Global Ecology and Biogeography, (Global Ecol. Biogeogr.) (2008) 17, 284–296
Blackwell Publishing Ltd
RESEARCH
PAPER
Revealing biogeographical patterns by
nonlinear ordinations and derived
anisotropic spatial filters
Miguel D. Mahecha1,2* and Sebastian Schmidtlein3
1
Max Planck Institute for Biogeochemistry, PO
Box 10 01 64, 07701 Jena, Germany, 2Ecological
Modelling, University of Bayreuth, 95440
Bayreuth, Germany, 3Vegetation Geography,
Institute of Geography, University of Bonn,
53115 Bonn, Germany
ABSTRACT
Aim Spatial floristic and faunistic data bases promote the investigation of biogeographical gradients in relation to environmental determinants on regional to continental scales. Our aim was to extract major gradients in the distribution of vascular
plant species from a grid-based inventory (the German FLORKART data base) and
relate them to long-term precipitation and temperature records as well as soil conditions.
We present an ordination technique capable of coping with this complex data array.
The goal was also to sort out the influence of spatial autocorrelation, assuming
floristic autocorrelation is anisotropic.
Location Germany, at a spatial resolution of 6′ × 10′.
Methods Isometric feature mapping (Isomap) was applied as a nonlinear ordination method. Isomap was coupled to ‘eigenvector-based filters’ for generating spatial
reference models representing spatial autocorrelation. What is novel here is that the
derived filters are not based on the assumption of equidirectional autocorrelation.
Instead, the so-called ‘principal coordinates of anisotropic neighbour matrices’ build
filters to test the influence of geographical vicinity in directions of high similarity
among observations.
Results The Isomap ordination of floristic data explained more than 95% of the
data variance in six dimensions. The leading two dimensions (representing about
80% of the FLORKART data variance) revealed clear spatial gradients that could be
related to independent effects of temperature, precipitation and soil observations.
By contrast, the third and higher FLORKART dimensions were dominated by an
antagonism of anisotropic spatial autocorrelation and soil conditions. A subsequent
cluster analysis of the floristic Isomap coordinates educed the spatial organization of
the floristic survey, indicating a considerable sampling bias.
Conclusions We showed that Isomap provides a consistent methodical framework
for both ordination and derived spatial filters. The technique is useful for tracing the
often nonlinear features of species occurrence data to environmental drivers, taking
into account anisotropic spatial autocorrelation. We also showed that sampling
biases are a conspicuous source of variance in a frequently used floristic data base.
*Correspondence: Miguel D. Mahecha,
Max Planck Institute for Biogeochemistry,
PO Box 10 01 64, 07701 Jena, Germany.
E-mail: [email protected].
Web site: http://www.bgc.mpg.de/bgc-mdi/
Keywords
Anisotropic spatial autocorrelation, floristic survey, FLORKART, Germany,
hierarchical variance partitioning, Isomap, nonlinear dimensionality reduction,
PC(An)NM, sampling artefacts.
INTRODUCTION
Over the course of the last decades, a variety of floristic and
vegetation data bases have been set up comprising species
occurrence data on regional to continental scales (e.g. Jalas et al.,
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1972–1999; Mucina et al., 1993; Myklestad & Birks, 1993; Chytry
& Rafajová, 2003; Fridley et al., 2005; Bekker et al., 2007). Such
data collections have provided an important insight, revealing
how major biogeographical patterns are related to environmental
determinants (e.g. Kühn et al., 2006; Whittaker et al., 2007). The
DOI: 10.1111/j.1466-8238.2007.00368.x
© 2007 The Authors
Journal compilation © 2007 Blackwell Publishing Ltd www.blackwellpublishing.com/geb
Spatial pattern extraction by Isomap
dimensionality of macroecological data bases is typically high,
ranging from several hundred to several thousand taxonomic
units on spatial entities. Owing to redundancies in this information,
an assessment of the underlying environmental patterns is
potentiated by steps of data aggregation or dimensionality reduction.
The latter is possible, for example by means of ordination
methods (McCune & Grace, 2002).
The most popular ordination methods are principal components
analysis (PCA), classical multidimensional scaling (CMDS, also
called principal coordinate analysis or PCoA), and correspondence analysis (CA, and derived algorithms), all of which
are restricted to linear feature extraction (Legendre & Legendre,
1998). From a mathematical perspective, these methods follow
very similar analysis schemes (Borg & Groenen, 1997): ecological
data are usually organized in matrices describing observations
of species occurrences or values of other variables on spatial
samples. Thus, the starting point is always the construction of a
matrix expressing sample distances, correlations or covariances.
In biogeography this usually refers to the ecological site-to-site
distances or β-diversity inferred from varying species compositions. The resulting distances are then used to represent the
samples by a set of a few independent eigenvectors, permitting
further direct interpretations and avoiding redundancies and
collinearities. Differences between ordination methods are
primarily due to different metrics of sample distances and different
subsequent transformation procedures before mapping the data
to the so-called ordination space (Legendre & Legendre, 1998;
Cox & Cox, 2001).
It was recognized early on (e.g. Kendall, 1975) that ordinations
based on linear distance matrices are plagued by artefacts, often
summarized as the ‘horseshoe effect.’ Especially for data sets with
overall high values of β-diversity or ‘gradient length’ (McCune &
Grace, 2002), linear ordinations of vegetation data, based on
subsequent eigenvector embeddings as well as their derivates
(e.g. DCA), are known to produce unsatisfactory results
(Minchin, 1987). One problem is that sites that do not comprise
common species are uniformly set to maximum distance
(Williamson, 1978). This does not take into account the fact that
such sites could be considered more or less similar in terms of
composition: two places without actual co-occurrences may
feature more or fewer taxa associated in other places. Ignoring
these indirect affinities is a problem, in particular when data
from large areas include distinct floras.
Williamson (1978) proposed a very simple heuristic alternative
to this problem. According to his approach, it is only justified to
express the distance between two sites by a linear measurement if
they comprise at least one common species occurrence. All other
sites could be connected across shared, linked sites. The distance
between two sites without common species is then represented
by the smallest sum of distances across connected sites. Using this
so-called ‘step-across’ approach, Williamson (1978) computed a
largely nonlinear distance matrix from all shortest-path connections of the sample set, which allows two sites without common
species occurrences to still be ‘close’ in terms of their β-diversity.
The resulting nonlinear distance matrix is subjected to CMDS.
Since CMDS finds a low-dimensional ordination space that best
preserves the input metric (de Silva & Tenenbaum, 2002),
Williamson found an easy way for nonlinear dimensionality
reduction. The ‘step-across’ idea was later taken up and slightly
modified by Bradfield and Kenkel (1987) and reformulated
by De’ath (1999a) as ‘extended dissimilarities’. Unfortunately,
these methodological advances derived from graph-theoretical
considerations have not yet found their way into the literature on
ordination in applied ecological studies.
Tenenbaum (1998) and Tenenbaum et al. (2000) developed
‘isometric feature mapping’ (Isomap), which can be seen as a
generalized reinvention of Williamsons’ method, as it relies on
similar assumptions. Although there are no cross-relationships
documented in the literature, Isomap can be classified a successor
of ‘step-across’ ordination by providing the natural extension
to any distance metric in arbitrary fields of applications.
This general formulation implies several advantages in terms of
mathematical elegance and convergence (Bernstein et al., 2000).
The strengths of Williamson’s method are preserved: the method
is non-iterative and based on a spectral decomposition of the
estimated nonlinear distance matrix.
Isomap has proved to be very useful for analysing highdimensional vegetation data at the site level (Mahecha et al.,
2007). The primary goal of this study is to investigate the potential
of Isomap in the context of a typical macroecological problem: to
extract and to relate major patterns from both biogeographical
and environmental data bases. Throughout this analysis it is
crucial to take into consideration that the target data (which are
species occurrences here) are spatially autocorrelated. This effect
is partly induced by the spatially structured environment and
partly an intrinsic property due to space-dependent biotic strategies
(Legendre, 1993, Dray et al., 2006). Separating the fraction
related to known environmental determinants from other
sources of spatial variation deserves special attention (Borcard &
Legendre, 2002; Diniz-Filho & Bini, 2005; Wagner & Fortin,
2005; Kühn, 2007). For this, the spatial structure of the analysed
data collection needs to be specifically addressed in all data-driven
modelling tasks to avoid both an overestimation of the explanatory
power of environmental determinants and insufficient model
performance inducing spatially autocorrelated model residuals
(Griffith & Peres-Neto, 2006). One successful method for
generating such reference filters for biogeographical data is
‘principal coordinates of neighbour matrices’ (PCNM; Borcard
& Legendre, 2002), also known as ‘eigenvector-based spatial
filters’ (Diniz-Filho & Bini, 2005). PCNM and recent extensions
(e.g. Dray et al., 2006; Griffith & Peres-Neto, 2006) make use of
CMDS (or PCoA), which performs the dimensionality reduction
in Isomap.
The PCNM method assumes that the intensity of intrinsic
autocorrelations acts independently of geographical directions;
this assumption is referred to as ‘isotropic’. Instead, we propose
here a novel way of constructing eigenvector-based spatial filters
by deriving them from a preceding Isomap ordination. This
extended PCNM variant is capable of capturing the spatial
anisotropy of autocorrelation. This means that the derived filters
are not merely based on distances between neighbouring
observations but on distances in the direction of maximum
© 2007 The Authors
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M. D. Mahecha and S. Schmidtlein
similarity among observations. Together with the Isomap ordination, the method provides a coherent methodical framework
for biogeographical data analysis.
DATA AND METHODS
Data
The present study investigates the FLORKART data base, which
is the outcome of a cumulative mapping project involving
literature reviews and thousands of voluntary surveyors in
several organizational subunits (Haeupler & Schönfelder, 1989).
FLORKART contains grid-based vascular plant species counts
for Germany, which have proved to be of significant value for
biogeographical analyses in terms of derived attributes (e.g.
Kühn et al., 2004; Schmidtlein, 2004; Kühn et al., 2006). The
floristic records include species data at all taxonomic levels,
including subspecies and aggregates. For this study, the data base
was completely revised and records were considered at the
species level. The resulting data contained presence–absence data
for 3917 vascular plant species recorded between 1950 and 2000.
The majority of the data were collected between 1980 and 1990.
The array is denoted here as an (S × N) matrix X, where S is the
number of spatial elements and N is the number of species. The
vectors Xi represent spatial polygons, covering Germany at a
resolution of 6′ × 10′. For our purposes, cells with less than 50%
of their area within Germany or fewer than 100 records were
considered insufficiently mapped and therefore excluded. The
remaining 2762 cells contained 1,794,758 records, each indicating
the presence of one species in a polygonal cell.
We used the so-called CRU–PIK climate data for obtaining
long-term records of temperature and precipitation. These data
rely on a data set from the Climate Research Unit of the University of East Anglia, UK (CRU; New et al., 2002) and were adjusted
in order to remove temporal inconsistencies at the Potsdam
Institute for Climate Impact Research (PIK; Österle et al., 2003).
The spatial resolution of the data is 0.5° × 0.5° in space. For each
spatial grid cell, the data comprise monthly mean air temperatures
and monthly mean precipitation for the period 1900 – 2003. The
CRU–PIK data were transformed to zero mean and unit variance.
A soil data matrix on the FLORKART grid was derived from
the digital version of the German soil map 1:1,000,000 (Hartwich
et al., 1995), which differentiates 72 categorical soil–landscape
units characterized by individual combinations of soil types.
Fractions of spatial cover of these units were derived for each
FLORKART grid cell.
Isometric feature mapping: Isomap
Tenenbaum (1998) and Tenenbaum et al. (2000) introduced
Isomap as a nonlinear generalization of CMDS. As already
established by Williamson (1978), CMDS preserves a given input
metric in the course of reduction of dimensionality. Thus, the
derivation of an appropriate distance matrix from the original
data array is pivotal. In accordance with the ‘step-across’ method
of Williamson (1978), the basic idea in Isomap is that the
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relationship between similar points can be described by their
linear distances, whereas that between faraway points connected
by underlying nonlinearities cannot. Consequently, the initial
step in Isomap is to construct a matrix of linear interpoint
(X)
distances, here denoted as D(X) = { di , j } (Fig. 1a,b). These initial
(X)
distances di , j are normally the Euclidean interpoint distances,
e.g. for the environmental data sets in the present case. However,
when dealing with binary data such as the FLORKART species
records, it is customary in ecology to use the square-root of a
Sørensen-based distance function defined by
(X)
di , j = 1 −
2a
.
2a + b + c
Here a denotes the number of species that occur at both sites i
and j, b is the number of species that occur only at site i, and c is
the number of species that occur only at site j. The introduction
of the square root is necessary to obtain a fully Euclidean distance
metric (Legendre & Legendre, 1998).
Now, because linear distances underestimate the ‘true’ sample
distance in the presence of nonlinearities, the main goal is to
approximate the distance on the underlying manifold. For this,
a threshold parameter is introduced that can be a number of
nearest neighbours (k-NN) or a radius (ε) in the Euclidean
space. Two points are connected linearly only if one is considered
a nearest neighbour of the other or if their distance is equal to or
less than ε. The identified locally linear connectivities are stored
in an S × S adjacency matrix E (where ei,j = 1 indicates a firstorder link between the points (i and j) and ei,j = 0 the opposite;
Fig. 1c). Interpoint distances that are not directly linked
through an edge ei,j are subsequently connected across other
samples. Isomap sets up an undirected graph were the edges are
(X)
weighted by the locally linear distances, di , j . The final distance
(G)
(G)
matrix D = { di , j } is computed as shortest-path connections
on the constructed neighbourhood graph, usually identified by
Dijkstra’s algorithm (Dijkstra, 1959). Figure 1(d) shows how the
distance between two points (n, m) in a sample data set produced
by a one-dimensional manifold (nonlinearly embedded in a
two-dimensional space) might be drastically underestimated if
the points are linearly connected. It can be seen that the shortest
path on the respective Isomap graph could provide an acceptable
proxy for the ‘geodesic’ distance. The geodesic distance matrix
(D(G); Fig. 1d) is subjected to CMDS to map the data points to a
low-dimensional ordination space. CMDS double-centres the
distance matrix and decomposes the new matrix to its eigenvalues
and eigenvectors. The ordination axes Yi, also called ‘principal
coordinates’, are determined as eigenvectors scaled by the square
root of their corresponding (non-negative) eigenvalues (Borg &
Groenen, 1997; Cox & Cox, 2001).
The critical step in Isomap is the heuristic setting of the
above-mentioned threshold parameter ε or k. A trade-off has to
be made between a small threshold value, reducing connectivity
between entities (and maximizing nonlinearity), and a sufficiently
large value to warrant the overall graph connection. The optimal
threshold parameter emerges as a maximum of explained
variance when the threshold parameter is varied systematically
© 2007 The Authors
Global Ecology and Biogeography, 17, 284–296, Journal compilation © 2007 Blackwell Publishing Ltd
Spatial pattern extraction by Isomap
Figure 1 The k-Isomap method visualized as a methodological flow chart. (a) The initial data array is denoted here as an S × N matrix X, where
S is the number of spatial elements and N is the number of species. Based on the matrix X a symmetric S × S distance matrix D(X) is constructed
(b). The introduction of a nearest-neighbour (NN) parameter k then allows constructing an undirected k-NN graph, where the edges E are
shown in (c). The graph weights are set to the linear interpoint distance. Finding the shortest path of all point-wise connections, (d), creates a
new matrix of geodesic interpoint distances D(G). Based on this nonlinear distance structure, a low-dimensional representation of the data metric
in the ordination space is found by means of classical multidimensional scaling (CMDS). The ordination space is spanned by the Isomap
coordinates Yi, where the index i indicates the rank order of the sorted (non-negative) eigenvalues (e).
from its minimum to maximum (Mahecha et al., 2007). The
choice of the scale invariant k is usually easier than identifying a
good ε, and was also used here. Note that during this parameter
identification, the special cases k = S – 1 or ε = max (di,j),
respectively, are included, where Isomap equals conventional
CMDS. The established criterion for estimating the explanatory
power of the low-dimensional modes on the ambient space is
given by R2(D(G), D(Y )), where R is Pearson’s product–moment
correlation coefficient and D(Y ) are the Euclidean interpoint
distances in the low-dimensional embedding space (Tenenbaum
et al., 2000). For computing the Isomap embedding, Tenenbaum
included a free Matlab® code in the accompanying original
publication (Tenenbaum et al., 2000), which is available at http://
isomap.stanford.edu/.
Integrating Isomap and PCNM
The connectivity structure of spatial samples obtained by an
Isomap ordination can be used to construct anisotropic
eigenvector-based spatial filters. For a better explanation, the
original PCNM approach is first summarized as a ‘how-to’ recipe
(Borcard & Legendre, 2002; Dray et al., 2006):
I. Compute a matrix of geographical distances of the georeferenced
sampling points, D(PCNM). This is the space-explicit equivalent to
the Isomap step in Fig. 1(b).
II. Set a threshold value for interpoint connections; analogous to
Isomap, this is usually a defined distance radius ε or a k-NN
value (comparable to Fig. 1 c). The edges of this graph E(PCNM)
can also be defined using other criteria.
III. Calculate the weights of the edges by a point-wise product of
the connectivity matrix Ε(PCNM) and the matrix of spatial interpoint distances, D(PCNM). All further analyses are based on this
weighted graph.
IV. Set interpoint distances beyond the threshold value to an
(arbitrarily chosen) maximum value. This step is known as
‘matrix truncation’, where it has been established empirically that
a ‘truncation’ value set to four or more times the maximum distance
or higher provides reasonable results (Borcard & Legendre, 2002).
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M. D. Mahecha and S. Schmidtlein
V. Compute a CMDS embedding of this truncated distance
matrix, where the leading coordinates corresponding to nonnegative eigenvalues are representations of the spatial structure
at varying spatial scales (analogous to Fig. 1e).
As anticipated above, we would like to stress that this approach
(particularly when based on an ε threshold) includes one critical
assumption: the created spatial filters only model equidirectional
(isotropic) spatial autocorrelation. However, we see that there is
no reason to assume that spatial autocorrelation in biogeography
is described well by assuming isotropy, since both induced and
intrinsic causes of spatial correlation are directed processes (an
example is seed dispersal along nonlinear trajectories).
In order to take relevant spatial anisotropies into account,
PCNM can be linked directly to Isomap. Recall that Isomap
establishes first-order links (E(Isomap); Fig. 1c) for nearby points
expressed through short distances, measured in terms of βdiversity. Replacing the conventional PCNM connectivity structure
(the matrix E(PCNM) in step II) by Isomap edges (E(Isomap)) constructs
the desired anisotropic filters. In the following, this procedure
will be referred to as ‘principal coordinates of anisotropic neighbour
matrices’ or PC(An)NM. In Step III this means that the Isomap
adjacency matrix is weighted by the spatial interpoint distances.
The emerging graph is anisotropic in geographical space if
species resemblances are anisotropic. Steps IV and V in PCNM
are preserved.
The extracted set of spatial filters represents space as it matters
to plant distributions at different spatial scales. For example,
barrier effects causing different floras in close vicinity will be
represented by filters and the lack of connectivity between both
sides of the barrier is preserved. Applying such filters will check
for the influence of geographical neighbourhood only where
autocorrelation effectively occurs. This may be interpreted as a
modification of the truncation procedure used by the original
PCNM approach, where only distances between neighbouring
spatial units are included in the derivation of spatial filters. In the
PC(An)NM, distances are truncated based on autocorrelation
itself.
partitioning of four explanatory variables considers a set of
24 = 16 multiple regression models (MRM). These are set up in a
nested manner, so that four levels of complexity are obtained.
The first hierarchical level considers each explanatory variable
individually, while the second hierarchical level constructs
MRMs out of all possible combinations of two predictors.
Similarly, higher levels contain more predictors (in all combinations)
until the entire set of predictors is combined into one MRM at
the highest level. The hierarchical partitioning theorem (Chevan
& Sutherland, 1991) states that it is possible to estimate
independent effects of a particular variable by averaging the
additional explanatory power induced by this predictor on the
MRMs across all hierarchical levels. In a complementary way, the
joint effect of a certain variable with its covariables can be
estimated (MacNally, 2000; Walsh et al., 2004). The independent
effects are thought to exclude all ‘explanatory’ power, which can
be attributed to collinearities between and redundancies in the
sets of explanatory variables and allows identification of the
origin of a fraction-explained variance (MacNally, 2000). In
contrast, the rest of the spatial autocorrelation (biotic effects and
effects of environmental variables not included in the driving
variables) is represented by the independent effects of the
PC(An)NM filters. In the present study, we encounter the situation
that each factor considered in the hierarchical portioning is
multivariate, being set up by its leading Isomap modes (for
details on the selection of Isomap modes see Appendix S1 in
Supplementary Material) or spatial filters. In MRMs, where the
PC(An)NM filters are considered, further test statistics are
required in order to warrant that the model residuals are not
significantly autocorrelated in space (Griffith & Peres-Neto,
2006). A detailed description of the application of hierarchical
variance partitioning to a mixed set of several (multivariate)
environmental predictors and anisotropic spatial filters including
test statistics is given in Appendix S1 in Supplementary Material.
RESULTS
Dimensionality reduction
Relating environmental determinants to
biogeographical patterns: hierarchical variance
partitioning
The recovered modes, which represent floristic pattern, environmental drivers or structures of anisotropic spatial autocorrelation,
provide the basis for data-driven modelling. In the present case
we were less interested in creating predictive models than gaining
insight into how the biogeographical patterns can be traced back
to environmental determinants and intrinsic spatial autocorrelation.
Exploring the relation between the different factors is possible
by means of hierarchical variance partitioning (Chevan &
Sutherland, 1991; see also Appendix S1 in Supplementary
Material). This allows estimating the independent, joint and
overall effects of each explanatory variable on a dependent structure.
Here, we tried to analyse the impact of the four predictors
(temperature, precipitation, soil and the PC(An)NM filters) on
each leading FLOKART Isomap dimension. Hierarchical variance
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The variation of the Isomap threshold parameter k for identifying
an optimal low-dimensional embedding space is shown in
Fig. 2(a) for the FLORKART data. Highest values of explained
variance were achieved by setting k = 13. The cumulative amount
of explained variance in higher dimensions then converged to
93% in the sixth mode and 95% in the ninth mode (Fig. 2b).
Linear CMDS (Isomap k = S – 1) was clearly inferior to Isomap,
with k = 13 recovering only 70% in the first 10 dimensions. In
contrast, the optimal embedding dimension for the CRU–PIK
temperature–time series was identified to be exactly k = S – 1,
which leads to a linear embedding. Recall that the time series
were transformed to zero mean and unit variance. In this particular
case, CMDS becomes equivalent to PCA (cf. Gámez et al., 2004,
for a comprehensive derivation). The explanatory power of the
CMDS/PCA embedding of the temperature records exceeds the
floristic data where the values of explained variance converged to
99% after a clear cut-off in the third dimension (Fig. 2b). The
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Spatial pattern extraction by Isomap
‘envelope’ of the very low-dimensional climate data (two or
three), the higher-dimensional manifold of the soil data (around
nine) indicating a data set of intermediate intrinsic dimensionality.
Pattern visualization
Figure 2 (a) Cumulative explained variances for the first eight
Isomap coordinates extracted from the FLORKART data base for
a varying threshold parameter k. After a critical k-value is reached,
all consecutive parameter choices lead to a decreasing explanatory
power of the Isomap modes. Here, k = 13 was identified as the
optimal parameter. The nonlinear ordination approach explained
70% of the variance in the first dimension, 78% in the second
dimension and 85% in the third dimension. The amount of
explained variance converged in the subsequent dimensions to 93%
in the sixth dimension and 94% in the ninth. (b) The explained
variances for all analysed data bases (species data in FLORKART,
CRU–PIK temperature and precipitation, soil data) after the optimal
k-parameter was identified.
precipitation data were again best represented by a highly
nonlinear Isomap embedding with k = 11, where the first two
dimensions together explained more than 98% of variance. The
pedological embedding was found to be optimally represented
by Isomap k = 10. While the major patterns of the precipitation
data were well expressed by two dimensions, the pedological data
required many more dimensions in order to achieve a similar
explanatory power on the input data (Fig. 2b). The overall view
suggests that we are dealing with a set of variables with varying
dimensionality of their respective manifolds. It might be
worthwhile mentioning that the floristic data lie precisely in the
Following Perry et al. (2002), the very first approach for an
effective evaluation of spatial analysis in ecology is to provide
meaningful visualizations. After recovering the independent
principal coordinates, score values for each of the sites were
available. Hence, visualizations of the extracted coordinates can
be provided by projecting them directly onto the geographical
space. Figure 3(a)–(l) shows how the first three recovered dimensions derived from the different data sets behave in geographical
space. A clear north-north-west to south-south-east gradient was
found on the first modes of all data sets. The second axis shows a
curved east–west gradient for all data except in the representation
of the soils. Generally, the climate data show much smoother
gradients than the floristic and pedological data. The spatially
more differentiated structures found in the representations of the
floristic data can also be found in the soil Isomap modes, which
produced the most heterogeneous picture. The latter confirms
the findings of the explained variances in the course of dimensionality reduction, suggesting a far more complicated set of
patterns inherent in the soil data compared with the other data
sets. At the given spatial scale, climate data features simpler
patterns and less regional individuality than floristic data and
soil regions. The soil data tend to be unable to be explained by
coarse-scale trends. These differences may in part be attributed
to the phenomena themselves, and in part to differences in
sampling schemes and spatial resolutions.
Plotting the Isomap interpoint connections in geographical
space provides an impression of the local spatial orientation of
β-diversity (see Fig. 4a). As expected from methodological
considerations, the FLORKART graph indicates that the biogeographical similarity structure is highly anisotropic in the
geographical space where areas of dense site connectivity are
found (e.g. the southernmost Alpine regions) and also lacking
connectivity at some neighbouring sites (e.g. central Germany).
Figure 4(b)–(g) shows examples of anisotropic spatial filters
derived by means of PC(An)NM from the georeferenced
Isomap graph. It can be seen that PC(An)NM uncovers a whole
range of highly anisotropic spatial structures, including steep local
gradients.
Relating biogeographical patterns and environmental
determinants
Hierarchical variance partitioning confirmed that the leading
FLORKART Isomap coordinates can be associated with both
environmental drivers and anisotropic spatial autocorrelation.
The leading FLORKART dimensions were strongly related to
independent effects of temperature, precipitation and soil data as
well as intrinsic spatial autocorrelation (Fig. 5a). Each of these
variables had an independent effect of c. 20 –30% of the first
FLORKART Isomap dimension, where the climate data depicted
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289
M. D. Mahecha and S. Schmidtlein
Figure 3 The leading Isomap coordinates
of the analysed data set projected onto the
geographical space. The images (a), (b) and
(c) show consecutively the first three Isomap
coordinates extracted from the species data
Y1–3(sp) with their respective values of
explained variances. The panels (d), (e) and
(f) show the values of the first three Isomap
coordinates for the long-term temperature
records Y1–3(T) spatially resampled on the
grid of the FLORKART data in space.
Similarly, (g), (h) and (i) show the first three
Isomap coordinates for precipitation time
series, Y1–3(P). Finally, (j), (k) and (l) are the
first three dimensions extracted from the soil
survey, Y1–3(soil). The colour code was
adjusted to the range of the respective
Isomap coordinates.
the strongest statistical influence. For higher FLORKART coordinates
(Y3+), the independent ‘contributions’ of climate declined, which
could be expected since these data are almost entirely represented
by very few dimensions, precipitation being explained by two
and temperature by three dimensions (Fig. 2b). In contrast,
higher-dimensional representations of soil conditions with
strong local variability (Figs 2b, 3j–l) maintained a stable independent ‘contribution’ of c. 20% in all five leading FLORKART
dimensions. The independent effects of the anisotropic spatial
autocorrelation instead increased systematically with each
FLORKART coordinate. Thus, we maintain that the leading
dimensions and the most fundamental patterns of FLORKART
can be ascribed to environmental drivers, whereas the remaining
spatial autocorrelation in high dimensions is attributable to
intrinsic biotic factors or unconsidered external drivers.
However, these higher Isomap coordinates (Y3+) are of sub-
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ordinate meaning, allocating only a very small fraction of the
biogeographical data variance (< 2% each, Fig. 2b).
An estimate of overall variation attributed to the environment
and intrinsic autocorrelation (Fig. 5b), which is the sum of
independent and joint ‘contributions’ of the different variables,
completes this picture: here, the first FLORKART mode appears
to be similarly related to climate, soil and anisotropic spatial
effects, each achieving more than 85% of explained variance. The
overall variance ‘explained’ by environmental variables decreases
in higher FLORKART dimensions. Again, in the third and higher
dimensions, no precipitation effects are detectable; from the fifth
dimension onward, temperature also does not appear to be
related to the uncovered biogeographical patterns. The decrease
in the higher floristic dimensions is less prominent for soil data
than for climate variables. Thus, the relatively coarse climate data
can be related to the first two FLORKART modes representing
© 2007 The Authors
Global Ecology and Biogeography, 17, 284–296, Journal compilation © 2007 Blackwell Publishing Ltd
Spatial pattern extraction by Isomap
Figure 4 The Isomap graph, weighted by and projected onto the geographical space is shown in (a). This graph provided the basis for
constructing spatial filters by means of principal coordinates of anisotropic neighbour matrices, PC(An)NM. Examples of the derived
anisotropic spatial filters are shown in (b)–(g), which are the PC(An)NM modes Yi: i = 4, 8, 12, 16, 20, 24. The colour code was adjusted to the
range of the respective filter Yi.
Figure 5 The hierarchical variance
partitioning traces biogeographical patterns
back to environmental determinants and
intrinsic spatial autocorrelation. Bars in
(a) show the independent effects of the leading
Isomap modes extracted from temperature
time series (T), precipitation time series (P),
the soil data (soil) and the anisotropic spatial
filters (PC(An)NM) on the respective
FLORKART Isomap dimensions. The overall
explained variance by the driving sets of
variables is shown in (b).
variation at high-scale levels; additional FLORKART dimensions
instead can be related to a mixture of autocorrelation and soil
effects, which are generally found at lower spatial scales. All these
findings are visually anticipated in Figs 3 and 4: a clear relationship can be seen between the broad biogeographical gradients
and climate, whereas soil structures and spatial PC(An)NM
filters express local gradients.
The estimates of the independent and overall effects are
instructive because they separate out redundancies from individual explanatory powers of the variables. The hierarchical
partitioning was performed to obtain a broad view of the variance
allocation of the FLORKART data. This does not mean that one
cannot find MRMs of higher explanatory power than found by
the summarized variance allocation of the individual variables.
Indeed, some of the 16 MRMs provided very high explanatory
powers on the respective floristic patterns, exceeding 95%. Yet,
as stated above, the goal was not to find an optimal regression
model. Instead, we extracted rules of influence of the environmental drivers on the floristic pattern by averaging across all
scales of MRM complexity.
DISCUSSION
Isomap in the context of nonlinear ordination
methods
Linear relationships in ecological data may occur as special cases
of underlying nonlinearities, but they cannot be expected to be a
standard in real-world data. As shown here, optimal Isomap
performance on three of four data bases was achieved by highly
nonlinear embeddings (low threshold parameters). This was
particularly true for the floristic data.
At this point, the methodological innovation of Isomap
renders the method superior to Williamson’s (1978) original
© 2007 The Authors
Global Ecology and Biogeography, 17, 284–296, Journal compilation © 2007 Blackwell Publishing Ltd
291
M. D. Mahecha and S. Schmidtlein
‘step-across’ method. The latter would have produced an entirely
linearly linked species space (and thus been indistinguishable
from CMDS), since many common species like Plantago major
are found throughout the whole area. Thus, the application of
Williamson’s (1978) method might be suitable for studies on
sampling units with very distinct characteristics, but it counteracts
the goal of adjusting the nonlinearity to the inherent properties
of a continuous macroecological data set. The search for
improved nonlinear ordination methods has a strong tradition
in ecology. In this context ‘non-metric multidimensional scaling’
(NMDS; Shepard, 1962; Kruskal, 1964), which has recently been
extended in an analogous way to Isomap by coupling it to a
geodesic distance matrix (NMDS-G; Mahecha et al., 2007), has
become widely acknowledged. Although this new NMDS-G
variant appears to be able to achieve an embedding success similar
to Isomap, it has some intrinsic problems, for example because of
its iterative learning strategy, it does not allow an accurate reproduction of an ordination result. In fact, NMDS is currently often
replaced by self-organized maps (SOMs; Kohonen, 2001; for an
example application in ecology see, e.g., Hopton & Mayer, 2006).
Nevertheless, comparative studies found few advances provided
by the neighbourhood-preserving SOMs with respect to NMDS:
both methods lack trustworthiness when dealing with highdimensional manifolds (De Backer et al., 1998; Giraudel & Lek,
2001) and require pre-definition of the embedding space.
Another example for nonlinear dimensionality reduction was
introduced to ecology by De’ath (1999b), who explored the
potential of principal curves (Hastie & Stuetzle, 1989) for
ordination tasks in vegetation sciences. The method searches an
optimal one-dimensional curved manifold in the high-dimensional
data set as good proxy for the analysis. Several different
approaches were developed for finding optimal curved solutions
for dimensionality reduction and manifold approximations,
which are suitable for dealing with nonlinear problems. Despite
this progress, Hadid and Pietikäinen (2003) found that principal
curves run into complex analyses due to the fact that they require
several free parameters, which in turn implies a lack of convergence.
For practical purposes, this means that principal curves (like
NMDS and SOMs) are limited to manifolds of extremely low
dimensionality (Saul & Roweis, 2003). However, a comparative
study of advanced principal curve-related methods and Isomap
showed that ‘very experienced users’ are able to tune curvilinear
distances in such a way that they might outperform Isomap (Lee
et al., 2004). The argument against this finding is that the
mathematical elegance of Isomap simply circumvents problems
of sophisticated parametrization procedures. The mathematical
advances of Isomap were demonstrated by Bernstein et al.
(2000), who provided proof of convergence showing that the
geodesic metric found by the single threshold parameter leads to
a proper approximation of the ‘true’ underlying distance structure.
Similar theoretical foundations are lacking for the other methods
of nonlinear dimensionality reduction.
The advantages of Isomap, being non-iterative, fast,
mathematically sound and easy to apply, might be reasons for its
successful application, for example in the related field of climate
time series (Gámez et al., 2004) or hyperspectral data analysis
292
(Bachmann et al., 2005). Furthermore, the integrated PC(An)NM
approach presented here provides further justification for using
Isomap, since we found that spatial models in these contexts rely on
fewer (regression) parameters than their respective linear equivalents,
depicting higher explanatory powers on the ambient space.
Extracted patterns
We found that the relevant floristic patterns can be summarized
in very few components, of which only two are directly linked to
climate data, whereas higher modes are related to a mixture of
pedological structures and (intrinsic or induced) effects of
anisotropic spatial autocorrelation. As already mentioned, these
findings may in part be attributed to the phenomena themselves,
and in part to differences in sampling schemes and techniques.
For example, topographic effects on climate are not represented
in the coarse climate data, while existing coarse-scale features are
not adequately reflected in soil data. The latter are based on a
categorical map that emphasizes individual characteristics of soil
landscapes rather then general gradients.
The first floristic Isomap component (Fig. 3a) was found to be
dominated by the antagonism between lowland and upland or
northern and southern Germany. Typical for the lowland is the
occurrence of species preferring more temperate climates. We
observed high point-biserial correlation coefficients with this
axis for Hydrocotyle vulgaris, Hottonia palustris, Carex arenaria,
Sium latifolium or Ornithopus perpusillus (nomenclature according
to Wisskirchen & Haeupler, 1989). Other species, such as Daphne
mezereum, Colchicum autumnalis, Luzula luzuloides or Ranunculus
nemorosus, avoid coastal plains. The most extreme antagonists in
this gradient are the North Sea islands and the Bavarian Alps.
The second Isomap component (Fig. 3b) reflects an east–west
gradient. Species with a more continental distribution have high
correlation coefficients with this axis (such as Helichrysum
arenarium or Peucedanum oreoselinum); others are more
inclined to the west (such as Digitalis purpurea and Ilex aquifolium).
The most extreme values are found in the north between the
Lower Rhine valley and the border to Poland.
The third axis (Fig. 3c) differentiates a cluster of grid cells
including the Alps (in the very south) and the Schleswig-Holstein
area (the northernmost parts). This group of grid cells is
determined by a lack of data in Schleswig-Holstein and a corresponding lack of species in the Alps. Only the latter is caused by
real distribution gaps. Neophytes such as Robinia pseudoacacia,
which are rare in the Alps, were not considered in the mapping
scheme for Schleswig-Holstein. Plants like Linaria vulgaris, with
sparse occurrence in the Alps, are not recorded at the taxonomic
species level in Schleswig-Holstein.
There are further examples for sampling biases recovered by
the Isomap embedding. Figure 6(a,b) features distributions of
Tripleurospermum perforatum, which is highly correlated with
the second Isomap coordinate, and Tripleurospermum maritimum,
which coincides well with the third coordinate. It can be seen that
while T. perforatum has been recorded everywhere in western
Germany, it only occurs in the southern part of eastern Germany
(Saxony). In contrast, T. maritimum seems to appear exclusively
© 2007 The Authors
Global Ecology and Biogeography, 17, 284–296, Journal compilation © 2007 Blackwell Publishing Ltd
Spatial pattern extraction by Isomap
Figure 6 The figure shows the occurrences of Tripleurospermum perforatum (a) and of Tripleurospermum maritimum (b) in the FLORKART
data base across Germany. The distribution patterns reflect political boundaries, which are at the same time the boundaries of organizational
units of the mapping project. In western Germany, T. perforatum was interpreted as a species in its own right; in eastern Germany it was
apparently considered a subspecies of T. maritimum. Clustering eight Isomap components of FLORKART resulted in the pattern shown to the
right (c), where the grid cells are colour coded according to their cluster affiliation. The map is overlaid with the state boundaries of Germany
(white lines). This figure shows that in many cases both the clusters and the state boundaries coincide.
in eastern Germany. From a botanical point of view, this is
odd, since T. perforatum can be interpreted as a subspecies of
T. maritimum (Tripleurospermum maritimum subsp. inodorum
[K. Koch] Var.). However, it can also be described as a species
(T. perforatum [Mérat] Lainz) in its own right. Apparently, the
last variant was the predominant taxonomical concept in western
Germany, where T. maritimum was rarely reported at the species
level.
Artefacts like this emerge even more frequently if additional
Isomap components are considered. Clustering eight components resulted in the striking pattern shown in Fig. 6(c). The
diagram was overlaid with the state boundaries of Germany. This
figure shows that in many cases, both the clusters and the state
boundaries coincide. It has to be mentioned that the state
borders are often identical with the spatial organizational units of
the floristic survey and rarely reflect landscape unit boundaries.
One could make the criticism that we identified these patterns by
setting two parameters (the Isomap k and the number of
clusters) based on heuristic decisions. However, we argue that the
existence of one parameter combination that identifies this data
structure is enough to cause one to be suspicious of the data base.
It is clearly discernible that the ‘Iron Curtain’, which divided East
Germany from the capitalist West, is one central property of the
recovered structure. An east–west gradient in German floras is
well known in plant geography (Ellenberg, 1988), but species
reach their limits at different positions along this gradient. Thus,
a more or less continuous transition should be expected. In the
data base, taxonomic inconsistencies similar to those previously
mentioned instead cause coinciding boundaries that are identified
by the applied data mining techniques as false gradients or as the
most prominent features of true gradients.
CONCLUSIONS AND OUTLOOK
The presented Isomap ordination and derived anisotropic spatial
filters provide an effective approach for extracting the main
patterns of high-dimensional macroecological data bases. We
found further evidence that attention has to be paid to the strong
influence of sampling biases. This also changes the interpretation
of the unexplained spatial autocorrelation, which is usually
attributed to biotic effects and missing environmental data. Steep
local gradients in the spatial filters may partly be attributed to
sampling bias (Fig. 4b–g). The present approach, coupling
dimensionality reduction and clustering, was well suited to the
identification of the effect of the latter; however, the bias could
not be quantified. This problem cannot be completely solved
unless accurately sampled data are available for validation. The
astonishing coincidence of political and floristic boundaries,
which can also be observed in other data bases such as the Atlas
Florae Europaea (Jalas et al., 1972–1999, results not shown),
underscores the need for more consistent sampling schemes for
future data collections (for further discussions see Roleèek et al.,
2007). We showed that it is very important to perform crosscalibrations of the involved mapping projects, establishing
consistent and traceable taxonomical concepts. Although difficult
to realize in the case of volunteer surveyors, a randomized
sampling approach would be useful to ensure a spatially equalized
level of expertise.
The limitations discussed above are in no way meant to
downgrade the obvious merits of FLORKART or similar data
bases: the biases are well known in the scientific community, and
in most analyses, a great deal of effort is made to cope with the
problem. For instance, the use of derived attributes such as
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Global Ecology and Biogeography, 17, 284–296, Journal compilation © 2007 Blackwell Publishing Ltd
293
M. D. Mahecha and S. Schmidtlein
relative proportions of plant functional groups (Schmidtlein &
Ewald, 2003; Schmidtlein, 2004; Kühn et al., 2006) is thought to
be largely unaffected. One reason is that these derived attributes
are insensitive to sampling intensity and taxonomic confusion,
which is apparently one major factor in the sampling bias. The
technique applied here allows checking geodata like FLORKART
for areas of consistent sampling. In the southernmost part of
Germany (Bavaria and Baden-Württemberg), for example, no
administrative frontiers were reflected in the visualization of the
ordination space. In these areas, cluster limits follow known
landscape boundaries, which can also be seen very nicely in the
leading modes of the soil data (Fig. 3j–l).
The overall findings of this study indicate that the application
of modern machine-learning methods is very effective for the
recovery of the global nonlinear properties of biogeographical
data. We showed that nonlinear dimensionality reduction is a
powerful tool for the detection of both biogeographical and
environmental gradients as well as sampling artefacts. In other
biogeographical surveys, Isomap and related algorithms could be
applied for similar evaluations. This is especially important now
that there are extensive data bases either already available or still
being compiled (Dirzo & Loreau, 2005) and computers are fast
enough to put new developments in the area of nonlinear dimensionality reduction into practice (Hinton & Salakhutdinov, 2006).
ACKNOWLEDGEMENTS
The authors are indebted to the thousands of volunteers
mapping the flora of Germany and to the German Federal
Agency for Nature Conservation (Bundesamt für Naturschutz)
for providing the FLORKART data, in particular Rudolf May.
We would also like to thank Hermann Österle and his colleagues
at the Potsdam Institute for Climate Impact Research (PIK)
for providing the CRU–PIK data. We gratefully acknowledge
Michael Hauhs, Holger Lange and Gunnar Lischeid for their
methodological comments. The quality of the manuscript was
substantially improved following suggestions by Rebecca van
Dyck, the editors David Currie and José Alexandre Felizola
Diniz-Filho, and the anonymous referees. Finally, M. D. Mahecha
would like to thank the Max Planck Society for supporting
the independent junior group on ‘Biogeochemical Model Data
Integration’.
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SUPPLEMENTARY MATERIAL
BIOSKETCHES
Miguel D. Mahecha is a PhD candidate at the Max
Planck Institute for Biogeochemistry in Jena, Germany.
He is interested in learning from ecological data by means
of empirical pattern extraction and data-driven
modelling. He has worked with biogeographical and
hydrological data in the past, and now focuses on
biogeochemical fluxes.
Sebastian Schmidtlein is Professor of Vegetation
Geography at the University of Bonn. He is interested in
vegetation ecology, macroecology, biogeographical
modelling and in the remote sensing of vegetation
patterns.
The following supplementary material is available for this article:
Appendix S1 From multiple regressions to hierarchical variance
partitioning.
This material is available as part of the online article from:
http://www.blackwell-synergy.com/doi/abs/10.1111/
j.1466-8238.2007.00368.x
(This link will take you to the article abstract.)
Please note: Blackwell Publishing is not responsible for the
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Editor: José Alexandre F. Diniz-Filho
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