Download Fuzzy-Mapping-Rules

Fuzzy If-Then Rules
Adnan Yazıcı
Dept. of Computer Engineering,
Middle East Technical University
Ankara/Turkey
Fuzzy If-Then Rules
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There are two different kinds of fuzzy rules: Fuzzy
mapping rules and Fuzzy implication rules.
A fuzzy mapping rule describes an association;
therefore, its fuzzy relation is constructed from the
Cartesian product of its antecedent fuzzy condition
and its consequent fuzzy condition.
A fuzzy implication rule, however, describes a
generalized logic implication; therefore, its fuzzy
relation needs to be constructed from the semantics
of a generalization to implication in multi-valued
logic.
Fuzzy If-Then Rules
The difference between the semantics of fuzzy mapping rules
and fuzzy implication rules can be seen from the difference in
their inference behavior. Even though these two types of rules
behave the same when their antecedents are satisfied, they
behave differently when their antecedents are not satisfied.
Example:
Implication rule,
Mapping rule
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(procedural representation)
Given:x ∈ [1,3] Æ y ∈ [7,8], Stm: If x∈[1,3], Then y∈[7,8]
Input: x=5
Variable value: x = 5
Infer: y is unkown (y ∈ [0,10]) Execution result: no action
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Fuzzy Mapping Rules
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The needs to approximate a function of interest is often due
to one or more of the following reasons:
1) The mathematical structure of the function is not
precisely known at al.
2) The function is so complex that finding its precise
mathematical form is either impossible or practically
infeasible due to its high cost.
3) Even if finding the function is not impractical,
implementing the function in its precise mathematical
form in a product or service may be too costly. This is
particularly important for low cost high volume products
(e.g., automobiles, cameras, and many other consumer
products).
Fuzzy Mapping Rules
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Fuzzy rule-based function approximation is a
partition-based technique.
The
partition-based
approximation
techniques approximate a function by
partitioning the input space of the function
and approximate the function in each
partitioned region separately (e.g., piecewise
linear approximation).
Fuzzy Mapping Rules
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Because each fuzzy rule approximates a small segment of
the function, the entire function is approximated by a set of
fuzzy mapping rules.
We refer to such a collection of fuzzy mapping rules as
fuzzy rule-based models or simply fuzzy models
A fuzzy model describes an (approximate) mapping (i.e.,
function) from a set of input variables to a set of output
variables.
Examples:
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A fuzzy model of the stock market can be used to predict future
changes of the IMKB average.
A fuzzy control model of a petrochemical process can be used to
predict the future state of the process.
Fuzzy Mapping Rules
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A fuzzy model can be defined as a model that is
obtained by fusing multiple local models that are
associated with fuzzy subspaces of the given input
space.
The result of fusing multiple local models is
usually a fuzzy conclusion, which is converted to a
crisp final output through a defuzzification process.
The main difference between fuzzy and nonfuzzy
rules for function approximation lies in their
interpolative reasoning capability, which allows the
output of multiple fuzzy rules to be fused for a
given input.
Fuzzy Mapping Rules
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The four major concepts in fuzzy rule-based
models thus are as follows:
1.
2.
3.
4.
Fuzzy partition,
Mapping of fuzzy subregion to local models,
Fusion of multiple local models,
Defuzzification.
1- Fuzzy partition
A fuzzy partition of a space is a collection of
fuzzy subspaces whose boundaries partially
overlap and whose union is the entire space.
¾ Formally, a fuzzy partition of a space as a
collection of fuzzy subspace Ai of S that
satisfies the following condition:
∑ μAi(x) = 1, ∀x ∈ S.
¾ That is, for any element of the space, its
membership degree in all subspaces always
adds up to 1.
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1- Fuzzy partition
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We call a collection of fuzzy subspaces Ai of S a weak fuzzy
partition of S if and only if it satisfies the following
condition:
0< ∑ μAi(x) ≤ 1, ∀x ∈ S.
The “greater than 0” condition requires each element in the
space S to be covered by at least one fuzzy subspace in the
partition.
The “sum to 1” condition of a fuzzy partition can be relaxed
to the “sum to less or equal to 1” condition because the
interpolative reasoning of fuzzy models includes a
normalization step.
Research Note: It has been shown that ∑ μAi(x) = 1 is a
desirable property in a framework for analyzing the stability
of fuzzy logic controllers.
2- Mapping a Fuzzy Subspace to a Local Model
y
large
small
x
small
medium
large
Fuzzy mapping
2- Mapping a Fuzzy Subspace to a Local Model
A local model for a subspace of the entire
input space describes the system’s inputoutput mapping relationship in the small
subspace.
¾ In contrast, a global model for an input space
describes
the
system’s
input-output
relationship for the entire input space.
¾ Because the scope of the local model is
smaller than that of a global model, it is
usually easier to develop a local model.
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Mapping a Fuzzy Subspace to a Local Model
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In particular, a nonlinear global model (i.e., whose inputoutput mapping function is not linear) can often be
approximated by a set of linear local models. This can be
understood by remembering the well-known approximation
technique called piecewise linear approximation, which
approximates an arbitrary nonlinear function using segments
of lines.
The following figure shows such an approximation
technique, where dotted line indicates the function being
y
approximated.
x
Mapping a Fuzzy Subspace to a Local Model
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Piecewise linear approximation has two major components:
1.
Partitioning the input space to crisp regions
2.
Mapping each partitioned region to a linear local model.
The main difference between fuzzy modeling and piecewise
linear approximation is that the transition from one local
subregion to a neighboring one is gradual rather than abrupt.
Generally, the mapping from a fuzzy subspace to a local model
is represented as a fuzzy if-then rule in the form of:
If ⎯x is in FSi Then yj = LMi (x)
where ⎯x and yj denote the vector of input variables and output
variable, respectively, FSi and LMi denote ith fuzzy subspace
and the corresponding local model, respectively.
Mapping a Fuzzy Subspace to a Local Model
The local model can be one of four different types:
¾ 1. Crisp constant: This type of local model is simply a crisp
(nonvisual) constant. For example;
If xi is Small Then y = 4.5
¾ 2. Fuzzy constant: A local model that is a fuzzy constant (e.g.,
Small) belong to this type. For example;
If xi is Small Then y is Medium
¾ 3. Linear Model: this describes the output as a linear function of
the input variables, such as:
If x1 is Small And x2 is Large Then y = 2x1 + 5x2 + 3.
¾ 4. Non-Linear Model: Theoretically, a local model can be more
complex than a linear model. In practice, however, there is
rarely such a need. These models have been introduced in a
hybrid neuro-fuzzy system that uses neural networks to
represent nonlinear local models associated with the rule.
Fusion of local models through interpolative reasoning
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Fuzzy models use interpolative reasoning to fuse
multiple local models into a global model.
The basic idea behind interpolative reasoning is
analogous to drawing a conclusion from a panel of
experts, each of whom is specialized in a subarea of
the entire problem.
Each expert’s opinion is associated with a weight,
which reflects the degree to which the current
situation is in the expert’s specialized area.
These weighted opinions are combined to form an
overall opinion.
Fusion of local models through interpolative reasoning
In this analogy, an expert corresponds to a
fuzzy if-then rule, the specialized subarea of
the expert corresponds to the fuzzy subspace
associated with the if-part of the rule.
¾ The weight of an expert’s opinion is
determined by the degree to which the current
situation belongs to the subspace.
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Defuzzification
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We may interpret a possibility distribution either
through linguistic approximation, or through
defuzzification.
The former gives a qualitative interpretation, while the
latter gives a quantitative summary and is more
commonly used in fuzzy logic applications, i.e.,
industrial applications.
Given a possibility distribution of a fuzzy model’s
output, defuzzification amounts to selecting a single
representative value that captures the essential
meaning of the given distribution.
Defuzzification
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There are three common defuzzification techniques: mean of
maximum, center of area, and height.
Mean of Maximum (MOM): This calculates the average of
those output values that have the highest possibility degrees.
Suppose “y is A” is a fuzzy conclusion to be fuzzified. We can
express the MOM defuzzification method using the following
formula:
MOM (A) = ∑y*∈P y* / |P|
Where P is the set of output values y* with highest possibility
degree in A.
If P is an interval, the result of MOM defuzzification is
obviously the midpoint in that interval.
This technique does not take into account the overall shape of
the possibility distribution.
Defuzzification
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Center of Area (COA): This method (also referred to as the
center-of-gravity, or centroid method) is the most popular
defuzzification technique.
Unlike MOM, the COA method takes into account the entire
possibility distribution in calculating its representative point.
This method is similar to the formula for calculating the center of
gravity in physics, if we view μA(x) as the density of mass at x.
If x is discrete, the fuzzification result of A is:
COA(A) = ∑x μA(x) * x / ∑x μA(x).
The main disadvantage of the COA method is its high
computational cost. However, the calculation can be simplified
for some fuzzy models.
Defuzzification
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The Height Method: This method can be viewed as a two step
procedure.
First we convert the consequent membership function Ci into
crisp consequent y = ci where ci is the center of gravity of Ci.
The centroid defuzzification is then applied to the rules with
crisp consequents with the following formula:
y = ∑Mi=1 wici / ∑Mi=1 wi
where wi is the degree to which ith rule matches the input data.
This method reduces the computation cost and facilitates the
application of neural networks learning to fuzzy systems;
hence, many well-known neuro-fuzzy models use this type of
defuzzification method.
The main disadvantage of this method is that it is not well
justified and is often considered an approximation to the
centroid defuzzification.
An example for a Fuzzy Model
Types of Fuzzy Rule-Based Models
short
long
Types of Fuzzy Rule-Based Models
low
high
Types of Fuzzy Rule-Based Models
maintain speed
increase speed
decrease speed
Types of Fuzzy Rule-Based Models
Types of Fuzzy Rule-Based Models
Types of Fuzzy Rule-Based Models
Types of Fuzzy Rule-Based Models
Types of Fuzzy Rule-Based Models
Rudimentary Flow Mixing Controller
R1:
IF the target temperature T is Low
THEN set the voltage to V (i.e., turn on the cold flow).
R2:
IF the target temperature T is High
THEN set the voltage to V’ (i.e., turn on the hot flow).
Membership functions of the taget temperature are;
μ
μ
High
1
1
Low
T
T
Rudimentary Flow Mixing Controller