Simple Fuzzy Logic Models to Estimate the Global Temperature

Change Due to GHG Emissions

Carlos Gay García

, Oscar Sánchez Meneses

, Benjamín Martínez-López

, Àngela Nebot

and Francisco Estrada

Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México, Ciudad Universitaria, D.F., Mexico

Grupo de Investigación Soft Computing, Universitat Politècnica de Catalunya, Barcelona, Spain

Keywords: Fuzzy Inference Models, Greenhouse Gases Future Scenarios, Global Climate Change.

Abstract: Future scenarios (through 2100) developed by the Intergovernmental Panel on Climate Change (IPCC)

indicate a wide range of concentrations of greenhouse gases (GHG) and aerosols, and the corresponding

range of temperatures. These data, allow inferring that higher temperature increases are directly related to

higher emission levels of GHG and to the increase in their atmospheric concentrations. It is evident that

lower temperature increases are related to smaller amounts of emissions and, to lower GHG concentrations.

In this work, simple linguistic rules are extracted from results obtained through the use of simple linear

scenarios of emissions of GHG in the Magicc model. These rules describe the relations between the GHG,

their concentrations, the radiative forcing associated with these concentrations, and the corresponding

temperature changes. These rules are used to build a fuzzy model, which uses concentration values of GHG

as input variables and gives, as output, the temperature increase projected for year 2100. A second fuzzy

model is presented on the temperature increases obtained from the same model but including a second

source of uncertainty: climate sensitivity. Both models are very attractive because their simplicity and

capability to integrate the uncertainties to the input (emissions, sensitivity) and the output (temperature).

1 INTRODUCTION

There is a growing scientific consensus that

increasing emissions of greenhouse gases (GHG) are

changing the Earth's climate. The IPCCs Fourth

Assessment Report (IPCC, 2007) states that

warming of the climate system is unequivocal and

notes that eleven of the last twelve years (1995-

2006) rank among the twelve warmest years of

recorded temperatures (since 1850). The projections

of the IPCCs Third Assessment Report (TAR)

(IPCC, 2001) regarding future global temperature

change ranged from 1.4 to 5.8 °C. More recently, the

projections indicate that temperatures would be in a

range spanning from 1.1 to 4 °C, but that

temperatures increases of more than 6 °C could not

be ruled out (IPCC; 2007). This wide range of

values reflects the uncertainty in the production of

accurate projections of future climate change due to

different potential pathways of GHG emissions.

There are other sources of the uncertainty preventing

us from obtaining better precision. One of them is

related to the computer models used to project future

climate change. The global climate is a highly

complex system due to many physical, chemical,

and biological processes that take place among its

subsystems within a wide range of space and time

scales.

Global circulation models (GCM) based on the

fundamental laws of physics try to incorporate those

known processes considered to constitute the climate

system and are used for predicting its response to

increases in GHG (IPCC, 2001). However, they are

not perfect representations of reality because they do

not include some important physical processes (e.g.

ocean eddies, gravity waves, atmospheric

convection, clouds and small-scale turbulence)

which are too small or fast to be explicitly modeled.

The net impact of these small scale physical

processes is included in the model by means of

parameterizations (Schmidt, 2007). In addition,

more complex models imply a large number of

parameterized processes and different models use

different parameterizations. Thus, different models,

using the same forcing produce different results.

One of the main sources of uncertainty is,

518

Gay García C., Sánchez Meneses O., Martínez-López B., Nebot À. and Estrada F..

Simple Fuzzy Logic Models to Estimate the Global Temperature Change Due to GHG Emissions.

DOI: 10.5220/0004164905180526

In Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (MSCCEC-2012), pages

518-526

ISBN: 978-989-8565-20-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

however, the different potential pathways for

anthropogenic GHG emissions, which are used to

drive the climate models. Future emissions depend

on numerous driving forces, including population

growth, economic development, energy supply and

use, land-use patterns, and a variety of other human

activities (Special Report on Emissions Scenarios,

SRES). Future temperature scenarios have been

obtained with the emission profiles corresponding to

the four principal SRES families (A1, A2, B1, and

B2) (Nakicenovic et al., 2000). From the point of

view of a policy-maker, the results of the 3rd and 4th

IPCC’s assessments regarding the projection of

global or regional temperature increases are difficult

to interpret due to the wide range of the estimated

warming. Nevertheless this is an aspect of

uncertainty that scientists and ultimately policy-

makers have to deal with. Furthermore, most of the

available methodologies that have been proposed for

supporting decision-making under uncertainty do not

take into account the nature of climate change’s

uncertainty and are based on classic statistical theory

that might not be adequate. Climate change’s

uncertainty is predominately epistemic and,

therefore, it is critical to produce or adapt

methodologies that are suitable to deal with it and

that can produce policy-useful information. The lack

of such methodologies is noticeable in the IPCC’s

AR4 Contribution of the Working Group I, where

the proposed best estimates, likely ranges and

probabilistic scenarios are produced using

statistically questionable devices (Gay and Estrada,

2009).

Two main strategies have been proposed for

dealing with uncertainty: trying to reduce it by

improving the science of climate change a feat tried

in the AR4 of the IPCC, and integrating it into the

decision-making processes (Schneider, 2003). There

are clear limitations regarding how much of the

uncertainty can be reduced by improving the state of

knowledge of the climate system, since there

remains the uncertainty about the emissions which is

more a result of political and economic decisions

that do not necessarily obey natural laws.

Therefore, we propose that the modern view of

climate modelling and decision-making should

become more tolerant to uncertainty because it is a

feature of the real world (Klir and Elias, 2002).

Choosing a modelling approach that includes

uncertainty from the start tends to reduce its

complexity and promotes a better understanding of

the model itself and of its results. Science and

decision-making have always had to deal with

uncertainty and various methods and even branches

of science, such as Probability, have been developed

for this matter (Jaynes, 1957). Important efforts have

been made for developing approaches that can

integrate subjective and partial information, being

the most successful ones Bayesian and maximum

entropy methods and more recently, fuzzy set theory

where the concept of objects that have not precise

boundaries was introduced (Zadeh, 1965). Fuzzy

logic provides a meaningful and powerful

representation of uncertainties and is able to express

efficiently the vague concepts of natural language

(Zadeh, 1965). These characteristics could make it a

very powerful and efficient tool for policy makers

due to the fact that the models are based on

linguistic rules that could be easily understood.

In this paper two fuzzy logic models are

proposed for the global temperature changes (in the

year 2100) that are expected to occur in this century.

The first model incorporates the uncertainties related

to the wide range of emission scenarios and

illustrates in a simple manner the importance of the

emissions in determining future temperatures. The

second incorporates the uncertainty due to climate

sensitivity that pretends to emulate the diversity in

modelling approaches. Both models are built using

the Magicc (Wigley, 2008) model and Zadeh´s

extension principle for functions where the

independent variable belongs to a fuzzy set. Magicc

is capable of emulating the behaviour of complex

GCMs using a relative simple one dimensional

model that incorporates different processes e.g.

carbon cycle, earth-ocean diffusivity, multiple gases

and climate sensitivity. In our second case we intend

to illustrate the combined effects of two sources of

uncertainty: emissions and model sensitivity. It is

clear that we are leaving out of this paper other

important sources of uncertainty whose contribution

would be interesting to explore. The GCMs are,

from our point of view, useful and very valuable

tools when it is intended to study specific aspects or

details of the global temperature change.

Nevertheless, when the goal is to study and to test

global warming policies, simpler models much

easier to understand become very attractive. Fuzzy

models can perform this task very efficiently.

2 FUZZY LOGIC MODEL

OBTAINED FROM IPCC DATA

The Fourth Assessment Report of the IPCC shows

estimates of emissions, concentrations, forcing and

temperatures through 2100 (IPCC, 2007). Although

there are relationships among these variables, as

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519

those reflected in the figure 1 (upper panel), it would

be useful to find a way to relate emissions directly

with increases of temperature. A more physical

relation is established between concentrations and

temperature because the latter depends almost

directly upon the former through the forcing terms.

Concentrations are obtained integrating over time

the emissions minus the sinks of the GHGs. One

way of relating directly emissions and temperature,

could be achieved if the emission trajectories were

linear and non-intersecting as illustrated in figure 1

(upper panel). Here, we perform this task by means

of a fuzzy model, which is based on the Magicc

model (Wigley, 2008) and Zadeh´s extension

principle (see Appendix).

Figure 1: Upper panel: Emissions scenarios CO2,

Illustrative SRES and Linear Pathways. (-2) CO2 means -

2 times the emission (fossil + deforestation) of CO2 of

1990 by 2100 and so for -1, 0, 1, to 5 CO2. All the linear

pathways contain the emission of non CO2 GHG as those

of the A1FI. 4scen20-30 scenario follows the pathway of

4xCO2 but at 2030 all gases drop to 0 emissions or

minimum value in CH4, N2O and SO2 cases. Lower

panel: CO2 Concentrations for linear emission pathways,

4scen20-30 SO2 and A1FI are shown for reference. Data

calculated using Magicc V. 5.3.

Using as input for the Magicc model the

emissions shown in the previous figure we calculate

the resulting concentrations (figure 1 lower panel);

forcings (figure 2 upper panel) and global mean

temperature increments (figure 2 lower panel).

Figure 2: Upper panel: Radiative forcings (all GHG

included) for linear emission pathways and A1FI SRES

illustrative, the 4scen20-30 SO2 only include SO2. Lower

panel: Global temperature increments for linear emission

pathways, 4scen20-30 SO2 and A1FI as calculated using

Magicc V. 5.3.

The set of emissions shown in figure 1 (upper

panel) has been simplified to linear functions of time

that reach by the year 2100 values from minus two

times to 5 times the emissions of 1990. The

trajectories labelled 5CO2 and (-2) CO2 contain the

trajectories of the SRES. We observe that the

concentrations corresponding to the 5CO2 and the

A1FI trajectories, by year 2100 are very close. The

choice of linear pathways allows us to associate

emissions to concentrations to forcings and

temperatures in a very simple manner. We can say

than any trajectory of emissions contained within

two of the linear ones will correspond, at any time

with a temperature that falls within the interval

delimited by the temperatures corresponding to the

linear trajectories. This is illustrated for the A1FI

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trajectory, in figure 2 (lower panel) that falls within

the temperatures of the 5CO2 and 4CO2 trajectories.

We decided to find emission paths that would lead to

temperatures of two degrees or less by the year

2100, this led us to the -2CO2, -1CO2 and 0CO2.

The latter is a trajectory of constant emissions equal

to the emissions in 1990 that gives us a temperature

of two degrees by year 2100.

From the linear representation, it is easily

deduced that very high emissions correspond to very

large concentrations, forcings and large increases of

temperature. It is also possible to say that large

concentrations correspond to large temperature

increases etc. This last statement is very important

because in determining the temperature the climate

models directly use the concentrations which are the

time integral of sources and sinks of the green house

gases (GHG). Therefore the detailed history of the

emissions is lost. Nevertheless the statement, to

large concentrations correspond large temperature

increases still holds. These simple observations

allows us to formulate a fuzzy model, based on

linguistic rules of the IF-THEN form, which can be

used to estimate increases of temperature within

particular uncertainty intervals. Fuzzy logic provides

a meaningful and powerful representation of

measurement of uncertainties, and it is able to

represent efficiently the vague concepts of natural

language, of which the climate science is plagued.

Therefore, it could be a very useful tool for decision

makers. The basic concepts of fuzzy logic are

presented in Appendix.

The first fuzzy model one input one output

defined for the global temperature change is

(quantities between parenthesis were used with

Zadeh’s principle to generate the fuzzy model, the

number 1 means the membership value (μ) of the

input variables used in formulating the fuzzy

model):

1. If (concentration is very low

(about -2CO2)) then (deltaT is very low

(1)

2. If (concentration is low (about -

1CO2)) then (deltaT is low (1)

3. If (concentration is medium-low

(about 0CO2) then (deltaT is medium-low

(1)

4. If (concentration is medium

(about 1CO2)) then (deltaT is medium

(1)

5. If (concentration is medium-high

(about 2CO2)) then (deltaT is medium-

high (1)

6. If (concentration is high (about

3CO2)) then (deltaT is high (1)

7. If (concentration is very high

(about 4CO2)) then (deltaT is very high

(1)

8. If (concentration is extremely

high (about 5CO2)) then (deltaT is

extremely high (1)

The 8 rules for concentration are based on 8 adjacent

triangular membership functions (the simplest form)

corresponding to linear emission trajectories (-2CO2

to 5CO2). The concentrations were obtained from

Magicc model and cover the entire range (210 to

1045 ppmv). The apex of each membership function

(μ=1) corresponds with the base (μ=0) of the

adjacent one, as we show below:

(μ=0, μ=1, μ=0)

1. -2CO2 (210, 213, 300)

2. -1CO2 (213, 300, 401)

3. 0CO2 (300, 401, 513)

4. 1CO2 (401, 513, 633)

5. 2CO2 (513, 633, 762)

6. 3CO2 (633, 762, 899)

7. 4CO2 (762, 899, 1038)

8. 5CO2 (899, 1038, 1045)

The global temperature changes were obtained

through Zadeh’s extension principle applied to data

from Magicc model.

From the point of view of a policy maker, a

fuzzy model as the one represented by the previous

rules is a very useful tool to study the effect of

different policies on the increases of temperature.

The fuzzy rules model can be evaluated by

means of the fuzzy inference process in such a way

that each possible concentration value is mapped

into an increase of temperature value by means of

the Mamdani’s defuzzification process (see

Appendix). The resulting increases of temperature at

year 2100 for each possible concentration (emission

in the case of our linear model) value (solid line) are

shown in the upper panel of figure 3.

The lower panel illustrates the formulation of the

rules by showing the fuzzy set associated with the

different classes of concentrations, the antecedent of

the fuzzy rule, the IF part and the consequent fuzzy

set temperature, the THEN part. The figure 3 lower

panel also illustrates the uncertainties of one

estimation: If the concentration is of 401 ppmv (it

fires rule number 3) within an uncertainty interval of

(300 to 513 ppmv) 4 then the temperature increment

is 1.95 degrees within an uncertainty interval of

(1.23 to 2.63 deg C) in this case the temperatures

will have uncertainties of one or two times the

intervals defined by the expert or the researcher.

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521

Figure 3: Fuzzy model based on linguistic rules and

Zadeh´s principle. Upper panel: increases of temperature

at year 2100 for each possible concentration (emission in

the case of our linear model) value (solid line). Lower

panel: Fuzzy rules associated with the different classes of

concentrations. (Calculated with MATLAB).

The fuzzy model is simpler and obviously less

computationally expensive than the set of GCM’s

reported by the IPCC. The most important benefit,

however, is its usefulness for policy-makers. For

example, if the required increase of temperature

should be very low or low (-2CO2, -1CO2), then the

policy-maker knows, on the basis of this model, that

concentrations should not exceed the class small.

3 A SIMPLE CLIMATE MODEL

AND ITS CORRESPONDING

FUZZY MODEL

Here, we again use the Magicc model but this time

we introduce a second source of uncertainty, the

climate sensitivity. The purpose is to illustrate the

effects of the combination of two sources of

uncertainty on the resulting temperatures. The

climate model is driven by our linear emission paths.

The relationship between concentrations and

sensitivity and increases of temperature at year 2100

is then used to construct a fuzzy model following the

extension principle of the fuzzy logic approach (see

Appendix).

The set of fuzzy rules obtained in this case is the

following.

1. If (concentration is very very

low) and (sensitivity is low) then

(deltaT is low) (1)

2. If (concentration is very low)

and (sensitivity is low) then (deltaT

is low) (1)

3. If (concentration is very low)

and (sensitivity is high) then (deltaT

is med) (1)

4. If (concentration is medium-low)

and (sensitivity is low) then (deltaT

is low) (1)

5. If (concentration is medium-low)

and (sensitivity is high) then (deltaT

is high) (1)

6. If (concentration is medium) and

(sensitivity is low) then (deltaT is

med) (1)

7. If (concentration is medium) and

(sensitivity is high) then (deltaT is

high) (1)

8. If (concentration is medium-high)

and (sensitivity is low) then (deltaT

is med) (1)

9. If (concentration is medium-high)

and (sensitivity is high) then (deltaT

is high) (1)

10. If (concentration is high) and

(sensitivity is low) then (deltaT is

med) (1)

11. If (concentration is high) and

(sensitivity is high) then (deltaT is

high) (1)

12. If (concentration is medium-low)

and (sensitivity is med) then (deltaT

is med) (1)

Note that we have used the same nomenclature as

before and the very high and extremely high

concentrations are not considered.

And the fuzzy sets for the temperature and

sensitivity are shown in figure 4. We used this figure

to build the rule above in combination with 6 fuzzy

sets for concentration similar to those from our first

model described in section 2:

(μ=0, μ=1, μ=0)

1. -2CO2 (100, 213, 300)

2. -1CO2 (213, 300, 401)

3. 0CO2 (300, 401, 513)

4. 1CO2 (401, 513, 633)

5. 2CO2 (513, 633, 762)

6. 3CO2 (633, 762, 899)

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For sensitivity we built 3 triangular fuzzy sets

corresponding to sensitivity values of 1.5, 3 and 6

deg C/W/m2, showed below:

(μ=0, μ=1, μ=0)

1. 1.5 (low) (1.5, 1.5, 3.0)

2. 3.0 (medium) (1.5, 3.0, 6.0)

3. 4.5 (high) (3.0, 6.0, 6.0)

Similarly, for global temperature change we have 3

triangular fuzzy sets built with data obtained from

Magicc model and Zadeh’s extension principle for

each sensitivity value; the apex of each fuzzy set is

the value of global temperature change for the 0CO2

linear emission path according to the value of

sensitivity, the base of the fuzzy sets range from -

2CO2 to 3CO2 (assuming global temperature

changes below 6 deg C) for each sensitivity value

(see figure 4):

( μ=0, μ=1, μ=0)

1. Low (0.07, 1.07, 2.13)

2. Medium (0.36, 1.98, 3.70)

3. High (0.92, 3.27, 5.75)

Figure 4: ΔT Global and sensitivity fuzzy sets for six

linear emission pathways at 2100. The dashed lines show

the membership functions.

The Mamdani’s fuzzy inference method is used

also here as the defuzzification method to compute

the increase of temperature values. The results are

shown in figure 5. The upper panel of figure 5 shows

the surface resulting from the defuzzification

process. The lower panel illustrates again that for the

case of a concentration of 401 ppmv and a

sensitivity of 3 (medium sensitivity) the temperature

is about 2 degrees within an uncertainty interval of

(0.36 to 3.70 deg C) where the membership value is

different from 0. When we compare our previous

result with this one we find that the answers are very

close in fact the fall within the uncertainty intervals

of both. The uncertainty of concentrations and

sensitivity are respectively (300 to 513 ppmv) and

(1.5 to 6 deg C/W/m2). The result is to be expected

since in our first experiment we used the Magicc

model with default value for the sensitivity and this

turns to be of 3.

Figure 5: Fuzzy model for concentrations and sensitivities.

Upper panel: ΔT surface. Lower panel: Fuzzy rules.

(Calculated with MATLAB).

4 DISCUSSION AND

CONCLUSIONS

In this work, simple linguistic fuzzy rules relating

concentrations and increases of temperatures are

extracted from the application of the Magicc model.

The fuzzy model uses concentration values of GHG

as input variable and gives, as output, the increase of

temperature projected at year 2100. A second fuzzy

model based on linguistic rules is developed based

on the same Magicc climate model introducing a

second source of uncertainty coming from the

different sensitivities used by the Magicc to emulate

more complicated GCMs used in the IPCC reports.

These kind of fuzzy models are very useful due to

their simplicity and to the fact that include the

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523

uncertainties associated to the input and output

variables. Simple models that, however, could

contain all the information that is necessary for

policy makers, these characteristics of the fuzzy

models allow not only the understanding of the

problem but also the discussion of the possible

options available to them. For example going back

to the question of stabilizing global temperatures at

about 2 degrees or less, we can see the fuzziness of

the proposition; we could estate that we should stay

well below 400 ppmv by year 2100. The observed

emission pictured in figure 6 where the IPCC

scenarios are also shown are contained within A1F1

and the A1B therefore we could say that they point

to a temperature increase that will surpass the two

degrees. In fact to keep temperatures under 2

degrees we have already stated we should remain

under 400 ppmv and we are very very close (fuzzy

concept) to this concentration.

Figure 6: Observed CO2 emissions against IPCC AR4

scenarios (taken form http://www.treehugger.com/clean-

technology/iea-co2-emissions-update-2010-bad-news-

very-bad-news.html).

ACKNOWLEDGEMENTS

This work was supported by the Programa de

Investigación en Cambio Climático (PINCC,

www.pincc.unam.mx) of the Universidad Nacional

Autónoma de México.

REFERENCES

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Gay, C., Estrada, F. 2009. Objective probabilities about

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4].

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Science Basis. Contribution of Working Group I to the

Fourth Assessment Report of the Intergovernmental

Panel on Climate Change [Solomon, S., D. Qin, M.

Manning, Z. Chen, M. Marquis, K. B. Averyt, M.

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NY, USA, 996 pp.

IPCC-WGI, 2001: Climate Change 2001: The Scientific

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Assessment Report of the Intergovernmental Panel on

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D.J.; Noguer, M.; van der Linden, P.J.; Dai, X.;

Maskell, K.; and Johnson, C.A., ed.] Cambridge

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01495-6).

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Mechanics: Phys. Rev., 106, 620-630.

Klir, G. and Elias, D. 2002. Architecture of Systems

Problem Solving, 2nd ed., NY: Plenum Press.

Nakicenovic, N., J. Alcamo, G. Davis, B. de Vries, J.

Fenhann, S. Gaffin, K. Gregory, A. Grübler, T. Y.

Jung, T. Kram, E. L. La Rovere, L. Michaelis, S.

Mori, T. Morita, W. Pepper, H. Pitcher, L. Price, K.

Riahi, A. Roehrl, H.-H. Rogner, A. Sankovski, M.

Schlesinger, P. Shukla, S. Smith, R. Swart, S. van

Rooijen, N. Victor, Z. Dadi, 2000. Special Report on

Emissions Scenarios: A Special Report of Working

Group III of the Intergovernmental Panel on Climate

Change. Cambridge University Press, Cambridge, 599

pp.

Ross, T. J. 2004. Fuzzy Logic with Engineering

Applications (2nd ed.) John Wiley & Sons.

Schneider, S. H., 2003. Congressional Testimony: U.S.

Senate Committee on Commerce, Science and

Transportation, Hearing on “The Case for Climate

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Schmidt, G. A. 2007. The physics of climate modeling,

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Vol. 8(3) p. 338-353.

APPENDIX

A.1 Fuzzy Logic Basic Concepts

As Klir stated in his book (Klir and Elias, 2002), the

view of the concept of uncertainty has been changed

in science over the years. The traditional view looks

to uncertainty as undesirable in science and should

be avoided by all possible means. The modern view

is tolerant of uncertainty and considers that science

should deal with it because it is part of the real

world. This is especially relevant when the goal is to

construct models. In this case, allowing more

uncertainty tends to reduce complexity and increase

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credibility of the resulting model. The recognition

by the researchers of the important role of

uncertainty mainly occurs with the first publication

of the fuzzy set theory, where the concept of objects

that have not precise boundaries (fuzzy sets) is

introduced (Zadeh, 1965).

Fuzzy logic, based on fuzzy sets, is a superset of

conventional two-valued logic that has been

extended to handle the concept of partial truth, i.e.

truth values between completely true and completely

false.

In classical set theory, when A is a set and x is an

object, the proposition “x is a member of A” is

necessarily true or false, as stated on equation 1,

⎩

⎨

⎧

∉

∈

Axfor

)(

(1)

whereas, in fuzzy set theory, the same proposition is

not necessarily either true or false, it may be true

only to some degree. In this case, the restriction of

classical set theory is relaxed allowing different

degrees of membership for the above proposition,

represented by real numbers in the closed interval

[0,1], i.e.

[]

1,0: →XA

. Figure A.1 presents this

concept graphically.

Figure A.1: Gaussian membership functions of a

quantitative variable representing ambient temperature.

Figure A.1 illustrates the membership functions

of the classes: cold, fresh, normal, warm, and hot, of

the ambient temperature variable. A temperature of

23°C is a member of the class normal with a grade

of 0.89 and a member of the class warm with a grade

of 0.05. The definition of the membership functions

may change with regard to who define them. For

example, the class normal for ambient temperature

variable in Mexico City can be defined as it is

shown in figure A.1. The same class in Anchorage,

however, will be defined more likely in the range

from -8°C to -2°C. It is important to understand that

the membership functions are not probability

functions but subjective measures. The opportunity

that brings fuzzy logic to represent sets as degrees of

membership has a broad utility. On the one hand, it

provides a meaningful and powerful representation

of measurement uncertainties, and, on the other

hand, it is able to represent efficiently the vague

concepts of natural language. Going back to the

example of figure A.1, it is more common and useful

for people to know that tomorrow will be hot than to

know the exact temperature grade.

At this point, the question is, once we have the

variables of the system that we want to study

described in terms of fuzzy sets, what can we do

with them? The membership functions are the basis

of the fuzzy inference concept. The compositional

rule of inference is the tool used in fuzzy logic to

perform approximate reasoning. Approximate

reasoning is a process by which an imprecise

conclusion is deduced from a collection of imprecise

premises using fuzzy sets theory as the main tool.

The compositional rule of inference translates the

modus ponens of the classical logic to fuzzy logic.

The generalized modus ponens is expressed by:

Rule: If X is A then Y is B

Fact: X is A'

Conclusion: Y is B'

where, X and Y are variables that take values from

the sets X and Y, respectively, and A, A' and B, B'

are fuzzy sets on X and Y, respectively. Notice that

the Rule expresses a fuzzy relation, R, on X x Y.

Then, if the fuzzy relation, R, and the fuzzy set

A' are given, it is possible to obtain B' by the

compositional rule of inference, given in equation 2,

[

]

),(),('minsup)(' yxRxAyB

Xx∈

(2)

where sup stands for supremum (least upper bound)

and min stands for minimum. When sets X and Y

are finite, sup is replaced by the maximum operator,

max. Figure A.2 illustrates in a simplified way the

compositional rule of inference graphically.

Figure A.2: Simplified graphical representation of the

compositional rule of inference.

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The compositional rule of inference is also useful

in the general case where a set of rules, instead of

only one, define the fuzzy relation, R.

A.2 Extension Principle

Zadeh says that rather than regarding fuzzy theory as

a single theory, we should regard the conversion

process from binary to membership functions as a

methodology to generalize any specific theory from

a crisp (discrete) to a continuous (fuzzy) form. The

extension principle enables us to extend the domain

of a function on fuzzy sets, i.e., it allows us to

determine the fuzziness in the output given that the

input variables are already fuzzy. Therefore, it is a

particular case of the compositional rule of

inference. Figure A.3 gives a first idea of the

extension principle showing an example of two input

variables with 3 fuzzy sets each.

Figure A.3: Extension principle example for two input

fuzzy variables A and B with 3 fuzzy sets each.

The extension principle is applied to transform

each fuzzy pair (A

, B

), in a fuzzy set of the C

output variable. Notice that in the example of figure

A.3 we have 9 pairs of fuzzy input sets and,

therefore, 9 fuzzy sets are obtained representing the

conclusion as shown in the right hand side of figure

A.3. The extension principle when two input

variables are available is presented in equation 3. C

is the k

output fuzzy set extended from the two

input fuzzy sets A

and B

. In the example at hand, as

illustrated in figure A.3, the extension principle is

applied 9 times, to obtain each of the output fuzzy

sets associated to each fuzzy input pair.

(, )

max min ,

kij

CfAB

CAB

⎡⎤

⎣⎦

(3)

For instance, the output fuzzy set C

, is obtained

when using the extension principle of equation 3

with the input fuzzy sets A

and B

(Klir and Elias,

2002); (Dubois and Prade, 1980); (Ross, 2004).

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