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
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
The concept of Fuzzy Theory addresses the epistemology or theory of knowledge, as this section explains.
Applications are surveyed now and in the future; first in Japan, where fuzzy logic took hold, and elsewhere
Table 1.2 creates a cross-section classification of the different areas where fuzzy logic is or can be applied
Table 1.3 Extensive survey of current research and future applications
Table 1.4 outlines current areas of interest
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
Most natural language contains ambiguity and multiplicity of meaning
A means of quantifying meaning is needed.
Definitions: fuzzy set, membership function a mapping from crisp set to the [0,1] interval
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
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
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
Operations with Fuzzy Numbers are an application of the extension principle.
Detailed example provided
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.
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.
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
Fuzzy matrix elements are the partial characteristic functions
Fuzzy graph vertices are the elements of the fuzzy set; arcs are the grades
Example provided.
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.
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.
Converse fuzzy relations, the Identity relation, the Zero relation, are all defined in terms of characteristic functions.
Table 3.1 provides a concise summary of 13 different types of fuzzy relations
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
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
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.
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.
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
The concept of finding an equivalence class by means of an equivalence relation can be applied to similarity relations.
Example provided.
The fuzzy relation "look alike" is reflexive and symmetrical but NOT transitive, so it is NOT a similarity relation ... but it is still interesting.
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.
The remaining sections of Chapter 3 are brief, some are just a sentence or two.
Aparently an extension of conventional Upper and Lower Bounds
Aparently an extension of conventional LUBs and GLBs
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.
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 )
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.
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.
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
Examples are provided, one for each type of data, standard and fuzzy, respectively.
Prices of Japanese housing is regressed against 5 variables, including a constant fuzzy number variable
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.
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.
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.
Formalizes a general fuzzy Bayes decision method.
8 page analysis of a fuzzy Bayes decision problem, given a priori probabilities.
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.
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.
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).
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].
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.
Fuzzy Quantification Theory IV (multi dimensional scaling) is a method which uses numerical values to express mutual intimacy among the members of a set.
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.
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:
(u1 and (u2 or u3))(x) = ((u1 and u2) or (u1 and u3))(x)
These axioms feed Theorem 7.1:
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.
Fuzzy linear systems have been formulated from various points of view.
Three different approaches to fuzzy linear systems are presented.
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.
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.
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.
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.
Evaluation models use fuzzy measures.
This is apparently a generalization of crisp measure theory.
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".
The medical diagnosis process is rife with ambiguous, subjective, un-standardized procedures, estimates, and assessments.
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.
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.
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.
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
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.
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.
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)
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
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 ]
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.
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
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".
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
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
Processing Outline
observation block
quantification block
inference block
interpretation block
robot control block
grasping block
Nice graphical documentation
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.
CCD camera with 256x256 image elements, 8 bits per element ( 64K Bytes frame memory required per image )
Image classification
Fuzzy C-means method, Fuzzy Isodata program, is an extension of the C-means hard clustering method.
Chapter 14 - 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.
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.
Chapter 16 - Expert system for damage assessment
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
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.
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.
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.
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