fuzzy0

fuzzy systems theory and applications

2007-08-03T11:14:00.000-07:00

"Fuzzy Systems Theory and Its Applications" , by , Terano, Asai, Sugeno, 1992 Academic Press, 1987; 266 pages

This is a thorough introduction to the field of fuzzy theory, covering both theory and applications

Chapter 1 - Outline
17 page comprehensive, high level top down description of the range and type of fuzzy systems applications

1.1 The How, What and Why of Conversion to Fuzzy Systems
These different areas are organized according to
- machine systems - concerned with artificial intelligence
- human systems- application of scientific methods to human problems
- human/machine systems - involve interaction with both man and machine
Table 1.1 surveys different areas of application

1.2 The Concept of Fuzzy Theory
The concept of Fuzzy Theory addresses the epistemology or theory of knowledge, as this section explains.

1.3 Applications Now and the Future Outlook
Applications are surveyed now and in the future; first in Japan, where fuzzy logic took hold, and elsewhere

1.3.1 Overview
Table 1.2 creates a cross-section classification of the different areas where fuzzy logic is or can be applied

1.3.2 Organization of Applications Research
Table 1.3 Extensive survey of current research and future applications

1.3.3 Current Japanese research and future outlook
Table 1.4 outlines current areas of interest

1.3.4 Current Overseas research and future outlook
Different branches of fuzzy logic are noted, in particular fuzzy systems control, first used in Denmark

Chapter 2 - Basics of Fuzzy Theory
This chapter explains the fundamentals required for the remainder of the book

2.1 Quantification of Ambiguity
Most natural language contains ambiguity and multiplicity of meaning
A means of quantifying meaning is needed.

2.2 Fuzzy Sets
Definitions: fuzzy set, membership function a mapping from crisp set to the [0,1] interval

2.3 Crisp Sets
basic ops: union, intersection, complement
basic nomenclature: ~ , ^ , V
basic sets: X:support set; A,B:subset
correspondence: union -> V
correspondence: intersection -> ^

equivalence relation: equal fuzzy sets <-> equal membership functions
equivalence relation: fuzzy set inclusion relation <-> membership function inequality relation

double negation law
De Morgan's law

2.4 Operations with Fuzzy Sets
Excellent union example, "ah-hah" moment

algebraic sum
algebraic product
bounded sum
bounded difference
lambda complement, degree of complementation
integral representation of union, intersection, complement
normal fuzzy set: grade of 1, max membership = 1
convex fuzzy set: membership function inequality
fuzzy set direct product

2.5 Alpha-Cuts and the Extension Principle
This is an important section that requires a close read ...

These important results depend on the a-cuts:
- Resolution Principle
- Extension Principle
- Representation Theorem

weak vs strong a-cuts

The Resolution principle enables the membership function to be expressed as an a-cut.
The Resolution principle enables a fuzzy set to be decomposed as a (union) of a-cuts without resorting to membership functions.

The extension principle enables the concept of an inverse image function.
Example provided

Fuzzy number defined: for a normal, convex fuzzy set, if a weak a-cut is a closed interval, then it is a Fuzzy Number.
Example provided

2.6 Operations with Fuzzy Numbers
Operations with Fuzzy Numbers are an application of the extension principle.
Detailed example provided

2.7 Fuzzy Propositions
Fuzzy Propositions are propositions that include also fuzzy predicates like, "x is small", or, x is A
A is a fuzzy predicate, a fuzzy variable, also known as a linguistic variable

Chapter 3 - Fuzzy Relations
This chapter explains *how* fuzzy reasoning plays a role in fuzzy control and diagnosis - in terms of compositions and fuzzy relations.

3.1 Fuzzy Relations
Fuzzy Relations are an extension of conventional relations.
Example: "x and y are almost equal"
Fuzzy Relations are widely used, in clustering, pattern recognition, inference systems, process control.

3.1.1 Fuzzy Relations
A fuzzy relation is a mapping from an n-ary direct product, X x Y x ... Z, to the real unit interval [0,1]
For n=1, this is simply a fuzzy set
For n=2. an example is provided
Fuzzy relations can also be represented as a matrix or a graph

3.1.2 Fuzzy Matrices and Fuzzy Graphs
Fuzzy matrix elements are the partial characteristic functions
Fuzzy graph vertices are the elements of the fuzzy set; arcs are the grades
Example provided.

3.2 Operations for Fuzzy Relations
Fuzzy Relations from X to Y are first class fuzzy sets in X x Y.
Therefore, all the usual fuzzy logic rules and operations follow: inclusion, union, intersection, complement.

3.2.1 Composition of Fuzzy Relations
In addition, if R is a fuzzy relation in X x Y, and, S is a fuzzy relation in Y x Z, the composition of R and S is defined as
R o S <-> characteristic function uRoS(x,y) = V { uR(x,y) ^ uS(y,z) }

This is known as max-min composition, where, as usual, V=max and ^=min
Other kinds of relations are possible, e.g., min-max composition, or max-product composition

A matrix example provided.

3.2.2 Converse Relations
Converse fuzzy relations, the Identity relation, the Zero relation, are all defined in terms of characteristic functions.

3.3 Basic Properties of Fuzzy Relations
Table 3.1 provides a concise summary of 13 different types of fuzzy relations

3.4 Fuzzy Relations and Fuzzy Reasoning
This 8-page section goes into the meat of fuzzy reasoning, that is, approximate reasoning.
Fuzzy/approximate reasoning utilizes the concepts developed in the previous sections.

The first fuzzy reasoning example goes like this
Premise 1 "if-then" statement
Premise 2 assignment statement
Conclusion assignment statement

Next, these linguistic statements are converted into fuzzy relational statements
This is the analytical part of constructing a fuzzy model

Definition. "modus ponens" is a common rule of inference, and takes the following form:
If P, then Q.
P.
Therefore, Q.

Table 3.2 lists seven logical implication rules frequently used in fuzzy reasoning

In other words, linguistic logical assertions can be translated into fuzzy relational statements.

Table 3.2 is entitled, "Results of Reasoning"

On page 51, a form of fuzzy reasoning commonly used in fuzzy control is provided.
The compositional parts of the fuzzy conditions are tied together with the "and" of the two fuzzy propositions:

Premise 1 If x is A and y is B then z is C
Premise 2 x is A' and y is B'
Conclusion z is C'

Fuzzy propositions and conditions are converted into fuzzy relations.
For example, "If x is A and y is B then z is C" in premise 1 is replaced by the fuzzy relation R(A,B;C) in direct product UxVxW.

The discussion gets verbose and concludes with two examples
These types of fuzzy reasoning are frequently used in fuzzy control and fuzzy diagnosis

3.5 Fuzzy Relational Equations

This section requires a careful read.

Evidently, fuzzy relational equations play an important role in planning fuzzy controllers. and systems analysis.

A o R = B

A: is a fuzzy set in X, A can be viewed as the fuzzy input
R: is a fuzzy relation in X x Y,
B: can be viewed as the fuzzy output

This is further developed as a fuzzy systems control language

3.6 Various Types of Fuzzy Relations
By adding restrictions to fuzzy relations, we can obtain various types of fuzzy relations. Examples of these are "similarity relations" and "fuzzy order relations". First, the basic properties of fuzzy relations are further developed.

For any x,y,z in X, and fuzzy relation R in X x X:
-1- reflexive: u(x,x) = 1
-2- non-reflexive: u(x,x) = 0 (?)
-3- symmetric: u(x,y) = u(y,x)
-4- anti-symmetric: u(x,y) > 0, u(y,x) > 0 -> x=y, OR, x != y, u(x,y) > 0 -> u(y,x) = 0
-5- transitive: V { u(x,y) ^ u(y,x) } <= u(x,z) These in turn can be expressed in terms of fuzy relations. Additional examples and properties provided.

3.7 - 7 pages - Similarity Relations and Fuzzy Order Relations
Similarity relations are extensions of ordinary equivalence relations.
Similarity classes and partition trees are derived from Similarity relations.
Fuzzy order relations are standard order relations.
Fuzzy partial order relations and linear orderings are derived from Fuzzy order relations.

3.7.1 Similarity Relations
A similarity relation is a fuzzy relation that meets three conditions:
-1- reflexivity
-2- symmetry
-3- transitivity

See figure 3.6 for a description of a "partition tree".

Similarity relations are also known as fuzzy equivalence relations

3.7.2 Similarity Classes
The concept of finding an equivalence class by means of an equivalence relation can be applied to similarity relations.
Example provided.

3.7.3 Resemblance Relations
The fuzzy relation "look alike" is reflexive and symmetrical but NOT transitive, so it is NOT a similarity relation ... but it is still interesting.

3.7.4 Fuzzy Partial Order Relations
An anti-symmetric fuzzy relation satisfies these 3 properties
-1- reflexivity
-2- anti-symmetry
-3- transitivity

An example and more properties are provided
The example illustrates a fuzzy partial order relation expressed as a Hasse diagram.

3.7.5 Dominating Class and Dominated Class
The remaining sections of Chapter 3 are brief, some are just a sentence or two.

3.7.6 Fuzzy Upper and Lower Bounds
Aparently an extension of conventional Upper and Lower Bounds

3.7.7 Least Upper Bounds and Greatest Lower Bounds
Aparently an extension of conventional LUBs and GLBs

3.7.8 Special Fuzzy Order Relations
Fuzzy linear ordering ... and fuzzy preorder relations.

Chapter 4 - Fuzzy Regression Models
With standard regression models, the difference between the data and the error inferred value obtained from the model is taken to be observational error, but with fuzzy regression models, it is assumed that the gap between the data and the model is an ambiguity in the structure of the system that gives the input and output.

4.1 Linear possibility systems
One explanation for fuzzy sets is that the membership function can be seen as a possibility distribution.
Because the regression coefficients, expressed as fuzzy numbers, give the possibilities for the coefficient, the system is called a linear possibility system. This type of model is called a fuzzy regression model.

Definition 4.2. Reference function L(x), defined implicitly: u(x) = L( (x-a)/c )

4.2 Linear possibility regression models
Discussion of the linear regression method now begins.
There are two types of regression data handled: standard data, and data for which the output are fuzzy numbers.

4.2.1 Standard Data
With standard regression models, the difference between the actual data and the inferred values is interpreted as observational error, and the regression analysis is performed by means of the probability model. The linear possibility regression analysis here, follows the possibility model. The uncertainty in the data is seen as originating in the system itself.

4.2.2 Fuzzy Data
With fuzzy data, the goal is to solve a different optimization problem.
This section considers 2 different problems ...

-1- Min Problem
-2- Max Problem

The upper/lower/super/sub script notation utilized in this section leaves something to be desired

4.3 Examples of Applications
Examples are provided, one for each type of data, standard and fuzzy, respectively.

4.3.1 Regression Analysis with Standard Data
Prices of Japanese housing is regressed against 5 variables, including a constant fuzzy number variable

4.3.2 Regression Analysis with Fuzzy Data
In this example, the relationship between the value of the Yen, and the "trading conditions" for businesses is ambiguous in a very complicated way.
Therefore, it can thought to be more appropriate to try and grasp the changes in the value of the Yen in terms of Possibility, not Probability.
Here we will identify a linear possibility system in which the input variables are the trading conditions and the output is the value of the Yen.

In this case we are using fuzzy data, so recall the Min and Max fuzzy data problems introduced in Section 4.2.2.
Solutions to the Min and Max problems are presented for each of the fuzzy data coefficients in this example.
However, the Author provides no details on the derivation of these Min and Max solutions.
These Min and Max solutions are crucial to creating the fuzzy data regression model.
The L(x) function is assumed to be a triangular fuzzy number.
The last two points in the sample data are used to double check the model.

4.4 Supplementary note
Methods for regression analysis using linear possibility were formulated in 1980. The formulation is still being discussed, it is still considered a new approach. The need to bring in expert advice on data analysis, we see that fuzzy data in one form of expert knowledge. In addition, since regression analysis brings us back to LP problems, solutions are obtained by means of constraint on the coefficients introduced by system experts.

Chapter 5 - Statistical Decision Making
Fuzzy statistical decision making methods are formulated based on the ideas resembling those of Bayes theory.

5.1 Fuzzy probability and fuzzy entropy
The concept of probability with fuzzy events is effective in circumstances involving ambiguous events and probabilistic events.

Define a standard probability space to be (Q, K, P) with
Q: sample space
K: complete subset of Q
P: probability measure
E: ordinary event in K
Xe: characteristic function

Define a fuzzy event
F: a fuzzy event
u: membership function

The probability of a fuzzy event is similar in form to a conventional event; namely, summation over discrete sets and integration over continuous sets.

Also, fuzzy events A, B are independent if and only if, P(AB) = P(A)P(B)
Furthermore, conditional probability extends to fuzzy events: P(A|B) = P(A,B)/P(B)

Various types of entropy,
- entropy for the ambiguity of a fuzzy set
- entropy for the probability of a fuzzy set
- entropy for the occurance of a fuzzy event in the fuzzy set

Example 5.1 - Consider the fuzzy event of a "large roll" of a dice ... good analysis.

5.2 Fuzzy Bayes Decision Making
Formalizes a general fuzzy Bayes decision method.
8 page analysis of a fuzzy Bayes decision problem, given a priori probabilities.

5.3 Fuzzy Discrimination Methods
A special case of fuzzy Bayes decision making.

Chapter 6 - Fuzzy Quantification Theory
It would be easier to compare (linguistic) qualitative judgments and to learn the evaluative structure underlying them if we could replace qualitative expressions with numerical expressions. A type of multivariate analysis is used to accomplish this.

In the 1950's Chikio Hayashi proposed such a quantification theory to achieve this, it consists of four methods I,II,III,IV.

This chapter describes the methods for handling qualitative data using fuzzy set theory, explained in terms of fuzzy events, using values on the [0,1] interval that express qualitative judgments.

6.1 Characteristics of Fuzzy Quantification Theory
Since sample sets are commonly called groups in multivariate analysis, we call the fuzzy sets that form samples: fuzzy groups.

A example is provided of two companies that sell razors. One question is to find the primary purchasing factor.

Degree of probability is defined, using equation 5.2. Definitions for fuzzy mean and fuzzy variance follows from that.

6.2 Fuzzy Quantification Theory I
The objective of Fuzzy Quantification Theory I (qualitative regression analysis) is to find the relationship between qualitative descriptive variables, which are given on the [0,1] interval, and numerical object variables in the fuzzy group samples.

Minimum variance equation (6.13).

6.3 Fuzzy Quantification Theory II
The objective of Fuzzy Quantification Theory II (qualitative discrimination analysis) is to express several fuzzy groups in terms of qualitative descriptive variables. These qualitative descriptive variables take the form of values on [0,1].

6.4 Fuzzy Quantification Theory III
Fuzzy Quantification Theory III is a method in which pattern classification is done; similar methods were developed independently in various countries. These are known by different names such as dual scaling, correspondence analysis, pattern classification, etc.

6.5 Fuzzy Quantification Theory IV
Fuzzy Quantification Theory IV (multi dimensional scaling) is a method which uses numerical values to express mutual intimacy among the members of a set.

6.6 Note on Applications

Chapter 7 - Fuzzy Mathematical Programming
In mathematical programming there are problems in which a real problem is described in terms of a mathematical model (model building) and problems in which an optimal solution is found from the model obtained (model solving).

If a model approaches the actual problem without limitations, it is said that the model becomes complicated and finding a solution is difficult; contradictory results are common. [my italics]

In addition, real problems include constrains and goals that are expressed in natural language. For example, the kind of ambiguity expressed in "we want to gain about A yen" or "we want to keep investments to about B yen or less" is included. Fuzzy mathematical programming, which expresses this kind of ambiguity in terms of fuzzy sets, comes close to our understanding. In this chapter we will discuss the formulation of fuzzy mathematical programming and its applications.

7.1 Basic concept and general formulation
In real problems, for a variety of reasons, constraints and objective functions are often flexible.
In the corporate world, for example, constraints and objectives must often follow some standard.
In other words, handling objectives to satisfy some standard to some degree, is better than maximizing some objective function.
Interesting question: Are objective functions essential, or are they introduced simply to limit the solution set to one.

Now we discuss fuzzy decisions obtained when fuzzy constraints and fuzzy objectives are given as fuzzy sets C and G, respectively.
Decision set D is then expressed as the intersection of C and G

* D = C intersection G
* mu-D = mu-C ^ mu-G

mu-D, mu-C, mu-G the membership functions for fuzzy sets D, C, G, respectively

Moreover, this section provides procedures for obtaining the decision set.
For example, the best decision, the maximum value of the membership function for fuzzy set D, can be derived from the application of the min operation upon the membership functions of C and G, respectively.
The maximized decision is defined by max and min operations, so these meanings are clarified further.

definitions: define fuzzy sets 1 and 2
definitions: define functions f and g to be functional representations of the "and" and "or" fuzzy set operations
definitions: define a and b to be the membership functions for fuzzy sets 1 and 2 respectively

Axiom Definitions:

f and g are continuous and non-decreasing

f and g are symmetrical for a and b, f(a,b) = f(b,a) and g(a,b) = g(b,a)

f(a,a) and g(b,b) are strictly increasing functions

f(a,b) <= min[a,b] and g(a,b) >= max[a,b]

f(1,1) = 1 and g(0,0) = 0

logically equivalent fuzzy sets agree with each other, e.g.:

(u1 and (u2 or u3))(x) = ((u1 and u2) or (u1 and u3))(x)

These axioms feed Theorem 7.1:

f(u1,u2) = u1 ^ u2

g(u1,u2) = u1 V u2

For proof of Theorem 7.1, see Bellman, Giertz, "On the Analytic Formalizm of the Theory of Fuzzy Sets", Information Sciences, 5, pp149-156 (1973)

Proof of Theorem 7.1, requires 2 previously developed fuzzy set resources: fuzzy sup, and fuzzy convexity.

Theorem 7.1 leads to the solution of the decision fuzzy set maximization problem

An algorithm is provided.

7.2 Fuzzy linear programming
Fuzzy linear systems have been formulated from various points of view.
Three different approaches to fuzzy linear systems are presented.

7.2.1 Fuzzy LP problems using fuzzy inequalities
Zimmermann's formulation expresses both the objectives and constraints as fuzzy inequalities [references].
A real fuzzy LP transport example is provided.
This and the previous sections in Chapter 7 require a very close read.

7.2.2 Fuzzy LP problems using the linear interval method
Negoita's formulation of fuzzy LP problems [references].
In this type of problem, the ambiguity of the coefficients of the linear constraints is expressed as a fuzzy set, and it uses the concepts of interval programming. When the LP problem can be characterized by convex fuzzy sets K, the powerful results of section 7.1 can be utilized. Using the concept of level sets, this problem is transformed into a linear interval programming problem. Fuzzy set K can now be expressed as a linear combination of r number of level sets. Sounds like a very interesting extension of section 7.1.

7.2.3 Possibilistic linear programming problems
Possibilistic linear systems.

In this case we only know that the coefficients for the LP problem are ambiguous, and we look at this ambiguity in term of fuzzy numbers. The fuzzy numbers are given as information from experts, and we consider their possibility distributions .. as fuzzy inequalities with fuzzy coefficients.

Interesting last sentence: [With this interval,] the decision maker can consider conditions that are not incorporated into the mathematical model and make a decision.

7.3 Supplementary note
Author mentions this was (back in 1987) an active area of research, such as unification schemes, by sum and product, and by integration.

The author also mentions that constraints and objectives are not divided up; meaning that both constraints and objectives are treated on equal footing. Then he says that multiple-objective programming problems are also being considered. Not sure what the former has to do with the latter.
Chapter 8 - Evaluation
This is a short chapter and ... a bit fuzzy.

This chapter considers evaluation within the scope of problems that do not involve any decision-making.
There are two ways in which fuzziness enters evaluation.
First, the ambiguity of the characteristics of the object to be measured.
Second, the ambiguity in the measurement method of the subject performing the evaluation.
This chapter is apparently concerned only with the second type of ambiguity.

8.1 Fuzzy measure
Evaluation models use fuzzy measures.
This is apparently a generalization of crisp measure theory.

8.2 Fuzzy integrals

Chapter 9 - Diagnosis
This chapter is geared towards the medical domain, no other references mentioned.
Relationships between symptoms and causes are expressed in terms of fuzzy relational equations.
A correspondence is asserted between the diagnostic process and solving inverse fuzzy relational equations.
A diagnostic method is discussed that involves the "degree of conformity".

9.1 Ambiguity in Diagnosis
The medical diagnosis process is rife with ambiguous, subjective, un-standardized procedures, estimates, and assessments.

9.2 Diagnosis Using Fuzzy Relations
The basic diagnostic fuzzy relational equation involves symptoms Y, factors X, relations R.
The relations are expressed using Boolean min max two-valued logic
Additional definitions and one simple example are included.

This section discusses the case in which it is acceptable to think that a disease has a cause, with the symptoms appearing as a result.

9.3 Diagnosis Using Symptom Patterns and Degrees of Conformity
This section discusses the case in which a specific group of symptoms is indicated and the disease is named.
In this case, diagnosis means investigating the degree of conformity of previously established symptom patterns, and actual observations.

9.4 Applications of Knowledge Engineering in Diagnosis
This section appears to be about expert systems, specifically medical expert systems.
There was one famous example in 1976 named MYCIN.
It is not clear whether this approach has actually been successful since 1977.

A useful table is presented of linguistic values, intervals, and representative fuzzy values.

Chapter 10 - Control

Fuzzy control was the first application of fuzzy theory to get attention.

10.1 The form of fuzzy control rules and inference methods
Fuzzy control describes the algorithm for process control as a fuzzy relation between information about the condition
of the process to be controlled, x and y, and the input for the process, z.
The control algorithm is given in "if-then" (antecedent-consequent) expressions,

The major difference among the methods is:
- fuzzy control permits single stage inference
- knowledge engineering is almost always multistage inference

Fuzzy control rules are characterized by 3 points:
- form of the antecedent and the consequent
- form of the fuzzy variables
- inference method

10.1.1 Inference Method 1
3 (high level) steps to this inference method ( apparently the same steps for the other methods too )
(1) determine w, the compatibility for each input and antecedent
(2) determine inference results for each rule
(3) determine overall inference result as a weighted mean of the inference results WRT their compatibilities

fuzzy variables can be continuous or discrete.
... this section provides a good example of the discrete form.
fuzzy variable domains are Usually normalized to [-1,1] or [0,1].

In inference method #1, it is common to have five to seven fuzzy variables.

10.1.2 Inference Method 2

This method is suited for monotonic membership functions.
There are only two types of variables: positive and negative.
The arctan() function is used as the membership function.

The overall inference result for 2 rules, is y = (w1y1 + w2y2) / (w1 + w2)

This method is well suited for many input variables.
This method is not well suited for translating expert knowledge from linguistic into logical form.

10.1.3 Inference Method 3
In this method, the antecedents are made up of fuzzy propositions and the consequents are standard relational equations of inputs and outputs.
This was conceived for fuzzy process modeling, rather than fuzzy control.

an example result for this method is y = (w1f2(x1,x2) + w2f2(x1,x2)) / (w1 + w2)
f is usually a linear relational equation.
If there is only one rule the antecedent parts are no longer necessary, and only the consequent part remains,
so the result is the same as having a linear expression.
If there is more than rule, the input interval is partitioned into subspaces and linear input/output relation is found for each subspace.

This method is NOT appropriate for linguistic expressions ... but it does exceed the other methods in descriptive capability.
The rules used in inference method 1 do not go beyond description of quantitative relations.

The antecedents in the rules of all three forms are most easily understood when interpreted as being ambiguous partitionings
of the input spaces, that is, specifiers of fuzzy subspaces, rather than descriptions of conditions
(see Figure 10.6 p165)

10.2 Planning of fuzzy controllers
To design a controller means to determine the form of the control rules, namely, determination of the antecedents and the consequents.

determination of the antecedents -
- input information x1, x2, x3, etc
- conditions, that is, fuzzy partitions of the input
- parameters for the fuzzy variables

determination of the consequents -
- the output is generally the control input for the process
- fuzzy parameters

3 design methods.

10.2.1 Expert Experience and Knowledge
Expert system approach; fuzzy control is the first real example of expert systems.
The experience of skilled operators and the knowledge of control engineers is first expressed qualitatively, and then formalized via fuzzy control rules.

The main problem is to derive the fuzzy partitions of the input space, via operator interviews and engineering expertise.
Parameters for the fuzzy variables is not an issue using inference method #1

10.2.2 Operator Models
Modeling the operator can be very difficult, no answers to this problem here

10.2.3 Fuzzy Models of Processes
The above 2 methods depend on the access to a human expert operator which is not always the case.
When the object is a process without experts or human operators, a better method is based on
a fuzzy model for the design of a controller aimed at high quality control.

[ ... confusing section ]

10.3 Features of fuzzy control

In sum, fuzzy control has 3 features
- logical control - meaning free expression of control algorithms using "if-then" form.
- parallel (dispersed) control - meaning control policies can work in a dispersed manner
- linguistic control - meaning it is possible to ambiguous linguistic variables, especially as rule antecedents

Chapter 11 - Human Activities
This is a GREAT chapter, as it focuses on the target areas that Zadeh raised in his 1973 paper,
yet seem to be over-looked by the majority of FL publishing by other researchers.

From Zadeh's 1973 paper:
"By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate
and yet effective means of describing the behavior of systems which are too complex or too ill-defined
to admit of precise mathematical analysis."

"It's main applications lie in economics, management sciences, artificial intelligence, psychology, linguistics,
information retrieval, medicine, biology, and other fields in which the dominant role is played by the
animate rather than the inanimate behavior of system constituents."

Now back to this book ...
Most plant and transport breakdowns happen because of some form of human error.
Up to know, human error and mechanical breakdowns have been approached in a similar manner and carried using probability techniques.
However, human beings are different from machines and are influenced by an extremely large number of factors;
their reactions are widely varied and so it is impossible to express human reliability in terms of probability
Other differences: humans can multi-task and collaborate with others to share tasks.
There are many aspects and functions such as learning, judgment, and reasoning that we cannot discuss on the same level as mechanics.

11.1 Human reliability models
Experimental models for testing physiological(amount of work) and psychological(concentration, tension).
Workload is very ambiguous and so it is expressed as a fuzzy set.
The experiment involves showing random numbers on a CRT and the subjects respond by pressing the key for the last digit of the sum.
Repeated 100 times, etc.
Five inter-related factors:
- workload, ability, reliability
- workload and stress
- stress and ability
- ability and distribution
- environment and stress
Membership functions for human reliability
Construction of Reliability Models

Stress testing humans is far more complex than stress testing machines

Diagrams of membership function identification, relations between workload and ability
Fuzzy modeling block diagrams, 1,2,3 jobs per person reliability models, factoring personality types

11.2 Data entry systems
Two problems arise:
- not enough information on the people side, inputting of ambiguous information cannot be avoided
- when, in spite of ambiguous information, a selection must be made from a number (usually 3-7) of choices.

Interesting application of information entropy techniques to solve problem two
The average entropy H/n is plotted for N choices, the maximum is determined to be 3
This makes intuitive sense, i.e., choices: yes, no, "don't know".

11.3 Multistage decision making using fuzzy dynamic programming
The diagram on page 179 in this section is on the cover of the book!

The reason for making mathematical programming fuzzy is to allow the model of the object or evaluation to have ambiguity, to come up with a solution that seems good.
Many aspects of the problem can be made fuzzy: state variables, control variables, state transitions, etc.
Fuzziness can be introduced into
(1) system state
(2) state transitions
(3) constraints
(4) final state
(5) evaluation values, and
(6) decision points (in time and place)

However, the basic problem structure should not be fuzzy.

In order to understand the usefulness of introducing fuzziness, it is best to first consider the problem in the domain of ordinary set theory.

There are many problems with fuzzy dynamic programming, so best to emphasize the most important points as follows:
(1) "if-then" rules that express state transitions do not normally include time explicitly
(2) quasi-optimal as well as optimal decisions are calculated at each stage, resulting in "curse of dimensionality"
(3) the principle of optimality must be brought out in definite microsystems
(4) constraints, evaluations, evaluation, etc, have tradeoff relations with each other, so fuzzy dynamic programming turns out to be a multi-objective optimization problem.

Since any of these are large problems by themselves, there are times when, they offset the usefulness of introducing fuzziness in the first place.
Best way of dealing with this is to reduce the number of decision stages, but then accuracy decreases.
Therefore, we must consider hierarchical decision making design instead. There are two ways of doing this.

A concrete fuzzy dynamic programming example is provided, a 26,000-ton bulk freighter
for which the operation of the sails is under automated control,
with wind and speed as fuzzy inputs.
The sails are used for 30% fuel savings maximum.

Current wind speeds and directions are expressed as a grid of fuzzy numbers
The calculation time using probabilistic dynamic programming techniques is 30 minutes;
Only 30 seconds with fuzzy dynamic programming.

Since this includes decision input stages, it is actually an expert system.

Chapter 12 - Robots
3 examples
* path-determining robot, moving robot with sensors
* industrial robot grasping moving objects on a conveyor belt using sensors, CCD camera
* industrial robot arm with touch sensors that infers position

12.1 Path-judging robot
Origin of word robot: Karel Capek, "Rossum's Universal Robots", 1920
Example includes human-machine information exchange functions, using a phonemic composition method, 6 LEDs

12.2 Object-grasping robot
Processing Outline
observation block
quantification block
inference block
interpretation block
robot control block
grasping block

12.3 Placement inference robot

12.3.1 Hierarchy of Decision Rules
Nice graphical documentation

12.3.2 Circumstantial understanding of the Command

12.3.3 Internal Knowledge or "Search Points"

12.3.4 Inference of the Object's Position
Example fuzzy logic formula for the degree of matching

Chapter 13 - image recognition

Two applications:
(1) Visual recognition by a robot's eye.
Methods for recognizing seven different objects, and the distance, using a CCD camera and 16-bit processor.
(2) Results of using fuzzy clustering techniques, based on FUZZY ISODATA, in area partitioning by means of texture analysis of LANDSAT images.
Also, we describe methods of interpretation using entropy.

13.1 Shape recognition and distance/direction information: extraction using a CCD camera
CCD camera with 256x256 image elements, 8 bits per element ( 64K Bytes frame memory required per image )
Image classification

13.2 Texture analysis of aerial photographs
Fuzzy C-means method, Fuzzy Isodata program, is an extension of the C-means hard clustering method.

Chapter 14 - databases

14.1 standard databases
Conventional SQL databases are implementations of conventional set theory relations
Fuzzy databases are implementations of fuzzy set theory relations.
Many if not most relational queries are vague and imprecise to begin with.
Conventional relational database servers are simply not well equipped to contend with the fuzzy nature of relational data.
As an after thought, the conventional SQL language has bolted the following features that are superficially "fuzzy-like""
the LIKE operator, the SOUNDEX operator, regular expressions, etc.

14.2 fuzzy databases

14.2.1 Ambiguous Queries

14.2.2 Extension of Data Models
An example is provided of a fuzzy database language based upon fuzzy sets, a fuzzy relational model, and a fuzzy query language.

Chapter 15 - information retrieval
Considering that this book is 20 twenty years old, I wonder if this particular chapter needs a post-Google update.

Other information retrieval research, Internet retrieval technology, the research of Ben Liu and others, may supercede this chapter.
This chapter does not really address fuzzy systems directly.

15.1 information retrieval and modeling of estimation processes using fuzzification functions

15.1.1 A Priori Knowledge Spaces

15.1.2 Recognition of Requested Concepts for Retrieval Using Dialogue

15.1.3 Choice of Query Attributes Based on Recognition Concepts

15.1.4 Adaptive Change of the Estimation Strength in the Recognition Concept Revision Process

15.1.5 Termination of Recognition Concept Revision and Output of Results

15.1.6 Results of a Simulation of the Recognition Concept Revision Process

15.2 prototype document retrieval system

15.3 Characteristics of request concepts and recognition efficiency

15.3.1 Extent of the Request Concept and Estimation Strength

15.3.2 Request Concept Coherency and Estimation Strength

15.4 the role of A Priori knowledge and intelligent interfaces

Chapter 16 - Expert system for damage assessment

16.1 Expert systems and fuzzy knowledge
p247 Here is succinct description of the relationship between
- expert systems (ES)
- knowledge engineering (KE), and
- artificial intelligence (AI)
The academic research area for the basic technology for the construction of these expert systems is called knowledge engineering (KE),
and what forms the basis of knowledge engineering is research into artificial intelligence.
In other words, knowledge engineering is can be called practical artificial intelligence,
since it is an area oriented toward applications of artificial intelligence,
and expert systems are concrete products of the up-loading of knowledge if specialists.
Since the mid-1970s advent of the famous expert system MYCIN, many others have been constructed.

expert system structure
- user interface
- knowledge base
- inference mechanism

Primary Components of knowledge engineering & expert systems
- knowledge representation methods
- knowledge utilization methods
- knowledge acquisition and management
- user interface

p249-250 ill defined, ambiguous problems
In damage assessment, the degree of damage is very hard to assess

16.2 Expression of problems that contain uncertainty
This is where the author attempts to connect fuzzy logic with the subject of this chapter
Emphasizing the ambiguity inherent in knowledge engineering and expert systems
Also introducing SPERIL. The remainder of the Chapter is one big SPRERL case study

Interesting discussion of AND/OR/COMB-logic binary trees representing problems accompanied by uncertainty.
The combination relation indicates a relation between sub-problems,
such as when a goal is supported independently by two or more pieces of evidence.

In the MYCIN example, an intuitive COMB function was used to attain this goal.

16.3 Dempster-Shafer theory and it's extension to fuzzy sets
p254 - Bayes theorem does not handle ignorance effectively, something else is needed:
Dempster-Schafer theory allows rational handling of uncertainty that is connected with subjectivity.
Dempster's rule of combination gives the method of combining the basic probabilities inferred from independent evidence.
this is used in the basis of the SPERIL inference engine.

16.4 SPERIL system
SPERIL is a rule-based expert system for damage assessment of buildings that have suffered earthquake excitation.
SPERIL-1 was implemented in the C language.
SPERIL-2 was implemented in the Lisp language.
Nice graphical representation of the expert system / inference network.

fuzzy modeling and control

2007-07-23T09:17:00.000-07:00

"Essentials of Fuzzy Modeling and Control" , by Ronald R. Yager and Dimitar P. Filev, New York, NY: John Wiley, 1994; 388 pages

Excellent and comprehensive guide to Fuzzy Modeling and Control. This book successfully bridges theory with practice by including many concrete examples. It provides detailed examples of Fuzzy Logic Control (FLC) systems. And it provides the theoretical background behind many fundamental aspects of the subject, including the theory of Approximate Reasoning (AR).

Here is the Table of Contents annotated with key points from the book

Chapter 1 - Basic Concepts of Fuzzy Set Theory

1.1 introduction

1.2 fundamental concepts of fuzzy subsets

1.3 operations on fuzzy sets

1.4 fuzzy relationships

1.5 the Extension Principle and fuzzy arithmetic

1.6 measures of fuzziness

1.7 linguistic values and possibility distributions

1.8 measures of specificity

1.9 references

Chapter 2 - Aggregation Operations on Fuzzy Sets
Intersection and union operations can be viewed in a more general setting as an aggregation of fuzzy subsets.
Many applications of fuzzy sets involve aggregations.
In this chapter we provide a number of different methods for aggregating fuzzy subsets.

We first look at the intersection and the union operation.
We introduce two classes of aggregation operators, the t-norm and the t-conorm, which generalize the intersection and union operations.

We then turn to the issue of weighted intersections and unions.
In some applications in artificial intelligence, a need arises for the ability to reason with default knowledge.
We then turn from intersection and union to other aggregation operations which lie between them, mean operators.
Then we turn to a specific type of mean operator, ordered weighted aggregation (OWA).
We then use OWA operators to help make a connection between natural language stipulations of aggregations
in terms of linguistic quantifiers and the mathematical realization of the stipulated aggregation.
Last, fuzzy measure and fuzzy integral are introduced.

2.1 introduction

2.2 intersection and union of fuzzy sets

2.3 weighted unions and intersections

2.4 non-monotonic fuzzy operations

2.5 mean aggregation operators

2.6 ordered weighted averaging (OWA) operators

2.7 obtaining OWA aggregation functions

2.8 linguistic quantifiers and OWA operators

2.9 fuzzy measures and integrals

2.10 references

Chapter 3 - Theory of Approximate Reasoning (AR)

3.1 introduction
Lotfi Zadeh introduced the concept of Approximate Reasoning in 1979.
Based upon fuzzy sets, it provides a powerful framework for reasoning in the face of uncertain information.
Central to this theory is the representation of propositions as statements assigning fuzzy sets as values to variables.
The formal AR framework is provided in this chapter.
The AR framework provides a mechanism for modeling and making inferences from imprecise functional relationships.
This mechanism forms the basis of the fuzzy systems modeling technique.

3.2 primary elements of the AR system
Definitions: atomic variables, universe of discourse, joint variable, canonical statement, possibility, tautology,
conjoin, cylindrical extension, proposition equivalence.

3.3 semantics of the AR system
Translation rules: proposition negation, proposition Anding, proposition Oring, Plying operation

3.4 deduction in AR
Definitions: deduction, Inference rules 1 and 2
Theorems: monotonicity

3.5 minimal solutions and projections
Definitions: projection, minimal value,
Theorems: 6 theorems

3.6 binary logic in AR
Binary propositional logic as a special case of AR

3.7 functional representations
Extension of the AR inference mechanism to (non-binary) functional relationships.

3.8 references

Chapter 4 - introduction to Fuzzy Logic Control
The best chapter of the book.

4.1 introduction
Introduction to conventional system control; feedback.
Definitions
C: controller
S: system
y: output
w: set point
u: controller action
e: error, w-y
f: non-linear function

Conventional controller algorithms are expressed via analytic algorithms based upon conventional set theory.
Fuzzy logic controllers (FLC) are expressed via knowledge-based algorithms based upon fuzzy set theory.
FLCs are expressed via a linguistic description, and a predicate rule-based inference algorithm.

4.2 fuzzy logic controller (FLC): basic concept
The general form of the conventional logic controller is
(1) u(k) = f(e(k),e(k-1),...,e(k-v),u(k-1),u(k-2),...,u(k-v))

A FLC is a knowledge=based system consisting of IF ... THEN rules with vague predicates and a fuzzy logic inference mechanism.
The general form of the fuzzy logic controller is
(2) u(k) = F(e(k),e(k-1),...,e(k-v),u(k-1),u(k-2),...,u(k-v))

where the function F is described by a rule-base.

An example using 5 predicate rules and linguistic labels represented as fuzzy sets.
The linguistic labels & predicate logic are what allow a heuristic implementation.
These are based upon human knowledge of the particular system.

4.3 reasoning with an FLC
Very important section ... and difficult to capture without a technical document editor (LaTex here we come):

The phrase 'reasoning' is used to indicate 'the process' in which the given input values and antecedent variables are used in conjunction with the knowledge-base to obtain the output value, the consequent. This is based upon the theory of Approximate Reasoning (AR).

The procedure for obtaining the fuzzy output value is:
1 - find the firing level of each of the rules
2 - find the output of each of the rules
3 - aggregate the individual rule outputs to obtain the overall system output.

The degree (level) of firing (DOF) for the ith is shown to be

DOF = B1(x1*) ^ B2(x2*)

for antecedent linguistic labels B1, B2 and inputs x1, x2

the ith output fuzzy set F is obtained by anding the DOF and the consequent set D:

F = DOF ^ D

Similarly, aggregation across ALL i individual rule outputs F gives the fuzzy output F

F(y) = V F(y) = V ( DOF ^ D )

This approach is known as the (constructive) Mamdani method

4.4 illustration of the basic reasoning algorithm
The procedure for obtaining the de-fuzzified output value is further detailed as follows:

1 - calculate the DOF of the rules for crisp inputs or for fuzzified noisy inputs
2 - find the membership function of output fuzzy set Fi inferred by the ith rule, for all i
3 - form the membership function of output fuzzy set F, inferred by the rule-base of FLC by aggregating the Fi
4 - calculate the crisp output of FLC y* by defuzzification of fuzzy set F using the center of area (COA) or mean of maxima (MOM) method

Extended example included

4.5 on the relationship to PI, PD and PID control
How Mamdani-type FLCs can be regarded as PI, PD and PID-like control algorithms
Now, the next two sections address 2 of the most important FLC implementation challenges

4.6 design of the FLC: determination of the linguistic values
Determination of the linguistic values, the membership functions of the associated reference fuzzy sets, can be a challenge.
This is a surprisingly complex topic.
A complex example of reference fuzzy set membership functions in analytic form is provided.
Creating the membership functions requires pairing skills and experience in both FL methodology and the particular problem domain.

4.7 design of the FLC: construction of the knowledge base
Construction of the knowledge base is the most crucial and difficult part of FLC design. In the past, knowledge base construction was an ad hoc process. FLC has been criticized for a lack of a general systematic tool for creating the knowledge base. However, there are a variety of (non-systematic) approaches. Building the FLC knowledge base requires both a system operator expert, as well as an FLC system specialist.

One more formal approach utilizes the concept of a knowledge template rule-base.
The MacVicar-Whelan template rule-base is described in this section.

4.8 design of the FLC: tuning
Summary of practical design rules in FLC design

step 1 - determine the input and output variables of the FLC, and any PI, PD, or PID-like similarities
step 2 - define the FLC parameters - scaling factors, normalized universe of discourse, linguistic variables, and membership functions
step 3 - determine the FLC rule-base
step 4 - computational realization of the FLC

4.9 extension of the Mamdani FLC
Originally, the Mamdani FLC was considered a knowledge-base alternative to conventional system control methods.
The concept of fuzzy control can be applied (with appropriate rules) not only to calculating the control variable, but also for updating the control strategy itself.
For example, modifying the setpoint.
An example of this is provided, for modifying the setpoint of a non-linear, non-stationary biochemical process.

4.10 references

Chapter 5 - Fuzzy System Models

5.1 introduction
The complexity of the real world is most dramatically reflected in "soft" area such as social, economic, ecological, and biological systems.
Multi-dimensionality, hierarchical structures, mutual interactions, internal feedback mechanisms, and unpredictable dynamics are only a part of the characteristics of such complex systems.
This complexity accounts for some of the reasons for the difficulties in the attempts to transfer the powerful systems and control techniques to these disciplines.
The weakness of the traditional quantitative techniques to adequately describe complex phenomena was summarized in the well-known principle of incompatibility, formulated by L. Zadeh. This principle states that "as the complexity of a system increases, our ability to make precise and yet significant statements about its behaviors diminishes, until a threshold is reached beyond which precision and significance or relevance) become almost mutually exclusive characteristics."

Fuzzy system models fall into 2 categories
(1) Linguistic Models (LM) based upon collections of IF-THEN rules with fuzzy predicates, fuzzy quantities, and natural language system description
(2) Takagi-Sugeno-Kang (TSK) models, formed by logical rules with a fuzzy antecedent and a functional consequent.
TSK models combine fuzzy and non-fuzzy models.
[[ note that this 60-page chapter devotes only 13 pages to TSK models ]]

5.2 linguistic models as a tool for complex systems representation

5.3 general representation for inference from a fuzzy model

According to the theory of approximate reasoning, each model rule: IF U is B THEN V is D
can be translated into a canonical proposition of the form (U,V) is R
where R is a fuzzy relationship defined on the Cartesian product universe X x Y

5.4 Mamdani-type (constructive) linguistic models
There are 2 types of Linguistic models
(1) Mamdani-type (constructive) LM
(2) Logical-type (destructive) LM

Mamdani-type models are built by constructing the union of all the individual fuzzy relations.

5.5 logical-type (Destructive) linguistic models
Under this approach the aggregation of individual rules (ALSO connectives) is accomplished by the fuzzy intersection.

5.6 linguistic models and the problem of defuzzification

5.7 multiple variable linguistic models

5.8 linguistic models of dynamic systems: state space approach

5.9 input-output fuzzy models of dynamic systems

5.10 Takagi-Sugeno-Kang (TSK) fuzzy models

5.11 quasi-linear and quasi-nonlinear fuzzy models

5.12 references
[28] Box, Jenkins, "Time Series Analysis, Forcasting and Control", Holden Day 1970

Chapter 6 - Developing Fuzzy Models

6.1 introduction
Three fuzzy modeling principals according to Zadeh for complex/ill-defined systems
1 - linguistic variables
2 - conditional fuzzy statements
3 - fuzzy algorithms

an important issue in model design is ... how to build the model, the methodology. Here are two approaches.
*1* (direct approach) early examples inspired by expert systems (ES) extracted the model from the operator expert describe the system linguistically using natural language, then translate into formal structure using approximate reasoning (AR)
*2*(system identification approach) inspired by classical systems theory, and more recent neural network development, this is a two phase approach.
phase 1: structure identification
phase 2: parameter identification

p205

Structure identification ... template based models combine expert knowledge the system expert provides template linguistic values which allow us to partition the input-output (I/O) space
fuzzy sets are given a priori the template values are used to define 'potential' rules; I/O data is used to generate weights or probabilities

If template linguistic values are not available, then the system structure can be obtained by clustering the IO space. Mountain-clustering provides a systematic approach to identifying the most important rules from the IO data.

Parameter identification is closely related to member function estimation. Real success in parameter id was achieved with the TSK method, which combines Kalman filtering. The TSK method also demonstrates a way of developing fuzzy models though a simplification of the Mamdani reasoning.

The coupling of FL and NN is a new and powerful tool for parameter identification of FL models with the use of back-propagation

6.2 direct approach to constructing linguistic models

6.3 learning of LM based on fuzzy relational equations

6.4 template-based fuzzy systems modeling

6.5 template fuzzy modeling via belief structures

6.6 learning fuzzy model parameters via back-propagation

6.7 LMS method and Widrow-Hoff rule for learning consequents

6.8 Kalman filter method of learning consequents

6.9 the mountain-clustering method

6.10 developing fuzzy knowledge bases by mountain clustering

6.11 references
[35] Aoki, "Optimization of Stochastic Systems", 1967

Chapter 7 - Theoretic Analysis of FLC

7.1 introduction
p265 FLC does not require a precise mathematical model of the controller object because it models the experience & knowledge of the operator.
But, thee is more ... recent developments into FLC self tuning
Comparison to (classical) PD controllers, like a virtual non-linear PD controller ...
Lots of discussion of sliding mode ... complex ...

7.2 knowledge-based design of FLC

7.3 the FLC as a virtual PI (PD) controller

7.4 tuning FLC as a PID controller

7.5 the FLC as a variable structure system

7.6 on the relationship between sliding mode control and FLC

7.7 sliding mode control by FLC

7.8 references
p311 references include [9], [10] design of self-tuning fuzzy controllers.

Chapter 8 - Defuzzification Problem

8.1 introduction
p311 Compare & contrast the various defuzzification methods
p315 Defuzzification is a "selection" problem. First, a fuzzy set must be converted into a "probability" distribution. Then, a crisp value is selected either:
(a) by random experiment, or
(b) by expected value

8.2 the probabilistic nature of the selection process based on a fuzzy set

8.3 general view of the defuzzification process

8.4 BDD transformations and the BADD dufuzzification method

8.5 the SLIDE defuzzification method

8.6 the M-SLIDE defuzzification method

8.7 the RAGE defuzzification method

8.8 defuzzification under constraints: the RAGE approach

8.9 defuzzification by constrained optimization

8.10 references

Chapter 9 - The Flexible Structure of Fuzzy Systems

9.1 introduction
p357 introduce a more powerful & flexible parameterized fuzzy reasoning framework that unifies both Mamdani-type and logical-types models.
More powerful ... how ?

9.2 main fuzzy reasoning paradigms

9.3 soft fuzzy implications

9.4 soft rule aggregating

9.5 soft rule firing

9.6 parameterized fuzzy reasoning methods

9.7 compromise fuzzy reasoning method (CFR)

9.8 confidence in the partitioning of the input and output space

9.9 application of the structure flexibility

9.10 references

fuzzy computing references

2007-07-05T09:16:00.000-07:00

These references fall into one of 3 categories:

fuzzy logic soft computing methods, theory & applications

conventional (hard) computing methods, for comparative purposes

background to some problems possibly amenable to a fuzzy logic approach

BISC - Berkeley Initiative for Soft Computing
This site was updated .... it now includes Zadeh's papers, including the original ones from the 1960's & 70's, that were previously difficult to obtain (without paying publisher fees):. More current papers through 2005 are also included.
- Zadeh’s 1965 paper on fuzzy sets
- Zadeh’s 1973 paper on the analysis of complex systems and decision processes
- Zadeh’s 1976 paper on fuzzy-algorithmic approach
- Zadeh’s 2002 paper on computing with words
- Zadeh’s 2002 paper on computational theory of perception (CTP)
- Zadeh’s 2005 paper on a Generalized Theory of Uncertainty (GTU)—An Outline, Information Sciences
robert.engle-Financial Econometrics - A New Discipline With New Methods.394.pdf
robert.engle.hi-frequency-data-econometrica-ucsd9615.pdf
Econometrics is financial signal processing. These articles illustrate standard Econometrics methods - ARCH, GARCH, time series analysis.
Quantitative analysis of Energy Markets - world.sci-6352_chap01.pdf
Quantitative analysis of Energy Markets - world.sci-6352_toc.pdf
Lukewarm treatment of a hot topic
2002 - Society for Industrial and Applied Math - Ross,Booker,Parkinson.pdf
Attempting to bridge the gap between fuzzy logic and classical probability methods
199X-Arnold-Shapiro-Actuarial-Dept-Penn-State.pdf
2003-Arnold-Shapiro-Actuarial-Dept-Penn-State.pdf
Interesting, a Penn State financial actuarial professor's view on fuzzy neural methods
Genetic Programming of Fuzzy Logic Rules Financial Trading-scientio.com.pdf
Plausible application of fuzzy logic methods to trading problems
2005-TAC-Adaptive Trading Agent for Supply Chain.pdf
not convinced
2006-Italian-economists-ALTERNATIVE APPROACH TO FIRMS’ EVALUATION.pdf
Interesting application of fuzzy logic to financial statement analysis
2001-Agent bidding strategy based on fuzzy logic in a continuous double auction-ACM.pdf
Needs a second read
1999-Preist-Commodity trading using an agent-based iterated double auction.pdf
Not convinced
1985-Tutorial on fuzzy logic in simulation-NASA-grant.pdf
Intro level, not much on simulation
Kandel,Meyer,Rundus-the triad of fuzzy theory.pdf
Fuzzy logic, set theory, and mathematics ... discusses confusion and the burden of proof
1994-Fuzzy systems - an overview-ACM.pdf
Good introduction, factual tables and graphs; perhaps at times over-exhuberant
1998-Object oriented process modeling with fuzzy logic-ACM.pdf
Not pursuaded
1995-Dubois,Prade-What does fuzzy logic bring to AI-ACM.pdf
Discussion of fuzzy logic and AI; highlights the burden of proof on the former
1999-Hierarchical fuzzy configuration of implementation strategies-ACM.pdf
First read: Does this added complexity subtract from the essence and the value proposition of fuzzy logic.
Requires a second read
Mamdani-Application of fuzzy logic to approximate reasoning using linguistic synthesis.pdf
Mamdani contributed the major concepts of Fuzzy systems control and linguistic synthesis.
Review - Essentials of Fuzzy Modeling and Control by Ronald R. Yager and Dimitar P. Filev (John Wiley, 1994).pdf
Bought the book based upon this review
1994-Comparison of fuzzy to Proportional-Derivative control systems.pdf
Comparison of fuzzy systems control to conventional PD systems control
1994-fuzzy model-based optimal control strategy-Schlumberger-ACM.pdf
Schlumberger Corporation research on fuzzy control systems
Siemens-Fuzzy signals in control loops.pdf
Siemens Corporation research on fuzzy control systems
Ontonix Stochastic (optimal) vs Fuzzy (robust) Methods.pdf
interesting presentation - comparison of stochastic and fuzzy logic methods
2002-FuzzyDamageDetection.pdf
Fuzzy logic expert systems
2007 - FUZZY LOGIC FOR BUSINESS, FINANCE - Bojadziev, World Science chap01.pdf
My introduction to fuzzy logic - consistent notation, good delivery - free download
2003-Fuzzy-Modeling Imprecise Factors in Environmental Policy.pdf
Application of fuzzy logic to environmental policy
1976-Fuzzy reasoning and the logics of uncertainty - p179-gaines.pdf
Includes 100 references, need to re-read
2000-fuzzy logic and OOD- ACM-marcelloni.pdf
A fuzzy logic solution to the problem of object oriented design. Not pursuaded.
2005-Fuzzy Logic Controller (FLC) for shortening the TCP.pdf
Nice example of a TCP/IP fuzzy logic controller.
2005 Larsen Lecture Notes - Fundamentals of fuzzy sets and fuzzy logic.pdf
Excellent - my first introduction to fuzzy set theory
NickolasTufillaro-nonlinear systems-quadratic_map_lab.pdf
Example of non-linear, non-stable systems behaviour (stabiity is not a given ...)
2007 - StreamBase Algo_Trading1.07.pdf
Algorithm trading vendor solution ... questionable
2005-Online Trading Algorithms and Robust Option Pricing.pdf
Stanford Business School options analysis ... questionable
2005-Mertacor A Successful Autonomous Trading Agent.pdf
Fuzzy logic online trading application ... questionable
LOTFI A. ZADEH, Fuzzy Logic, Neural Networks, and SoFt Computing, Communications of the ACM, March 1994, Volume 37 No. 3.pdf
Great article from the inventor of Soft Computing and fuzzy logic.
wilmott - Stochastic Volatility and Mean Variance analysis.pdf
Topics in Stochastic Volatility and Mean Variance ...
2007 Cisco - trading_floor_architecture.pdf
High frequency signal processing vendor solution
Monte Carlo -Long winded diatribe.pdf
Meditation and intuition of the Monte Carlo method.
Can soft computing provide an alternative, approximate solution to the Monte Carlo method.
black-scholes - laplaceTransform - 020310_skachkov.pdf
Interesting ... alternative LaPlace Transform solution to Black Scholes
2007-Grid Computing Financial Services Report.pdf
Grid processing solutions in the financial services
BC_FPGA_Datasheet.pdf
Field programmable gate array (FPGA) vendor
2001-Hellmann fuzzyLogic Introduction.pdf
Good fuzzy logic introduction and great color pictures

Priemer,Nigam Murmur_Detection_2006.doc
Simplicity-Based Fuzzy Clustering Approach for Detection and Extraction of Murmurs from the Phonocardiogram
Priemer,Kolodzjc-Stable and Dynamic Fuzzy Controller.pdf
Design of a Stable and Dynamic Fuzzy Controller to Achieve Linguistic Closed Loop System Performance Specifications
fuzzy author index 06004490.pdf
2006 Author Title index of fuzzy systems papers
Intel - multi-core programming - 2006 MCP_SampleCh01.pdf
Concurrent systems programming, Intel multi-core
parallel programming with GCC - rhs2006.pdf
Concurrent systems programming, parallel programming GCC extensions
XtremeData-NathanWoods_FPGA_2007_presentation.pdf
FPGA vendor, systems architecture overview
XtremeData-FPGA_Acceleration_in_HPC.pdf
FPGA vendor, systems architecture overview, financial analytics case Study
Information (Quantum) Mechanics - John L. Haller Jr. - 0106081.pdf
Information mechanics ... is hard to relate to soft computing
2003-java-messaging-p2p-junginger-masters-thesis.pdf
An M.S. thesis on HPC and java messaging.
2006-nonextensive approach to the dynamics of financial observables.pdf
Application of statistical mechanics to finance ... close to information mechanics
2004-robert.engle Analysis of High Frequency Financial Data.pdf
High frequency financial signal processing
2003 - Kalman filtering for fuzzy discrete time dynamic systems.pdf
Integration of fuzzy systems and Kalman filters, Dan Simon
thesis - Stefano Mazzocchi.pdf
Thesis from the creator of Apache Cocoon, JServ and other open source projects [nothing to do with FL]
Kalman filter Taygeta Scientific
Taygeta Scientific Inc ... Kalman filtering, non-linear systems
ALEXANDER LIPTON-LIFSCHITZ
UIC professor and financial engineering leader
Kalman Filtering for Fuzzy Dynamic Systems
Kalman Filtering for Fuzzy Dynamic Systems, Academic, Dan Simon
Kalman filtering
Kalman filtering development, Corporate, Dan Simon
wikipedia fuzzy_logic
wikipedia fuzzy_set
wikipedia control_theory
wikipedia triangular_norm
wikipedia princile_of_bivalence
wikipedia multi-valued_logic
wikipedia fourier_transform
wikipedia fast_fourier_transform
wikipedia fuzzy_system
wikipedia Uncertainty_principle
wikipedia Characteristic_function
wikipedia Indicator_function
wikipedia Maximum_likelihood
wikipedia Multivariate_random_variable

2007 fuzzy book review

2007-07-01T09:13:00.000-07:00

"Fuzzy Logic for Business, Finance, and Management" Bojadziev, 2007, worldscibooks.com

Four decades have passed since Lotfi Zadeh delivered fuzzy set theory in 1965.
Hundreds of fuzzy set theory books and papers have been written since then.
Here is the latest ... the publisher gives away Chapter One for free download.

Here is the outline of Chapter One.

(1.1) - classical set theory review: relations and functions

bi-variate membership rule, universal set, empty set, interval, subset, complement, intersection, union, disjoint, convex, complementary

cartesian product, relation, function

indicator (or characteristic) function

membership function

tall example

(1.2) - fuzzy set theory introduction

fuzzy set is a generalization of Cantor crisp set

fuzzy set members have a degree of membership - is formally equal to it's membership function

fuzzy set normalization

fuzzy singleton set

alpha-level interval, or a-cut defined- is a crisp set, is a confidence level indicator

fuzzy set convexity - iff all alpha-level intervals are convex

tall example, revisited

(1.3) - basic operations on fuzzy sets

most classical set operations hold, except: the principle of the excluded middle does *NOT*

(1.4) - fuzzy numbers

fuzzy number is a fuzzy set that is convex and normalized (concave != convex)

fuzzy number have an interval [a1; a2] called the supporting interval, as well as a maximum

piecewise-quadratic fuzzy number - bell-shape with 2 parameters: peak point and bandwidth

(1.5/6) - triangular / trapezoidal fuzzy numbers

(1.7) - fuzzy relations

2 notions of generalization .... (? intentional or accidental)

fuzzy relations generalize fuzzy sets from 2-space to 3-space

fuzzy relations generalize crisp relations (more expressive domain:-> range mapping )

... enter 'linguistic relations'

(1.8) fuzzy relation operations

R1 = { (x,y), ur1(x,y) } , (x,y) in A x B

previous operations confirmed, plus

direct min prouduct, and

direct max prouduct

Concluding Notes

examples: Heap, Tall

fuzzy sets are not a new probability type, there are substantial differences.

For instance, grade or degree of membership is not a probablistic concept.