- 3.1. Introduction to
fuzzy inference

3.2. Model 1

3.3. Model 2

3.3. General comments and ideas for improvements

Fuzzy set theory is synonymous to fuzzy logic and was formalised by Lofti Zadeh in 1965 and heralded as a new mathmatical paradigm. The philosophy behind the theory has been bitterly contested throughout history, especially by Aristotelian philosophers who view logical propositions about the world as either true or false and never both. In recent times the study of paradox revitalised interest in alternative philosophies and despite the somewhat enevitable opposition, fuzzy logic championed. The reason it became a success is usually credited to the engineers who applied it to complex control problems in engineering outperforming more conventional methods in practice with ease. Fuzzy logic based modelling or fuzzy inference is now recognised and applied in a whole variety of disciplins for a whole variety of different kinds of complex control and modelling tasks.

Fuzzy sets form the basis for fuzzy logic based reasoning or fuzzy inference.
They can be thought of as a multi-valued version of classical or crisp
sets which define an infinite range of partial truths between true and
false. Fuzzy set theory and fuzzy sets are analogous to classical set theory
and crisp sets, the technical difference is that the fuzzy approach disregards
'the law of the excluded middle'. Its ignorance of that one law is like
that which defines the difference between Euclidean and non-Euclidean geometry.
All forms of logic represent and infer knowledge via conditional propositions.
In classical logic, these propositions must be either true or false there
is no inbetween. In fuzzy logic there is a continuous range of truth values,
for example, a proposition could be nearly always true meaning that it
is also false to a degree although it may be a very small amount. Multi-valued,
vague or fuzzy logic is more similar to the reasoning mechanism behind
human thought processes. It is by thinking in a fuzzy way that us humans
are able to summarise massive amounts of vague information in order to
estimate or decide what might be best to do. When a crisp yes/no decision
or true/false answer is required fuzziness is removed via a defuzzification
process which aggregates a whole range of fuzzy values into a bi-valued
domain and comparing with an average threshold an extreme is selected.s
nd compares tll the results our thinking usually by aggregating selecting
the most probable taskingwhat from complex patterns. and vague , does not
appear to be

based on classical logic, but rather on a fuzzy or multi-valued logic
with a continuous spectrum of partial truth values

and corresponding rules of fuzzy inference.

nference based on fuzzy logic can be used to incorporate existing expert
understanding of general knowledge to develop a model geographical processes.
a powerful expression of uncertainty and is

Fuzzy set membership functions are the building blocks of a fuzzy model.
They are used to define the model variables,

which are based upon imprecise and abstract concepts, for example,
a short or long distance. The shape and form of the

membership functions does not matter too much so long as the different
classes overlap sufficiently.

There are a number of fuzzy set operators one of which is the fuzzy
hedge. Fuzzy hedges extend the modelling

vocabulary to include additional everyday words such as approximately,
mostly and very. The application of a hedge

operator acts to modify the meaning of a fuzzy set similar to the way
in which adjectives and adverbs modify the

meaning of nouns and verbs in a natural language. There are four main
types of fuzzy hedge operator: concentration,

dilation, intensification and diffusion. A concentration operator reduces
the degree of membership of each of the

elements in the set by an amount relative to its present degree. The
dilation operator is the opposite. Intensification acts

as a combination of concentration and dilation by increasing the degree
of membership of those elements in the set

with high membership values and decreasing those with low membership
values. This has the effect of making the

boundaries of the membership function steeper and moves the membership
values closer to 1 and 0, thereby reducing

overall fuzziness.

Hedges also take precedence over the logical operators And, Or and Not
unless parentheses are used to indicate

otherwise. As with membership functions, customised hedges can be developed
that may provide better representations

of natural language modifiers than the standard operators. Although
fuzzy hedges are not widely used in fuzzy control,

they have the potential to provide added flexibility in terms of a
larger linguistic modelling vocabulary, which renders it

possible to incorporate additional subjectivity into the meaning of
a linguistic term. Cox (1994) has used hedges

successfully in several commercial applications, and they may prove
to be particularly relevant to future research

concerned with qualitative hypothesis testing.

Fuzzy variables are analogous to numerical variables in a conventional
model in terms of the role they play, but they

differ completely in their respective definitions. Fuzzy variables
consist of a series of overlapping membership

functions, where each individual function defines a semantic label
that describes a different state of the variable. The

overlapping between membership functions ensures that a gradual transition
exists when moving between variable

states. The image below provides a hypothetical definition for the
fuzzy variable distance, where each of the

membership functions is associated with a different linguistic label.
Fuzzy variables are sometimes called linguistic

variables since they provide a systematic way of capturing vague and
complex linguistic or expert knowledge into a

format that is computable.

There are several different ways of deciding where and how many membership
functions to place on the variable

domain. The most important requirement is that they overlap, but the
optimal number will be determined by

experimentation and the complexity of the problem.

To represent imprecise and abstract knowledge, fuzzy variables are embedded
into linguistic statements, propositions or

rules of the form, If x is A Then y is B, where x and y are fuzzy variables
and A and B are fuzzy set membership

functions. The rule expresses an inference, the If part of the rule
is also known as a premise or antecedent, while the

Then part is called a conclusion or a consequent. A simple example
of a linguistic rule relating to land degradation risk

might be; If rainfall decreases Then land degradation risk is high.
To make inferences of this type and hence process the

knowledge expressed by this rule, a formal method of reasoning, known
as approximate or fuzzy reasoning is used.

Fuzzy reasoning is based upon the compositional rule of inference.
Armed with these rules of reasoning, a fuzzy model

can be used to manipulate information represented as a set of fuzzy
If-Then rules and process the knowledge.

The models that we are create in an attempt to represent reality essentially
perform a basic function; they map inputs,

such as predicted land-use, to a outputs, such as land degradation
risk. In mathematical modelling, this mapping is

performed by a series of equations, where the formulae directly represent
the behaviour of the system, such as the

dynamics of hydrological flow. In a generic fuzzy model, the mapping
process between inputs and outputs consists of

three main components: (1) a fuzzy rulebase, which is a collection
of fuzzy variables embedded in a set of fuzzy

If-Then rules; (2) a database of model-specific vocabulary, which defines
the membership functions used in the fuzzy

rules; and (3) a fuzzy logic inference engine that drives the model
by execution of the rulebase in response to a set of

system inputs.