On the Art of Classification in Spatial Ecology: Fuzziness as an Alternative for Mapping Uncertainty

Abstract

In this piece, we wanted to highlight the limitations associated with classification techniques that are based on Boolean logic (i.e., true/false) and that impose discrete boundaries to systems. We proposed to shift practices toward techniques that learn from the system under study by adopting soft classification to support uncertainty evaluation and we give reason why this is so recommended.

Full text

OPINION
published: 21 December 2018
doi: 10.3389/fevo.2018.00231
Frontiers in Ecology and Evolution | www.frontiersin.org 1December 2018 | Volume 6 | Article 231
Edited by:
Miles David Lamare,
University of Otago, New Zealand
Reviewed by:
Zhi Huang,
Geoscience Australia, Australia
*Correspondence:
Dario Fiorentino
dario.fiorentino@awi.de
Specialty section:
This article was submitted to
Biogeography and Macroecology,
a section of the journal
Frontiers in Ecology and Evolution
Received: 28 June 2018
Accepted: 10 December 2018
Published: 21 December 2018
Citation:
Fiorentino D, Lecours V and Brey T
(2018) On the Art of Classification in
Spatial Ecology: Fuzziness as an
Alternative for Mapping Uncertainty.
Front. Ecol. Evol. 6:231.
doi: 10.3389/fevo.2018.00231
On the Art of Classification in Spatial
Ecology: Fuzziness as an Alternative
for Mapping Uncertainty
Dario Fiorentino 1
*, Vincent Lecours 2and Thomas Brey 1,3
1Helmholtz Institute for Functional Marine Biodiversity at the University Oldenburg (HIFMB), Oldenburg, Germany, 2Fisheries
and Aquatic Sciences, School of Forest Resources and Conservation, University of Florida, Gainesville, FL, United States,
3Alfred Wegener Institute Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany
Keywords: uncertainty, spatial ecology, discrete classification, soft boundaries, mapping transitions
INTRODUCTION
Classifications may be defined as the result of the process by which similar objects are recognized
and categorized through the separation of elements of a system into groups of response (Everitt
et al., 2011). This is done by submitting variables to a classifier, that first quantifies the similarity
between samples according to a set of criteria and then regroups (or classifies) samples in order to
maximize within-group similarity and minimize between-group similarity (Everitt et al., 2011).
Classifications have become critical in many disciplines. In spatial ecology, for example,
grouping locations with similar features may help the detection of areas driven by the same
ecological processes and occupied by same species (Fortin and Dale, 2005; Elith et al., 2006),
which can support conservation actions. In fact, classifications have been used with the aim of
investigating the spatial distribution of target categories such as habitats (Coggan and Diesing,
2011), ecoregions (Fendereski et al., 2014), sediment classes (Hass et al., 2017), or biotopes (Schiele
et al., 2015). Sometimes such classifications were found to act as surrogates for biodiversity in
data-poor regions (e.g., Lucieer and Lucieer, 2009; Huang et al., 2012), some class being known
for supporting higher biodiversity. Many of the traditional classification methods were developed
in order to reduce system complexity (Fortin and Dale, 2005) by imposing discrete boundaries
between elements of a system; it is easier for the human mind to simplify complex systems by
identifying discrete patterns (Eysenck and Keane, 2010), and grouping similar elements together
(Everitt et al., 2011). However, in natural environments, spatial and temporal transitions between
elements of a system are often gradual (e.g., an intertidal flat transitioning from land to sea) (Farina,
2010). Those transitions may display distinct properties from those of the two elements they
separate. Despite the particularities and importance of such transitions, they are often disregarded
in ecological research (Foody, 2002), leading to the adoption of approaches that, by defining sharp
boundaries, may fail to appropriately describe natural patterns and groups of a system. Such
approaches have become the norm, despite the existence of approaches such as, fuzzy logic (Zadeh,
1965) and machine learning (Kuhn and Johnson, 2013) that are able to offer a more representative
description of those natural transitional zones. In ecology for instance, machine learning approaces
have gained some traction because of their ability to predict classes distribution (area-wide) in
data-poor conditions (e.g., sparse punctual information) with a relative high performance and
with no particular assumption in building the relationship between targetted classes and physical
parameters (e.g., Barry and Elith, 2006; Brown et al., 2011; Fernández-Delgado et al., 2014).
In the present contribution, we aim at highlighting the limitations associated with classification
techniques that are based on Boolean logic (i.e., true/false) and that impose discrete boundaries to
systems. We propose to shift practices toward techniques that learn from the system under study
by adopting soft classification to support uncertainty evaluation.

Fiorentino et al. Soft Classification and Uncertainty
DISCRETE CLASSIFICATIONS
The increasing availability of tools and software to semi-
automatically perform classifications has reduced the amount
of critical thoughts put into the exercise. Classification results
are sensitive to a variety of decisions made when establishing
the methodology of a particular application (Lecours et al.,
2017). For instance, (1) the method to compare the objects
to be classified (e.g., distance-based method), (2) whether or
not the method assigns each object to one single class (i.e.,
Boolean approach) or assigns a membership for one or multiple
classes (e.g., fuzzy logic approach), (3) whether or not the
method uses samples to train the classification (i.e., supervised
or unsupervised approach), and (4) the evaluation methods
(see Foody, 2002; Borcard et al., 2011). Furthermore, a number
of other potential sources of errors resulting in potentially
misleading classifications remains: variation in the data collection
methods, the spatial, temporal, and thematic scales (i.e., the way
the data are categorized/identified), the spatial and temporal
stability of the observations and the goodness of model fit (Barry
and Elith, 2006).
One of the most challenging parts of using classifications is to
evaluate how representative of real patterns the classification is.
Due to the cumulative effect of the factors listed above (Rocchini
et al., 2011), classification results may not adequately represent
natural patterns, making the classified patterns artificial, and
misleading, for instance when used to assist decision-making
(Lecours et al., 2017; see Fiorentino et al., 2017). Even
when a robust method is developed to reduce the impact of
these factors, the concept of discrete classes may in itself be
misleading; this type of classes can provide an incomplete,
oversimplified representation of complex natural patterns, and
thus misrepresent the reality to be described. In spatial ecology,
using discrete classes involves establishing “hard” boundaries
between them, which has been shown to cause misclassification
errors (Sweeney and Evans, 2012; Lecours et al., 2017). For
instance, an object may not always belong to one of the defined
classes, thus being forced into one of those defined classes by the
classifier. When doing so, the interpretation that is often made
is that the object was misclassified or that there was an error
of the algorithm, while in fact, it is a consequence of the often-
erroneous assumption that all objects must belong to a specific
class. A solution that has been proposed to avoid those errors is
to shift practices toward “soft” classifications instead of “discrete”
classifications.
SOFT CLASSIFICATIONS
It has been argued in the literature that soft classification
approaches better represent and describe natural patterns,
including transitional areas (Ries et al., 2004). Soft classification
approaches, which include fuzzy logic (Zadeh, 1965) Bayesian,
neural networks, support vector machines, decision trees,
boosting, bagging, generalized linear models, and multiple
adaptive regression splines, among others (see Fernández-
Delgado et al., 2014 for a comprehensive review on
classification and associated problems), acknowledge that
one object may belong to more than one class This enables
the recognition of elements of the real world that are
between classes (i.e., do not belong to one specific pre-
defined class), such as transitional zones and fine-scale natural
heterogeneity.
While hard classifiers assign an object to a class following
a Boolean, binary system (true/false, in a class or not), soft
classifiers assign memberships to objects. Memberships are the
estimated probabilities of objects to belong to a class (Everitt
et al., 2011). Each object will have as many membership values
as there are classes. Membership varies along the continuum
ranging from 0 (not a member) to 1 (definitely a member). An
object that has relatively high membership values for more than
one class is thus said to be not clearly associated with one specific
class. This may be an indication that no class adequately describes
this particular object, which could inform and guide further
analyses in order to explain that pattern. Those analyses may for
instance highlight that this data object is an error, or if there are
many objects in the same situation, that a new class needs to be
defined.
CAN SOFT CLASSIFICATIONS BE USED
TO ESTIMATE UNCERTAINTY?
In remote sensing, traditional pixel-based classifications use
image pixels as objects to be classified. The use of discrete
classifiers in land cover studies often oversimplifies the actual
land cover (Foody, 2000). For instance, forested wetlands, which
in nature mark the transition between forested areas and waters
bodies, would most likely be classified as a “forest” land cover
or a “water” land cover by a hard classifier. For the purpose
of this example, we assume that it classifies it as “forest.” On
the other end, a soft classifier might assign to that same pixel a
“forest” membership of 0.65 and a “water” membership of 0.35.
As a result, different users could interpret those classifications
according to the following statements:
A) Based on the hard classification result, the pixel represents
“forest” land cover. In the absence of a ground-truth point data
for that particular pixel, it could be assumed that this result is
100% certain.
B) Based on the soft classification result, the pixel represents a
mixture of “forest” land cover and “water” land cover.
C) Based on the multiple memberships from the soft
classification, the pixel has an associated level of ambiguity
(e.g., it is 35% ambiguous that the pixel represents a “forest”
land cover and 65% ambiguous that it represents a “water”
land cover).
D) Based on the multiple memberships from the soft
classification, it is possible that the pixel belongs to a
distinct class characterized by a mixture of forest and water
(the transition), perhaps “wetland.” After validation or a
re-run of the soft classification with a new class, it could
potentially be possible to say that the pixel represents a
“wetland” land cover with much less ambiguity. While we
argue that soft classifications are more appropriate than hard
classifications, we note that a re-run of the hard classification
Frontiers in Ecology and Evolution | www.frontiersin.org 2December 2018 | Volume 6 | Article 231

Fiorentino et al. Soft Classification and Uncertainty
TABLE 1 | Methods for synthesizing membership values (p) resulting from soft classifications at each location (i) for a given number of classes (K).
Name Confusion index (CI) Pielou evenness index (J) Red Green Blue (RGB)
Formula CIi=
pmax−1
pmax
Ji=
−PK
i=1pilog pi
log K
Ri=255 (pmax)i
Gi=255 pmax−1i
Bi=255 pmax−2i
Interpretation 0 1 0 1 red, green and
blue
yellow, violet and
cyan
white
full dominance of one
class
classes have
similar probability
full dominance of one
class
classes have similar
probability
full dominance of
one class
two classes evenly
dominate
all classes have
similar probability
Number of classes 2 k 3
References (Burrough et al., 1997; e.g., in Lucieer and
Lucieer, 2009)
(Pielou, 1975; e.g., in Fiorentino et al., 2017) (Boughen, 2003; e.g., in Hass et al., 2017)
We note that the Red Green Blue (RGB) method only provides a visual representation and it is not quantitative. It combines the first three highest memberships into a composite three-band RGB image, that is the geographic overlay of
the three bands.
with a class corresponding to wetlands would also be more
accurate than the initial hard classification.
The interpretation in “A” is not fully representative of the
reality as the water component of the pixel is not reported
or acknowledged at all. In “B” a nuance is added to the
interpretation as the potentially mixed nature of the pixel is
acknowledged. In “C,” that nuance is interpreted as a measure of
ambiguity, i.e., that it is acknowledged that the classifier could
not distinguish between the two classes. Finally, in “D,” the mixed
nature of the pixel is used to redefine the classification based
on ecological knowledge (e.g., that wetlands can be a mixture
of forest and water from a remote sensor’s perspective), and to
guide further analyses about the nature of the land cover. That
simple example illustrates uncertainty as defined by Zhang and
Goodchild (2002), i.e., as the ambiguity of a classification. Since
membership quantifies the probability of an object to belong to
multiple classes, it can be used as a measure of ambiguity, and
therefore as a measure of uncertainty (Yager, 2016).
Depending on the nature of the analysis, the spatial
representation of membership can be used to display spatial
uncertainty of classifications, or to identify transitions between
existing classes. Traditional fuzzy and model-based approaches
thus acknowledge transitions among classes.
However, such approaches have the limitation to be
algorithmic, i.e., not explicitly accounting for data statistical
properties such as randomness (Warton et al., 2015) and mean-
variance trends (Warton et al., 2012)–although resampling
or finite mixture approaches may provide likelihood-based
foundations to the clustering (Pledger and Arnold, 2014). In
fact, methods based on a single classifier offer only one piece
of evidence, which does not provide any confidence interval
of the membership estimation (Huang and Lees, 2004; Liu
et al., 2004). In turn, ensemble modeling approaches overcome
such problems in a consistent environment (Elder IV, 2003).
Despite some criticisms to ensemble modeling approaches like
the lack of interpretation capability and some tendency to
overfit (see Elder IV, 2003 for a discussion on the topic),
ensemble modeling approaches were shown to outperform other
methods (Elder IV, 2003; Fernández-Delgado et al., 2014). The
confidence interval around the membership values provided by
such models may be used to acknowledge natural transitions
in addition to the error estimates around those membership
values.
THE ADVANTAGE OF MAPPING
UNCERTAINTY
The concepts of uncertainty and error have often mistakenly
been used interchangeably (Zhang and Goodchild, 2002; Jager
and King, 2004). As a consequence, uncertainty is often perceived
negatively because of its association with the idea of error and
inaccuracy. Uncertainty is inherent to any and all data and
cannot necessarily be removed or minimized to get closer to
the truth the same way errors can (Zhang and Goodchild,
2002; Beale and Lennon, 2012). Uncertainty is part of our
perception of natural patterns (e.g., temporal dynamics, spatial
Frontiers in Ecology and Evolution | www.frontiersin.org 3December 2018 | Volume 6 | Article 231

Fiorentino et al. Soft Classification and Uncertainty
FIGURE 1 | Example of learning classifier workflow using a fuzzy clustering. The learning phase is linked to the data acquisition because fuzziness may highlight data
weakness, thereby areas where new data acquisition is required. Note that the same workflow can be translated to any other methods that allow uncertainty.
variability), data representation (e.g., positional uncertainty,
measurement uncertainty, thematic uncertainty), and modeling
(e.g., error introduced by the model) (e.g., Barry and Elith,
2006).
Maps of uncertainty, or maps of ignorance, help to identify
areas where the classification has a stable, consistent, and distinct
solution and can be used further to target the areas where
uncertainty is higher, thus highlighting the need to deepen the
investigation in those areas (Rocchini et al., 2011). It has been
demonstrated that maps of uncertainty enhance decision-making
in conservation contexts by solving issues that often appear when
hard classifications are applied (Regan et al., 2005; Langford et al.,
2009). In fact, maps of uncertainty permit a more realistic and
natural delineation of boundaries between classes (Zhang and
Goodchild, 2002).
Soft classifications and ensemble models allow users’
knowledge of the system to grow by providing an estimation
of uncertainty. When classes cannot provide an appropriate
description of the system under study (for example in cases
of high fuzziness), the investigator can choose to change the
classifier or deepen the investigation to better understand
the patterns in the system and how it translates into the
data representation. On the opposite, Boolean (true/false)
approaches only offer a static view, and while measures of
accuracy can be calculated to quantify misclassifications, they
have been shown to sometimes misrepresent the amplitude of
the misclassifications (e.g., Lecours et al., 2016) and often cannot
guide further investigations to better understand the dynamics
at play.
To assist with the interpretation of uncertainty resulting from
the application of a soft classifier, at least three solutions based
on membership assignations of objects to classes can be used
(Table 1). These solutions can be used to display uncertainty
spatially in a map, which can be interpreted, discussed, and then
used to further enhance the classification in an iterative process
(Figure 1).
CONCLUSION
We acknowledge that classification methods need to be reliable
and appropriate for the intended use, and adequately represent
the natural complexity of the systems under study. However, we
think that the main challenge of classifications is the proper and
meaningful interpretation of the associated uncertainty rather
than the method itself.
The use of soft classifiers to provide the visual and spatial
display of classification uncertainty enhances the value of
classifications. Providing a measure of uncertainty associated
with classes leads to the fulfillment of classifications’ potential,
which goes beyond the simple identification and separation of
classes. Classifications based on concepts of fuzzy logic, model-
based approaches, and ensemble modeling approaches, will help
move away from classes with hard, discrete boundaries, yielding
better solutions to represent accurately and better understand
complex systems. Acknowledging uncertainty encourages
learning from the classification process by encouraging
further investigation and hypothesizing about its causes (e.g.,
inappropriate spatial resolution, data quality, inappropriate
number of classes).
Whether the aim of an exercise is to communicate results
or to start the investigation of a system through exploratory
analyses, the spatial display of uncertainty provides directions
on which actions need to be undertaken. While stakeholders
may use map of uncertainty to find for instance the proper
conservation measure and therefore to better handle areas where
high uncertainty is displayed, scientists may use the same
information to build new hypotheses and shed light on processes
that underpin such uncertainty.
Frontiers in Ecology and Evolution | www.frontiersin.org 4December 2018 | Volume 6 | Article 231

Fiorentino et al. Soft Classification and Uncertainty
AUTHOR CONTRIBUTIONS
DF conceived and organized the manuscript. DF, VL, and TB
wrote and reviewed the manuscript.
ACKNOWLEDGMENTS
We would like to thank the reviewer and Dr. Casper Kraan whose
comments helped to improve the manuscript.
REFERENCES
Barry, S., and Elith, J. (2006). Error and uncertainty in habitat models. J. Appl. Ecol.
43, 413–423. doi: 10.1111/j.1365-2664.2006.01136.x
Beale, C. M., and Lennon, J. J. (2012). Incorporating uncertainty in predictive
species distribution modelling. Philos. Trans. R. Soc. Lond. B Biol. Sci. 367,
247–258. doi: 10.1098/rstb.2011.0178
Borcard, D., Gillet, F., and Legendre, P. (2011). Numerical Ecology With R, 1st Edn.
New York, NY: Springer.
Boughen, N. (2003). LightWave 3D 7.5 Lighting, 500th edn. Plano, TX: Wordware
Publishing, Inc.
Brown, C. J., Smith, S. J., Lawton, P., and Anderson, J. T. (2011). Benthic habitat
mapping: a review of progress towards improved understanding of the spatial
ecology of the seafloor using acoustic techniques. Estuar. Coast. Shelf Sci. 92,
502–520. doi: 10.1016/j.ecss.2011.02.007
Burrough, P. A., van Gaans, P. F. M., and Hootsmans, R. (1997). Continuous
classification in soil survey: spatial correlation, confusion and boundaries.
Geoderma 77, 115–135. doi: 10.1016/S0016-7061(97)00018-9
Coggan, R., and Diesing, M. (2011). The seabed habitats of the central english
channel: a generation on from holme and cabioch, how do their interpretations
match-up to modern mapping techniques? Cont. Shelf Res. 31, 132–150.
doi: 10.1016/j.csr.2009.12.002
Elder IV, J. F. (2003). The generalization paradox of ensembles. J. Comput. Graph.
Stat. 12, 853–864. doi: 10.1198/1061860032733
Elith, J., H., Graham, C., P., Anderson, R., Dudík, M., et al. (2006). Novel methods
improve prediction of species’ distributions from occurrence data. Ecography
29, 129–151. doi: 10.1111/j.2006.0906-7590.04596.x
Everitt, B. S., Landau, S., Leese, M., and Stahl, D. (2011). Cluster Analysis, 5th Edn.
London: John Wiley & Sons. Available online at: http://onlinelibrary.wiley.com/
book/10.1002/9780470977811 (Accessed April 24, 2013).
Eysenck, M. W., and Keane, M. T. (2010). Cognitive Psychology: A Student’s
Handbook, 6 Student Edn. New York, NY: Taylor & Francis.
Farina, A. (2010). Ecology, Cognition and Landscape: Linking Natural and Social
Systems, 2009 edn. Dordrecht ; New York, NY: Springer.
Fendereski, F., Vogt, M., Payne, M. R., Lachkar, Z., Gruber, N., Salmanmahiny, A.,
et al. (2014). Biogeographic classification of the Caspian Sea. Biogeosciences 11,
6451–6470. doi: 10.5194/bg-11-6451-2014
Fernández-Delgado, M., Cernadas, E., Barro, S., and Amorim, D. (2014). Do
we need hundreds of classifiers to solve real world classification problems? J.
Mach. Learn. Res. 15, 3133–3181. Available online at: http://jmlr.org/papers/
v15/delgado14a.html
Fiorentino, D., Pesch, R., Guenther, C.-P., Gutow, L., Holstein, J., Dannheim,
J., et al. (2017). A ‘fuzzy clustering’ approach to conceptual confusion: how
to classify natural ecological associations. Mar. Ecol. Prog. Ser. 584, 17–30.
doi: 10.3354/meps12354
Foody, G. M. (2000). Mapping land cover from remotely sensed data with a
softened feedforward neural network classification. J. Intell. Robotic Syst. 29,
433–449. doi: 10.1023/A:1008112125526
Foody, G. M. (2002). Status of land cover classification accuracy assessment.
Remote Sens. Environ. 80, 185–201. doi: 10.1016/S0034-4257(01)00295-4
Fortin, M.-J., and Dale, M. R. T. (2005). Spatial Analysis: A Guide for Ecologists.
Cambridge; New York, NY: Cambridge University Press.
Hass, H. C., Mielck, F., Fiorentino, D., Papenmeier, S., Holler, P., and Bartholom,ä,
A. (2017). Seafloor monitoring west of Helgoland (German Bight, North Sea)
using the acoustic ground discrimination system RoxAnn. Geo Mar. Lett. 37,
125–136. doi: 10.1007/s00367-016-0483-1
Huang, Z., and Lees, B. G. (2004). Combining non-parametric models for
multisource predictive forest mapping. Photogr. Eng. Remote Sens. 70, 415–425.
doi: 10.14358/PERS.70.4.415
Huang, Z., Nichol, S. L., Siwabessy, J. P. W., Daniell, J., and Brooke, B. P.
(2012). Predictive modelling of seabed sediment parameters using multibeam
acoustic data: a case study on the Carnarvon Shelf, Western Australia.
Int. J. Geograph. Inform. Sci. 26, 283–307. doi: 10.1080/13658816.2011.
590139
Jager, H. I., and King, A. W. (2004). Spatial uncertainty and ecological models.
Ecosystems 7, 841–847. doi: 10.1007/s10021-004-0025-y
Kuhn, M., and Johnson, K. (2013). Applied Predictive Modeling. New York, NY;
Heidelberg; Dordrecht; London: Springer.
Langford, W. T., Gordon, A., and Bastin, L. (2009). When do conservation
planning methods deliver? Quantifying the consequences of uncertainty. Ecol.
Inform. 4, 123–135. doi: 10.1016/j.ecoinf.2009.04.002
Lecours, V., Brown, C. J., Devillers, R., Lucieer, V. L., and Edinger,
E. N. (2016). Comparing selections of environmental variables for
ecological studies: a focus on terrain attributes. PLoS ONE 11:e0167128.
doi: 10.1371/journal.pone.0167128
Lecours, V., Devillers, R., Edinger, E. N., Brown, C. J., and Lucieer, V. L.
(2017). Influence of artefacts in marine digital terrain models on habitat maps
and species distribution models: a multiscale assessment. Remote Sens. Ecol.
Conserv. 3, 232–246. doi: 10.1002/rse2.49
Liu, W., Gopal, S., and Woodcock, C. E. (2004). Uncertainty and confidence in land
cover classification using a hybrid classifier approach. Photogram. Eng. Remote
Sens. 70, 963–971. doi: 10.14358/PERS.70.8.963
Lucieer, V., and Lucieer, A. (2009). Fuzzy clustering for se afloor classification. Mar.
Geol. 264, 230–241. doi: 10.1016/j.margeo.2009.06.006
Pielou, E. C. (1975). Ecological Diversity. New York, NY: Wiley & Sons.
Pledger, S., and Arnold, R. (2014). Multivariate methods using mixtures:
correspondence analysis, scaling and pattern-detection. Comput. Stat. Data
Anal. 71, 241–261. doi: 10.1016/j.csda.2013.05.013
Regan, H. M., Ben-Haim, Y., Langford, B., Wilson, W. G., Lundberg, P., Andelman,
S. J., et al. (2005). Robust decision-making under severe uncertainty for
conservation management. Ecol. Appl. 15, 1471–1477. doi: 10.1890/03-5419
Ries, L., Robert, J., Fletcher, J.,B attin, J., and Sisk, T. D. (2004). Ecological responses
to habitat edges: mechanisms, models, and variability explained. Annu. Rev.
Ecol. Evol. Syst. 35, 491–522. doi: 10.1146/annurev.ecolsys.35.112202.130148
Rocchini, D., Hortal, J., Lengyel, S., Lobo, J. M., Jiménez-Valverde, A.,
Ricotta, C., et al. (2011). Accounting for uncertainty when mapping species
distributions: the need for maps of ignorance. Prog. Phys. Geogr. 35, 211–226.
doi: 10.1177/0309133311399491
Schiele, K. S., Darr, A., Zettler, M. L., Friedland, R., Tauber, F., von Weber, M., et al.
(2015). Biotope map of the German Baltic Sea. Mar. Pollut. Bull. 96, 127–135.
doi: 10.1016/j.marpolbul.2015.05.038
Sweeney, S. P., and Evans, T. P. (2012). An edge-oriented approach
to thematic map error assessment. Geocarto Int. 27, 31–56.
doi: 10.1080/10106049.2011.622052
Warton, D. I., Foster, S. D., De’ath, G., Stoklosa, J., and Dunstan, P. K. (2015).
Model-based thinking for community ecology. Plant Ecol. 216, 669–682.
doi: 10.1007/s11258-014-0366-3
Warton, D. I., Wright, S. T., and Wang, Y. (2012). Distance-based multivariate
analyses confound location and dispersion effects. Methods Ecol. Evolut. 3,
89–101. doi: 10.1111/j.2041-210X.2011.00127.x
Yager, R. (2016). Uncertainty modeling using fuzzy measures. Knowl. Based Syst.
92, 1–8. doi: 10.1016/j.knosys.2015.10.001
Zadeh, L. A. (1965). Fuzzy sets. Inform. Control 8, 338–353.
doi: 10.1016/S0019-9958(65)90241-X
Zhang, J., and Goodchild, M. F. (2002). Uncertainty in Geographical Information.
London; New York, NY: Taylor & Francis.
Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2018 Fiorentino, Lecours and Brey. This is an open-access article
distributed under the terms of the Creative Commons Attribution License (CC BY).
The use, distribution or reproduction in other forums is permitted, provided the
original author(s) and the copyright owner(s) are credited and that the original
publication in this journal is cited, in accordance with accepted academic practice.
No use, distribution or reproduction is permitted which does not comply with these
terms.
Frontiers in Ecology and Evolution | www.frontiersin.org 5December 2018 | Volume 6 | Article 231