indices are associated to each model. The presence-absence model was applied in calcareous grasslands of Belgium. The dispersion of flowering plants was observed in transects meter wide and to 3 meters long, cut into 5 cm square plots. In each plot, the presence of the species was noted. The following indices were used (CHESSEL, 978): The conventional dispersion index Dp, - the true contagion index in presence absence-ep, - the local variance Ep*. The variable p(i,j) taes the value when the species is present in the j th unit of the i th bloc of size ; when the species is absent, p(i,j) is. p (i,.) Is the total number of presences in the i th bloc of size. p i,. p i, j j The dispersion index Dp is calculated as follows: P B i p i, j If is the number of units, 7

8 8 B i p m i p D,. with P P m and 3 P P P P Dp is the measure of the total variability, characterizing the differences between the contents of all the blocs. Dp is also the only index that allows an overall test for all bloc sizes (CHESSEL & GAUTIER, 984); it means that from the moment a value is significant for a bloc size, the whole of the graph is significant. The heterogeneity between two blocs is measured by the variable,.,. i p i p H The mean and variance of H are calculated as follows: E(H ;, ) =, ), ; (., ; H E H E H E H E, ;., ;, ;, ; P H E P P P H V The true contagion index Ep is given by J i i i i p H V H E H J E

9 The following types of dispersion were observed (BOUXI and GAUTIER, 979 and 98): - POISSO dispersion, most of the tests are not significant; this does not mean that the dispersion is random, it simply means that on the scales of observation used, no dispersion pattern appears; - dispersion into pacets or aggregates, the non-parametric dispersion curve and the true contagion curve exhibit a net maximum in small bloc sizes; - dispersion in clumps, the presences are grouped significantly at certain places of the grid, the remaining space being empty; the non-parametric dispersion curve generally assumes high values for the intermediate bloc sizes; - gaps and absences in clumps; these dispersion patterns involve frequent species; they are characterized by the existence of significant empty zones; statistically, results are similar to the existence of aggregates and plates; - density gradient, the non-parametric dispersion curve is increasing; the values of true contagion are high for the largest blocs; - stationary variation of density, the species is present throughout the transect, a net maximum of heterogeneity is observed for small bloc sizes; - non-stationary variation of density, the histograms show a non-monotonic mode of variation of the presence densities, relatively continuous and not very repetitive; - complex structures, with two or more scales of heterogeneity. With the frequency data, several indices were used (CHESSEL, 978, CHESSEL & GAUTIER, 984): Let n (i,j) be the number of individuals belonging to the unit of the i th bloc of dimension. If n (i,.) Is the total number of individuals in the i th bloc of dimension, the grand total is n i,. n i, j j Dn is the measure of total variability: - the conventional dispersion index Dn, D n B i n B i,. n n in wich 9

10 n B n,. i B i The non-parametric test is constructed from the equiprobability model of the M allocations of M = n (i,.) individuals to blocs. A parametric test also exists. If M/ is sufficiently large (> = 5), D n is a chi-square variable with (-) degrees of freedom. If (-)> 3, then the quantity ( ) D 3 n can be compared with a unit normal distribution. The statistics are not independent for the different bloc sizes. It is often useful to use the index D p after substituting n (i, j) for or depending on whether its value is below or above a fixed threshold (for example, the median or the arithmetic mean). A global test is then possible. However, the conditions of application of the test are not always fulfilled and the parametric test is then preferable. - The true contagion index E n, E n L i n i,. n i,. m3i i 3 in which m P i j j 3 j P IT n i,. n i,. / and i n i,. n i m i 3,. 3 The summation is for L pairs of blocs, each containing more than two individuals. In the general equiprobability model of M allocations of M individuals to blocs, the null hypothesis specifies that the distribution of individuals in two blocs of a pair is random. En(K) is useful when the number of pairs of blocs fulfilling this condition is greater than or equal to. En(K) does not require a large number of individuals. Under these conditions, the index follows a normal distribution.

11 E n* - The local variance E n*, i n n i,. n i,. i,. n i,. The summation relates to the L* pairs of blocs containing at least individuals. En* is applicable with a small number of pairs of blocs. The distribution law is a chi-square variable with L* degrees of freedom (MEAD 974). egative values of En() and En*() for small, reveal regular dispersions. - The local heterogeneity L no and L n*. L no with R i V i m5 i i m 5 ( i) ( ) n ( i,.) m ( n ( i,.) 5 ( i) ( )( ) m5 ( i) 5 i) and L n* R i T i m6 i i with i n i,.. n i,.. V (i) being the number of empty units in the i th bloc of size and T i n i, j j L no and L n* are tested on the basis of a unit normal distribution. These indices reveal locally regular dispersions, whatever the other structures of heterogeneity. L no is powerful in case of empty units. Further information is given by CHESSEL (978 & 979), CHESSEL & DE BELAIR (973) and DEBOUZIE et al. (975).

12 With abundance data, the spatial autocorrelation index Dx is used (CHESSEL 98). This is the GEARY index, as described by CLIFF & ORD (973), adapted to data collected in grids or transects. j i x x j i,, x j i x x ij ), (, A D, x H i j i t A x x H B i j j j i j i V,, t H V H Z Z E X X X Z Var A.. 3 A B X B A X B A D X

13 B i, j x i, j 4 i, j, j i As a result, the autocorrelation index is Z D x Var Z When a spatial autocorrelation exists, i.e. there is a significant resemblance between the values of two neighboring points, the observed value of Z decreases and D x becomes significantly positive. The distribution of D x is approximately normal. If the abundance data is replaced by frequency data, then the variable (B - ).D n () is a chi - square variable with B - degrees of freedom. With presence-absence data, D x () is equivalent to D p (). The conversion of abundance and frequency data then allows an overall test, which is not the case with D x. Similar patterns to those encountered in limestone grasslands have been observed in other vegetations such as in the Aagera savannah (BOUXI, 983) or along streams (BOUXI & Le BOULEGE (983), BOUXI (99a & 999) but with of course other sizes. Finally, three variabilities are measured: - total variability, characterizing the differences between the contents of the blocs, - the variability at a given scale, lined to the difference between two neighboring blocs of the same size, - local variability, measuring the heterogeneity between units of the same bloc. on-parametric methods can also detect specific patterns such as density gradient, presence of autocorrelation, homogeneous blocs (CHESSEL, 978). The use of grids of plots or juxtaposed segments has been criticized because the detection of certain patterns of dispersion is liely to be strongly influenced by the starting point of the grid, mainly as regards the regular patterns (that are rare). This problem is highlighted by some authors and various solutions have been proposed by USHER (969, 975 & 983), HILL (973), GALIAO (98a) and GALIAO et al. (987). We decided to adopt the GALIAO index called "new two-terms local variance" or LV. This index, tested on numerous occasions, revealed its high sensitivity and accurately highlights the size of the patterns detected. For example, the size of the clumps is detected independently of the distance separating them. The size of the clumps, when there are several in the same transect, is in any case variable and the detection of the precise size of these plates is not always a priority objective. 3

14 of: If numerical values are represented by X, X, X 3,...X n in a set of n sampling units, the index is the average 3, X 3 X 4 X X 3, X X X, X X X X X X X n n n n for bloc size and X X X X X X X X X X X X X X X X X X n3 3 X n 3 4 X 4 5 n n n4 3 n3 4,..., n n These two expressions apply to blocs of larger sizes. The LV index was used by HARPER et al. (6). Unfortunately, as we have seen, the conditions for testing indices are sometimes very restrictive. The test only applies to frequent species, large bloc sizes or large datasets. The argument can be developed with other indices. We have therefore turned to the permutation or Monte Carlo methods to solve these difficulties (BOUXI, 99), as recommended by UPTO & FIGLETO (985) inter alia. The principle of permutation testing is simple (MALY, 997). Consider the value of a test calculated for a set of observed data; the level of significance of the test is obtained by comparing this value with the distribution of those obtained by randomly rearranging the set of observed data. It is a technique involving many calculations; the theoretical distribution being constructed either from all possible rearrangements or from a very large number of rearrangements (, or,). If Pr is the proportion of all possible permutations for which the test value is greater than or equal to the observed value, an estimate of Pr based on a random sample of q permutations (for large sets of data), is unbiased with a Pr ( - Pr) / q variance (DIETZ, 983). Permutation tests have two interesting properties: first, they are valid with non-random sampling schemes (such as plot grids); second, many non-standard tests can be used. A permutation test is specific to the processed data and informs us whether a certain observed pattern can or cannot occur by chance. Thus, such a test is perfectly adapted to the great variability of the biological data. ow, the existence of spatial autocorrelations between the data is a case of non-application of the tests. These spatial autocorrelations are mostly possible in small plot sizes and are very difficult to verify because they are sometimes dependent on many biological or environmental factors. The Monte Carlo methods are used to test hypotheses: the significance of a statistical test is estimated by comparing it with the sample of tests obtained by generating random samples on the basis of models such as random 4

15 dispersion of plants. The random model is therefore a reference tool and the important thing here is to see to what extent we deviate from it. A Monte Carlo test for spatial dispersion of plants was presented by GALIAO et al. (987), based on the random permutation of abundance data in transects. The permutation of the observed data is repeated times and the test is calculated each time for different bloc sizes. test values are thus calculated for each bloc size and the null hypothesis is the average of the values. Confidence intervals are deduced. The application of Monte Carlo tests is as follows. For each test, a value s of S for the observed values is compared with the distribution of S which is obtained by assigning random assignments to the relevés (POISSO model). If we consider that all the possible orders of the data have the same probability of occurrence, the order of observed data is only one of the orders equally possible and the place of s in the distribution of S indicates whether the null hypothesis is rejected or not. The significance level of s is the proportion or percentage of values that are extreme or more extreme than the value in the Monte Carlo distribution. Depending on whether the value is less than 5%, % or.%, the null hypothesis is rejected in a significant, very significant or very highly significant way. ote, however, with COX (987) that individuals of some species mix intimately or merge when they grow. In this case, some biases can appear in the recognition of patterns. The study of the dispersion maes sense only for bloc sizes including the plant masses or tufts. Example with a relevé strip and presence-absence data For the presence-absence model, we give a first example with a set of 3 contiguous relevés along a stream, from the source to the confluence. It is the Maillen stream located ilometers south of the Sambre-et-Meuse furrow, east of the city of amur (see chapter 3). Our method of observation of the broolets is as follows (BOUXI, 987): the stream is divided into a number of juxtaposed sections (here 3), defined by a set of ecological factors such as the current velocity (or general slope), the nature of the bed and the bans (vases, gravels, pebbles, blocs, roc slabs), the nature and degree of disturbance of the crossed woods. In each section, the list of Monocotyledons, herbaceous Dicotyledons, Pteridophytes and species of two genera of Bryophytes (Fontinalis and Sphagnum) living on the bottom, edge or ban of the broolet is established. Only the most hygrophilous species were recorded. The length of the sections is variable. The general appearance of vegetation (photo ) only justified presence-absence data. 5

16 Photo. Winter view of the Maillen broolet. The presence is noted and the absence. The data and results are shown in the Maillen, MaillenPat and MaillenGal tables. The D p and E p indices were calculated for bloc sizes of, 4, 8 and 6 relevés and the new local variance calculated for all bloc sizes between and 6 relevés. The number of permutations is, each time. The results are shown in tables and and figures to 4. Angelica sylvestris has the following dispersion : The results are given in table and figure : D p D p E p E p lv lv

17 Tableau. Indices D p et E p and Monte Carlo test for the species Angelica sylvestris. : bloc sizes; D p : nonparametric dispersion index; E p : true contagion index. lv : "new two-terms local variance". Index followed with : percentages of Monte Carlo values greater than or equal to the observed value..,5,5,5 Angelica sylvestris Dp Ep lv -,5 - -, Bloc sizes Photo. Angelica sylvestris Figure. Indices D p, E p et lv according to bloc size. D p and E p are not significant for any bloc size; lv is significant for the sizes 5 to and for the size. The pattern is therefore a dense clump of size 5 accompanied by some other presences. lv is here more sensitive and more flexible than the other two indices. Athyrium filix-femina has the following dispersion : The results are given in figure : 7

18 5 4 3 Athyrium filix-femina Dp Ep lv Bloc sizes - Figure. Indices D p, E p et lv according to bloc size Photo 3. Athyrium filix-femina D p is significant for sizes 8 and 6 and Ep is significant for size 6; lv is significant for the sizes, 3, 5 and 6. The pattern is therefore a gradient. Chrysosplenium oppositifolium has the following dispersion : The results are given in figure 3 : Chrysosplenium oppositifolium Dp Ep lv Bloc sizes Figure 3. Indices D p, E p and lv according to bloc sizes. D p is significant for sizes 4, 8 and 6 and E p is significant for sizes 4 et 6; lv is significant for sizes 7 to and 3 to 6. The pattern is thus a clump of aggregates located in the lower course. Ranunculus repens has the following dispersion : The results are given in figure 4 : 8

19 3,5,5 Ranunculus repens Dp Ep lv,5 Bloc sizes Figure 4. Indices D p, E p et lv according to bloc sizes. Photo 5. Ranunculus repens D p and E p are both significant for size 4; lv is significant only for size 3. The pattern is therefore a set of aggregates. Lysimachia nemorum has the following dispersion : The results are given in figure 5 : 3,5 Lysimachia nemorum,5 Dp Ep lv,5 Bloc sizes Figure 5. Indices D p, E p et lv according to bloc size. Photo 6. Lysimachia nemorum In order to evaluate the necessary number of permutations to arrive at an accurate calculation of the significance level of the indices, we repeated the test several times with, 5 and permutations. The results are given in table : Perm. Dp Dp Dp3 Dp4 Dp5 Dp6 Dp7 Dp8 Dp9 Dp

20 K Ep Ep Ep3 Ep4 Ep5 Ep6 Ep7 Ep8 Ep9 Ep Table. Repetitions of the same sets of Monte Carlo permutations for the observed D p and E p indices (Lysimachia nemorum, same data as above) with,, 5, and, permutations respectively). : bloc sizes; D p : nonparametric dispersion index; E p : true contagion index; : bloc size; D pi and E pi : percentages of Monte Carlo values above the observed values. It appears here that a number of permutations limited to a few thousand is not enough. As far as possible, we recommend. permutations at least. Example with a relevé strip and frequency data For frequency data, we present data from 36 contiguous streams from the source (left) to the mouth (right). This is the Crupet stream, located a few ilometers from Maillen broolet (Map ), about 8 ilometers long, whose relevés were carried out in the same way. Each relevé consists of 4 sub-relevés (in each sub-relevé, only presence is noted), with frequencies varying from to 4. The number of permutations is, each time. The complete data and results can be found in the Crupetb4, Crupetb4Pat and Crupetb4Gal tables in the database. The results are presented in tables 3 and 4 and figures 6 to. The first species considered is Agrostis stolonifera (table 3 and figure 6). It has the following dispersion:

21 The results are given in table 3 and figure 6 : 3,5,5,5 Agrostis stolonifera Dn En lv -, Bloc size -,5 Figure 6. Indices D n, E n and lv according to bloc size. Photo 7. Agrostis stolonifera Dn Dn En En En* En* Ln Ln Ln* Ln* lv lv Table 3. Indices D n and E n, E n*, L n, L n*, lv and permutation tests for species Agrostis stolonifera. : bloc sizes; D n : dispersion index; E n : true contagion index; E n* : local variance; L n et L n* : local heterogeneities. lv : "new twoterms local variance". Index followed with : percentages of permutation values equal or above the observed values. Only D n, E n et E n* are significant for bloc size 6; lv is never significant. Agrostis stolonifera is therefore a widely dispersed species, with a locally regular dispersion but having two zones of lower density of size 6. This is therefore a pattern with gaps. Alnus glutinosa has the following dispersion :

22 The results are in figure 7 : 5 5 Alnus glutinosa Dn En lv Bloc size Figure 7. Indices D n, E n et lv according to bloc size Photo 8. Branch of Alnus glutinosa D n has significant values for all bloc sizes; E n is significant for sizes et 8 but indicates a trend to regularity for small bloc sizes; E n* reflects the gap of 6 blocs and L n et L n*, the trend to local regularity however without being significant ; lv is significatif for all bloc sizes higher or equal to pour 6 blocs. Bidens tripartita has the following distribution : The results are given in figure Bidens tripartita Dn En lv Bloc size Photo 9. Bidens tripartita Figure 8. Indices D n, E n and lv according to bloc size. D n has significant values for bloc sizes 3, 4, 6 and 8; E n is significant for sizes 6 and 8 but indicates a trend to regularity for sizes, 4 and ; E n* cannot be computed and L n et L n* are not significant; lv is significant for all

23 bloc sizes higher or equal to 4 blocs, with two maxima for sizes 7 and 8. The pattern is here a clump located in the upper half course. Epilobium obscurum has the following dispersion : 3 The results are given in figure 9: 7 Dn Epilobium obscurum 6 En 5 lv 4 3 bloc size Figure 9. Indices D n, E n and lv according to bloc size. Photo. Epilobium obscurum D n has significant values for all bloc sizes; E n is significant for sizes 6, and 8 but indicates a trend to regularity for size 4; E n* cannot be computed and L n et L n* are significant except for size 6; lv is significant for size and for all the sizes higher or equal to 6 blocs. The pattern is here a clump of aggregates located in the upper half course. Calystegia sepium has the following dispersion : 3 The results are given in figure : Calystegia sepium Dn En lv Bloc sizes Photo. Calystegia sepium 3

24 Figure. Indices D n, E n and lv according to bloc size. D n has significant values for all bloc sizes; E n is significant for sizes and ; E n* cannot be computed and L n and L n* are significant only for size 8; lv is significant for sizes to 5 and for sizes and. The pattern is formed of small irregularly dispersed small clumps; two of the three clumps are located in the lower part of the course. Cirsium palustre has the following dispersion: 333 The results are given in figure : 3 Cirsium palustre Dn En lv Bloc size Figure. Indices D n, E n eand lv according to bloc size. Photo. Cirsium palustre D n has significant values for sizes and only; E n is significant for sizes and 3; E n* cannot be computed and L n et L n* are significant only for sizes 4, 6 and 8; lv is significant for sizes to 4 and for sizes 8 to. The pattern is composed of aggregates and of a large clumps of aggregates plate. Epilobium hirsutum has the following dispersion : The results are given in figure : Photo 3. Epilobium hirsutum 4

25 3 Epilobium hirsutum Dn En lv Bloc size Figure. Indices D n, E n and lv according to bloc size. D n has only one significant value for size 3; E n and E n* are only significant for size and L n is significant for sizes 4, and 8; lv is significant for size 5 only. The species is widely distributed with two less dense areas on average, respectively in the upper and lower courses of the brooelet. Example with a relevé strip and abundance data For abundance data, we also present data from 38 contiguous stream relevéss. The species considered is Apium nodiflorum. The data and results can be found in the ApiumPat and ApiumGal tables in the database The results are given in figure 3 and in table Apium nodiflorum Dx lv Bloc size Figure 3. Indices D x eand lv according to bloc size. Photo 4. Apium nodiflorum 5

26 D x D x lv lv< lv Table 4. Indices D x and lv and permutation tests for species Apium nodiflorum. : bloc size; D x : dispersion index; lv : "new two-terms local variance". Index followed with < or : percentages of permutation values respectively lower or higher than the observed value. D x has significant values for all sizes with three local maxima; the column lv< of table 4 indicates a regular dispersion for size (rare pattern) and the column lv indicates that the index is significant for sizes equal to or higher than 4. This is a complex pattern with clumps of aggregates forming a gradient. Example with a grid of contiguous quadrats Examples are now given with a grid of quadrats from a limestone grassland typical of a region called Calestienne, located in ismes, in the southern half of Belgium (see Chapter 3). Globularia bisnagarica has the following distribution : The results are given in figures 4 and 5. 6

27 lv 4 3 Globularia bisnagarica Dp Dp Ep Ep Bloc size Photo 6. Globularia bisnagarica. Figure 4. Indices D p and E p according to the bloc sizes. Suffixes and indicate that the indices were calculated in two perpendicular directions. Sizes = x et x ; size = x ; sizes 3 = 4 x et x 4; size 4 = 4 x 4; size 5 = 4 x 8; size 6 = 4 x ; size 7 = 4 x ; size 8 = 4 x 4; size 9 = 4 x 3; size = 4 x 4; size = 4 x 6. D p has significant values for sizes x, 4 x 8 and 4 x and E p for sizes x and 4 x. The infrequent species therefore presents a low density pattern but the indices reveal an aggregate structure with aggregates distributed in clumps, with empty spaces leaving. There is an asymmetry between the two halves of the transect. The lv index was calculated for three bloc widths each time.,45,4,35,3 Globularia bisnagarica width width width4,5,,5,,5 Bloc length Figure 5. Index lv according to bloc length, computed for three bloc widths. The width of the bloc has a net effect on the value of the index but the profiles are similar for the three widths. The lv index is not significant for small bloc lengths. With the blocs of width, the index is significant only for the blocs of length 54, which reflects an asymmetry in the transect. With the blocs of width and 4, many values are significant for lengths equal to or greater than 9, which reflects the existence of a low density clump and the asymmetry of the transect. 7

28 lv Lotus corniculatus has the following dispersion: The results are in figures 6 and 7: Lotus corniculatus Dp Dp Ep Ep Bloc size Photo 7. Lotus corniculatus subsp. corniculatus Figure 6. Indices D p and E p according to bloc size. Suffixes and indicate that the indices were calculated in two perpendicular directions. Sizes = x and x ; size = x ; sizes 3 = 4 x and x 4; size 4 = 4 x 4; size 5 = 4 x 8; size 6 = 4 ; size 7 = 4 ; size 8 = 4 x 4; size 9 = 4 x 3; size = 4 x 4; size = 4 x 6. 4 x 4. D p has significant values for small blocs up to size 4 x 4 and then 4 x and 4 x and E p for sizes 4 x to The pattern is a set aggregates, some of which are grouped into low density clumps.,6,5,4 Lotus corniculatus width width width4,3,, Bloc length Figure 7. Indice lv according bloc length, computed for three bloc widths. 8

29 lv It is the smallest bloc sizes that are highlighted by the lv index which is significant for most bloc lengths between and 6 (also for length 5 for blocs of unit width and lengths 5 and 5 for width ). The same conclusions are reached as with the other two indices. Rosa rubiginosa has the following dispersion: The results are given in figures 8 et 9: Rosa rubiginosa Dp Dp Ep Ep Bloc 7 size 8 9 Photo 8. Rosa rubiginosa Figure 8. Indices D p and E p according to bloc size. Suffixes and indicate that the indices were calculated in two perpendicular directions. Sizes = x and x ; size = x ; sizes 3 = 4 x and x 4; size 4 = 4 x 4; size 5 = 4 x 8; size 6 = 4 ; size 7 = 4 ; size 8 = 4 x 4; size 9 = 4 x 3; size = 4 x 4; size = 4 x 6. D p has significant values for all but the largest sizes and E p for sizes x, 4 x 4 and 4 x 4. The species forms a low density clump between columns 56 to 7.,,9,8,7,6,5,4,3,, Rosa rubiginosa width width width4 Bloc length

30 Figure 9. Index lv according to bloc length, computed for three bloc widths. The lv index has many significant values with the lengths between 9 and 55 for the width, between 5 and 55 for the width and for the lengths, 3 and between 9 and 55 for the width 4. It corresponds to the dispersion with an aggregate, presences scattered in a dense plate and most of the presences in the upper half of the transect. Thymus praecox has the following reflexion : The results are given in figures et : 5 5 Thymus praecox Dp Dp Ep Ep 5 Bloc size Photo 9. Thymus praecox Figure. Indices D p and E p according to bloc size. Suffixes and indicate that the indices were calculated in two perpendicular directions. Sizes = x and x ; size = x ; sizes 3 = 4 x and x 4; size 4 = 4 x 4; size 5 = 4 x 8; size 6 = 4 ; size 7 = 4 ; size 8 = 4 x 4; size 9 = 4 x 3; size = 4 x 4; size = 4 x 6. D p has significant values for all bloc sizes execept for the highest and E p for sizezq 3 to 8. It is a set of aggregates and dense clumps. 3

31 lv 3,5 3,5 Thymus praecox width width width4,5,5 Bloc length Figure. Index lv according to bloc length, computed for three bloc widths. The lv index is significant for almost all bloc lengths. lv shows peas located for long blocs of 4 to plots or even 59 or 6. This corresponds very well to a clump structure with asymmetry between the two halves of the transect. Brachypodium pinnatum has the following dispersion: The results are given in figures et 3 : Brachypodium pinnatum Dp Dp Ep Ep Bloc size Photo. Brachypodium pinnatum Figure. Indices D p and E p according to bloc size. Suffixes and indicate that the indices were calculated in two perpendicular directions. Sizes = x and x ; size = x ; sizes 3 = 4 x and x 4; size 4 = 4 x 4; size 5 = 4 x 8; size 6 = 4 ; size 7 = 4 ; size 8 = 4 x 4; size 9 = 4 x 3; size = 4 x 4; size = 4 x 6. 3

32 lv D p has significant values for all bloc sizes except sizes 9 and and E p for sizes 3 to 5 and 8. The pattern is a set of clumps of varying sizes. 3,5 3,5 Brachypodium pinnatum width width width4,5,5 Bloc length Figure 3. Index lv according to bloc length, computed for three bloc widths. The lv index is significant for the sizes 4 to 58 for the blocs of width and from 3 to 59 for the blocs of sizes and 4. The three curves have the same shape. There are four peas, for bloc lengths of 8, 7, 48 and 58 plots. There is therefore a mixture of dense aggregates and clumps of different sizes, irregularly distributed. Anthyllis vulneraria has the following dispersion : The results are presented in figures 4 et 5 : 3

33 lv 5 5 Dp Dp Ep Ep Anthyllis vulneraria -5 Bloc size Photo. Anthyllis vulneraria Figure 4. Indices D p and E p according to bloc size. Suffixes and indicate that the indices were calculated in two perpendicular directions. Sizes = x and x ; size = x ; sizes 3 = 4 x and x 4; size 4 = 4 x 4; size 5 = 4 x 8; size 6 = 4 ; size 7 = 4 ; size 8 = 4 x 4; size 9 = 4 x 3; size = 4 x 4; size = 4 x 6. D p has significant values for all bloc sizes except sizes x et 4 x 6. E p is not significant for the calculable sizes. The pattern is complex, with a global density variation.,5 Anthyllis vulneraria width width width4,5,5 Bloc length Figure 5. Index lv according to bloc length, computed for three bloc widths. For blocs of widths and, the values of lv are significant for most sizes between 9 and 39 and for size 39. For blocs of width 4, most of the significant values are found for sizes between 3 and 57. The curves have the same appearance. There are therefore peas at sizes 3 and 5, a plateau between and 5, and some other peas, notably between 56 and 6. There is therefore a density variation, but aggregation is also detected for small sizes (are mainly absences which have an aggregate structure) and an asymmetry between the two halves of the transect. Potentilla neumanniana has the following dispersion : 33

34 lv The results are given in figures 6 et 7 : 5 4 Potentilla neumanniana 3 - Dp Dp Ep Ep Bloc size Photo. Potentilla neumanniana Figure 6. Indices D p and E p according to bloc size. Suffixes and indicate that the indices were calculated in two perpendicular directions. Sizes = x and x ; size = x ; sizes 3 = 4 x and x 4; size 4 = 4 x 4; size 5 = 4 x 8; size 6 = 4 ; size 7 = 4 ; size 8 = 4 x 4; size 9 = 4 x 3; size = 4 x 4; size = 4 x 6. D p has significant values for all bloc sizes. E p is significant for size 4 x 4. The pattern is typically a gradient. 7 6 Potentilla neumanniana width width 5 width4 4 3 Bloc length Figure 7. Index lv according to bloc length, computed for three bloc widths.. lv is significant for bloc lengths between 9 and 6 (width ) or between 4 and 6 (widths and 4). Index lv clearly indicates a gradient. 34