Using Fuzzy Cognitive Mapping and Nonlinear Hebbian Learning for

Modeling, Simulation and Assessment of the Climate System,

Based on a Planetary Boundaries Framework

an Paz-Ortiz

and Carlos Gay-Garc

ıa

Departament de Llenguatges i Sistemes Inform

atics, Univertitat Polit

ecnica de Catalunya, Barcelona, Spain

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

 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

,...,C

) in an interconnected network.

Each node C

represents a variable concept, and the

edges e

i j

which connect C

and C

(denoted as C

→

) are causal connections and express how much C

causes C

. Edges can have either a numerical or a lin-

guistic value. w

i j

> 0 indicates a positive causality

between concepts C

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

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)

at each time t are

represented in a row state vector A

(t)

= [C

(t)

,...,C

(t)

]

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:

t+1

= f

∑

i=1,i6= j

i j

(t)

(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

deﬁned by Kosko

(1986) as:

CEN(C

) = IN(C

) + OU T (C

) (2)

where:

IN(C

) =

∑

k=1

(3)

OU T (C

) =

∑

k=1

(4)

IN(C

) represents the number of concepts that

causally act on concept C

. Similarly, the row sum

OU T (C

) is the number of concepts in which C

causally acts (Kosko 1986). Then the conceptual cen-

trality represents the importance of C

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

). 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

) and OUT (C

), 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

(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

concentration and the radiative forcing.

Changes in atmospheric CO

concentration

Deﬁned as the increase in the parts per million of CO

molecules in the atmosphere (IPCC, 2007, 2014).

Most models suggest that, as atmospheric CO

in-

creases also does global temperature. For example,

doubling atmospheric CO

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

emission level. When the atmosphere

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

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

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

. 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

depletion is estimated according to the ozone

concentration in the atmosphere in Dobson Units.

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

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

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

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

) 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

). 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

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

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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

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

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Table 2: Basic simulation of the cognitive map.

Initial values of nodes Initial weights Resulting vector Number

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

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Table 3: Planetary Boundaries.

Earth-system processes Parameters Proposed boundary Current status

Climate change CO

concentration, changes in radiative forcing 350 ppm, 1W/m

280 ppm, 1.5W/m

Rate of biodiversity loss Extinction rate 10 > 100

Nitrogen (with P cycle) N

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

concentration 276 283

Ocean acidiﬁcation Global saturation of aragonite in surface sea 2.75 2.90

Global fresh water use Consumption (km

/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

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

) and the initial ad-

jacent matrix of weights W

Step 2: For each iteration k.

Calculate A

by using Eq (1).

Update W

for edges coming out Industrialization by

using the equation:

(t+1)

i j

= w

(t)

i j

+ ηA

(t)



(t)

− sgn(w

(t)

i j

(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

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

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

concentration positively weak, 0.2, -

7 - 10 Biodiversity loss - CO

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

concentration - Ocean acidiﬁcation positively medium, 0.5, -

10 - 1 CO

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

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

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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

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

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

exico (UNAM).

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