Using Fuzzy Cognitive Mapping and Nonlinear Hebbian Learning for
Modeling, Simulation and Assessment of the Climate System,
Based on a Planetary Boundaries Framework
Iv
´
an Paz-Ortiz
1
and Carlos Gay-Garc
´
ıa
2
1
Departament de Llenguatges i Sistemes Inform
`
atics, Univertitat Polit
`
ecnica de Catalunya, Barcelona, Spain
2
Programa de Investigaci
´
on en Cambio Clim
´
atico, Universidad Nacional Aut
´
onoma de M
´
exico, Mexico City, Mexico
Keywords:
Climate System, Fuzzy Cognitive Maps, Nonlinear Hebbian Learning, Planetary Boundaries, System
Analysis.
Abstract:
In the present work a fuzzy cognitive map for the qualitative assessment of the Earth climate system is devel-
oped by considering subsystems on which the climate equilibrium depends. The cognitive map was developed
as a collective map by aggregating different experts opinions. The resulting network was characterized by
graph indexes and used for simulation and analysis of hidden pattens and model sensitivity. Linguistic vari-
ables were used to fuzzify the edges and were aggregated to produce an overall linguistic weight for each edge.
The resulting linguistic weights were defuzzified using the “Center of Gravity”, and the current state of the
Earth climate system was simulated and discussed. Finally, a nonlinear Hebbian Learning algorithm was used
for updating the edges of the map until a desired state. The overall results are discussed to explore possible
policy implementation, environmental decision making and management.
1 INTRODUCTION
Nowadays, the widespread concerns over the climate
stability and the resilience of natural ecosystems un-
der pressure have pointed out the necessity of devel-
oping new tools to monitor present conditions, asses
future scenarios and explore possible policy imple-
mentations oriented to environmental decision mak-
ing and management.
Currently a strong tendency to explore the use of
Fuzzy cognitive maps for the development of models
in the environmental sciences is emerging. Examples
include FCM as methodological framework for envi-
ronmental decision making and management (Papa-
georgiou and Kontogianni 2012), to evaluate cases of
study, like the future of water in the Seyhan Basin in
Turkey (Cakmak et al. 2010), and the description of
current system dynamics together with the develop-
ment of land cover scenarios in the Brazilian Amazon
(Soler et al. 2011).
This approach is useful by its capability for in-
cluding quantifiable and non-quantifiable concepts in
a model (Papageorgiou and Kontogianni 2012), but
also due to the fact that it doesn’t require neither large
capacity of computation nor to have numeric equa-
tions of the analyzed phenomena. FCM are also a
suitable framework for integrating information that
is scattered in several places, as is the case in envi-
ronmental systems in which, to built integrated mod-
els, the information must be taken in several disci-
plines (e.g atmospheric sciences, biology, and geo-
physics). Moreover, the latest models in cognitive
mapping seek not only to simulate systems, but to
control and thus to establish recommendations and
explore possible policy implementation, environmen-
tal decision making and management. The training
and tuning of FCM is performed by using Hebian
learning algorithms (Papakostas et al, 2011). These
algorithms aim to find appropriate weights between
the concepts of the map so the model equilibrates to a
desired state.
In the present work we used a FCM to analyze the
dynamic of the climatic system based on a planetary
boundaries framework, with Earth subsystems iden-
tified by Rockstr
¨
om et al. (2009) as those on which
the Earth’s climate equilibrium depends. The paper is
structured as follows: In Section 2 we present the ba-
sis of the FCMs theory. Section 3 describes the plan-
etary boundaries framework. Section 4 presents the
FCM simulation and a comparison with the current
state of the results versus the current state, Section 5
describes the implementation of a nonlinear Hebian
852
Paz-Ortiz I. and Gay-Garcia C..
Using Fuzzy Cognitive Mapping and Nonlinear Hebbian Learning for Modeling, Simulation and Assessment of the Climate System, Based on a
Planetary Boundaries Framework.
DOI: 10.5220/0005140608520862
In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (MSCCEC-2014), pages
852-862
ISBN: 978-989-758-038-3
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
learning algorithm to adjust the weights of the map
until the map reaches a desired state. And finally, Sec-
tion 6 includes the conclusions and further work.
2 FUZZY COGNITIVE MAPS
Fuzzy cognitive maps are directed graph structures for
representing causal reasoning between variable con-
cepts (Kosko 1986). The concepts are represented as
nodes (C
1
,C
2
,...,C
n
) in an interconnected network.
Each node C
i
represents a variable concept, and the
edges e
i j
which connect C
i
and C
j
(denoted as C
i
C
j
) are causal connections and express how much C
i
causes C
j
. Edges can have either a numerical or a lin-
guistic value. w
i j
> 0 indicates a positive causality
between concepts C
i
and C j, w
i j
< 0 indicates a in-
verse (or negative) causality, and w
i j
= 0 indicates no
causality. Linguistic quantifiers (such as low, medium
or high) are also used to represent the value of the
weights, they indicate the qualitative relation among
concepts.
Figure 1 shows a FCM containing four concepts (or
nodes) numbered I, II, III, and IV. The table shows the
assigned weights for the relationships between con-
cepts.
Figure 1: A FCM of four concepts (numbered I, II, III, IV).
The table shows the weights for the edges associated with
the relationships between concepts.
FCM are built using expert’s opinion. They de-
fine, considering their expertise, the main components
of the system as the concepts of the network. Once
these are defined, the (causal) relations among them,
as well as the weights for each relation, are estab-
lished.
2.1 The FCM Inference Process
The dynamic of a fuzzy cognitive map can be simu-
lated analytically through a specific inference process
(Papageorgiou and Kontogianni 2012). For this pro-
cess, we considered FCM concepts having values in
a specific numerical interval (e.g [0,1]). The value of
each C
i
at time t is called the “activation level”, and
is interpreted as the state of activation (or quantity) of
this concept.
The values of the concepts C
(t)
i
at each time t are
represented in a row state vector A
(t)
= [C
(t)
1
,...,C
(t)
n
]
which describes the state of the system at each time
step or iteration t (Kosko, 1991). Given a particular
input A
(0)
, the state of the system during the iterations
can converge into an equilibrium point, into a limit
cycle or diverge. The inference process is calculated
using the following equation:
A
t+1
i
= f
N
i=1,i6= j
W
i j
A
(t)
j
!
(1)
2.2 FCM Graph Indexes
The structural properties of cognitive maps can be
analyzed by using graph theory indexes. Through
this analysis we gain intuition of the general structure
and quality of the network. This is important since
cognitive maps are a subjective representation of the
real system, hardly dependent on the expert’s opinion.
Graph indices allow us to analyze how experts are re-
garding and structuring the system. Those include,
among others, the number of concepts, connections,
the ratio concepts/connections, and the density, de-
fined as the ratio of the number of connections in the
map and the square value of concepts.
The complexity of the concepts is analyzed
through structural measures like the conceptual cen-
trality index for a particular node C
i
defined by Kosko
(1986) as:
CEN(C
i
) = IN(C
i
) + OU T (C
i
) (2)
where:
IN(C
i
) =
n
k=1
w
ik
(3)
OU T (C
i
) =
n
k=1
w
ki
(4)
IN(C
i
) represents the number of concepts that
causally act on concept C
i
. Similarly, the row sum
OU T (C
i
) is the number of concepts in which C
i
causally acts (Kosko 1986). Then the conceptual cen-
trality represents the importance of C
i
to the causal
UsingFuzzyCognitiveMappingandNonlinearHebbianLearningforModeling,SimulationandAssessmentoftheClimate
System,BasedonaPlanetaryBoundariesFramework
853
flow on the map. When fuzzy weights values are used,
the causal amount counts, i.e, a node can be connected
to fewer nodes than another and still have greater con-
ceptual centrality (Kosko 1986).
The density index (D) shows how connected the
network is. It is defined as the number of existing
connections divided by the maximum number of pos-
sible connections among all nodes (given by N
2
). A
map with high density will have a large number of
causal relations.
The structure analysis of a cognitive map also
includes the transmitter, receiver and ordinary vari-
ables. These are defined by their associate values of
IN(C
i
) and OUT (C
i
), called indegree and outdegree,
respectively. Nodes whose outdegree is positive and
their indegree is 0 are defined as transmitter variables.
Receiver variables, on the contrary, are those nodes
whose outdegree is 0 and their indegree is positive.
Variables with non zero outdegree and indegree are
called ordinary variables. (Eden et al. 1992, Ozesmi
& Ozesmi, 2004). Receiver and transmitter variables
show the structure of the map, for example, transmit-
ter variables can be seen as forcing functions, which
influence the system but cannot be controlled by other
system’s variables. The complexity of the system, or
sometimes the degree of elaboration, can be analyzed
by considering the number of receiver and transmit-
ter variables. A great number of receiver or transmit-
ter variables could be interpreted as the map is not
being well elaborated or the relationships among its
variables are not well known, which means that the
causal relations among components are not clear for
the experts (Eden, 1992, Papageorgiou and Konto-
gianni 2012).
Collective maps are used to integrate different per-
spectives of a particular system. When collective cog-
nitive maps are developed each expert creates a map
and then the different versions are condensed either
by grouping subgraphs in a single node, or by main-
taining the nodes when several experts coincide. The
centrality is used to decide which concepts will be
represented in the collective map.
3 SUBSYSTEMS OF THE
CLIMATE SYSTEM: A
PLANETARY BOUNDARIES
FRAMEWORK
The stability analysis of the Earth climate system is
based on the planetary boundaries framework pro-
posed by a group of scientist heading by Rockst
¨
om
(2009), and later discussed by Foley (2010). They
established a planetary boundaries frame by identi-
fying and quantifying the boundaries associated with
the planet’s biophysical subsystems that must not be
transgressed in order to prevent an unacceptable envi-
ronmental change. Based on the review of those re-
ports we created an expert’s opinion knowledge base
for the construction of the fuzzy cognitive map. To
do this, we grouped in a single map the concepts and
relations identified in each article by its authors. They
mentioned nine processes suggested as those in which
is necessary to define planetary boundaries: climate
change; rate of biodiversity loss (terrestrial and ma-
rine); interference with the nitrogen and phosphorus
cycles; stratospheric ozone depletion; ocean acidi-
fication; global freshwater use; change in land use;
chemical pollution; and atmospheric aerosol loading.
These concepts and their relationships are briefly de-
scribed below.
Climate change refers to the increase in the mean
temperature of the earth. More precisely, to changes
in climate variability in terms of the extreme and
mean values (IPCC, 2007, 2014). Specifically we re-
fer to antropogenic climate change, which is a con-
sequence of the human activity. Climate change
causes changes in vegetation distribution, and so
threat the ecological live-support systems as well as
human activities. The climate change is described in
terms of two variables having critical thresholds that
qualitatively separate different climate system states
(Rockst
¨
om et al., 2009), these are the atmospheric
CO
2
concentration and the radiative forcing.
Changes in atmospheric CO
2
concentration
Defined as the increase in the parts per million of CO
2
molecules in the atmosphere (IPCC, 2007, 2014).
Most models suggest that, as atmospheric CO
2
in-
creases also does global temperature. For example,
doubling atmospheric CO
2
will lead to a rise about
3C (with a probable uncertainty of 2-4.5C).
Changes in radiative forcing
The radiative forcing is the rate of energy change per
unit area of the globe as measured at the top of the
atmosphere.
Rate of biodiversity Loss
Refers to the extinction rate, the number of species
loss per million per year. Mace and collaborators
(2005), define biodiversity as the variability of liv-
ing organisms, included terrestrial and marine ecosys-
tems, other aquatic ecosystems and the ecological
systems in which they reside. It comprises the diver-
sity within species, among species and within ecosys-
tems. Mace emphasizes three levels of biodiversity:
genes, species, and ecosystems. Biodiversity loss dur-
ing the industrial period has grown notably. The
species extinction rate is estimated against the fos-
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Applications
854
sil record. The extinction rates per million per year
varies for marine life between 0.1 and 1 and for mam-
mals between 0.2 and 0.5. Today, the rate of extinc-
tion of species is estimated to be 100 to 1,000 times
more than what could be considered natural (Rock-
str
¨
om, 2009).
Ocean Acidification
Defined as the ocean pH increase, mainly in the sur-
face layer. The acidification process is closely related
with the CO
2
emission level. When the atmosphere
CO
2
concentration increases, the amount of carbon
dioxide dissolved in water as carbonic acid increases,
which in turn, modifies the surface pH. Normally, the
ocean surface is basic with a pH of approximately 8.2.
Nevertheless, the observations show a decline in pH
to around a value of 8. These estimations are made
using the levels of aragonite (a form of calcium car-
bonate) that is created in the surface layer. This con-
cept has an important relation with biodiversity loss as
many organisms (like corals and phytoplankton), ba-
sic for the food chain, use aragonite to produce their
skeletons or shells. As the aragonite value decreases,
the ocean ecosystems weaken. (Foley et al., 2010).
Phosphorus and Nitrogen Cycles
The human activity at the planetary scale is perturbing
the global cycles of phosphorus (P) and nitrogen (N).
The agriculture activity convert around 120 million
tones of N
2
from the atmosphere per year into reac-
tive forms (Rockstr
¨
om, 2009). The pressure that this
changes exerted over the environment threats the bal-
ance of natural equilibrium. For example the nitrous
oxide is one of the most important greenhouse gases
and its grown directly increases the radiative forcing.
In the case of phosphorus around 20 million tonnes
are mined every year and around 8.5 million - 9.5 mil-
lion tonnes flow into the oceans perturbing the marine
ecosystems. To establish boundaries, the changes in P
and N cycles are estimated with the quantity of P go-
ing to the oceans, measured in million tones per year,
and with the amount of N
2
removed from the atmo-
sphere for human use, also in million tones per year.
Change in land use (Urban Growth and Agri-
culture Use)
The IPCC defines the change in land use as the
percentage of global land converted into cropland.
A general definition of land use change includes
any type of human use. This transformation, either
to cropland or urban, increases the biodiversity
loss, which is associated with the destruction of
ecosystems. In order to establish the difference in
land use between urban growth and agriculture use,
and their different consequences, we include both
concepts as nodes in the map.
Chemical pollution
It refers to the emitted quantity, persistence, or con-
centration of organics pollutants, plastics, heavy met-
als, chemical and nuclear residues, etc., which affect
the dynamic of ecosystems.
Global Freshwater use
Defined as the increase in its current use. Today,
the annual use of freshwater from rivers, lakes and
groundwater aquifers is of 2,600 km
3
. From that, 70%
is destined for irrigation, 20% for industry, and 10%
for domestic use. This extraction causes the drying
and reduction of body waters. (IPCC, 2007, 2014).
Atmospheric aerosol loading
Referred as the concentration of particles in the atmo-
sphere. These can be lead, copper, magnesium, iron,
traces of fire, ashes, etc.
Stratospheric Ozone Depletion
O
3
depletion is estimated according to the ozone
concentration in the atmosphere in Dobson Units.
1
With the previous subsystems and the highlighted
relations among them we built the cognitive map.
Even though FCM are a subjective representation
of the reality, this representation is not arbitrarily
because its constructions is reviewed and processed
carefully to extract the system knowledge of the ex-
perts.
4 A COGNITIVE MAP OF THE
EARTH CLIMATE SYSTEM
The FMC constructed was based on four different
cognitive maps. Two of them based on the analysis
of Rockstr
¨
om et al, (2009) and Foley (2010), and
others proposed by Paz-Ortiz (2011), and Gay-Garc
´
ıa
(2012). Then, we analyzed the maps separately and
a collective cognitive map was created. For maps
based on Rockstr
¨
om et al, (2009) and Foley (2010)
we extracted the concepts mentioned by the authors
as those who described the Earth climate stability
system, as well as the relations among the concepts.
For each relation, we assigned a value of 1 or -1 (ac-
cording to the positive or negative causality described
by the author) to be the weight of the arc representing
it. If an author did not mentioned a concept, it was
included in its map’s matrix without relation (writing
0 in the adjacent matrix) with other concepts. So all
the matrices had the same dimensions. The adjacent
1
Dobson unit is a measure of the ozone layer thick-
ness, equal to 0,01 mm of thickness in normal conditions
of pressure and temperature (1 atm and 0 C respectively),
expressed as the molecule number. DU represents the exis-
tence of 2.69 x 10
16
molecules per square centimeter.
UsingFuzzyCognitiveMappingandNonlinearHebbianLearningforModeling,SimulationandAssessmentoftheClimate
System,BasedonaPlanetaryBoundariesFramework
855
matrix of the collective map was defined as the
sum of all adjacent matrices of the individual maps.
Figure 2 and 3 show the cognitive maps derived
from Rockstr
¨
om (2009), and Foley (2010). Cognitive
maps Paz-Ortiz (2011) and Gay-Garc
´
ıa & Paz-Ortiz
(2012) are not shown. Figure 4 shows the collective
cognitive map created considering the four cognitive
maps. Figures were developed in pajek software
[http://vlado.fmf.uni-lj.si/pub/networks/pajek/]).
Table 1. shows the average (±SD) graph theoretical
indices (Papageorgiou & Kontogianni 2012) of the
individual FCMs and the indices of the collective
FCM. The indices allowed us to evaluate the descrip-
tive strength of the model.
Table 1: Cognitive map index.
Indices Individual Maps Collective
CM
Maps 4 1
Variables (N) 11.5 ± 0.5 14
Number of connections (W) 18.5 ± 6.42 40
No. of transmitter variables (T) 2 ± 1.22 1
No. of receiver variables (R ) 1.25 ± 0.43 0
Connection/Variable (W/N) 1.84 ± 0.26 2.85
Density (D = W/N
2
) 0.13 ± 0.04 0.20
Figure 2: Rockstr
¨
om et al, (2009) Cognitive map of the
Earth climate system.
In Table 1 the index for the number of connec-
tions (W) shows values of 40 for the collective map
and 18.5 for the individual ones. Also the number
of variables (N) is 11.5 for the individual and 14 for
the collective. This illustrate how experts have dif-
ferent perspectives respecting to nodes and relations.
The integration of these different perspectives in one
model is another powerful characteristic of FCMs.
As said, a measure of how a cognitive map is con-
nected or sparse, is the density, expressed as the num-
ber of connections divided by the maximum number
of connections possible (N
2
). The collective cogni-
Figure 3: Foley, (2010) Cognitive map of the Earth climate
system.
tive map is highly connected compared with individ-
ual maps.
The transmitter and receiver variables indicate
how the experts structured causal relations in a cog-
nitive map. As referred by Papageorgiou & Kon-
togianni (2012), cognitive maps containing a larger
number of receiver variables consider many outcomes
that are a result of the system, while maps contain-
ing many transmitter nodes show the ”flatness” of a
cognitive map where causal arguments are not well
elaborated.
The number of transmitter and receiver variables
diminish in the collective map in comparison with in-
dividual maps. The indices ratio Connection/Variable
(W/N) and Density also show that the collective map
provides a strongest model in comparison with indi-
vidual ones.
4.1 Simulation
To simulate the dynamic of the map, the inference
process described in Equation 1 was used. For the
simulation, we represented the earth process in a
state vector ordering the nodes as follows: 1 Climate
Change, 2 Ocean acidification, 3 Stratospheric ozone
depletion, 4 Nitrogen and Phosphorus cycles, 5 Fresh
water use, 6 Land use, 7 Biodiversity loss, 8 Aerosil
loading, 9 Chemical pollution, 10 CO
2
atmospheric
concentration, 11 Radiative forcing, 12 Polar sheets,
13 Agriculture, 14 Industrialization. For the basic
simulation process, in order to analyze the general
behavior of the network, we considered three cases.
These will give us information about hidden patterns
(when forcing the network), and the net sensitivity
when we use random values. Knowing the hidden
patterns (or feedback process) allows policy makers to
pay attention in the processes that can be irreversible.
While network sensitivity shows how much the out-
put of the network depends on the initial values, and
SIMULTECH2014-4thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
Applications
856
Figure 4: Collective cognitive map.
therefore how much our approximations for the edges
values will influence the results. This allow the stake-
holders to consider the model taking this perspective
into account.
Case 1: Initializing the network with random val-
ues in the interval [0,1] for nodes and in the interval
(0,1] for edges. This, since we assumed that the links
established by the experts cannot have a value of zero,
in which case there would be no such link. In the pro-
cess, all randomly generated values were generated
independently.
Case 2: Initializing the network with the same ran-
dom values used in Case 1 for nodes and edges. And
forcing Industrialization (Node 14) by giving a value
of 1 to this node after each iteration.
Case 3: Forcing the Industrialization (Node 14),
by inducing a recurrent value of 1 in this node after
each iteration, and considering an initial value of 0
for all nodes.
Table 2, summarizes the three cases: Initial values
of the nodes, initial weights used, the resulting vec-
tor, and the number of the iterations where the con-
vergence is reached.
Case 1: Cycles and feedback processes.
The resulting vector in the first simulation shows the
feedback processes of the network. Nodes that appear
with non zero values. We can see that Climate change
(Node 1), Ocean acidification (2), Fresh water use (5),
Biodiversity loss (7), and CO
2
atmospheric concen-
tration (10) converge into a value of 1. While node
polar sheets (12), only activated in the map by node
Climate change (and that got an edge random value
of 0.3), remain “ON” with this value. All these nodes
are inside feedback processes in the network, for this
reason they remain “ON” even when the driver has
gone out. These nodes are key to the design of poli-
cies, since disturbances in these systems could trigger
irreversible processes. In this type of simulation, i.e
initializing some nodes without forcing the network,
the resulting values will depend on the values of the
nodes and weights (Gay-Garc
´
ıa & Paz-Ortiz, 2012).
In some cases, where the weights are small enough,
the value of the concepts is damped until zero as the
system is iterated.
Case 2: Dependency of initial conditions (values
of nodes and edges).
The resulting vector in Case 2 shows the effect of the
weights of the edges in the resulting vector. Even
though the vector converges, the values to which each
node converge will depend on the strength of the
causal connections between them. This hypothesis is
supported by Case 3. In which we have forced Indus-
trialization (14) but considering a value of 1 for all
connections. Cases 2 and 3 show the sensitivity of the
network.
4.2 Fuzzy Weights
In order to have more information form the model,
we considered fuzzy weights for the edges. These are
UsingFuzzyCognitiveMappingandNonlinearHebbianLearningforModeling,SimulationandAssessmentoftheClimate
System,BasedonaPlanetaryBoundariesFramework
857
Table 2: Basic simulation of the cognitive map.
Initial values of nodes Initial weights Resulting vector Number
of
iterations
Case 1
random values [0,1] random values (0,1] 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0.3, 0, 0 6
Case 2
Same random values
of Case 1 [0,1]
Industrialization =1 for all t random values (Case 1) (0,1] 1, 1, 1, 0.83, 1, 0.81, 1, 0.6, 1, 1, 0.08, 0.3, 0.3, 1 5
Case 3
Industrialization = 1 for all t 1 for all edges 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 3
fuzzy linguistic variables associated with the relation-
ship between two concepts that determine the grade
of causality between them. We created and associated
these variables as proposed by the Papageorgiou and
Kontogianni (2012) methodology. We used thirteen
fuzzy quantities taken from the strength suggested for
each relation by each one of the expert in its orig-
inal map. The quantities used were T{influence}=
{negatively very very strong, negatively very strong,
negatively strong, negatively medium, negatively
weak, negatively very weak, zero, positively very
weak, positively weak, positively medium, positively
strong, positively very strong, positively very very
strong}. Then, the results were aggregated by us-
ing the SUM method (Papageorgiou & Stylios, 2008)
to produce an overall linguistic weight for each edge.
The resulting linguistic weights were defuzzified us-
ing the Center of Gravity (CoG) (Zadeh, 1986) that
takes all the linguistic weights and creates numerical
values within the interval [-1, 1]. The overall linguis-
tic values are shown in Table 4.
To obtain initial values for the edges we consid-
ered the current state of the earth system processes re-
ferred by Rockstr
¨
om et al (2009) (shown in Table 3).
We can see that subsystems: Climate change, Rate of
biodiversity loss and Nitrogen cycle (i.e three of nine
interlinked planetary boundaries), have already trans-
gressed their proposed limits. Stratospheric ozone de-
pletion and Ocean acidification are just above. Phos-
phorus cycle, Global fresh water use, and Change in
land use are close to the boundary. While Atmo-
spheric aerosol loading and Chemical pollution sys-
tems have not yet an established boundary. We as-
signed for each concept a fuzzy weight by consider-
ing its current position with respect to the proposed
boundary. For the Industrialization, Agriculture, and
Polar sheets processes, which are not considered in
this frame, we used the description given by Rock-
str
¨
om et al (2009) and Gay-Garca & Paz-Ortiz (2012)
to set the initial values. Again, the overall fuzzy quan-
tifiers were defuzzified by using the CoG method into
a numerical values for the system’s simulation. The
initial defuzzified numerical values for the concepts
are shown in the Initial Vector at the top of Table 4.
For this simulation the map was iterated while keep-
ing the forcing in node Industrialization (14) with its
initial value of 0.4. The resulting vector converged af-
ter 10 iterations to:
0.69, 0.21, 0.25, 0.27, 0.62, 0.32, 0.90, 0.17, 0.56,
0.43, 0.08, 0.42, 0.24, 0.4 (button Table 4).
It can be seen that nodes Climate change (1) and Bio-
diversity loss (7) referred as concepts who have al-
ready crossed the boundary appeared with the highest
values of 0.69 and 0.90, respectively. However, the
node representing Nitrogen and Phosphorous cycles
(4) exhibited a value of 0.27. This can be a conse-
quence that although the threshold is established con-
sidering both processes, the distance of each process
in respect to its proposed boundary is slightly differ-
ent, so when the concepts were defuzzified the model
could have lost accuracy. Also nodes Fresh water use
(5), and Chemical pollution (9) exhibited high values
of 0.62 and 0.56 respectively. In the first case, it is
possible that the established edges, as well as the re-
lations, are overestimating the state of the system. In
the second case, since there are no data on the system
state, this simulation is interesting since it allows us
to estimate the state of this node.
4.3 Training the Fuzzy Cognitive Map
by using Nonlinear Hebbian
Learning
Up to this point, the simulation qualitatively describes
the state of the climate system. However, in order to
establish action strategies that allow for planning and
support decision processes in terms of environmen-
tal policy, it is necessary to know what might be the
changes of the weights in order to obtain a desired
state for the system. To do this, we used the Nonlin-
ear Hebbian Learning (NHL) algorithm proposed by
Papageorgiou et al, (2006) to adjust the weights of the
SIMULTECH2014-4thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
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Table 3: Planetary Boundaries.
Earth-system processes Parameters Proposed boundary Current status
Climate change CO
2
concentration, changes in radiative forcing 350 ppm, 1W/m
2
280 ppm, 1.5W/m
2
Rate of biodiversity loss Extinction rate 10 > 100
Nitrogen (with P cycle) N
2
removed for human use 35 121
Phosphorus (with N cycle) Quantity of P flowing into the oceans 11 8.5-9.5
Stratospheric ozone depletion O
3
concentration 276 283
Ocean acidification Global saturation of aragonite in surface sea 2.75 2.90
Global fresh water use Consumption (km
3
/year 4000 2600
Change in land use Percentage of land cover converted to cropland 15 11.7
Atmospheric aerosol loading Particulate concentration, on a regional basis to be determined -
Chemical pollution Amount emitted to, or concentration to be determined -
map. This algorithm was used because it is referred
by Papakostas et al, (2011) in a comparative study
as the one which exhibits a satisfactory behavior in
control systems, having at the same time a low algo-
rithmic complexity when it is compared with similar
algorithms (Papakostas et al, 2011).
Two restrictions for the algorithm were used.
First: Don’t change the edges values that describe re-
lationships between concepts that operate over natural
processes. For example, it cannot modify the depen-
dency of climate change with respect to the increase
in the atmospheric CO
2
concentration, since doing so
it would modify a natural process. Therefore, the al-
gorithm was restricted for to only change the weights
of the edges starting from the node Industrialization.
The values of the other edges were kept equal after
each iteration. Second: the weight of these edges
couldn’t be less than zero, since the causal relations
were positively established by experts. Therefore, the
value of the concepts were kept >0.
The algorithm operates by using targets for the
value of the concepts to be adjusted. For this, only
targets over variables Climate change (1) and biodi-
versity loss (7) were used. This is because we were
interested in reducing the concepts that are operating
over the proposed boundary that were simulated by
the model. The targets were established to perform a
reduction in this two variables by 0.2 under its initial
value. The values of the other concepts, in the state
vector, were allowed to variate freely. Then, the algo-
rithm operated as follows:
Step 1: Given the initial vector (A
0
) and the initial ad-
jacent matrix of weights W
0
.
Step 2: For each iteration k.
Calculate A
k
by using Eq (1).
Update W
k
for edges coming out Industrialization by
using the equation:
w
(t+1)
i j
= w
(t)
i j
+ ηA
(t)
j
A
(t)
i
sgn(w
(t)
i j
A
(t)
j
w
(t)
i j
)
(5)
Where η is the learning rate.
Calculate error associated with Climate change and
Biodiversity loss, defined as the difference between
the current value of each concept and its established
target value.
Stop when error acceptable difference (0.05).
Step 3: Return the final weights.
The final weights of the concepts are shown in Ta-
ble 4. The system converges into a final vector:
0.49, 0.14, 0.17, 0.20, 0.49, 0.22, 0.67, 0.13, 0.45,
0.29, 0.06, 0.29, 0.2, 0.4 after 24 iterations. We found
the nodes Climate change and Biodiversity loss, 0.2
and 0.23 below its original values, respectively. This
coincides with the established targets of 0.2 units be-
low for each one.
A first remarkable result is that the greater reduc-
tion appears in the relation Industrialization - CO
2
at-
mospheric concentration, that goes from 0.7 to 0.45.
This can be interpreted from the point of view of the
possible action policies as the most important pro-
cesses that must be taken into account.
In general we can arrange our results in four
classes, depending on the reduction of the weights.
Relations with minimal reduction (0.05), Industrial-
ization - Fresh water use, Industrialization - Chem-
ical pollution, and Industrialization - Biodiversity
loss). Relations with reduction of 0.1,Industrializa-
tion - Stratospheric ozone depletion and Industrial-
ization - agriculture. Relations with a reduction of
0.2, Industrialization - Land use, and concepts with
weight reduction of 0.25 Industrialization - CO
2
at-
mospheric concentration. This results can be useful
to establish different levels of action. It is important
to say that declaring different levels of action does not
neglect other processes while serving one. It is nec-
essary to take into account that the systems are inter-
related. However, these results allow us to establish
hierarchies for a policy design.
UsingFuzzyCognitiveMappingandNonlinearHebbianLearningforModeling,SimulationandAssessmentoftheClimate
System,BasedonaPlanetaryBoundariesFramework
859
Table 4: Comparison between initial and final vector (with initial and final weights). Nodes-relation and linguistic (initial and
final) quantifiers and its respective associated numerical value.
INITIAL VECTOR 0.6, 0.1, 0.3, 0.8, 0.1, 0.1, 0.7, 0.2, 0.3, 0.7, 0.8, 0.2, 0.3, 0.4 Iterations: 0
Nodes - Relation Linguistic value - Initial weight - Final weight
1 - 5 Climate change - Fresh water use positively very weak, 0.3 -
1 - 12 Climate change - Polar sheets positively strong, 0.6, -
1 - 7 Climate change - Biodiversity loss positively weak, 0.3, -
2 -1 Ocean acidification - Climate change positively very weak, 0.1, -
2 - 7 Ocean acidification Biodiversity loss positively very weak, 0.1, -
3 - 1 Stratospheric ozone depletion - Climate change positively weak, 0.2, -
3 - 7 Stratospheric ozone depletion - Biodiversity loss positively very weak, 0.1, -
4 - 9 N P cycles - Chemical pollution positively medium, 0.5, -
4 - 7 N P cycles - Biodiversity loss positively very weak, 0.1, -
4 - 1 N P cycles - Climate change positively very weak, 0.1, -
4 - 11 N P cycles - Radiative forcing positively weak, 0.3, -
5 - 7 Fresh water use - Biodiversity loss positively very weak, 0.1, -
5 - 1 Fresh water use - Climate change positively very weak, 0.1, -
6 - 9 Land use - Chemical pollution positively strong, 0.6, -
6 - 4 Land use - N P cycles positively medium, 0.4, -
6 - 7 Land use - Biodiversity loss positively strong, 0.6, -
6 - 1 Land use - Climate change positively weak, 0.2, -
6 - 5 Land use - Fresh water use positively weak, 0.3, -
6 - 10 Land use - CO
2
concentration positively weak, 0.2, -
7 - 10 Biodiversity loss - CO
2
concentration positively very weak, 0.1, -
7 - 1 Biodiversity loss - Climate change positively very weak, 0.05, -
8 - 1 Aerosol loading - Climate change positively very weak, 0.05, -
9 - 8 Chemical pollution - Aerosol loading positively weak, 0.3, -
9 - 7 Chemical pollution - Biodiversity loss positively medium, 0.5, -
9 - 3 Chemical pollution - Stratospheric ozone depletion positively weak, 0.3, -
10 - 2 CO
2
concentration - Ocean acidification positively medium, 0.5, -
10 - 1 CO
2
concentration - Climate change positively very strong, 0.7, -
11 - 1 Radiative forcing - Climate change positively medium, 0.4, -
12 - 1 Polar sheets - Climate change positively weak, 0.2, -
13 - 4 Agriculture - NP cycles positively strong, 0.6, -
13 - 6 Agriculture - Land use positively medium, 0.5, -
13- 5 Agriculture - Fresh water use positively medium, 0.5, -
14 - 13 Industrialization - agriculture positively strong, 0.6 , 0.5
14 - 6 Industrialization - Land use positively medium, 0.5, 0.3
14 - 5 Industrialization - Fresh water use positively medium, 0.5, 0.45
14 - 10 Industrialization - CO
2
concentration positively very strong, 0.7, 0.45
14 - 9 Industrialization - Chemical pollution positively strong, 0.6, 0.55
14 - 7 Industrialization - Biodiversity loss positively weak, 0.2, 0.15
14- 3 Industrialization - Stratospheric ozone depletion positively weak, 0.2, 0.1
FINAL VECTOR WITH INITIAL WEIGHTS 0.69, 0.21, 0.25, 0.27, 0.62, 0.32, 0.90, 0.17, 0.56, 0.43, 0.08, 0.42, 0.24, 0.4 Iterations: 10
FINAL VECTOR WITH FINAL WEIGHTS 0.49, 0.14, 0.17, 0.20, 0.49, 0.22, 0.67, 0.13, 0.45, 0.29, 0.06, 0.29, 0.2, 0.4 Iterations: 24
5 CONCLUSIONS AND FURTHER
WORK
The methodology presented for evaluation and simu-
lation of the climate system showed qualitatively con-
sistent results, both with climate system state as well
as with the expected scenarios. Moreover, the adjust-
ments to the weights obtained through the implemen-
tation of the algorithm (where we clearly observed
different levels of adjustment) allow the design and
assessment of environmental policies, helping at the
same time, to their planning.
Given that the system is highly sensitive to changes in
the strength of interactions between subsystems, and
considering the feedback processes (Section 4.1), it
is clear that more research must be developed to in-
crease the description accuracy of the model in the
first case, and to analyze possible irreversible degra-
dation processes within the feedback loops. However,
since the system allows to asses and differentiate the
importance between these relationships, further re-
search can be prioritized. In that sense, this model can
also help to the elaboration of the research agenda.
The methodological approach presented (based on
Papageorgiou & Kontogianni, 2012) clearly allows
the construction of stronger collective cognitive maps
when compared with the capacity of descriptions of
the individual ones. As cognitive maps highly depend
SIMULTECH2014-4thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
Applications
860
on the expert’s opinion, this methodology also allows
us to evaluate the strength and descriptive capacity
of the model. The performed simulation was capa-
ble to identify the feedback processes, and when lin-
guistic variables were used and aggregated to produce
an overall linguistic weight for each edge with the as-
sociated defuzzification by using Center of Gravity,
the resulting model matched with the current descrip-
tion of the climate system referred by Rockstr
¨
om et al
(2009). Although the algorithm for the adjustment of
the weights was restricted, the adjustments where the
principal reduction occurs in the relation Industrial-
ization - CO
2
atmospheric concentration, were plau-
sible in the context of the current reports on climate
change. This, together with the results of the simula-
tions, support the idea that the developed model can
be used for the planning, implementation, and eval-
uation of policies. A possible further work, in order
to analyze the adjustments performed, could imple-
ment migration or evolutionary algorithms to adjust
the weights (Va
ˇ
s
ˇ
c
´
ak 2012) and evaluate the perform
of each type from the point of view of the stakehold-
ers.
ACKNOWLEDGEMENTS
The present work was developed with the support of
the Programa de Investigaci
´
on en Cambio Clim
´
atico
(PINCC) of the Universidad Nacional Aut
´
onoma de
M
´
exico (UNAM).
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