Sandrine Pavoine
email: sandrine.pavoine@mnhn.fr

An ordination approach to explore similarities among communities

Keywords:

Analysis of similarities among communities can help to decipher the biogeographical, evolutionary, and ecological factors that drive local diversity. Recent indices of similarity among communities incorporate not only information on species presence and abundance but also information on how similar species are in their traits and how closely related they are in terms of taxonomy or phylogeny. Towards this aim, trait-based, taxonomic or phylogenetic similarities among species have been defined and bounded between 0 (species are maximally distinct) and 1 (species are similar). A required property for an index of similarity between two communities is that it must provide minimum similarity (0) where communities have maximally distinct species, as well as maximum similarity (1) where communities are equivalent in their trait, taxonomic or phylogenetic compositions. Here, I developed a new ordination methodology that conforms to the requirement: double similarity principal component analysis (DSPCA). DSPCA summarizes multidimensional trait-based, taxonomic or phylogenetic similarities among communities into orthogonal axes. The species that drive each similarity pattern can be identified together with their traits or with their taxonomic or phylogenetic positions. I applied this methodology to theoretical examples and to empirical data sets on bird and bat communities to illustrate key properties of DSPCA. I compared the results obtained with DSPCA with those provided by related approaches. Theoretical and empirical case studies highlight the following additional properties of DSPCA: (i) axes are orthogonal and identify independent (dis)similarity patterns between communities; (ii) the more functionally, taxonomically or phylogenetically similar communities are, the closer they are on an axis; (iii) the coordinate of a species on an axis expresses how representative the species is of the pattern identified by the axis; and (iv) a species is representative of x communities if the functional, taxonomic or phylogenetic characteristics of this species are very common within each of these x communities. DSPCA is an efficient approach to visualize functional, taxonomic and phylogenetic similarities between communities. It is also a useful alternative to recent methods dedicated to phylogenetic diversity patterns. It will be an asset for all studies that aim to compare functional, taxonomic, genetic and phylogenetic diversity.

Introduction

In ecology, similarities among communities are considered to pinpoint in space and time where and when patterns of community structure change. These changes might be driven, for example, by abiotic and biotic environments, geographic barriers, and dispersal limitations. Similarities among communities depend on which species they contain and potentially on the relative abundances of these species. Recent developments of similarity coefficients also include taxonomic, phylogenetic or trait-based similarities among the species that compose the communities (e.g., [START_REF] Pavoine | From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis[END_REF][START_REF] Ferrier | Using generalized dissimilarity modelling to analyse and predict patterns of beta diversity in regional biodiversity assessment[END_REF]Bryant et al., 2008, Graham and[START_REF] Graham | Phylogenetic beta diversity: linking ecological and evolutionary processes across space and time[END_REF][START_REF] Webb | Phylocom: software for the analyses of phylogenetic community structure and trait evolution[END_REF][START_REF] Ricotta | Diversity partitioning of Rao's quadratic entropy[END_REF][START_REF] Pavoine | Functional and phylogenetic similarity among communities[END_REF][START_REF] Ricotta | A family of functional dissimilarity measures for presence and absence data[END_REF]. In species characterization, the traits selected for a given study may be qualified as functional when they are associated with the ability of species to gain resources, disperse, reproduce, respond to loss and generally persist [START_REF] Weiher | C = cornfield. Codes for species: Ajam= Artibeus jamaicensis; Alit= A. lituratus; Bdub= Bauerus dubiaquercus; Cbre= Carollia brevicauda; Cper= C. perspicillata; Dpha= Dermanura phaeotis; Dwat= D. watsoni; Gcom= Glossophaga commissarisi; Gsor= G. soricina; Mkea= Myotis keaysi; Mmeg= Mormoops megalophylla; Ppar= Pteronotus parnellii; Slil= Sturnira lilium; Ttri= Thyroptera tricolor; n27= all species descending from node named n27 in the phylogenetic tree (see Appendix C in Supplementary material); these include Chiroderma villosum, Platyrrhinus helleri; Vampyressa pusilla, Vampyrodes major, and Uroderma bilobatum[END_REF] or when they influence ecosystem properties or species responses to environmental conditions [START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: revisiting the Holy Grail[END_REF][START_REF] Hooper | Effects of biodiversity on ecosystem functioning: a consensus of current knowledge[END_REF]. Functional traits lead to measures of functional similarity between species and between communities. Two levels of similarities are thus nested: one among the species and one among the communities.

Estimating trait-based similarities among communities can reveal, for example, that some species are filtered out from an environment because of their traits, while others can expand, being adapted or tolerant to the environmental conditions (environmental filtering). This approach can also reveal that species with differences in fitness but similarities in niches rarely co-exist within the same community (competitive exclusion) [START_REF] Mayfield | Opposing effects of competitive exclusion on the phylogenetic structure of communities[END_REF]. Estimation of the phylogenetic similarities among communities -especially when the lineages driving these similarities are clearly identified -can provide insights into historical and evolutionary mechanisms, including the potential for allopatric and ecological speciation [START_REF] Graham | Phylogenetic beta diversity: linking ecological and evolutionary processes across space and time[END_REF].

Referring to [START_REF] Jost | Entropy and diversity[END_REF] observations on more traditional indices, [START_REF] Ricotta | Diversity partitioning of Rao's quadratic entropy[END_REF] observed that two communities should be completely distinct (similarity=zero) if they have no species in common and if their species have no (trait-based, taxonomic or phylogenetic) similarities. The absence of trait-based similarities among species can be observed if these species have maximally distinct trait states. The absence of phylogenetic similarity would be obtained relative to a given delimited clade if the species of the first community diverged from the species of the second community at the root of the clade without any subsequent shared history. This point of view assumes that previously shared history outside the clade is discarded. In all cases, Ricotta and Szeidl's viewpoint assumes that the differences between species have a maximum that cannot be exceeded. [START_REF] Pavoine | Functional and phylogenetic similarity among communities[END_REF] responded to this definition of completely distinct communities by developing a new family of indices for measuring the trait-based, taxonomic and phylogenetic similarity between two communities. Let S spe =( spe kl s ) be a matrix where spe kl s is the similarity between species k and species l; spe 1 kk s  for all k, and spe 01 kl s  for all k and l. The matrix is non-negative definite [START_REF] Seber | A matrix handbook for statisticians[END_REF], so that for any real vector   1 ... . In addition, when p ik =1/n i , where n i is the number of species in community i, then S Ochiai is equivalent to [START_REF] Ochiai | Zoogeographic studies on the soleoid fishes found in Japan and its neighbouring regions[END_REF] index of similarity that uses species presence and absence in communities:

ij i j a n n
, where a ij is the number of species shared by communities i and j. The problem raised by [START_REF] Jost | Entropy and diversity[END_REF], concerning completely distinct communities, was known by quantitative ecologists: with certain dissimilarity indices centered on species' identity only, two sites without any species in common may be attributed a smaller dissimilarity than another pair of sites sharing species [START_REF] Orloci | An agglomerative method for classification of plant communities[END_REF][START_REF] Legendre | Numerical ecology[END_REF]. [START_REF] Orloci | An agglomerative method for classification of plant communities[END_REF] therefore developed an index derived from the chord distance to circumvent this paradox. This issue was extended to phylogenetic and functional diversity by [START_REF] Ricotta | Diversity partitioning of Rao's quadratic entropy[END_REF].

Let S com =( com ij s ) be the matrix of similarities between communities obtained from eqn.

1.1 (i.e., com ij s = ( , )

Ochiai i j S pp ). The objective of this study is to develop a new ordination method that analyzes and summarizes the information driven by matrix S com of similarity among communities into independent one-dimensional axes that can be directly explained by the composition of species communities, by species' trait, taxonomic or phylogenetic positions. These methodological advances are illustrated with: 1) theoretical examples; 2) a case study where the taxonomic and trait-based (dis)similarities between bird communities are depicted along environmental gradients under Mediterranean and temperate bioclimates;

and 3) a case study on the phylogenetic dissimilarities between bat communities along a disturbance gradient in Selva Lacandona of Chiapas, Mexico.

Materials and Methods

DSPCA

As highlighted above, for the matrix S spe to be used in index S Ochiai , it needs to have a special mathematical property, i.e., non-negative definite. [START_REF] Pavoine | Functional and phylogenetic similarity among communities[END_REF] described various ways of obtaining a non-negative definite matrix S spe from trait-based, taxonomic and phylogenetic data and demonstrated that, in that case, the matrix S com has values bounded between 0 and 1. I show in Appendix A that if S spe is non-negative definite, S com is also non-negative definite. These mathematical properties common to S spe and S com are exploited in DSPCA.

DSPCA can be related to the analysis of correlation matrices in normed principal component analysis [START_REF] Corsten | Graphical exploration in comparing variance matrices[END_REF][START_REF] Seber | Multivariate observations[END_REF]. The approach can be described in four main steps: (1) obtaining a space in which species are positioned according to their similarities, [START_REF] Power | Challenges in the quest for keystones[END_REF] positioning the communities in this space according to the species they contain and the abundances of these species, (3) obtaining new axes which successively optimize the representation in few dimensions of the similarities among the communities, and

(4) projecting species and communities on these new axes.

The details of the approach are as follows. For the first step, similarities among species are described on a series of independent axes obtained from the eigen-decomposition of S spe : spe n×1 and m×1 vectors of units, respectively). For the second step, communities are positioned at the center of their species; the rows of t  Y P X thus provide coordinates for the communities. These coordinates are normalized as follows: . This third step allows switching from a space where the axes successively describe similarities among species to a space where the axes successively best describe similarities among communities in light of their species composition. In the fourth step, the final coordinates of the species are presented in the rows of final  X XB , and those of the communities in the rows of

1 t   Y Q P X ,
1 final final t   Y
YB Q P X . The columns of matrices X final and Y final are principal components and the rows within each matrix represent the species and the communities, respectively. A community point is located on the axes in the direction of the (abundance-weighted) center of its species; its exact position satisfies the requirement that the norm of the community coordinates is 1 (community and species are located in a ball of radius 1 such as variables in a normed principal component analysis). In the final multidimensional space, entities (species and communities) can be displayed by arrows starting from the origin of the space to the vertices defined by the rows of Y final and X final , respectively. A community arrow is thus unit length and points to a direction defined by a weighted mean of species' arrows; weights are the proportions (e.g., relative abundance) of 2D-graphics can be displayed using any two principal components of the communities. The first principal component contains the largest part of the similarities among communities, the second is orthogonal to the first and contains the second largest part, and so on. These 2D-graphics optimize the visualization of the similarities among communities while explaining these similarities with their species. In the multidimensional space, the arrows of any two communities i and j form an angle. The cosine of this angle is com ij s . This means that, in this graphical approach, two communities are similar if their arrows form a very acute angle. The larger the angle, the more dissimilar they are. Community and species coordinates are bounded between -1 and 1. In 2D-graphics, they can thus be represented within a circle of unit radius. The coordinate of a species in a principal component expresses how representative the species is of the similarity pattern identified by the principal component (see Appendix B in the Supplementary material and the case studies below).

The sum of all eigenvalues in Ψ is equal to the number of communities. The number of axes examined in an analysis depends on these eigenvalues. Several coefficients can be used to evaluate the quality of the graphical representation of the similarities obtained by retaining the first k out of K axes, including

  11 / 100% kK k i i ii      
(see [START_REF] Seber | Multivariate observations[END_REF] for indices developed in other contexts). The first eigenvalue, λ 1 , reflects the amount of overall similarity among all communities. Its value is approximately equal to 1 + (m-1) s [START_REF] Friedman | Interpreting the first eigenvalue of a correlation matrix[END_REF], where s is the mean similarity between any two communities and m the number of communities. If communities are not completely distinct, the last eigenvalue expresses the full dissimilarities between the communities (what is left when all similarities have been described). Intermediate eigenvalues detail multivariate similarity patterns, that is to say the fact that some similarities concern only part of the compared communities. In the extreme case where the similarities between communities are equal, say to s, then λ 1 = 1 + (m-1) s (Morrison, 1978, p. 289). For example, if communities are completely distinct, then s = 0 and λ 1 =1, which is the lowest possible value for λ 1 . In that case, all m eigenvalues are equal to 1. When the similarity between any two communities is positive, then at least λ 1 is higher than 1 and at least λ m lower than 1. If there are only two communities compared, then s is the similarity between these two communities, λ 1 = 1 + s and λ 2 the second and last eigenvalue equals 1 -s, expressing thus the dissimilarity between the two communities.

Case studies

Calculations were performed with R (R Core Team, 2018) as described in Appendices C and D of the Supplementary material.

Theoretical data set #1

Within-community diversity influences the length of the species arrows; for example, if the functional diversity of a community is high, then the constitutive species have low similarity in terms of their functional traits. Each species of the community is thus unlikely to be representative of others. More generally, if PCi, the ith axis of DSPCA, represents a certain similarity between x communities, then the contribution of a species shared by the x communities to the identified similarity pattern is high if the functional, taxonomic or phylogenetic characteristics of this species are very common within each of these x communities. To illustrate this point, I use three simple examples as described in Fig. 1.

Theoretical data set #2

The second theoretical data set aims to highlight the main discrepancies between DSPCA and another ordination approach: double principal coordinate analysis (DPCoA) developed by [START_REF] Pavoine | From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis[END_REF]. First, DSPCA uses similarities among species and communities whereas DPCoA focuses on dissimilarities. Second, DPCoA and DSPCA differ in their treatment of completely distinct communities. DPCoA was not defined to be restricted to bounded dissimilarities between communities. In this particular case, however, the distance between completely distinct communities in DPCoA maps depends on the diversity within each community. By contrast, DSPCA always provides zero similarity between completely dissimilar communities. To highlight these main differences between DPCoA and DSPCA, I applied both approaches to the following theoretical data set: 110 species, named s1 to s110, have no similarities with each other. S spe is thus a diagonal matrix with 110 rows and 110 columns, with unit values on the diagonal and 0s elsewhere. Four communities have no species in common. The first community c1 has species s1 to s50; the second, c2, has species s51 to s100; the third, c3, has species s101 to s105; and the fourth, c4, has species s106 to s110. Species' proportions within communities are even.

Theoretical data set #3.

A common practice when analyzing pair-wise dissimilarities between communities is to use non-metric (nMDS) or metric (MDS) multidimensional scaling depending on the Euclidean properties of the dissimilarity matrix of interest. For example, MDS can be applied to a matrix of dissimilarities obtained with 1 Ochiai S 

. When MDS and nMDS are used, however, information about species is lost, and it may not be possible to identify which species, trait, or phylogenetic position contributed to the dissimilarities among communities a posteriori. Placing species a posteriori at the barycenter of their communities in MDS or nMDS maps may be misleading. In doing so, the position of the species will reflect their abundance within communities, but not their functional, taxonomic or phylogenetic dissimilarities. To illustrate this fact, I used the theoretical data set described in Fig. 2a [START_REF] Pavoine | From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis[END_REF]. The data set [START_REF] Blondel | Is there ecomorphological convergence among mediterranean bird communities of Chile, California, and France[END_REF] contains bird communities living in different parts of the world under Mediterranean bioclimates: central Chile, California (United States), and Provence (France). These regions were compared to a control region under a temperate bioclimate: Burgundy (France). [START_REF] Blondel | Is there ecomorphological convergence among mediterranean bird communities of Chile, California, and France[END_REF] determined equivalent habitats among the four regions in terms of structure, height and physiognomy of vegetation. Overall, the habitats form a gradient of vegetation complexity from habitat#1 (the least complex) to habitat#4 (the most complex). The data set contains data on species' foraging substrate (multichoice nominal variable), morphometry (quantitative variable) and taxonomy. The effects of species abundance and species-to-species similarities on the results of DSPCA can be analyzed by considering both presence-absence data and abundance data, and by considering species as maximally dissimilar in addition to analyzing trait and phylogenetic information on species (see Appendix E in the Supplementary material for a pedagogic illustration). Here I explored the effect of species-to-species similarities by considering four matrices of species similarity: 1) spe MAX S contains 1 on the diagonal and 0 elsewhere, which means that species are maximally dissimilar; 2) spe FOR S was defined as a function of the substrates where species forage using the Ochiai index of similarity; 3) spe MOR S was obtained by applying [START_REF] Gower | A general coefficient of similarity and some of its properties[END_REF] similarity to species morphometric traits; and 4) spe TAX S has 1 on the diagonal, 3/4 between species of the same genus, 1/2 between species of the same family but distinct genera, 1/4 between species of similar order but distinct families, and 0 between species of different orders, families and genera. The method used to calculate taxonomic similarities is also related to the Ochiai coefficient. Indeed the taxonomic similarity between two species can be expressed as kl kk ll t t t , where t kl is the number of taxonomic levels shared by the two species and t kk is the total number of taxonomic levels that describe any species k (here 4 levels: species, genus, family, and order). This leads to t kk being equal to 4 for all k. The taxonomic similarity between two species k and l is thus t kl / 4. The calculation of all similarity matrices is detailed in this Appendix C of the Supplementary material. to the root of the tree, and c kk is the sum of branch lengths on the shortest path that connects species k to the root of the tree [START_REF] Pavoine | Functional and phylogenetic similarity among communities[END_REF]. This coefficient is thus also related to the Ochiai index. Because the phylogenetic tree is ultrametric, c kk = H, the height of the tree, for all species k, and the phylogenetic similarity between two species k and l reduces thus to kl cH . I compared the results obtained with DSPCA with those produced by evoPCA Chord , an ordination approach I developed in [START_REF] Pavoine | A guide through a family of phylogenetic dissimilarity measures among sites[END_REF] to specifically analyze phylogenetic tree data.

Results

Theoretical data set #1

When communities are maximally dissimilar (Fig. 1a), the species within a community are linked only to this community in DSPCA. Their arrows superimpose that of the community. The lengths of species arrows, however, depend on how representative each species is of the community. The more numerous species are within the community and the more distinct they are (from a functional, taxonomic or phylogenetic perspective), the less representative each species is of the community composition. When a community is nested within another, the similarity between these two communities depends on the number of species shared and on the number of similarities between these species and between unshared species (Fig. 1b). The lengths of species arrows also depend on these two factors. When communities do not share species, they can still be similar if the most representative species of each community are similar (Fig. 1c). In any case, the species arrows tend towards the communities where they occur and their length depends on how well they represent the composition of each community.

Theoretical data set #2

DSPCA identifies the absence of similarity between communities, placing them on orthogonal axes, with unit eigenvalues (Fig. 3). The arrows for species point to the direction of the communities in which they occur. However, their sizes change depending on the diversity within the associated community. As observed above, the size of a species arrow expresses how representative a species is of the similarity pattern. The example in Fig. 3 is extreme, so that each axis represents a community, and species are all maximally dissimilar.

In that case, the size of a species arrow associated with community i is 1/ i n , where n i is the number of species in community i. The size of a species arrow is thus inversely linked with the number of species within the community. By contrast, DPCoA identifies higher similarity between the most diverse communities.

Theoretical data set #3

I analyzed the data set presented in Fig. 2a using DSPCA (Fig. 2b) and MDS (Fig. 2c). MDS places the communities at the vertices of a regular tetrahedron (Fig. 2c). As communities do not share species, positioning species on the map of MDS due to their distribution in communities places them on the point of their community as shown in Figure 2c and thus independently of their traits. With DSPCA, the directions of species arrows indicate which community(ies) each species belongs to, and the size of a species arrow indicates how representative the species is of the(se) community(ies) compared to other communities (Fig. 2b). For example, species s1 with a low value for trait t1 and a medium value for trait t2 is the most characteristic of community c1 compared to other communities.

Species s9, s10, s27 and s28, with medium values for the two traits, are the least original species and have close-to-zero coordinates on the axes. They are the four species that discriminate the least among the four communities.

Bird data set

When bird species were considered maximally dissimilar, DSPCA identified four main principal components (axes) (Fig. 4): the first one for the similarities between Burgundy and Provence; the second for the similarities between habitats in Chile; the third for similarities between habitats in California; and the fourth for the distinction between habitats in Provence and those in Burgundy. The fifth and sixth principal components then highlight the gradient of vegetation complexity in Chile and California, respectively. The length of species arrows on these six axes increases with the number of habitats in which they were observed (from 1 to 4 per region) and decreases with the number of species in the region and each of its habitats. The orthogonal patterns highlight that California, Chile and France do not share species.

When applied to foraging substrate, the first principal component of DSPCA highlighted high similarities between all communities (Fig. 5). Species coordinates reveal that the species most representative of the study area forage on the ground solely or in addition to other substrates. The second and third principal components highlight the environmental gradient within each region, from species foraging on the ground in open habitats, to a large diversity of foraging substrates in closed habitats. These principal components are close, but not equal, to the first and second axes of DPCoA applied to the same data set [START_REF] Pavoine | From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis[END_REF].

When applied to morphometric data, DSPCA identified the most common morphological shapes for a bird species in the data set and, inversely, the most original shapes (Fig. 6a). The species with the highest coordinates on the first principal component, Sylvia hortensis, is the most representative of bird morphology in the study area (considering that a species that occurs in many places also increases similarities among these places). The five species with the lowest coordinates and thus the most morphometrically original species are Ammodramus sandwichensis with a relatively short tail, Sylviorthorhynchus desmursii with a relatively very long tail, and the three hummingbirds, notably with their unique beak shape, Archilochus alexandri, Calypte costae, Calypte anna. The eigenvalues of other axes were very low, which indicates low morphometric differences between communities within and across regions.

With taxonomic information, DSPCA underlined on the first principal component the dominance, in terms of species occurrences, of Passeriformes in all habitats of all regions (Fig. 6b). The second and third principal components highlighted minor differences discriminating the four regions from each others: e.g., the more frequent presence of

Bat case study

I applied DSPCA to the phylogenetic similarities between bat communities in Selva Lacandona of Chiapas, Mexico. The first principal component highlighted high similarities between all habitats (high eigenvalue and close-to-1 scores for all habitats) (Fig. 7a). The sets of the most abundant species in each habitat are closely related. The least representative species in the study area (Thyroptera tricolor, Bauerus dubiaquercus and Myotis keaysi, with close-to-zero scores) are the most isolated on the phylogenetic tree. They are also among the least abundant. The results obtained on the second and third principal components are close (Fig. 7c,d), but not equal, to those obtained with evoPCA Chord [START_REF] Pavoine | A guide through a family of phylogenetic dissimilarity measures among sites[END_REF]. Compared with evoPCA Chord , DSPCA does not directly position the nodes of the phylogenetic tree on the factorial maps. DSPCA distinguishes cornfields with high abundance of Sturnira lilium from old fields with high abundance of Carollia brevicauda and C. perspicillata and the rainforest, which is distinguished by the higher relative abundance of 10 species including Artibeus jamaicensis, A. lituratus, Dermanura watsoni and D. phaeotis (Fig. 7c,d). On the third principal component, Glossophaga commissarisi and G. soricina characterize both cornfields and old fields compared to other habitats (Fig. 7c,d). This pattern was not revealed by evoPCA Chord .

Discussion

Connections exist between ordination analyses and diversity measurements (e.g., [START_REF] Pélissier | Consistency between ordination techniques and diversity measurements: two strategies for species occurrence data[END_REF]. While measures value biodiversity, ordination analyses use these values to depict structures in the diversity of communities. They identify, for instance, which communities are similar. Some can also identify which species, taxa, clades or traits are responsible for these similarities (e.g., [START_REF] Pavoine | From dissimilarities among species to dissimilarities among communities: a double principal coordinate analysis[END_REF]. Recent approaches have focused on describing the phylogenetic patterns of communities (e.g., [START_REF] Duarte | Phylogenetic habitat filtering influences forest nucleation in grasslands[END_REF][START_REF] Pavoine | A guide through a family of phylogenetic dissimilarity measures among sites[END_REF].

DSPCA can describe how functionally or phylogenetically similar communities are. It is flexible in the type of similarities measured between species. DSPCA orders communities along axes, the number of which depends on the complexity of the similarity matrix among communities. The axes are orthogonal, provide independent information and are organized from the main to the most residual pattern of similarity. The strength of the similarity pattern provided by an axis is represented by a numerical value, which is an eigenvalue. It is thus possible to describe a pattern of similarity and to provide a value of its importance compared with the pattern of similarity expressed by all other available axes. If patterns are not presented per axis but for a set of axes, a coefficient is provided to evaluate the amount of information extracted by these axes (e.g., [START_REF] Seber | Multivariate observations[END_REF]. The methodology offers direct solutions for explaining the pattern of similarities among communities with their compositions in species and the functional, taxonomic or phylogenetic links specified between them.

DSPCA analyzes both similarities and dissimilarities between communities. For example, in the bat dataset, DSPCA revealed low effects of habitat disturbance on the phylogenetic structure of bat communities: the measured phylogenetic similarity between the four compared habitats was high and the only identified differences between habitats concerned young clades and terminal branches of the phylogenetic tree. The bird data set

showed that in all regions, the species composition changed along the gradient of vegetation complexity, from species foraging on the ground to species using a large diversity of foraging substrates in close habitats. Despite identified changes in species identity, in particular, despite the absence of species shared between California, Chile and the two French regions, DSPCA revealed high similarities between all regions and habitats in terms of species taxonomy and morphometry. DSPCA can thus be usefully applied to communities that share no species, because different species may have similarities due to their traits, phylogenetic or taxonomic positions. This shows that DSPCA could also be applied to entities that are systematically unshared by communities such as individuals and populations. DSPCA could thus be applied in the future to explore within-species variation by focusing on individuals or populations considering that species trait may vary from community to community.

Here, I analyzed trait-based (dis)similarities and phylogenetic (dis)similarities separately. Further applications of the approach could explore new ways of measuring the similarities among species to analyze trait-based diversity in light of phylogeny. For example, new approaches could be considered to apportion a matrix of species traits into a matrix of phylogenetically explained variations in traits among species, and inversely, a matrix of traitbased information independent of phylogeny (see, e.g., [START_REF] Diniz-Filho | An eigenvector method for estimating phylogenetic inertia[END_REF][START_REF] Desdevises | Quantifying phylogenetically structured environmental variation[END_REF][START_REF] Giannini | Canonical phylogenetic ordination[END_REF]. Using the latter matrix to calculate similarities among species in DSPCA could reveal trait-based similarities among communities not driven by phylogeny.

An alternative would be to follow [START_REF] Cadotte | The ecology of differences: integrating evolutionary and functional distances[END_REF] by developing similarities between species that are nonlinear combinations of trait-based similarities and phylogenetic similarities.

DSPCA also allows identification of the most representative species of one or several communities compared to other communities. In the bird data set, for example, DSPCA identified the species S. hortensis (the western Orphean warbler) as the most representative of the morphometric aspects of birds in the whole data set. DSPCA also allows the identification of the species with the rarest characteristics, such as the hummingbirds in the bird data set, with their unique beak shape. In the bat data set, DSPCA identified the most phylogenetically isolated species with the lowest abundance as the least representative species in the study area. Compared with other ordination approaches, DSPCA is thus able to identify not only original species in an original, species-poor community but also original species within a diverse and otherwise common community. The identification of original species may be important if these species are keystone, being rare while having important functions in the ecosystem [START_REF] Mouillot | Rare species support vulnerable functions in high-diversity ecosystems[END_REF][START_REF] Power | Challenges in the quest for keystones[END_REF]. Inversely, the most representative species may represent the species most adapted to their biotic and abiotic environments. The amount of functional redundancy in an assemblage, for instance, may enhance the resilience of the assemblage after a disturbance if functionally similar species differ in their response to disturbance [START_REF] Walker | Biodiversity and ecological redundancy[END_REF]. DSPCA thus allows a complete evaluation of the trait-based, taxonomic or phylogenetic diversity within and between communities due to its description of (dis)similarities between species and communities and as a result of the identification of original and redundant species.

DSPCA ensures, via index S Ochiai , that two completely distinct communities always have zero similarity, as recommended by [START_REF] Ricotta | Diversity partitioning of Rao's quadratic entropy[END_REF]. By contrast, in DPCoA, the similarity between two communities is considered high whenever the average similarity between an individual from the first community and an individual from the second community is approximately the same as the average similarity between two individuals drawn from the same community. DPCoA should be preferred over DSPCA when dissimilarities among species do not have to be bounded between 0 and 1. In that case, maximally dissimilar species cannot exist and neither can maximally dissimilar communities.

The use of DSPCA or DPCoA relates to how the biological dissimilarities and similarities among communities have to be defined considering the objective of the study at hand. Compared with DSPCA and DPCoA, evoPCA Chord is dedicated to phylogenetic data expressed by a hierarchical tree describing the evolutionary relationships between species.

Both DSPCA and DPCoA can handle a variety of data including functional, taxonomic and phylogenetic data. A common feature of all three approaches, however, is that the (dis)similarity indices they use are rooted in traditional literature on biodiversity. Indeed, if species have no similarities with each other, the index used by DPCoA is the Euclidean distance between the vectors of species proportions of the two compared communities, which corresponds to the index of β diversity developed for Gini-Simpson diversity [START_REF] Lande | Statistics and partitioning of species diversity, and similarity among multiple communities[END_REF] see also Appendix A). If species have no similarities with each other, the indices used by DSPCA and evoPCA Chord are both related to the [START_REF] Orloci | An agglomerative method for classification of plant communities[END_REF] index. If these species also have equal proportions within communities, these indices reduce to the [START_REF] Ochiai | Zoogeographic studies on the soleoid fishes found in Japan and its neighbouring regions[END_REF] index.

Compared with simply applying MDS or nMDS to a matrix of dissimilarity between communities, DSPCA, DPCoA and evoPCA Chord all permit the identification of the species, taxonomic groups, clades or traits responsible for the identified patterns of (dis)similarity between communities.

Conclusion

DSPCA summarizes multidimensional similarities into individual similarity patterns represented by orthogonal axes. These individual similarity patterns are ordered and their relative strength evaluated. Applied to the phylogenetic distribution of a group, DSPCA has the potential to raise hypotheses about historical processes such as colonization processes and dispersal limitation. Applied to (morphological, behavioral or life-history) traits of the species, DSPCA could also reveal the influence of the environment on the evolution of labile species functional traits or on the impact of conserved functional traits on the dispersal abilities of these species. If no information on the phylogeny and functional traits is given, this approach is still valid. In that case, it evaluates similarities in species abundances between sites. A comparison of the results obtained with DSPCA applied to species, functional, and phylogenetic data could increase the chance of identifying key ecological and evolutionary mechanisms that shape community assembly (e.g., [START_REF] Pavoine | Measuring biodiversity to explain community assembly: a unified approach[END_REF][START_REF] Stegen | Inferring ecological processes from taxonomic, phylogenetic and functional trait β-diversity[END_REF]; see also [START_REF] Swenson | The assembly of tropical tree communitythe advances and shortcomings of phylogenetic and functional trait analyses[END_REF].

Let A=  

1 t t  Q P R , com t  S
A A . The matrix S com that contains S Ochiai (p,q) for any number of communities is thus non-negative definite.

A.2. Conservation of the similarities among species and among communities in DSPCA

Y is the matrix with m rows and r columns defined in the main text. Matrices YY is by definition a non-negative definite matrix, the previous equations (Seber, 2008, theorem 10.8, p. 220) If these species also have equal proportions in each of the compared communities and if a is the number of species shared by communities i and j , b is the number of species found in community i only, and c is the number of species found in community j but not i , then

    2 22 1 1 1 1 1 2 DISC a b c a b a c a b a c             
From this equation, it can easily be noted that DISC depends on the diversity within communities even if the two communities have no species in common. If a = 0, the Ochiai index equals 0 and it does not depend on b and c. Species arrows are frequently superimposed. For example, s1-8 means that the arrows associated with these species, from s1 to s8, are identical. show that the higher the redundancy between the species that drive the similarity structure between communities, the longer the species arrows.

Single-column figure

2-column figure

An ordination approach to explore similarities among communities by S. Pavoine

Appendix C. Manual for R scripts 1 Functions

Function dspca

The R function dspca performs the ordination approach DSPCA. It will become part of package adiv of R. The reader can refer to the package for updated versions of the function. dspca has the following usage:

> dspca(com, S, tol=1e-8)

The parameters are defined as follows:

Parameter Explanation com Data frame or matrix with communities as rows, species as columns and abundances, proportions or presences/absences (1/0) as entries. S Matrix of similarities among species (species as rows and columns in the same order as in df). tol a tolerance threshold: an absolute value is zero if it is lower than tol.

The result is a list of the following objects:

Parameter Explanation eig Final eigenvalues (diagonal values of Ψ in the main text): positive eigenvalues of the matrix of similarities among communities. X Final coordinates of the species (X final ): matrix with the coordinates of the species on the principal components associated with the matrix of similarities among communities. The names of the matrix start with "CPC" indicating "communities' principal component". Y

Final coordinates of the communities (Y final ): matrix with the coordinates of the communities on the principal components associated with the matrix of similarities among communities. The names of the matrix start with "CPC" indicating "communities' principal component".

Scom

Matrix of similarities among communities (obtained with coefficient S Ochiai ).

Coordinates can be visualized with graphic tools available in R. Examples are provided in the next section entitled "Applications".

Applications

Load the R function contained in Appendix D. For that, you can use:

> source(file.choose()) 1 Install packages ade4, adiv, cluster, phylobase, adephylo and ape of R > install.packages("ade4") > install.packages("adiv") > install.packages("cluster") > install.packages("phylobase") > install.packages("adephylo") > install.packages("ape")

Load the packages:

> library(ade4) > library(adiv) > library(cluster) > library(phylobase) > library(adephylo) > library(ape)

bird case study

Load the data set on bird communities:

> data(ecomor)

Species are coded in this data set. Latin names associated with codes are available in object labels of the list ecomor:

> head(ecomor$labels) latin abbr E033 "Archilochus alexandri" "Arc|ale" E034 "Calypte anna" "Cal|ann" E035 "Calypte costae" "Cal|cos" E070 "Patagona gigas" "Pat|gig" E071 "Sephaniodes sephaniodes" "Sep|sep" E001 "Columba palumbus" "Col|pal"

Here are the instructions needed to reproduce the analyses done in the main text:

> com <-t(ecomor$habitat)
Species are maximally dissimilar Bu1 Bu2 Bu3 Bu4 Ca1 Bu1 1.000000e+00 8.576900e-01 5.003702e-01 4.728054e-01 3.412386e-18 Bu2 8.576900e-01 1.000000e+00 6.685032e-01 6.316762e-01 9.412505e-19 Bu3 5.003702e-01 6.685032e-01 1.000000e+00 9.449112e-01 1.250755e-18 Bu4 4.728054e-01 6.316762e-01 9.449112e-01 1.000000e+00 7.037151e-19 Ca1 3.412386e-18 9.412505e-19 1.250755e-18 7.037151e-19 1.000000e+00

> Stax <-diag(rep(1,129)) > # DSPCA > pcatax <-dspca(com, Stax) > # Eigenvalues > pcatax$eig [1] 3.
> pcatax$Scom[1:5,1:5] Bu1 Bu2 Bu3
Bu4 Ca1 Bu1 1.0000000 0.8576900 0.5003702 0.4728054 0 Bu2 0.8576900 1.0000000 0.6685032 0.6316762 0 Bu3 0.5003702 0.6685032 1.0000000 0.9449112 0 Bu4 0.4728054 0.6316762 0.9449112 1.0000000 0 Ca1 0.0000000 0.0000000 0.0000000 0.0000000 1

Similarities between species according to the place where they forage:

> Sfor <-dsimFun(ecomor$forsub, "M", method=4, type="similarity") > # DSPCA > pcafor <-dspca(com, Sfor) > # Eigenvalues > pcafor$eig A close square in the graph means that the species forage on the specified substrate. Legends for substrates are available with the following instruction: ?ecomor.

Species are characterized according to morphometrical traits:

To remove redundancies between morphometric traits, I performed a principal component analysis (PCA) on the morphometric traits. Then, I applied Gower similarity index to the normed coordinates of the species in PCA: > pcamorpho <-dudi.pca(log(ecomor$morpho), scann=FALSE, nf=8) > Dmor <-dsimFun(pcamorpho$l1[colnames(com), ], "Q", type="dissimilarity") > Smor <-1-as.matrix(Dmor/max(Dmor)) > #DSPCA > pcamor <-dspca(com, Smor) > # eigenvalues > pcamor$eig > # Axes 1 and 2 > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcataxo$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcataxo$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Axes 2 and 3 > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcataxo$X, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcataxo$Y, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Similarities between communities (sample) > (pcataxo$Y%*%t(pcataxo$Y))[1:5,1:5] Bu3 0.9201980 0.9512201 1.0000000 0.9932061 0.6221710 Bu4 0.9113432 0.9425491 0.9932061 1.0000000 0.6202234 Ca1 0.6740227 0.6616646 0.6221710 0.6202234 1.0000000

Bat data set

Load the data set on bat communities.

> data(batcomm) > phy <-read.tree(text=batcomm$tre) # phylogenetic tree > ab <-batcomm$ab # abundances of species within habitats Species abundances in front of the phylogenetic tree (log-transformed abundance):

> # Axes 1 to 3 > bat.4d <-phylo4d(phy, log(t(ab[, phy$tip.label])+1)) > table.phylo4d(bat.4d, center = FALSE, scale = FALSE, cex.symbol=2) Legend: F=rainforest; P=cacao plantation; O=oldfields; C=cornfields > # Phylogenetic similarities between species > Sphy <-dsimTree(phy, method=4) > # DSPCA > pcaphy <-dspca(ab[, rownames(Sphy)], Sphy) > # Axes 1 and 2 > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcaphy$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcaphy$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Axes 2 and 3 > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcaphy$X, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcaphy$Y, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) I calculate similarities between species applying to the trait data [START_REF] Gower | A general coefficient of similarity and some of its properties[END_REF] distance scaled between 0 and 1, as follows:

Strait <-dsimFun(trait, "Q", type="similarity")

The resulting matrix of species-species similarities has the following values: Strait s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s1 1.000 0.750 0.625 0.600 0.500 0.475 0.375 0.350 0.250 0.000 s2 0.750 1.000 0.875 0.850 0.750 0.725 0.625 0.600 0.500 0.250 s3 0.625 0.875 1.000 0.975 0.875 0.850 0.750 0.725 0.625 0.375 s4 0.600 0.850 0.975 1.000 0.900 0.875 0.775 0.750 0.650 0.400 s5 0.500 0.750 0.875 0.900 1.000 0.975 0.875 0.850 0.750 0.500 s6 0.475 0.725 0.850 0.875 0.975 1.000 0.900 0.875 0.775 0.525 s7 0.375 0.625 0.750 0.775 0.875 0.900 1.000 0.975 0.875 0.625 s8 0.350 0.600 0.725 0.750 0.850 0.875 0.975 1.000 0.900 0.650 s9 0.250 0.500 0.625 0.650 0.750 0.775 0.875 0.900 1.000 0.750 s10 0.000 0.250 0.375 0.400 0.500 0.525 0.625 0.650 0.750 1.000

Now I run DSPCA on this dataset: The first eigenvalue indicates high average similarities between the 5 communities.

The similarities between communities can be obtained as follows:

dspca1$Scom c1 c2 c3 c4 c5 c1 1.0000000 0.8253012 0.6865797 0.6434091 0.3351955 c2 0.8253012 1.0000000 0.9530516 0.9340369 0.6434091 c3 0.6865797 0.9530516 1.0000000 0.9759833 0.7389547 c4 0.6434091 0.9340369 0.9759833 1.0000000 0.8253012 c5 0.3351955 0.6434091 0.7389547 0.8253012 1.0000000

The average similarity is: mean(as.dist(dspca1$Scom))

[1] 0.7561222 this value is close to:

(dspca1$eig[1]-1)/4 [1] 0.7673151 = (λ 1 -1)/(m-1), where λ 1 is the first eigenvalue and m the number of communities.

The first two axes of DSPCA show that c3 is the community with the highest similarities with other species and that c1 and c5 are the most different. Indeed, although c3 do not share species with the other communities, its dominant species has close trait values with at least one of the species of the other communities. Although c1 an c2 have exactly the same species and c4 and c5 also have exactly the same species, the most abundant species of communities These graphs also show that species s5 is the most representative species of communities c2 and c4, while s1 is the most characteristic species of c1 and s10 the most characteristic species of c5.

The coordinates of the communities on the five axes are as follows:

dspca2$Y CPC1 CPC2 CPC3 CPC4 CPC5 c1 -3.781743e-01 7.000956e-01 1.136607e-17 6.006916e-01 -7.758901e-02 c2 -9.065444e-01 1.955914e-01 -4.110979e-17 -2.505847e-01 2.777204e-01 c3 -1.126169e-20 -6.254382e-19 1.000000e+00 1.664908e-18 1.597530e-16 c4 -9.065444e-01 -1.955914e-01 4.070810e-17 -2.505847e-01 -2.777204e-01 c5 -3.781743e-01 -7.000956e-01 -9.948896e-18 6.006916e-01 7.758901e-02

This shows that community c3 has a coordinate equal to zero on all axes except axis 3. Axis 3 indicates the complete dissimilarity between c3 and the other communities, because c3 does not share species with the other communities and because information on species traits was ignored.

Then, axis 4 indicates the differences between c1-c5 and c2-c4, and axis 5 the differences between c2 and c4, which are the lowest differences between any two of the communities.

# Axes 4 and 5 par.mar <-par()$mar par(mar=rep(0.1,4)) par(mfrow=c(1,2)) # Species ade4::s.arrow(dspca2$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species", xax=4, yax=5) symbols(0,0,1, inch=F, add=TRUE) # Communities ade4::s.arrow(dspca2$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities", xax=4, yax=5) symbols(0,0,1, inch=F, add=TRUE) par(mar=par.mar) Now I evaluate the effect of species abundance on community-to-community similarities. By transforming abundance data into 0/1 data (0 for the absence, 1 for the presence of a species in a community).

The new matrix of species presence/absence in communities is obtained as follows:

comPA <-com comPA[comPA>0] <-1 comPA s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 c1 1 1 1 0 1 0 0 0 0 0 c2 1 1 1 0 1 0 0 0 0 0 c3 0 0 0 1 0 1 0 1 0 0 c4 0 0 0 0 1 0 1 0 1 1 c5 0 0 0 0 1 0 1 0 1 1 I apply DSPCA to this matrix and the species-to-species trait similarities: The first eigenvalue of this new application of DSPCA indicates high similarities between the 5 communities.

The similarities between communities can be obtained as follows:

dspca3$Scom c1 c2 c3 c4 c5 c1 1.0000000 1.0000000 0.8787559 0.7058824 0.7058824 c2 1.0000000 1.0000000 0.8787559 0.7058824 0.7058824 c3 0.8787559 0.8787559 1.0000000 0.9183618 0.9183618 c4 0.7058824 0.7058824 0.9183618 1.0000000 1.0000000 c5 0.7058824 0.7058824 0.9183618 1.0000000 1.0000000

The average similarity is: mean(as.dist(dspca3$Scom))

[1] 0.8417765 this value is close to:

(dspca3$eig[1]-1)/4 [1] 0.8425449 = (λ 1 -1)/(m-1), where λ 1 is the first eigenvalue and m the number of communities.

With presence/absence data, communities c1 and c2 become similar to each other; c4 and c5 are also similar to each other. The overall similarities between the five communities (evaluated by the first eigenvalue) increases compared to the DSPCA applied to abundance data. Indeed, considering presence/absence data increases the similarities between c1 and c5 and the other communities. This pattern of similarity is shown on the first two axes of DSPCA where the points of communities c1 and c2 are superimposed and the points of communities c4 and c5 are also superimposed: dspca3$Y CPC1 CPC2 CPC3 c1 0.9171264 -0.39655411 -0.04029864 c2 0.9171264 -0.39655411 -0.04029864 c3 0.9837132 0.04131551 0.17493265 c4 0.9274282 0.37023776 -0.05292358 c5 0.9274282 0.37023776 -0.05292358 # Axes 1 and 2 par.mar <-par()$mar par(mar=rep(0.1,4)) par(mfrow=c(1,2)) # Species ade4::s.arrow(dspca3$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") symbols(0,0,1, inch=F, add=TRUE) # Communities ade4::s.arrow(dspca3$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") symbols(0,0,1, inch=F, add=TRUE) par(mar=par.mar)

2 

 2 where the columns of U contain eigenvectors and the diagonal values of Λ contain eigenvalues. The rows of 1/XU Λ provide coordinates for the species. The axes on which these coordinates are defined are called principal components in the context of multivariate analyses of correlation matrices. The expression "principal component" is also retained here although similarities replace correlations. Let   12 | | ... | m  P p p p be the n×m matrix with the proportions of n species in m communities (

2. 2 . 5 .

 25 Bat data set I also applied DSPCA to data from Medellín et al. (2000) on bats in four habitats in the Selva Lacandona of Chiapas, Mexico, with Fritz et al. (2009) phylogeny pruned for retaining only the species present in the Medellín et al. data set. The four compared habitats were distributed on a disturbance gradient from an active cornfield (the most disturbed), through old fields and cacao plantations, to rainforests (the least disturbed). The phylogenetic similarity between two species k and l was defined as kl kk ll c c c : c kl is the sum of branch 14 lengths on the shortest path that connects the most recent common ancestor of the two species

  Emberizidae species in open habitats of California and Chile, Piciformes in close habitats in California, Chile, and Burgundy, Paridae species in close habitats of Burgundy, species of the genus Sylvia in open habitats of Provence and more generally, Sylviidae and Turdidae in Provence and Burgundy.

  DSPCA uses the S Ochiai index (eqn. 1.1) while DPCoA relies on[START_REF] Rao | Diversity and dissimilarity coefficients: a unified approach[END_REF] DISC index,(trait-based, taxonomic or phylogenetic) dissimilarities between two species k and l. An advantage of DPCoA over DSPCA is that it has been extended, for instance, to evaluate how two interacting factors (e.g., habitat and geography) affect the compositions of communities in terms of the functions or lineages they contain[START_REF] Pavoine | A new technique for analysing interacting factors affecting biodiversity patterns: crossed-DPCoA[END_REF]. Such developments for DSPCA are directions for future research.

ΨBΨ

  have the same s non-zero eigenvalues, where s = rank( is the diagonal matrix with all eigenvalues on the diagonal including potential zero eigenvalues. Because t YY is real symmetric, A m is an m × m orthogonal matrix satisfying is a matrix with eigenvectors (in columns) associated with all eigenvalues of t YY, and r is the diagonal matrix with all eigenvalues including potential zero eigenvalues on the diagonal. Because t YY is real symmetric, r B is an r × r orthogonal matrix satisfying tt r r r r r  B B B B I , where r I is the r × r identity matrix (spectral decomposition theorem).

Fig. 1 .

 1 Fig. 1. Results of DSPCA applied to theoretical data set #1. The data set is described in the

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. Analysis of theoretical data set #3: (a) the data set with the matrix of species'

Fig. 5 .Fig. 6 .Fig. 7 . 40 Figure 2 Figure 5 Fig. B. 1 .

 56740251 Fig. 5. Result of DSPCA applied to the bird data set considering similarities between species

  :s.arrow(pcatax$X, xax=1, yax=4, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcatax$Y, xax=1, yax=4, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Axes 2 and 6: > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcatax$X, xax=2, yax=6, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcatax$Y, xax=2, yax=6, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Axes 3 and 5: > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcatax$X, xax=3, yax=5, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcatax$Y, xax=3, yax=5, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Similarities between communities (sample): > (pcatax$Y%*%t(pcatax$Y))[1:5,1:5]

  :s.arrow(pcafor$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcafor$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) :s.arrow(pcafor$X, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcafor$Y, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar)

  [1] 1.525938e+01 2.631015e-01 1.438653e-01 1.176506e-01 7.967625e-02[6] 4.805947e-02 3.345960e-02 1.412404e-02 1.368605e-02 9.434050e-03 [11] 6.596879e-03 3.757320e-03 3.048406e-03 2.132844e-03 1.305462e-03 [16] 7.234662e-04 > # Axes 1 and 2 > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcamor$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcamor$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Axes 2 and 3 > par.mar <-par()$mar > par(mar=rep(0.1,4)) > par(mfrow=c(1,2)) > # Species > ade4::s.arrow(pcamor$X, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") > symbols(0,0,1, inch=F, add=TRUE) > # Communities > ade4::s.arrow(pcamor$Y, xax=2, yax=3, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") > symbols(0,0,1, inch=F, add=TRUE) > par(mar=par.mar) > # Similarities between communities (sample) > (pcamor$Y%*%t(pcamor$Y))[1:5,1:5] Bu4 0.9749757 0.9855155 0.9988637 1.0000000 0.8979839 Ca1 0.9108033 0.9178688 0.8998454 0.8979839 1.0000000 Taxonomic similarities between species > Staxo <-dsimTaxo(ecomor$taxo[rownames(ecomor$habitat),], method=4) > # DSPCA applied to taxonomic data: > pcataxo <-dspca(com, Staxo)

  dspca1<-dspca(com=com, S=Strait) DSPCA leads to 5 orthogonal axes with the following eigenvalues:

  dspca3<-dspca(com=comPA, S=Strait) DSPCA leads to 3 orthogonal axes with the following eigenvalues: dspca3$eig [1] 4.37017950 0.59036930 0.03945121

  The diagonal values of Q are the square root of the diagonal values of YY t . If presences/absences are used, the proportion of a species present within a community i that contains n

	squared, diagonal matrix with the diagonal.	spe p S p t i	i	 	spe ik il kl kl p p s ,	at line i and column i and 0 out of where Q is a

i species is set to 1/n i (S Ochiai is not impacted by considering relative rather than absolute abundances, see Appendix A). The third step is determined by the eigendecomposition of t YY: tt  Y Y BΨB , with eigenvectors in B, and positive eigenvalues in Ψ

  the species in the community. It can be shown (Appendix A) that

	final final t Y Y		com S (with
	similarities among communities calculated with index S Ochiai ), so that the similarities among
	communities are preserved in the final space.

  imply that Consider that for any i and k, p ik = n ik / n i+ , where n ik is the absolute abundance of species k at site i (e.g., number of individuals from species k at site i), and n i+ is the total

	tt s s s  abundance at site i ( i and nn    ). ik k YY A ΨA 1/2 1/2 t t t s s s s s   YY YB Ψ Ψ Ψ B Y and thus that matrix s A can be chosen to be equal to The final coordinates of the communities in DSPCA are thus given by 1/2 ss  spe , spe spe ,, jl ik kl kl ij Ochiai jk jl ik il kl kl k l k l i i j j n n s  nn S nn nn ss n n n n        which yields YB Ψ . 1/2 final s s s  Y spe , spe spe ,, ik jl kl kl Ochiai ik il kl jk jl kl k l k l n n s S n n s n n s    YB A Ψ The similarities among communities are contained in com t  S Given that A.4. On the dissimilarity index used by DPCoA YY . 1/2 1/2 t t t s s s s s   and thus t t t ss  When species have no similarity and the dissimilarity ( spe kl d ) between any two species k and l is set equal to 1, then YY YB Ψ Ψ Ψ B Y YY YB B Y then, final final t  S The similarities among communities are conserved in the final space of DSPCA. A.3. Ochiai S treats relative and absolute abundances equally Y Y . com     2 , 1 , kl 2 i j ik il DISC p p   pp

  Table of species foraging substrates where species are ordered according to the first axis of DSPCA:

> table.value(ecomor$forsub[order(pcafor$X[,1]), ], ppoints.cex = 0.2, + labelsx = ecomor$labels[rownames(ecomor$forsub[order(pcafor$X[,1]), ]), 1])

Acknowledgements

I thank reviewers for their useful comments on the paper.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Appendix A. Mathematical proofs

The notations here are the same as in the main text.

A.1. If the matrix of similarities among species is non-negative definite, then the matrices of similarity among communities obtained with coefficient S Ochiai is also non-negative definite

For any matrix A, the matrixes A t A and AA t are non-negative definite (e.g., [START_REF] Albert | Conditions for positive and nonnegative definiteness in terms of pseudoinverses[END_REF]. By definition,

Because S spe is non-negative definite, there is a matrix R so that S spe = RR t (e.g., [START_REF] Seber | A matrix handbook for statisticians[END_REF]. Then,

Supplementary material

Supplementary material associated with this article can be found in the online version, at ---.

Appendixes B to E. Supplementary materials Species coordinates in front of the phylogenetic tree:

> # Axes 1 to 3 > bat.4d <-phylo4d(phy, pcaphy$X[phy$tip.

} dfunsimspe <-function(df, vartype=c("Q","N","M","P"), method=1:5, type=c("dissimilarity", "similarity")){ meantype <-method [1] if(!meantype%in%(1:5)) stop("Incorrect definition of method") fun0 <-function(i){ df0 <-as.matrix(df

return(daisy(df0, metric = "gower")*ncol(df0)) else return((1-as.matrix(daisy(df0, metric = "gower"))

if(type=="dissimilarity") return(daisy(df, metric = "gower")) else return(1-as.matrix(daisy(df, metric = "gower")))

Effects of abundance and species-species similarities in DSPCA -A theoretical example

Here I consider a theoretical example to illustrate how one can evaluate the effects of abundance and similarity data on community-to-community similarities thanks to DSPCA. R scripts used below are given in Appendix D; a manual is available in Appendix C. The scripts below also require that package adiv be loaded:

install.packages("adiv") library(adiv) I first define a matrix with the abundance of 10 species in five communities: com <-matrix(c(10, 1, 0, 0, 0, 5, 2, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0, 1, 0, 0, 1, 10, 0, 10, 1, 0, 0,10, 0, 0, 0, 0, 0, 5, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 5, 0, 0, 0, 1, 10), 5, 10) rownames(com) <-paste("c", 1:5, sep="") colnames(com) <-paste("s", 1:10, sep="") com s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 c1 10 5 2 0 1 0 0 0 0 0 c2 1 2 5 0 10 0 0 0 0 0 c3 0 0 0 1 0 10 0 1 0 0 c4 0 0 0 0 10 0 5 0 2 1 c5 0 0 0 0 1 0 2 0 5 10 Then I define trait values for the 10 species: trait <-c(-4,-2,-1,-0.8,0,0.2,1,1.2,2,4) names(trait) <-colnames(com) trait s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 -4.0 -2.0 -1.0 -0.8 0.0 0.2 1.0 1.2 2.0 4.0

The species traits are distributed on a segment from -4 to 4 with species s1 having the minimum value and species s10 the maximum value. s5 is in the middle of the segment. Species s3 and s4 have close trait values; same for s5 and s6 and s7 and s8. c2, c3, and c4 have trait values close or equal to zero. In contrast, the most abundant species of c1 has a trait value of -4 and that of c5 a trait value of 4. par.mar <-par()$mar par(mar=rep(0.1,4)) par(mfrow=c(1,2)) # Species ade4::s.arrow(dspca1$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") symbols(0,0,1, inch=F, add=TRUE) # Communities ade4::s.arrow(dspca1$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") symbols(0,0,1, inch=F, add=TRUE) par(mar=par.mar)

To evaluate the effect of species-to-species similarities on community-to-community similarities, I run again DSPCA considering that the species are maximally dissimilar (i.e. ignoring trait data). The obtained results are quite different.

The new dissimilarities between species are defined as follows:

Stax <-diag(rep(1,10)) rownames(Stax) <-colnames(Stax) <-colnames(com) Stax s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s1 1 0 0 0 0 0 0 0 0 0 s2 0 1 0 0 0 0 0 0 0 0 s3 0 0 1 0 0 0 0 0 0 0 s4 0 0 0 1 0 0 0 0 0 0 s5 0 0 0 0 1 0 0 0 0 0 s6 0 0 0 0 0 1 0 0 0 0 s7 0 0 0 0 0 0 1 0 0 0 s8 0 0 0 0 0 0 0 1 0 0 4 s9 0 0 0 0 0 0 0 0 1 0 s10 0 0 0 0 0 0 0 0 0 1 I now apply DSPCA to the community matrix and these new species-species similarities:

dspca2 <-dspca(com=com, Stax)

DSPCA leads to 5 orthogonal axes with the following eigenvalues: The first eigenvalue of this new application of DSPCA indicates much more moderate similarities between the 5 communities than the previous application of DSPCA where species trait values were considered.

The similarities between communities can be obtained as follows:

dspca2$Scom c1 c2 c3 c4 c5 c1 1.000000000 0.30769231 0 0.07692308 0.007692308 c2 0.307692308 1.00000000 0 0.76923077 0.076923077 c3 0.000000000 0.00000000 1 0.00000000 0.000000000 c4 0.076923077 0.76923077 0 1.00000000 0.307692308 c5 0.007692308 0.07692308 0 0.30769231 1.000000000

The average similarity is:

this value is lower than:

, where λ 1 is the first eigenvalue and m the number of communities. This is consistent with [START_REF] Friedman | Interpreting the first eigenvalue of a correlation matrix[END_REF] statement that the estimate 1 + (n-1) s "deteriorates slightly" as the variance of the similarities increases. [START_REF] Friedman | Interpreting the first eigenvalue of a correlation matrix[END_REF] actually analyzed correlation matrixes with positive values but their statement remain valid for similarity matrixes].

The first and second axes of DSPCA highlight similarity patterns between c1, c2, c4 and c5. Indeed these four communities share species s5. However s5 has the highest abundance in c2 and c4, whereas it has the lowest abundance in c1 and c5:

# Axes 1 and 2 par.mar <-par()$mar par(mar=rep(0.1,4)) par(mfrow=c(1,2)) # Species ade4::s.arrow(dspca2$X, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Species") symbols(0,0,1, inch=F, add=TRUE) # Communities ade4::s.arrow(dspca2$Y, ylim=c(-1.2,1.2), xlim=c(-1.2,1.2),sub="Communities") symbols(0,0,1, inch=F, add=TRUE) par(mar=par.mar)


Pierre Barberis

Céline Ly

Roland Zino

Roland Zino M5framatome

M5Framatome cladding high temperature oxidation behavior during simulated LOCA transients

Demonstrating the acceptable cladding oxidation behavior in LOCA conditions is critical for the fuel vendor and its customers. Key to this is the accurate characterization of the cladding's resistance to breakaway oxidation, which can lead to rapid hydrogen pick-up and loss of ductility. Framatome has developed a dedicated facility at its Paimboeuf plant in order to perform one-and two-sided oxidation tests in steam, with a tight monitoring (temperature, steam flow) .

Protective oxide and a low hydrogen pick-up were observed. These tests confirmed the excellent performance of M5 Framatome cladding and the absence of breakaway oxidation in LOCA relevant conditions, as already showed by V. Garat (TopFuel 2016).

Diffusion modeling

Conclusion & Perspectives

➢ M5 Framatome cladding does not show breakaway oxidation in these tests.

➢ Finite element modeling showed that iron is significantly pushed in the β-Zr phase, while Sn profile is almost flat and Nb migrates from oxide and α-Zr(O) towards β-Zr.

Future steps :

To include strong coupling between the chemical potential of the various alloying elements (phase stabilization) and computations of stress and strains in the various phases.

Experiments & results

Two-sided steam oxidation of M5 Framatome claddings for 5000 s at 800 -1050°C in Paimboeuf facility.

Optical microscopy of the samples • Interface undulation and porosity.

During oxide growth, the interface undulation reaches a maximum at 1030°C then decreases, with no sign of breakaway by 5000 s.

Oxide and α-Zr(O) thicknesses and hydrogen pick-up after steam oxidation for 5000 s.

Fick

  's equations were implemented in 1Daxisymmetry (whole cladding thickness) and coupled with in-house thermodynamic database. Diffusion coefficient of oxygen was fit to reproduce experimental data at 1000°C. Oxygen diffusion as a function of time during 1000 °C oxidation. Diffusion of Nb, Fe, (and Sn) (independently) in the three phases (oxide, α-Zr(O), β-Zr).

  a trademark or a registered trademark of Framatome or its affiliates, in the USA or other countries.

M5 Framatome cladding high temperature oxidation behavior during simulated LOCA transients 1. Context

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