E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

ava i l ab l e a t www.sc i enced i rec t . com

www.e l sev i e r. com/ l oca te /eco l i n f

Nonlinear dimensionality reduction: Alternative ordinationapproaches for extracting and visualizing biodiversity patternsin tropical montane forest vegetation data

Miguel D. Mahechaa,b,⁎, Alfredo Martínezc, Gunnar Lischeida, Erwin Beckc

aEcological Modelling, Bayreuth Centre for Ecology and Ecosystem Research BayCEER, University of Bayreuth, 95440 Bayreuth, GermanybMax Planck Institute for Biogeochemistry, 07701 Jena, GermanycPlant Physiology, Bayreuth Centre for Ecology and Ecosystem Research BayCEER, University of Bayreuth, 95440 Bayreuth, Germany

A R T I C L E I N F O

⁎ Corresponding author. Biogeochemical ModeJena, Germany. Tel.: +49 3641 576283.

E-mail address: miguel.mahecha@bgc-jenaURL: http://www.bgc-jena.mpg.de/bgc-md

1574-9541/$ - see front matter © 2007 Publidoi:10.1016/j.ecoinf.2007.05.002

A B S T R A C T

Article history:Received 25 October 2006Received in revised form7 May 2007Accepted 9 May 2007

Ecological patterns are difficult to extract directly from vegetation data. The respectivesurveys provide a high number of interrelated species occurrence variables. Since often onlya limited number of ecological gradients determine species distributions, the data might berepresented by much fewer but effectively independent variables. This can be achieved byreducing the dimensionality of the data. Conventional methods are either limited to linearfeature extraction (e.g., principal component analysis, and Classical MultidimensionalScaling, CMDS) or require a priori assumptions on the intrinsic data dimensionality (e.g.,Nonmetric Multidimensional Scaling, NMDS, and self organized maps, SOM).In this studywe explored the potential of Isometric FeatureMapping (Isomap). This newmethodof dimensionality reduction is a nonlinear generalization of CMDS. Isomap is based on anonlinear geodesic inter-point distance matrix. Estimating geodesic distances requires one freethreshold parameter, which defines linear geometry to be preserved in the global nonlineardistance structure.Wecompared Isomap to its linear (CMDS) andnonmetric (NMDS) equivalents.Furthermore, the use of geodesic distances allowed also extending NMDS to a version that wecalledNMDS-G. In additionwe investigated a supervised Isomap variant (S-Isomap) and showedthat all these techniques are interpretable within a single methodical framework.As an example we investigated seven plots (subdivided in 456 subplots) in different secondarytropical montane forests with 773 species of vascular plants. A key problem for the study oftropical vegetation data is the heterogeneous small scale variability implying large ranges of β-diversity. The CMDS and NMDSmethods did not reduce the data dimensionality reasonably. Onthe contrary, Isomapexplained 95%of thedata variance in the first fivedimensions andprovidedecologically interpretable visualizations; NMDS-G yielded similar results. Themain shortcomingof the latterwas the high computational cost and the requirement to predefine the dimension ofthe embedding space. The S-Isomap learning scheme did not improve the Isomap variant for anoptimal threshold parameter but substantially improved the nonoptimal solutions.We conclude that Isomap as a new ordination method allows effective representations ofhigh dimensional vegetation data sets. The method is promising since it does not require apriori assumptions, and is computationally highly effective.

© 2007 Published by Elsevier B.V.

Keywords:Vegetation dataOrdinationIsometric Feature Mapping (Isomap)Multidimensional Scaling (MDS)Nonlinear dimensionality reductionGeodesic distancesSecondary mountain tropical rainforestsEcuador

l-Data Integration Group, Max Planck Institute for Biogeochemistry, P.O. Box 10 01 64, 07701

.mpg.de (M.D. Mahecha).i (M.D. Mahecha).

shed by Elsevier B.V.

139E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

introduction of Principal Curves to ecology by De'ath (1999) or

1. Introduction

Data from vegetation surveys or floristic inventories are oftenused as the basis of ecological assessments in integratedstudies. Typically, the respective data are high dimensionalarrays, where the dimensionality equals the overall number ofrecorded species. The high number of species occurrence datamakes it difficult to derive ecological or biogeographicalinformation directly from the raw data. Techniques ofdimensionality reduction for extracting and visualizing theinherent properties of vegetation data are well established invegetation ecology since the late 1960s as “ordination meth-ods” or, more precisely, as “indirect gradient analysis” (Whit-taker and Gauch, 1982; Minchin, 1987; ter Braak, 1995). Thesemethods relate high dimensional vegetation data sets to asmall number of independent structures. Efficient descrip-tions of the structure of the data set are assumed to requirefewer free variables than provided by the observed number ofspecies (Camastra, 2003).

Traditionally, Principal Component Analysis (PCA), Classi-cal Multidimensional Scaling (CMDS), and CorrespondenceAnalysis (CA) are used as ordination approaches, whichmerely account for linear relationships within the data sets(Legendre and Legendre, 1998). However, especially in largedata arrays, linear structures may occur as special cases ofunderlying nonlinearities. Thus, these approaches are notappropriate for most real world data sets (Geng et al., 2005),including ecological data. Alternative nonlinear techniques ofdimensionality reduction, e.g., Nonmetric MultidimensionalScaling (NMDS, Shepard, 1962; Kruskal, 1964), or the methodsbased on Artificial Neural Networks (ANN, Bishop, 1995), useiterative learning schemes. This leads to high computationalrequirements for large data sets. In addition, these methodsoften get stuck in local minima of the error function. Hence,the solutions do not necessarily provide the best possible lowdimensional representation of the data manifold (de Silva andTenenbaum, 2003). NMDS and some of the ANN-approacheslike the Self-Organized Maps (SOM, Kohonen, 2001) share thedisadvantage of requiring a predefinition of the dimensional-ity of the reduced space. These a priori assumptions on theintrinsic data dimensionality are difficult to estimate directlyfrom the raw data (Camastra, 2003). Thereforemethods wherethe intrinsic dimensionality of the data emerges from theprojection are to be preferred.

In the last few years a whole series of new nonlinear di-mensionality reduction algorithms had been developed whichcircumvent most of the mentioned constraints. Examples ofthese so called manifold learning algorithms are IsometricFeature Mapping (Isomap, Tenenbaum et al., 2000), LocallyLinear Embedding (LLE, Roweis and Saul, 2000), LaplacianEigenmaps (Belkin and Niyogi, 2002), Hessian Eigenmaps(Donoho and Grimes, 2003), and Conformal Eigenmaps (Shaand Saul, 2005). These new nonlinear methods were mainlydeveloped and applied within the fields of cognitive sciences,artificial intelligence, and image processing. In ecologicalstudies new nonlinear methods are still rarely applied,although singular studies showed that transferring newdevelopments of machine learning and data mining toecological problems is very helpful. Examples are, e.g., the

the application of SOM's by Giraudel and Lek (2001). In thiscontext McCune and Grace (2002) suggested in their standardtext book that, e.g., Isomap and LLE would be ideally suited fordata analysis in vegetation ecology. Following this idea in thisstudy we investigated the Isomap-algorithm proposed byTenenbaum et al. (2000), as well as a supervised variant (S-Isomap, Geng et al., 2005). We aimed at introducing Isomap tothe ordination of species assemblages, exploring its potentialfor extracting global nonlinear properties of vegetation data.Isomap can be interpreted as a nonlinear extension of CMDS.Thus, Isomap can be investigated in the general framework ofMultidimensional Scaling methods. We focused on a compar-ison of Isomap and S-Isomap with the respective linear andnonmetric equivalents CMDS and NMDS and investigated alsopossible recombinations of these variants.

For exploring the potential of Isomap and derivates as newordination approaches we investigated an exemplary data setcollected in the tropical mountain forests of Ecuador. Theseecosystems are typically characterized by extreme high phyto-diversity. Apart from some areas protected as national parks,mostof theseecosystemsundergoconstant transitionscausedbyhuman impacts, and land use change. The data set consisted ofan inventory of more than 700 species in seven plots of afragmented Ecuadorian montane cloud forest, one of the globalhotspots of plant biodiversity (Brummitt and Lughagha, 2003).The surveyed forest plots were known to be secondary afterdifferent types and dates of human impact. The differentoverlapping stagesof successiondidnot allow identifyingdistinctplant communities in the field. This complexdata array (setupby773 species on 457 subplots in seven plots) is exceptional invegetation science since secondary tropical forests are not welldocumented in the botanical literature and the data collection islogistically difficult. The challenges working with this data were,first to find a way to deal with the high species richness (α-diversity), requiring a powerful dimensionality reduction meth-od,andsecondtosurmount theextremeheterogeneityexpressedas differences between the subplots. The latter is a problem ofdealing with a high degree of β-diversity.

2. Methods

2.1. Study area

The study area is located at 3°58′21ʺ S and 79°4′33ʺ W betweenthe province capitals Loja and Zamora in the south-eastern partof the Ecuadorian Andes, the “Cordillera Oriental”. The researcharea borders thenationalpark “Podocarpus”which is consideredas the protected area with the highest number of endemicspeciesof Ecuador (Valencia etal., 2000).The investigated forestsare at approximately 1950 (±50) m asl where the mean annualtemperature is 15.3 °C with an absolute measured maximum of29 °C, and the lowest reported temperature being 6 °C. Theaverage precipitation is 2067 mm yr−1, and the mean annualevaporation is estimated as 727 mm yr−1 (Bendix et al., 2006).

2.2. Sampling design

Seven different patches of secondary tropical mountain forest(C1, C2, C3, C4, L, F1, and F2) were examined (Fig. 1). Each plot

Fig. 1 –Orthophotography of the research area, according to the UTM projection based on the geoid WGS 84. White pointsindicate the centre of the investigated plots. The coordinate units are meters.

140 E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

was subdivided by a grid of contiguous rectangular subplots of5×5m each. Due to a strong relief the grid cells of plot C3 wereof 4×4m. Depending on the size of the forest patches, between60 and 91 subplots were established in each plot. Plots C1, C2,C3 and C4 are accessible by a trail along a water pipe. Theconstruction of the pipeline was the reason for clearing the

primary forest some 50 years ago. The history of impacts ateach investigated site could be reconstructed from aerialphotographs, together with reports from local people and fieldobservations. Aerial photographs (from 1962, 1976 and 1989,respectively) indicate that plot C1 was deforested between1962 and 1976 and was impaired in the same period, at least

141E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

partly by a land slide thatmay have been a consequence of thework on the pipeline. The aerial photographs show that plotC2 was deforested shortly before 1962, while C4 has notsuffered any direct clearing impact. Although not covered bythe aerial photographs the stand structure of plot C3 suggests,that the plot is partlymoremature than plot C2, but affected atthe edges by land slides. These observations coincide withfield observations on stand structure and with comments bylocals. The Plots L, F1 and F2 are at a comparable altitude onthe opposite side of the valley and are considered as remnantsof a ravine forest. The forest of plots F1 and F2was damaged byfires approximately 15 years ago. Site F2 suffered more impactsince it is also easier accessible for local farmers. By contrast,Plot L is a young succession (approximately 10 years) after acomplete destruction of the plant cover by a land slide, whereno organic soil layer has developed until now. All 3 plots aresurrounded by a fragmented agricultural area, showingremains of the former forest, e.g. charred tree trunks.

The recordings of the vegetation were done in several fieldcampaigns between 2001 and 2003. For species identificationthe Flora of Ecuador by Harling and Anderson (1973) was usedwhich is, however, not complete. The comparison of thecollected material with identified samples in the “HerbarioReinaldo Espinosa” of the Universidad Nacional de Loja, the“Herbario de la Pontifícia Universidad Católica del Ecuador,Quito”, and the “Herbario de la Universidad de Azuay, Cuenca”was another important method of identification. Somesamples could be identified using the data base “Visual Plants”(Dalitz, 2002, www.visualplants.de). Duplicates of the collect-ed samples were deposited in the mentioned collections andin the herbarium of the research station “Estacion CientíficaSan Francisco”. Scans of the identified species were includedinto the data base “Visual plants”. The total list contained 773species of vascular plants, from which 140 were identified atthe species level, and 358 at the genus level, while 213 couldonly be associated with plant families and 62 could not beidentified. Species which were not identified at species levelobtained working names, to allow differentiating them.

2.3. Data pre-processing

The examination of species–area-relationships at the subplotlevel showed that the effect of different subplot sizes (C3 vs.other plots) might influence the overall extracted patterns ofthe data. Consequently, Plot C3 (subplot size 4×4 m) wasresampled by overlaying a virtual 5×5 m grid and assigningthe species of each subplot to the spatially nearest resample-subplot. If an original subplot was equidistant to varioussubplots at the resample grid, the species of the former wererandomly assigned to those of the resample grid. Onlysubplots of the resample grid were considered whose areaswere covered by at least 50% of the original subplots. All thefollowing statistics had been applied to the data matrix basedon the 5×5 m grid (virtual for C3), leading to an overall of 456subplots.

The data were organized in a matrix X containing 456(subplots) by 773 (plant species) entries. As a quantitativeestimate on vegetation cover is not meaningful in a multi-strata tropical mountain rain forest the matrix X was set upfrom presence absence observations. In our nomenclature the

vectors xi: i=1, …, n describes the ith subplot and each entry{xi,j: i=1, …, n; l=1, …, m} denotes the presence (xi,j=1) orabsence (xi,j=0) of one out of m species at site i.

2.4. Classical Multidimensional Scaling (CMDS)

The idea of Classical Multidimensional Scaling (CMDS, Tor-gerson, 1952) is to project high dimensional data points into anordination space of much lower dimensionality. The goal ofthe projection is to preserve the linear inter-point distances asthey were measured between the data points in the inputspace (Borg and Groenen, 1997). Contrary to the well knownPCA which attempts preserving the covariance structure,CMDS searches for an optimal representation of the inter-point distances measured by any kind of metric. Thus,Minchin (1987) states that CMDS, other than PCA, is notlimited to data sets characterized by low β-diversity. TheCMDS-method is an effective and well suited ordination toolwhen the data matrix X is a high dimensional representationof a linear manifold (de Silva and Tenenbaum, 2003).

The initial step of CMDS is to set up a matrix of linear inter-point distances. Here, this was realized by a square-rooteddistance function based on Jaccard similarities. This was anappropriate approach, since the available data matrix X wasbinary {0,1}. The precise distance formula was given bydi;j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a

ðaþcÞq

, where a is the number of species co-occurrenceat the sites i and j, a ¼ Pm

l¼1 xi;lxj;l, where xi,l=xj,l, and c is thenumber of cases of unique occurrence of a species in i or j,c ¼ Pm

l¼1 xi;l þ xj;l, where xi,l≠xj,l.The resulting distances di,j are metric and completely embed-

ded in an Euclidean space Rm (Legendre and Legendre, 1998). Bycalculating the distances between all pair of points a symmetricmatrix Dwas obtained containing the distances {di,j: i, j,=1,…, n}.

After applying a double centering procedure the lowdimensional projection can be found by solving an eigenvalueproblem (for a detailed derivation see Borg and Groenen, 1997or Cox and Cox, 2001). The double centered distance matrixproblem is re-expressed by its eigenvectors Ei and eigenvaluesλi. The eigenvalues are sorted by convention in descendingorder. The principal coordinates are then defined as Eiλi1/2,which span the coordinate space of the low dimensionalrepresentation of the data (Cox and Cox, 2001). The CMDSapproach allows making use of any linear distance matrixwhich is one reason for its extreme flexibility. Although binarydata are typical for vegetation analysis, in many studiesinterval scaled data are available and allow constructing theinitial distance matrix based on classical Euclidean distances.Gámez et al. (2004) showed that if interval scaled data arecentered to zero mean and unit variance, CMDS based onEuclidean distances becomes numerically equivalent to PCA.Thus, CMDS can be seen also as a more general variant ofdimensionality reduction than PCA.

2.5. Nonmetric Multidimensional Scaling (NMDS)

The Nonmetric MDS approach (NMDS, Shepard, 1962; Kruskal,1964) is an alternative ordination method which aims atpreserving the rank ordering of the distances di,j in a lowdimensional space, expressed as a monotonic functioninstead of inter-point distances (Borg and Groenen, 1997; Cox

142 E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

and Cox, 2001). This nonmetric method differs fundamentallyfrom CMDS requiring the predefinition of the ordinationspace. The iterative algorithm minimizes a cost function, theso called stress function, which estimates the relationbetween the initial distance matrix D and a distance matrixset up from the vector position in the ordination space. A finalordination result is obtained after a maximum number ofiterations, or preferably, after a stress convergence criterion ismet. Here themaximumnumber of iterationswas fixed due tocomputational limits to 10000, and the stress convergencecriterion of the so called “stress formula one” was set to0.00001, which is a very accurate parameterization (McCuneand Grace, 2002). The predefined low dimensional represen-tation space is spanned by arbitrary axes which cannot beordered in function of their significance as indicated, e.g., bycorresponding eigenvalues in CMDS (Legendre and Legendre,1998).

We used different NMDS implementations in the Matlab®

7.2 environment, where the version provided byMark Steyvers(http://psiexp.ss.uci.edu/research//software.htm) performedbest and was used here.

2.6. Isometric Feature Mapping (Isomap)

The Isomap algorithm developed by Tenenbaum et al. (2000)can be interpreted as a nonlinear extension of CMDS. Insteadof trying to preserve the linear distances between each pair ofdata points while representing the data in a low dimensionalspace, Isomap tries to represent the geodesic distances as theywould bemeasured on an underlying nonlinear manifold. The“geodesic distance” is the shortest distance on any kind ofnonlinear structure. The concept is well known in geography,where the distance between two spatial points is estimated asgeodesic distance along the earth's nonlinear surface insteadof calculating the shortest linear distance through the globe.The key idea of Isomap is to recover the global nonlineargeometry of the manifold, and to project it to a lowdimensional embedding space. In the present case the lowdimensional manifold is unknown. For approximating thegeodesic distances Isomap connects all points xi within adefined radius ε in Rm as measured by a linear distancemetric. Based on these multiple inter-point connections anundirected neighbourhood graphG is set up. Now, the shortestpaths on the constructed graph G expressed by di,j(G) arecomputed, in this study by the Dijkstras-algorithm (Dijkstra,1959). This leads to a matrix D(G)= {di,j(G)}, where di,j(G) are theinter-point distances on the graph which are assumed tobe good approximations for the unknown “true” geodesicdistances. The approximation of the global nonlinear geode-sic distance structure of the data preserves the local linearinter-point geometry of the data points. An alternative way isto construct the neighbourhood graph G based on a numberof k-nearest neighbours of each point combination. Thechoice of k or ε is based on heuristics, where it must be at-tempted to find the smallest possible value, which still leadsto an overall connected graph. For example low thresholdvalues (k, or ε) in clumped data clouds will easily lead to graphdisconnections. In contrast, if the threshold parameter k, or εrespectively, is set too large, the approximation of thegeodesics suffers, since shortcuts on the approximated mani-

fold will underestimate the geodesic distance. This is thereason why in the case of a convergence of k to n−1, or ε tomax{dij}, the estimated geodesic distances will tend to equalthe linear ones: D(G)≈D.

After the neighbourhood connectivity has been estab-lished, the last step is to apply CMDS (as described above) onthe estimated geodesic distance matrix D(G). Recall that CMDSattempts to best preserve the input metric in a low dimen-sional ordination space. Thus, the principal coordinates Eiλi1/2

derived from a matrix of geodesic approximations areassumed to be good approximations for the underlingnonlinear structures. This is why Isomap is considered anonlinear generalization of CMDS; and given the prerequisitesfor an equality of PCA and CMDS, Isomap can be also seen asnonlinear generalization of the standard PCA (Gámez et al.,2004).

In vegetation science one often faces the problem that thestandard linear distance measure between plots which do notshare any species or between plots which share only a fewspecies tend to themaximum value of the respective measure(which is 1 in the case of square-rooted Jaccard distances).This does not permit to differentiate between sites which donot share species but which are of floristically comparablestructure and those which are not. The Isomap approachsearches for a defined number of neighbouring sites. Since theprobability is high that two comparable sites which do notshare species do have one ormore common neighbour a smalldistance between them could be established through, e.g., twoor very few links. This let us expect to find a betterrepresentation of inter-point distances which reflect the siteconditions and biogeographical history in data sets charac-terised by a high β-diversity.

The Isomap implementation at http://Isomap.stanford.edu/ (Tenenbaum et al., 2000) was used in this study for allIsomap and CMDS applications.

2.7. Supervised Isomap

Geng et al. (2005) proposed an extension to Isomap, whichdiffers fundamentally from the above described techniquesby following a supervised learning paradigm (S-Isomap). Theidea is that additional nominal scaled information, e.g., theplot affiliation of the subplots can be incorporated to theordination by assigning class labels. According to thisapproach, the subplots are described by (xi, yi), where xi: i=1,…, n are the data points (subplots) and yi are the class labels, inour case e.g. plot C3. The inter-point distances are thenweighted differently, depending whether the points belong toequal or distinct classes. The weight function is given by

dsi;j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e

�d2i;jb

sfor yi ¼ yjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e

d2i;jb � a

sfor yipyj

8>>>>><>>>>>:

. Thus, dsi,j is substantially increasing

with increasing di,j. The parameter β is used to reduce thisphenomenon at high values of di,j and normally set to theaverage distance of D. The parameter α is introduced towarrant that the distance between two subplots from differentclasses e.g. (xi, yi) and (xj, yj) where yi≠yj, could be less thanintra-class distances between (xi, yi) and (xk, yk), where yi=yk.

143E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

This is necessary to assure that the later constructed Isomap-graph remains connected; for details see Geng et al. (2005).After the supervised dissimilarities dsi,j are calculated, thematrix DSi,j={dsi,j} is used instead of D for all further cal-culations within the Isomap approach. However, the super-vised distances can also be used for other methods.

The implementation at http://lamda.nju.edu.cn/datacode/s-isomap/s-isomap.htm (Geng et al., 2005) was used for thesupervision of the distances before applying Isomap.

2.8. The general MDS framework

As it was shown above, the framework of multidimensionalscaling provides a variety of options, where, e.g., the goal topreserve the input metric (CMDS) has to be balanced with theattempt to preserve the rank order of the data points (NMDS).The presented methods can be combined in various ways. Forexample, having seen that CMDS is just a special variant ofIsomap (where k=n−1 or ε=max(dij)), it is possible to introduceS-CMDS in analogy to S-Isomap or to apply NMDS based ongeodesic distances which we call here NMDS-G. Thesevariants can be seen as an “MDS decision tree” (Fig. 2). Theanalyst has to take three basic decisions: First, he has to decidewhether or not to use additional information and to follow thesupervised learning approach. Second, he has to define thelocal geometry of the distance matrix by setting a k or εrelationship. The last question is whether or not to preservethe input metric in the projection or just their rank ordering.This decision tree comprises well known approaches (CMDS,NMDS, Isomap, and S-Isomap) as well as some new methodssuch as the supervised “S-CMDS” or “S-NMDS” variant or “S-NMDS-G”. In this studywe explored the performance of CMDS,Isomap, NMDS, NMDS-G, S-CMDS, and S-Isomap.

The chosen path in the decision tree requires a qualitativeevaluation criterion on the performance. One way to asses thequality of the dimensionality reduction is to apply a Manteltest (Mantel, 1967), which expresses the correlation betweenthe input distance matrix and a distance matrix derived fromthe point locations in the recovered low dimensional embed-

Fig. 2 –The decisions to be taken when perform

ding space. The null hypothesis is that the two matrices arenot related by any relationship. This hypothesis is rejected ifthe random permutation of the rows and columns of one ofthe two matrices has a lower correlation value (Legendre andLegendre, 1998). This procedure allows identifying the amountof explained variance principally analogous to the suggestionin the original Isomappaper by Tenenbaumet al. (2000), whereno randomization is performed and simply Pearsons' productmoment correlation coefficient is used as value of explainedvariance. This was also the way chosen here, since a pretest ofthis study indicated that all recovered values of explainedvariance were also highly significant in the Mantel-test at a plevel of 0.0001.

3. Results

3.1. Isomap and CMDS ordination

The aim of the present study was to explore the performanceof the Isomap algorithm in a real word data set. The crucialpoint in the application of Isomap is the identification of anoptimal threshold parameter, to determine the size of linearpatches to be preserved in the global nonlinear projection. Weopted to work with k instead of ε because the latter resulted inmany cases in a subdivision of the graph into unconnectedclusters and out ranges. The problem of graph-disconnectionwas not observed with k≥3. An optimal Isomap-solution wassought, by varying k systematically from 3 to n−1=456, whereIsomap equals CMDS. The values of cumulative explainedvariance were plotted versus the fraction of linear inter-pointconnection in the overall distance matrix (Fig. 3, A). Visualiz-ing the fraction of linear inter-point distances is favourable,since some points receive more than k linearly connectedneighbours especially at high k values. This effect is respon-sible for a nonlinear response of the fraction of linear inter-point distances for varying k-values (Fig. 3, D). The systematicvariation of the size of linear patches allowed identifying apeak of explained variance at k=7. This Isomap variant

ing an ordination in the MDS framework.

Fig. 3 –The panel A shows the values of explained variance, as recovered cumulatively by the leading modes. The Isomapperformance is presented for a varying k parameter, where themaximumk-value achieves a linear CMDS. Subplot B shows thecumulative explained variances for the supervised Isomap method (S-Isomap and S-CMDS). In C the deviations of Isomap andS-Isomap are shown, where a negative value indicates a better performance of S-Isomap. D shows the response of the effectivefraction of linear inter-point distances in the geodesic distance matrix versus the chosen k value (Isomap), and in thesupervised geodesic distance matrix (S-Isomap).

Fig. 4 –Cumulative values of explained variance for thedifferent MDS methods in the different embeddingdimensions.

144 E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

explained 71% of the variance in the first, 78% in the first two,and 85% in the first three dimensions. The explanatory powerconverged at 93% of explained variance for ten dimensions(Fig. 4). To the best of our knowledge no comparableordination results describing tropical high diversity patternswere reported in literature so far. The recovered values ofvariance behaved stable over a certain region of k-valuesclose to the optimal solution. After the stable region was left(setting k≥30 which equals a fraction of linear inter-pointconnections of 0.09) a further increase of the linear neighbour-hood patches leaded to drastically decreased values ofexplained variance (Fig. 3, A). Where all points are linearlyconnected Isomap (which equalled CMDS) recovered 12% ofthe variance in the firstmode, and 18% in the first two, and notmore in the first three coordinates (Fig. 3, A). More dimensionsdid not clearly improve the embedding results, indicating thatthe goal of dimensionality reduction was met properly (Fig. 3).Especially no reasonable cut-off criterion could be identified toestimate the required number of dimensions to describe thedata, which should emerge from the projection. By contrast,working with Isomap (k=7), we observed a convergence ofexplained variance after five dimensions, suggesting that thisis the number of required independent modes to describe thedata set. This is deducing that the intrinsic dimensionality ofthe data is five (Fig. 4).

3.2. NMDS and NMDS-G ordination

Because CMDS did not provide reasonable results in terms ofexplained variance we investigated whether the other classi-cal ordination technique NMDS could compete with Isomap.NMDS is generally assumed to lead to better ordination resultsthan CMDS since it is not constrained to a linear representa-tion (Legendre and Legendre, 1998). We estimated the

cumulative variance, recovered by the NMDS embeddingspaces of different dimensions. The one dimensional embed-ding space recovered 23% of the variance, the two dimensionalspace 31%, and the three dimensional NMDS 35%. Comparedto CMDS, this is a rather good result but still does not reach theexplanatory power of Isomap (Fig. 4).

We then applied NMDS-G to the geodesic distances estimat-edby thebest Isomapvariant (k=7). In theory,NMDS-Gshouldberun for varying k values as computed for Isomap. However,NMDS is based on an iterative learningmethod which leads to avery high computational demand. In our case it was practicallynot possible to perform a NMDS sensitivity analysis for allpossible k values. Instead, working with the optimal Isomap kparameter had the advantage to allow a direct comparison of

Fig. 5 –Two dimensional ordination plots for the different MDS variants. A is the CMDS ordination. B is the Isomap (k=7)embedding. C is the two dimensional NMDS ordination. D is the new developed NMDS-G variant (k=7). E is the S-CMDS, and Fthe S-Isomap (k=7) variant. The grey lines in B, and F is the neighbourhood graphwhich also is the basis of the geodesic NMDSin D. However, since NMDS-G does not preserve the linear distances the graph structure can not be visualized properly.

145E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

Isomap and NMDS-G. The results show that the latter recoveredmore of the variance in the first three embedding spaces thanIsomap. In theonedimensional space,NMDS-Grecovered72%ofthe variance, 82% were represented by the two dimensionalspace, and 86% in three dimensional embedding space. Thevalues of recovered variance converged in higher embeddingspaces to 91%. Interestingly, the convergence occurred again at

the fifth dimension, confirming the Isomap estimate on theintrinsic data dimensionality (Fig. 4).

3.3. Supervised CMDS and Isomap ordination

Using an α-value of 0.8 and β=mean(di,j), a distance matrixwas set up. The β parameter was chosen according to the

Fig. 6 –The Isomap embedding (k=7) of the subplots and therespective neighbourhood graph. The colour code indicatesthe number of species observed at each subplot. Thus, asimultaneous visualization of α-diversity and localβ-diversity was obtained. Compare Fig. 5, B for identificationof the subplots.

146 E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

suggestions provided by the sensitivity analysis of the originalpaper. The value of α was chosen to assure the overallconnectivity of the graph of geodesic distances even for lowk values (k≥3). The explanatory power of the S-Isomapembedding is shown in Fig. 3, B. For low k values whichrecovered high amounts of variance, almost no differencesbetween Isomap and S-Isomap were found. On the contrary,for large linear patches the supervision step improved theIsomap results. The deviation of the two methods expressedas differences in explained variance are shown in Fig. 3, C. Inthis graph, positive deviations indicate better performances ofthe standard Isomap. Negative deviations instead, quantifythe improvement of the Isomap embedding through thedistance supervision. The plot shows that the standardIsomap method could not be outperformed by S-Isomaparound the optimal k-value. However, at larger local linearpatches S-Isomap explained up to 20% more of the variance.This phenomenon is strongest near the global linear solution(CMDS). In other words, where Isomap performed best,additional supervision does not provide further improve-ments. This is the area where high values of explainedvariance let us expect having identified the “domain oftopological stability” (Balasubramanian et al., 2002). Bycontrast, in areas where Isomap did not provide convincingresults the supervision improved the ordination meaningly.

3.4. Visualizations

The “value of explained variance” is a very intuitive but not theexclusive criterion to evaluate the performance of an ordina-tion. Another essential feature of ordination is to providereasonable visualizations of a few leading patterns hidden inthe data space. Although CMDS recovered the fewest values ofvariance, already some interesting patterns emerge from thelow dimensional embedding space (Fig. 5, A). The subplotsform distinct clusters and the first axis clearly separates, e.g.,C1 from the others except from L. However, in many cases thesubplots are scattered in the same area and the plots are not

separated in higher dimensions (not shown). Isomap (k=7)performs much better, and a clear distinction between theplots is visible (Fig. 5, B). It is possible to identify a cleargradient from L, and C1, across C2 and C3 to C4. On the secondIsomap coordinate the L and F plots are oppositely arranged.This overall structure is confirmed by the two dimensionalNMDS embedding space, however the arrangement showsless distinct clusters (Fig. 5, C). A very well interpretableordination space was provided by NMDS-G (Fig. 5, D). Again aclear gradient though the C plots emerges and all plots arewellseparable from each other. The visualizations in the supposedregion of topological stability did not vary much, and thegeneral patterns in Fig. 5, B, and D can be reproduced by allembeddings of k≤30. In addition, the supervised variantsconfirmed the above described visualizations (Fig. 5, E and F).Here the ordering of the plots is already well visible in thelinear variant. This could be seen as a confirmation of thefindings in the domain of explained variance, where the firsttwo S-CMDSmodes improved CMDS bymore than 20% (Fig. 4).

All low dimensional visualizations based on a geodesicdistance matrix were well interpretable (Fig. 5B, D, and F).However, the Isomap results provide an additional interestingfeature. Since the projection preserves the input metric, thestructure of the neighbourhood graph is also preserved in theembedding space and can be visualized. As indicated before,the linear inter-point distances express the local β-diversity.Thus, a joint visualization of α-diversity and local β-diversity ina globalnonlinear projection is nowpossible (Fig. 6). In thiswayan Isomap ordination plot allows, e.g., to find out whether arecovered coordinate coincides with a gradient of biodiversity,or whether the β-diversity structure around a plot is related tothe effective number of occurring species etc. This might opena newperspective in theway of ordination complex vegetationdata sets.

4. Discussion

4.1. Comparison of indirect ordination methods

The main goal of ordination is to achieve a well interpretablelow dimensional visualization of patterns hidden in the rawdata. Ideally, this pattern extraction is purely empirical anddata adaptive, minimizing a priori input (Kirby, 2001). Isomaphas one free parameter, and its setting is usually based onheuristics. The performed analysis showed that the optimalthreshold parameter emerges from the respective value ofexplained variance as a very intuitive qualitative criterion.This approach is well justified since it is principally consistentwith the traditionally applied Mantel-test. One could arguethat using simply CMDS avoids this search for an optimal k.However, we showed that opting for CMDS is itself an implicitparameter choice within the Isomap framework. Hence, westate that the amount of explained variance is the crucialcriterion. In this way we found that the application of CMDSwas not justified here. Considering further the very poor CMDSvisualizations, one finding of this study was that (unless onehas well founded ecological reasons for assuming linear re-lationships) CMDS should not be used any more as an alonestanding ordination method.

Fig. 7 –The approximated geodesic distances are plottedversus the Jaccard based linear distances. At low geodesicdistances both measures are equal and scattered along adij(G )=dij line, which is the effect of preserving the local

geometry of the data and is consequently of concern for n·kinter-point distances. At high geodesic distances (dij

(G )N1) the

linear distances dij are all bound between 0.85 and 1 whilethe geodesics span over a range from 1 to 8, indicating thatIsomap provides a fare more detailed distance structure thanthe standard approaches.

147E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

The NMDS and NMDS-G methods proved to be goodalternatives for dimensionality reduction. Both variantsrecovered more variance in the first three dimensions andprovided better visualizations than their respective CMDS andIsomap equivalents. However, the application of NMDSconfirmed the findings of Legendre and Legendre (1998), whoindicated that NMDSmight be a useful tool for handling smalldata sets, but the application to huge distance matrices isconstrained by large computing times. By contrast, workingwith Isomap does not require considerable computationalpower, since the spectral decomposition of a distance matrixcan performed very effectively. This gives a good perspectivefor future Isomap applications, where large data bases mightrequire dimensionality reduction for further interpretations.Further shortcomings of NMDS are the need to predefine thedimension of the embedding space, the arbitrary sorting of theaxes, and especially that the method recovers arbitrarysolutions. This means that the ordination can not necessarilybe repeated and different runs might produce differentordination results. A further problem is that the optimizationsometimes gets stuck in local minima of the cost function.

Supervising the data by their plot affiliations leaves asidethe intention of working free of a priori input. Nevertheless,this is a justified alternative in vegetation science since therequired secondary information of plot affiliation is typicallyprovided by floristic surveys. The interesting observation inthis study was that supervising the ordination performanceonly improved unsatisfactory Isomap results. Even the best S-Isomap variant could not outperform the best standardIsomap embedding. This indicates that the method isdispensable, when only the optimal solution is considered inthe subsequent analysis. However, if one has reasons forworking with nonoptimal Isomap k-values, e.g., with CMDS,the results can be improved drastically. The supervisedIsomap variant was developed for stabilizing Isomap ordina-tion results for noisy data sets. An optimal Isomap requires awell sampled manifold. Whether a manifold is well sampledor not is not decidable in real world data sets. Only thepresence of “obvious” outliers in ecological data sets couldgive a hint on suspicious Isomap embeddings. Particularlywhenworkingwith a radius ε as threshold parameter, a strongsensitiveness to noise and outliers is observed and the presentstudy found that a threshold k (being scale free by definition)leads to more stable results. However, for data sets containinga whole range of outliers even the k parameter might notcompensate the induced distortions on the final embedding.In such cases a superimposed supervision of the distancemetric in terms of a S-Isomap or S-NMDS-G embedding isthought to provide appropriate solutions.

In general, the ordinations based on geodesic distancesweremuch better than on linear distance matrices. Fig. 7 provides areasonable explication. The plot of the estimated geodesicdistances (k=7) versus the linear distances shows that the lineardistances are allocated in a small range (0.85 to 1). In contrast,the geodesic distances show a highly differentiated distancestructure of several clusters. Considering that the inter-pointdistance is an estimate for β-diversity, the geodesic distanceadmits to representmore heterogeneous distance patterns. Thepreservation of these patterns in the course of dimensionalityreduction might be the reason why Isomap and NMDS-G

ordination are considerable improvements for ordination ofspecies assemblages of high β-diversity.

Summarizing, Isomap is an improvement of classicalordination techniques which are constrained either by theirlimitation to linear feature extraction (CMDS) or, by a highcomputational demand that renders the NMDS not to besuitable for large data sets.

4.2. Ecological interpretation of the nonlinear ordination

The objective of this study was to explore the Isomap methodfor its potential to find a low dimensional representation of ahigh dimensional vegetation data set. We did not aim attesting general hypotheses about driving ecological gradientsin tropical mountain forests. Nevertheless, the results canhelp to identify ecological structures for this specific area, as93% of the total variance could be mapped in five dimensionsby Isomap and Geodesic NMDS. The examination of therespective ordination plots (Fig. 5, B and D) with thebackground information from aerial photographs and com-ments from local people suggests the following interpretation.The order of the plots L, and C1 to C4 along the first axiscorresponds to the duration of recovery after severe impacts.Thus, we conclude that this dimension represents thematuration of the forest. The second Isomap dimensionclearly separates the fire affected plots F1 and F2 from thesuccession after a land slide in plot L. In tropical forestsystems fire is known to lead to temporary nutrient-inducedfertility flushes. A biogeochemical study in the same area byWilcke et al. (2003) compared a sequence of landslide affectedsoils of different ages. They found no significant differences in

148 E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

the nutrient availability of subsoils of the compared sites, butthe organic layers on landslide soils had significantly lessnutrients and the nutrient availability is supposed to be lessavailable to plants due to greater C/N, C/P, C/S ratios (Wilckeet al., 2002, 2003). The second Isomap axis can therefore beinterpreted as a representation of the nutrient inventory of theplots ranging especially from L (very low) to F1 and F2 (veryrich). The fact that plot C3 is partly affected by landslides andhas many subplots arranged at low levels of the second axissupports this assumption. The fact that C4-subplots havesimilarly low values of the second Isomap-coordinate can beinterpreted as an effect of forest maturation on the nutrientdepletion of the soil. To summarize, we assume that the firstIsomap dimension represents a gradient of aging, while thesecond extracted dimension represents a nutrient gradient.

4.3. Perspectives

Besides the results from this study, the application of Isomapis recommended by different authors due to a variety ofreasons. First, Isomap is mathematically well understood(Bernstein et al., 2000) but also intuitive reasonable. Atechnical advantage is the fact that there is no need forprevious assumptions on the inherent dimensionality, whatare key disadvantages of NMDS (Legendre and Legendre, 1998)but also of many other nonlinear methods. Second, Isomap islikely to find a global nonlinear representation for themanifold preserving the local linear structure (de Silva andTenenbaum, 2003; Geng et al., 2005), where the latter is notguaranteed in the presented NMDS-G variant. Third, differentpromising extensions had been developed recently, e.g., forbetter pattern classification (Yang, 2002), the multi-classmanifold learning (Wu and Chan, 2004), “Out-of-Sample”applications (Bengio et al., 2004), the analysis of space–timedata (Jenkins and Matarić, 2004), continuous usages (Zha andZhang, 2003), and the presented supervised performances(Geng et al., 2005). This broad and expansive range of fields ofapplication let us assume Isomap to be relevant to datahandling and data mining tasks in ecology, far beyond thepresented application to tropical vegetation survey.

5. Conclusions

In the near future, international vegetation survey programswill gather increasing amounts of data (Dirzo and Loreau,2005). For these large data sets nonlinear and computationallyefficient techniques of dimensionality reduction are required.The Isomap method is a promising approach of featureextraction, even in complicated data sets of high dimensionsand high β-diversity. Isomap has the advantage of being anoniterative method, of high computational performance. Forvery large data sets, the performance could be improved byworking with a subset of the data points in a landmark basedapproach (de Silva and Tenenbaum, 2003). Furthermore,Isomap provides a well interpretable qualitative criterion toestimate the goodness of the projection by estimating theamount of explained variance. The latter also permits toderive a cut-off criterion to identify the required number ofdimensions. The limiting factor for NMDS approaches is the

low computational performance and the requirement topredefine the projection space. The use of a supervisedordination approach requires a set of a priori assumptionswhich implies that the ordination results are anticipated tocertain extend. This contradicts the idea to learn the hiddenpatterns empirically. We think that the presented applicationof Isomap is only the first step toward future ecologicalordination methods, which must be able to handle theworldwide increasing data bases.

Acknowledgements

Thedata collection periodwas partly supported by theGermanResearch Foundation (DFG) in the project 402: “Functionality ina Tropical Mountain Rainforest”. We thank Jürgen Homeier forproviding the taxonomic help; Sebastian Schmidtlein andAlexandra Weigelt for the time they dedicated to the initialdata preparation. The authors gratefully acknowledge HolgerLange and Michael Hauhs for their valuable methodicaldiscussions. Anna Görner and two anonymous reviewersprovided helpful comments on the final manuscript.

R E F E R E N C E S

Balasubramanian, M., Schwartz, E.L., Tenenbaum, Y., de Silva,V., Langford, J.C., 2002. The Isomap algorithm and topologicalstability. Science 295, 7a.

Belkin, M., Niyogi, P., 2002. Laplacian Eigenmaps and spectraltechniques for embedding and clustering. In: Dietterich, T.G.,Becker, S., Ghahramani, Z. (Eds.), Advances in NeuralInformation Processing Systems, vol. 14. MIT Press, Cam-bridge, pp. 585–591.

Bendix, J., Homeier, J., Cueva-Ortiz, E., Emck, P., Breckle, S.-W.,Richter, M., Beck, E., 2006. Seasonality of weather and treephenology in a tropical evergreen mountain rain forest.International Journal of Biometeorology 50, 370–384.

Bengio, Y., Paiement, J.-F., Vincent, P., Delalleau, O., Le Roux, N.,Ouimet, M., 2004. Out-of-sample Extensions for LLE, Isomap,MDS, Eigenmaps, andSpectral Clustering. In: Thrun, S., Saul, L.K.,Schölkopf, B. (Eds.), Advances in Neural Information ProcessingSystems, vol. 16. MIT Press, Cambridge, pp. 2197–2219.

Bernstein, M., de Silva, V., Langford, J.C., Tenenbaum, Y.B., 2000.Graph Approximations to Geodesics on Embedded Manifolds.Technical report: http://Isomap.stanford.edu/BdSLT.pdf StanfordUniversity, Stanford.

Bishop, C.M., 1995. Neural Networks for Pattern Recognition.Oxford University Press, Oxford. 504 pp.

Borg, I., Groenen, P., 1997. Modern Multidimensional Scaling:Theory and Applications. Springer, New York. 471 pp.

Brummitt, N., Lughagha, E.N., 2003. Biodiversity: where's hot andwhere's not. Conservation Biology 17, 1442–1448.

Camastra, F., 2003. Data dimensionality estimation methods: asurvey. Pattern Recognition 36, 2945–2954.

Cox, T.F., Cox, M.A.A., 2001. Multidimensional Scaling, 2nd edition.Chapman and Hall/CRC, Boca Raton. 309 pp.

Dalitz, H., 2002. Visual plants— Bildbasierte Datenbank für dievegetationskundlicheoderökologischeForschung indenTropen.Berichte der Reinhold-Tüxen-Gesellschaft, RTG 14, 119–129.

De'ath, G., 1999. Principal curves: a new technique for indirect anddirect gradient analysis. Ecology 80, 2237–2253.

de Silva, V., Tenenbaum, Y.B., 2003. Local versus global methodsfor nonlinear dimensionality reduction. In: Becker, S., Thrun,

149E C O L O G I C A L I N F O R M A T I C S 2 ( 2 0 0 7 ) 1 3 8 – 1 4 9

S., Obermayer, K. (Eds.), Advances in Neural InformationProcessing Systems. MIT Press, Cambridge, pp. 705–712.

Dijkstra, E.W., 1959. A note on two problems in connexion withgraphs. Numerische Mathematik 1, 269–271.

Dirzo, R., Loreau, M., 2005. Biodiversity science evolves. Science310, 943.

Donoho, D.L., Grimes, C., 2003. Hessian Eigenmaps: locally linearembedding techniques for high-dimensional data. Proceedingsof the National Academy of Sciences 100, 5591–5596.

Gámez, A.J., Zhou, C.S., Timmermann, A., Kurths, J., 2004.Nonlinear dimensionality reduction in climate data. NonlinearProcesses in Geophysics 11, 393–398.

Geng, X., Zhan, D.-C., Zhou, Z.-H., 2005. Supervised nonlineardimensionality reduction for visualization and classification.IEEE Transactions on Systems, Man and Cybernetics. Part B.Cybernetics 35, 1098–1107.

Giraudel, J.L., Lek, S., 2001. A comparison of self-organizing mapalgorithm and some conventional statistical methods for eco-logical community ordination. Ecological Modelling 146, 329–339.

Harling, G., Anderson, L., 1973. Flora of Ecuador, vol. 1–68.University of Göteborg, Göteborg.

Jenkins, O.C., Matarić, M.J., 2004. A spatio-temporal extension toIsomap nonlinear dimension reduction. Proceedings, Interna-tional Conference on Machine Learning (ICML-2004), Banff,Canada, pp. 441–448.

Kirby, M., 2001. Geometric Data Analysis: An Empirical Approachto Dimensionality Reduction and the Study of Patterns. JohnWiley and Sons, Inc, New York. 390 pp.

Kohonen, T., 2001. Self-organized Maps, 3rd edition. Springer,Berlin. 501 pp.

Kruskal, J.B., 1964. Multidimensional scaling by optimizing good-ness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27.

Legendre, P., Legendre, L., 1998. Numerical Ecology, 2nd edition.Elsevier Science, Amsterdam. 853 pp.

Mantel, N., 1967. The detection of disease clustering and gener-alized regression approach. Cancer Research 27, 209–220.

McCune, B., Grace, J.B., 2002. Analysis of Ecological Communities.MjM Software Design, Glenden Beach, OR. 300 pp.

Minchin, 1987. An evaluation of the relative robustness oftechniques for ecological ordination. Vegetatio 69, 89–107.

Roweis, S.T., Saul, L.K., 2000. Nonlinear dimensionality reductionby locally linear embedding. Science 290, 2323–2326.

Sha, F., Saul, L.K., 2005. Analysis and extension of spectralmethods for nonlinear dimensionality reduction. Proceedings,International Conference on Machine Learning (ICML-2005),Bonn, Germany, pp. 785–792.

Shepard, R.N., 1962. The analysis of proximities: multidimensionalscaling with an unknown distance function. II. Psychometrika27, 219–246.

Tenenbaum, Y.B., de Silva, V., Langford, J.C., 2000. A globalgeometric framework for nonlinear dimensionality reduction.Science 290, 2319–2323.

terBraak,C.J.F., 1995.Ordination. In: Jongman,R.H.G., ter Braak,C.J.F.,van Tongeren, O.F.R. (Eds.), Data Analysis in Community andLandscape Ecology. Cambridge University Press, Cambridge,pp. 91–169.

Torgerson, W.S., 1952. Multidimensional scaling: I. Theory andmethod. Psychometrika 17, 401–419.

Valencia, R., Pitman, N., León-Yánez, S., Jorgensen, P.M., 2000.Libro rojo de las plantas endémicas del Ecuador. Herbario QCA,Pontificia Universidad Católica del Ecuador, Quito.

Whittaker, R.H., Gauch, H.G., 1982. Evaluation of ordinationtechniques. In: Whittaker, R.H. (Ed.), Ordination of PlantCommunities. Dr W. Junk Publishers, The Hague, pp. 279–336.

Wilcke, W., Yasin, S., Abramowski, C., Valarezo, C., Zech, W., 2002.Nutrient storage and turnover in organic layers under tropicalmontane rain forests in Ecuador. European Journal of SoilScience 53, 15–27.

Wilcke, W., Valladarez, H., Stoyan, R., Yasin, S., Valarezo, C., Zech,W., 2003. Soil properties on a chronosequence of landslides inmontane rain forest, Ecuador. Catena 53, 79–95.

Wu, Y., Chan, K.L., 2004. An extended Isomap algorithm forlearning multi-class manifold. Proceeding of 2004 Inter-national Conference on Machine Learning and Cybernetics,vol. 6, pp. 3429–3433.

Yang, M.-H., 2002. Extended Isomap for Pattern Classification. In:Dechter, R., Sutton, R. (Eds.), Proceedings of the EighteenthNational Conference on Artificial Intelligence (AAAI 2002),pp. 224–229. Edmonton.

Zha, H., Zhang, Z., 2003. Isometric embedding and continuum.Proceedings, International Conference on Machine Learning(ICML-2003). American Association for Artificial Intelligence(AAAI), Washington D.C., pp. 864–871.