3.1. Introduction to fuzzy inference

Fuzzy logic is arguably the best way of reasoning with vague, incomplete and often conflicting information and offers a sound basis for developing and integrating complex geographical models. Fuzzy logic based modelling is like more conventional mathematical and statistical modelling in that it operates by manipulating numerical data, but it is easier to think of it on a higher level as computing with words embedded in natural language statements. Language is inherently fuzzy because both words and statements are vague or imprecise, yet communicating using language allows humans to process vast amounts of information quickly and efficiently. Fuzzy logic can be used to help a computer cope with both vague and relative information by building both its vocabulary in a sensible way by defining words as fuzzy sets.

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.

3.2. Model 1

3.3. Model 2

3.3. General comments and ideas for improvements