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## Material Information- Title:
- A model validation methodology for isolating inconsistent knowledge between fuzzy rule-based and quantitative models using fuzzy simulation
- Creator:
- Kim, Gyooseok
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- vii, 112 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Boilers ( jstor )
Cognitive models ( jstor ) Fuzzy sets ( jstor ) Mathematical models ( jstor ) Membership functions ( jstor ) Modeling ( jstor ) Mortality rates ( jstor ) Point estimators ( jstor ) Quantitative modeling ( jstor ) Simulations ( jstor ) Computer and Information Science and Engineering thesis, Ph. D ( lcsh ) Dissertations, Academic -- Computer and Information Science and Engineering -- UF ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1998.
- Bibliography:
- Includes bibliographical references (leaves 108-111).
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- Typescript.
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- Vita.
- Statement of Responsibility:
- by Gyooseok Kim.
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A MODEL VALIDATION METHODOLOGY FOR ISOLATING INCONSISTENT KNOWLEDGE BETWEEN FUZZY RULE-BASED AND QUANTITATIVE MODELS USING FUZZY SIMULATION By GYGOSEOK KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 To my parents, my wife, Jun gsook, my daughter, Karyoon and my son, Seungyup ACKNOWLEDGEMENTS I wish to express my gratitude to my supervisor and committee chairman, Dr. P. A. Fishwick, for his invaluable guidance, encouragement and patience throughout the past four years of my graduate study. I would like to express my sincere appreciation to Dr. Douglas D. Dankel, Dr. L. M. Fu, Dr. S. Rajasekaran, and Dr. S. X. Bai for serving on my supervisory committee and providing invaluable suggestions for my research. I am grateful to all my colleagues at the Department of Computer and Information Science and Engineering for their help and friendship. I thank Dr. Jinjoo Lee who helped me a lot when I first joined the research group. Special thanks go to Youngsup Kim and Kangsun Lee in my group for their suggestions and companionship during my research. I wish all of them the best of luck in their future endeavors. Last but most importantly, I am deeply indebted to my parents, my wife Jungsook, my daughter Karyoon and my son Seungyup for their encouragement, love and understanding. My greatest gratitude is to them. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS...................................1iii ABSTRACT............................................ vi CHAPTERS............................................. 1 1 INTRODUCTION.......................................1 1.1 Problem Statement...................................2 1.2 Purpose of Research..................................5 1.3 Contribution to Knowledge..............................7 1.4 Outline.......................................... 7 2 BACKGROUND AND RELATED WORK.......................9 2.1 Modeling Process in Simulation and Fuzzy Set Literature..........9 2.1.1 Simulation Modeling Process........................9 2.1.2 Fuzzy Modeling Process..........................12 2.2 Fuzzy Set Theory and Its Application .. .. .. .. ... ... ... ..14 2.2.1 Notation, Terminology, and Basic Operations. .. .. .. ....15 2.2.2 Membership Function Construction. .. .. .. ... ... ....19 2.2.3 Fuzzy Controller. .. .. .. .... ... ... ... ... ....21 2.3 Fuzzy Set Theory in Computer Simulation. .. .. .. ... ... ....27 3 A NEW FUZZY SIMULATION APPROACH .. .. .. .. .... ... ..33 3.1 Expert Rule Format .. .. .. ... ... ... ... ... .... ....33 3.2 Fuzzy Simulation .. .. .. .. ... ... .... ... ... ... ....34 3.2.1 Fuzzy Simulation for Simplex Rules .. .. .. .. ... ... ..35 3.2.2 Fuzzy Simulation for Compound Rules with Arithmetic Operations. .. .. .. ... ... .... ... ... ... ... ....39 3.2.3 Fuzzy Simulation for Compound Rules with Logic Operations 42 4 A METHOD FOR ISOLATING INCONSISTENCY. .. .. .. .. .. ....46 4.1 Measurements of Inconsistency .. .. .. ... ... .... ... ...48 4.2 Checking Consistency When MFzprt is Available. .. .. .. .. ....49 4.3 Checking Consistency When MFxpert is Unavailable. .. .. ... ..50 4.3.1 Process in General. .. .. .. ... ... ..........50 4.3.2 Heuristic Function and Search Method for Generating Approximate M~uzy.. .. ..... ..... .... .........51 4.3.3 Various Forms of Expert's Estimates on Linguistic Terms . . . 54 iv 4.3.4 4.3.5 4.4 Time 4.4.1 4.4.2 Algorithm to generate M~..... .. .. .. .. .. .. .. .. Identify Inconsistent Rules.... .. .. .. .. .. .. .. .. .. Complexity........ .. .. .. .. .. .. .. .. .. .. .. .. Time Complexity for Fuzzy Simulation.. .. .. .. .. .. .. Time Complexity for M~uz Generation.... .. .. .. .. . 57 61 64 64 65 5 FULTON: STEAMSHIP MODELING .. .. .. ... ... ... ... ...67 5.1 5.2 5.3 5.4 Quantitative Model of Boiler Assembly... .. .. .. .. .. .. .. .. Qualitative Model of Boiler Assembly.... .. .. .. .. .. .. .. .. Checking Consistency When MFxpert is Available. .. .. .. .. .. Checking Consistency When MFxpert is Unavailable.. .. .. .. .. 5.4.1 The Case Where Approximate M~,,, is Successfully Generated 5.4.2 The Case Where M~,,, is Unsuccessfully Generated . ... 5.4.3 Human Intervention...... ..... . . . . .. .. .. .. .. 6 PREDATOR-PREY POPULATION... .. .. .. .. .. .. . 6.1 6.2 6.3 6.4 Qualitative Model .. .. .. .. .. Quantitative Model. .. .. .. .. Consistency Checking. .. .. .. . Mfuz Generation .. .. .. .. 7 FUTURE WORK....... . .... .. .. .. .. .. .. .. .. .. .. .. 7.1 Limitations and Improvements... .. .. .. .. .. .. .. .. .. .. . 7.1.1 Resolving Inconsistency... .. .. .. .. .. .. .. .. .. .. 7.1.2 Performance Index..... .. .. .. .. .. .. .. .. .. .. .. 7.1.3 Local Optimality During M~,,~ Generation. .. .. .. .. 7.2 Application..... ... . . . . .. .. .. .. .. .. .. .. .. .. .. . 7.2.1 Application in Control Industries.... .. .. .. .. .. .. .. 7.2.2 Application in MOOSE.... .. .. .. .. .. .. .. .. .. .. 8 CONCLUSION........ . . .... .. .. .. .. .. .. .. .. .. .. .. REFERENCES......... . .. . .... .. .. .. .. .. .. .. .. .. .. . BIOGRAPHICAL SKETCH......... .. .. .. .. .. .. .. .. .. .. .. 67 69 70 74 74 80 81 86 . . . . . . . 86 . . . . . . . 87 . . . . . . . 91 . . . . . . . 95 99 99 99 99 100 101 101 102 106 108 112 v Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A MODEL VALIDATION METHODOLOGY FOR ISOLATING INCONSISTENT KNOWLEDGE BETWEEN FUZZY RULE-BASED AND QUANTITATIVE MODELS USING FUZZY SIMULATION By GYOOSEOK KIM August, 1998 Chairman: Dr. Paul A. Fishwick Major Department: Computer and Information Science and Engineering Model validation is a complicated and multifaceted procedure in which all possible information such as real-world data, other model input-output data, compiled knowledge in the form of mathematical models and expert opinions can be made consistent. Even though validation using real-world data provides the ideal case, in reality, obtaining such data is not always possible. In this dissertation, we assume that expert opinion is presented in the form of fuzzy rules. The fuzzy simulation approach presented here provides a mechanism for directly encoding uncertainty from human reasoning into a computer simulation by mapping the fuzzy linguistic values of the expert's rules into simulation components. To perform a quantitative comparison between the two kinds of models, quantitative measures have been formulated to gauge the sources and the degree of inconsistency. Using the fuzzy simulation approach and the quantitative measures, this method provides an interactive environment for isolating inconsistency between the fuzzy rule-based model and the vi quantitative model allowing some degree of consistency. This environment also facilitates humans in the ioop resolving inconsistency by helping them to identify and revise the most inconsistent component rapidly and then analyze the effectiveness of that modification. Through checking consistency between these two knowledge representations, our approach serves as a method of checks and balances to enhance the complex model validation process. vii CHAPTER 1 INTRODUCTION While model verification deals with building the model right, model validation deals with building the right model [1]. Validation is concerned with determining whether the model is an accurate representation of the system under study [29]. Model validation is part of the total model development process, and it consists of performing a series of tests and evaluations within the model development process. This validation process is multifaceted and involves the minimal procedure of taking a set of real-system observations and rectifying these observations with an assumed mathematical model or vice versa. This process involves an estimation of parameters of the model that yields a model that best reflects the real-system behavior. In practice, validation is a complicated process, since there may be numerous sources of knowledge comparable with the model output that is under investigation. For many fields, most notably medicine, sociology and economics, there may be a plethora of knowledge and data. This knowledge may be quantitative or qualitative, or both, with experts providing opinions that must somehow be reconciled with quantitative proposed models known to characterize a similar real-world problem. However, currently no algorithm or procedure is available to identify suitable validation techniques [45]. Moreover, one does not always have a complete set of data and a set of models waiting to be identified. The data may be just as incomplete as the model suppositions. The difficulty of achieving the data validity for model validation is discussed in [29]. We have created a methodology that enhances the validation process for such situations. In particular, our method assumes that there is at least one fuzzy rule-based 1 2 model and one proposed quantitative model. The rule-based model is composed of a set of N IF-THEN statements, and the quantitative model is defined as a model whose parameters and state variables take real values. Our method locates inconsistencies between these two representations, and we created an interactive tool for partially resolving inconsistencies. Comparing and contrasting the expert rules with the quantitative model is viewed as being an integral part of an ongoing system validation procedure. 1.1 Problem Statement Knowledge about a given physical system is often obtained from experts in the form of rules. Although the rule-based model is occasionally associative or shallow in nature, this model can easily capture human heuristic and problem-solving knowledge in an efficient way [21, 5, 48]. Fuzzy set theory [57, 58, 59, 27, 14, 61] provides linguistic IF-THEN rules to make use of such experts' knowledge and experience naturally. Through a suitable fuzzy inference scheme, the rule-based system provides a solution without exploiting underlying causal relations on which the solution is based. In some cases, a quantitative model exists which represents all or part of the behaviors of the physical system. This model can be a mathematical model where the system behavior is characterized by one or a series of equations or inequalities. Or this model may be a simulation model where each change in the status of the system is captured over time. This kind of quantitative model provides deeper and more theoretical knowledge when expert system developers need to find solutions for technical problems [5, 16, 48]. To compare and contrast the qualitative model with the quantitative model to serve model validation, we should provide answers for the following fundamental questions: how much do the models differ? and how can one address the difference 3 Consistency Deduction Figure 1.1. A knowledge acquisition cycle between the two models? One way of handling the inconsistencies between the two different levels of models is to form a knowledge acquisition cycle as in Figure 1.1 [5]. Approaches for creating model bases are discussed within the context of computer simulation [15, 18]. For example, the model base represents compiled knowledge about many domains, such as a mathematical queuing model for waiting line problems. If a match is found during the model base search, then shallow rules (or input-output pairs) are generated by means of deduction process. For such a process, exhaustive simulations are executed based on this deep model. Since the size of the shallow rules in Figure 1.1 resulting from the deduction process is usually too large for a human to study and validate against the original expert's rules, fuzzy induction or fuzzy system identification methods [31, 51, 38, 50, 49] can be employed to obtain a more comprehensible and generalized set of linguistic rules. After completing these processes, the rules obtained as a final product become suitable for a human to study and compare against the expert's original rules. However, the fundamental problems of the knowledge acquisition cycle mentioned above can be described as 1. First and most importantly, forming the above knowledge acquisition cycle to check consistency requires a series of difficult tasks as well as a long process. 4 2. For the direct comparison between the generalized rules in Figure 1. 1 and the expert rules, the two rule sets should have the same rule structure, linguistic variables and values. This means that fuzzy induction or identification methods should take the expert's rules into account when they generate the rules. However, most approaches take numerical input and output data to generate fuzzy rules without considering the expert's prior knowledge about the system. Consequently, it is difficult to measure the inconsistency between the two models quantitatively. This could make a decision-making process quite complicated if one wants to resolve inconsistency, especially when a large portion of knowledge components shows inconsistency. 3. Identifying fuzzy rules using only numerical data without properly assumed rule-structure information easily suffers from the curse of dimensionality [32], in which the number of possible rules increases exponentially with the number of possible variables. Moreover, searching for influential variables among all possible variables based only on the numerical data causes the problem of combinatorial optimization. 4. For the deduction process in Figure 1.1, all fuzzy simulation approaches [4, 46, 17] require that proper linguistic definitions (i.e., fuzzy membership functions) be defined a priori before performing fuzzy simulation. Since such definitions of the linguistic terms are difficult to obtain even from experts because of the uncertainty arising from linguistic vagueness, searching for the proper membership functions belongs to the general problem area of knowledge acquisition within the underlying framework of fuzzy set theory [27]. Thus, if such definitions are not available a priori, this additional knowledge acquisition process will clearly become a bottleneck in the cycle shown in Figure 1.1. 5 1.2 _Purpose of Research This research is concerned with devising a method for isolating inconsistency between the two different levels of models in an efficient and systematic manner. To achieve this goal, we have developed a knowledge acquisition cycle as shown in Figure 1.2 [26, 24, 25]. Using this approach, we can alleviate the problems discussed in the previous section. " First of all, the knowledge acquisition cycle presented here forms a simple process compared to the one in Figure 1.1. The fuzzy simulation approach introduced here directly encodes uncertainty arising from human linguistic vagueness into simulation components and utilizes quantitative models for the deduction process. Since this method accepts the linguistic values in the expert's rule premises as simulation inputs and produces linguistic values as outputs using the same terminologies that the expert used in his (or her) rules, the direct comparison between these two models is possible without an additional induction step. " Next, by taking into account the expert's prior knowledge about the system, the major variables affecting the system don't need to be identified. This alleviates major problems such as high computational complexity and local optimality, which usually arise from the structure identification process. " Finally, to alleviate the knowledge acquisition bottleneck, this method allows users to employ various levels of estimates depending on the linguistic vagueness in their rules. As shown in Figure 1.2, this method isolates inconsistency through the two phases: 1) consistency checking and 2) interactive user control. In the consistency checking 6 Expert's Mondel. bas-esearch Deep rules model Isolating Deduction by fuzzy simulation inconsistency (consistency checking and interactive user control)Shlo (:rules Figure 1.2. A knowledge acquisition cycle using fuzzy simulation phase, depending on the expert's various levels of linguistic vagueness (for example, central points, intervals or fuzzy sets), the result of fuzzy simulations is directly compared against the expert's rules (when fuzzy sets are provided), or fuzzy set definitions, which properly fill in the expert's knowledge, are searched through incremental optimization over fuzzy space (when central points or interval estimates are provided). In both cases, the method suggests the source and the amount of inconsistency to users using quantitative measures we have formulated. If the amount of inconsistency exceeds a reasonable range, human intervention is possible via interaictive user control: users may modify either expert rules or simulation model components to reduce the amount of inconsistency. Even at this point, the quantitative measures mentioned above help users identify and revise the most inconsistent component rapidly and analyze the effectiveness of that modification. This humanmachine interaction allows the two models to gradually reach a consensus with a high resolution. O'Keefe [35] pointed out that this kind of visual interaction is one of the most promising validation techniques in expert systems. 7 1.3 Contribution to Knowledge The primary contribution of this work is that, through checking consistency and resolving inconsistency, our approach serves as a method of checks and balances during the model validation phase of system analysis. During this process, we provide benefits to expert systems from simulation and benefits to simulation modeling from expert knowledge. In particular, when expert system researchers are studying the acquisition of deep knowledge from an expert or validating the expert's knowledge against quantitatively compiled knowledge, the first type of benefits can be obtained from simulation models [16, 15, 17, 54, 34]. The advantage from the reverse process is also obtained when simulation model validations are performed during the simulation modeling process with the aid of the expert knowledge [54, 6, 44, 45]. Additionally, using this method, we can contribute a second benefit from automatically generating fuzzy membership functions where expert rules and quantitative models match maximally. Most methods for constructing fuzzy membership functions [27, 42, 41, 9, 52, 2, 3, 33, 19] rely on a set of sample data (i.e., pairs of an element and its membership degree) from an expert's (or experts') opinions before applying curve-fitting or learning methods. However, gathering such information is not a straightforward matter, even by domain experts, if the system's input, process and response are too complex. If a quantitative model exists for the system, the method presented in this paper allows us to utilize natural rules from the domain experts instead of the restricted sample data to construct approximate fuzzy membership functions. 1.4 Outline In Chapter 2, we discuss some issues arising from the modeling process which appears in simulation and fuzzy set literature. Here, we address the importance of 8 the expert's role for validating simulation models and for constructing fuzzy models. Then , we review the fuzzy set theory that is relevant to this research and its relation to computer simulation. First, we explain the basic definitions of fuzzy set theory and various approaches for constructing fuzzy membership functions. Since one of the most successful application areas of fuzzy systems is the fuzzy controller, we present its overview with its key components. Finally, we briefly discuss the fuzzy simulation approaches which appear in both fuzzy set and computer simulation literature. In Chapter 3, we present a new fuzzy simulation approach. We first assume three forms of experts' rules as the inputs to the fuzzy simulation. Then, we propose three different fuzzy simulation algorithms for directly encoding the fuzziness in the expert rules into computer simulation. In Chapter 4, based on the fuzzy simulation method discussed in Chapter 3, we describe an interactive environment that we've developed for isolating inconsistent knowledge. First, we introduce two measurements of inconsistency that we've employed for quantitative comparison. Then, we provide two consistency checking procedures to handle various forms of experts' estimates on linguistic rules. Finally, we analyze the time complexities of these procedures. In Chapter 5 and Chapter 6, we consider a steam-powered propulsion ship (FULTON) and a predator-prey population, respectively, to illustrate the applications of the methodology. Finally, future work and conclusion are presented in Chapter 7 and Chapter 8, respectively. CHAPTER 2 BACKGROUND AND RELATED WORK 2.1 Modeling Process in Simulation and Fuzzy Set Literature 2.1.1 Simulation Modeling Process Figure 2.1 [44, 45, 7] shows the general simulation modeling process and its relation to validation and verification. The problem entity is the real or proposed system to be modeled. The conceptual model represents the mathematical, logical or verbal representation of the problem entity, and this model is developed through an analysis or modeling phase. The computerized model represents the conceptual model implemented on a computer, which is developed through a computer programming and implementation phase. Inference about the problem entity is obtained by conducting computer experiments on the computerized model in the experimentation phase. The conceptual model validity determines the validity of the underlying assumptions and theories by using mathematical or statistical methods. Additionally, the process is concerned with whether this specific model's representation of the problem entity being modeled and its structure, logic and mathematical and causal relationships are reasonable for the intended use of the model [44]. For such a process, techniques such as face validation and traces are used. Face validation involves having domain experts evaluate the conceptual model to determine if they believe it is correct and reasonable for its purpose. This usually means examining the flowchart, graphical model or the set of model equations. In the traces the behavior of different types of specific entities in the model are traced through the model to determine if the model's logic is correct and if the necessary accuracy is obtained. 9 10 Operational Analysis Conceptual Validity Experimentation and Model Data Modeling Validity Validity) Computerized Computer Programming Conceptual Model ~---------------------- --Model and Implementation E Computerized ModelIVerification Figure 2.1. Modeling process and its relation to validation/verification Computerized model verification determines that the implementation of the conceptual model is correct. The verification techniques used here can be found in the software engineering field, such as formal correctness proof, structured walk-through, top-down and button-up testing [47]. Operational validity is defined as determining that the model output has sufficient accuracy for its intended purpose. Most of the validation efforts take place in this stage. Any discrepancy found may be due to an inadequate conceptual model or an improperly implemented conceptual model. In this case, one set of system data is used for calibration of the simulation model, and another independent set is used for validation. If the simulation output data agree with the system output data, the model can be considered valid. For this reason, data validity is needed for comparing 11 the problem entity's behavior with the model's behavior, as well as for building the conceptual model, for developing theories and for testing the underlying assumption. However, in determining operational validity, it is difficult to use the classical statistical test (t, two-sample chi-square, etc.) between the model output and the corresponding system output data, due to the nature of the data [29]. Specifically, " an observation may be nonstationary: the distribution of the successive observations change over time. " an observation may be autocorrelated: the observations are correlated with each other. Therefore, techniques commonly used for operational validity are [45] " face validation, where experts are asked to make subjective judgements on whether the model has sufficient accuracy, " statistical tests for confidence intervals and hypotheses and " turing test, where individuals knowledgeable about the system are asked if they can discriminate between system and the model outputs. However, the statistical methods mentioned above also have difficulties in some applications, such as military or manufacturing systems, due to the paucity of real-world data. Because of the difficulty in obtaining data validity, human knowledge about the system takes a relatively important role during the entire validation process, where approaches such as comparison to other models on graphical displays, intuition, opinions or past experience are usefully adopted [6]. The importance of human information and knowledge representation in problem solving tasks are discussed in the field of Systems Engineering [43]. Possibilities for expert aids in model validation are presented in [44]. 12 Figure 2.2 Fluzyerngiern 2.1.2 Fursy Modlin Processule gienphsialsyte s Tebscuiofteuzfuntion appoiainilh F THENruleanIthequaifthen aprxmtindpnsinhwsatterlsae makey tetnning rocess hadermtereafgtermnolg tihe hes andcotrtgruls coverng pyial inputoutpu Tae spce efine by the zucatin production of the Ipu pattern and the output pattern easily causes the rule explosion. For these reason, fuzzy engineering mainly concerns itself with developing efficient tuning algorithms or finding optimal set of rules. Figure 2.2 shows a general process for modeling fuzzy systems. The modeling process can start with numerical input-output data. Fuzzy system identification, often called black box modeling, is the most common approach for designing mathematical models of dynamic systems from numerical input-output data. The black box modeling contrasts with the white box modeling, where mathematical models are driven from already known physical laws [32]. The derivation of black box fuzzy models 13 from numerical data employs various clustering methods, neural networks and genetic algorithms to identify fuzzy rules with their membership functions. The recent achievements and comprehensive discussion of the black box modeling can be found in the literature [23, 36]. Their main tasks are structure and parameter estimations. However, the fundamental limitation in using numerical data comes from the curse of dimensionality problem [32]. That is, the number of possible rules, n, increases exponentially with the number of possible variables, r by the formula r n J H k, (2.1) k=1 where nk is the number of fuzzy values in the kth variable. Since all variables are not important to describe the behavior of the underlying system, searching for influential variables among all possible variables causes the problem of combinatorial optimization. Therefore, the modeling process using these techniques suffers from computational complexity or local optimality [28, 32]. In contrast, the modeling process can also start with a first cut fuzzy system as shown in Figure 2.2. In practice, the first cut fuzzy system can be directly obtained from the expert in the form of a verbal expression. The domain expert forms rules by performing the following three steps [28]: (1) selecting input and output fuzzy variables, (2) selecting fuzzy values of these variables and (3) constructing fuzzy rules by relating the input values to the output values. Rough fuzzy rules also can be obtained from numerical input-output data through unsupervised learning by clustering the data; this gives quick approximation, but is less accurate. The utilization of prior knowledge provided by experts alleviates many problems arising from using traditional system identification, because important variables of the system and their approximate forms of values are provided by the experts. Another advantage of utilizing the expert's prior knowledge is that some system phenomena such as those cannot be revealed by means of collecting the physically observed 14 input-output data can be captured. However, the first potential drawback of using the expert's knowledge is that it is difficult for the human expert to capture all causal system relations, especially in modeling complex systems. Even the expert is often unaware of all of the contextually dependent factors qualifying generalizations [13]. Secondly, obtaining such a good quality of first cut fuzzy rules is not easy in practice. Generally speaking, the gap between the implementation level (i.e., fuzzy rules) and the expert's knowledge is too wide. Specifically, both linguistic imprecision and uncertainty may exist in a fuzzy rule [22]. For example, given a fuzzy rule, IF wind is high THEN the sailing should be good (0.8), the linguistic imprecision comes from the fuzzy value "high" and the uncertainty comes from the numeric value (often called confidence factor) "0.8." The linguistic imprecision results from linguistic inexactness, specially, the linguistic vagueness in which the boundary is not clearly defined [22, 621. Uncertainty can arise because reliable information cannot always be gathered to assign a probablistic uncertainty; moreover, even an expert may be unsure of a particular piece of causal information. Much of the uncertainty in such cases is possiblistic rather then probabilistic in nature [60]. 2.2 Fuzzy Set Theory and Its Application This section presents a review of the relevant aspects of fuzzy set theory which form the basis of our fuzzy simulation. The theory of fuzzy sets can be found in References [57, 58, 59, 27, 14, 61]. Generally speaking, fuzzy sets may be viewed as an attempt to deal with a type of imprecision arising when the boundaries of classes are not sharply defined. A fuzzy set A of a universe of discourse X is characterized by a membership function AUA : -4 [0, 1] which associates with each element x of 15 X a number IIA(X) in the interval [0, 1] which represents the grade of membership of x in A. 2.2.1 Notation, Terminology, and Basic Operations " Definition 2.1: A fuzzy set A of the universe of discourse X is convex if and only if for all X1, X2 in X IIA(Ax1 + (1 - A)X2) >- Min(PI~A(X1), ,uA(X2)), where A E [0, 1]. " Definition 2.2: A fuzzy set A of the universe of discourse X is called a normal fuzzy set if ]xi E X, PA (xi) =1. " Definition 2.3: A fuzzy number is a fuzzy set in the universe of discourse X that is both convex and normal. To simplify the representation of fuzzy sets, a finite fuzzy set, A, of X is expressed as A = /IA(X1)XX + IIA(X2)/X2 +. + JLA(Xn)X, or A = E', PLA(XX)Xi, where + sign denotes the union rather than the arithmetic sum. If the fuzzy set, A, is not finite, A may be represented in the form A fXPLA(X)/X in which the integral sign stands for the union of the fuzzy singletons ILA(X)/x. " Definition 2.4: The height of fuzzy set A, heightA, in the universe of discourse X is the supremium Of JLA(X) over A. Formally, heightA =SUPXEX[LA(X). (2.2) " Definition 2.5: The complement of A is denoted by A and is defined by A= J(I - PA (X)) /X. (2.3) 16 The operation of complementation corresponds to negation. " Definition 2.6: The union of fuzzy sets A and B is denoted by A U B and is defined by A u B (PJ OI(X) V IB (X)) /X, (2.4) where V is a maximum operator. " Definition 2.7:- The intersection of fuzzy set A and B is denoted by A flB and is defined by AflB J(A(x) A ItB(X))/x, (2.5) where A is a minimum operator. " Definition 2.8: If f is an n-ary crisp function which is a mapping from a Cartesian product X, x ..x X,, to a space Y, and if A is a fuzzy set in X1 X ... X Xn, which is characterized by a membership function A(XI,... ,I with xi, i =1, . . ,n, denoting a generic point in Xj, then extension principle [59] says It is assumed that the membership function of A is expressed by where ItAj, i ,Z n, is the membership function of Aj. *Definition 2.9: Let A and B represent two fuzzy numbers and let * denote any of the four basic arithmetic operations. Then, using the extension principle 17 (2.6) under the assumption (2.7), we define fuzzy set, A *B on R?, where R is a set of all real numbers, as [LA*B(z) mTaxz=.*y(A(x) A I'B(Y)), (2.8) for all z E R1. Thus, for example, if A, B C 7?. are two fuzzy numbers with respective membership functions PA (x) and 'B (y), then the four basic arithmetic operations, i.e., addition, subtraction, multiplication and division give, for each x, y, z C R?, the following results: PLA+B (Z) = mnaxz~x+, (PA (x) A [AB(Y)), (2.9) IA-B(z) = maxz=._y(IA(x) A IIB(Y)), (2.10) ILAxB(Z) = maxzxxy(IIA(x) A IIB(Y)), (2.11) /IA+B (z) = Tnax,.,,_-y(PuA(x) A IPB(Y)). (2.12) *Definition 2. 10: Let P be a compound statement of the type (X is A) *(Y is B), where X, Y =fuzzy variables that take real numbers from some universal set X, Y, respectively, A, B =fuzzy values on X, Y, respectively, and * = a conjunction (and) or a disjunction (or). When * is a conjunction, the rule of conjunctive composition [61] states that P can be expressed by a possibility distribution 7r (x, y) which is defined by {ILAB (X, Y) /(X, Y) I X E X,y YG Y1, (2.13) where [Ax B (X,Y) = min(LA (X), PB (y)), and x = Cartesian product. 18 When * is a disjunction, the rule of disjunctive composition [61] states that P can be expressed by a possibility distribution 7r(x, y) which is defined by Equation (2.13), where LAX B (X, Y) =max (PuA(X), IiB (Y)) . " Definition 2. 11: A fuzzy relation [58] ft from a set X to a set Y is a fuzzy subset of the Cartesian product X x Y. R is characterized by a bivariate membership function IIR(X, Y) and is expressed RIfpR,(x, y) /(x, y). (.4 " Definition 2.12: A fuzzy conditional statement, IF X =A THEN Y =B, or for short, A =* B, in which A and B are fuzzy sets, can be defined as a fuzzy relation A x B. " Definition 2.13: If ft is a relation from X to Y and S is a relation from Y to Z, then the max-mmn composition [58] of ft and S is a fuzzy relation denoted by Rto S and defined by ft oS =fX XZ Vy ([PR(X, y) A us (y, z)) /(x, z), (2.15) where VY is the supremurn over the domain of y. * Definition 2.14I: In traditional logic, one of the main tool for reasoning is modus ponens, that is, (A A (A = B)) z:;> B. Then an approximate extension of the modus ponens, called generalized modus ponens can be used for approximate reasoning in the following type of inference Premise: X is A' Implication: IF X is A THEN Y is B Conclusion: Y is B', 19 where the conclusion B' can be derived by min-max composition rule of inference B' =A'o [A xB]. (2.16) 2.2.2 Membership Function Construction Numerous methods for constructing membership functions have been described in the literature [27, 42, 41, 9, 52, 2, 3, 33, 19]. All these methods can be classified into two approaches: direct and indirect methods. Both methods are further classified depending on whether one expert or multiple experts are involved. In the direct method [27], given a fuzzy set A, an expert is expected to assign a membership degree, IIA(X), to each element x, according to his or her opinion. This can be done by defining a complete membership function in terms of mathematical formula, or by exemplifying it by answering a question such as "what is the degree of membership of x in A?" In the indirect method, an expert is required to answer simpler questions which are easier to answer and less sensitive to the various biases of subjective judgement. The most common approach is based on pairwise comparisons [42, 41] of relevant elements, which replaces the direct estimates of membership degree. An example of typical question in this method is "which color, A or B, has the property of darkness more strongly, and how much more?" For all pairs of elements, comparisons are repeated by giving numerical scale to express the relative strength of the property. The relative weights are represented by a nonsymmetric full matrix. Then the membership degrees are the components of the eigenvector corresponding to the maximum eigenvalue. In this section, we briefly review basic methods for constructing membership functions from sample data gathered from the direct or indirect method. These include the curve-fiting method and learning from neural network. In these methods, 20 we assume that n sample data < xi a,>, Lagrange Interpolation Lagrange interpolation is a curve-fitting method in which the constructed membership function is represented by a polynomial form defined by f (x) =aiLi(x) + a2L2(X) + ... + anLn(X), (2.18) where Li(x) -(x - a) ... (x - ai)(x~ - ai+1) ... (x~ - an). (2.19) Since values f (x) need not be in [0,1], the following formula is applied to make the fuzzy set A normal: IIA(x) =max[0, min[1, f (x)]]. (2.20) Even though the membership function matches the sample data exactly, complexity increases with the number of sample data. Besides, for the values of x outside the given sample range, this method does not work well. This requires that sample data be well distributed over the fuzzy set A. Least-square Curve Fitting Given sample data (2.17) and a suitable function f (x; a, 1,.),where a,1, are parameters whose values distinguish function in its class from one another, this method selects a function f (x; a0o, io, ...) from the class for which E V f(xi; a, /3, .)-a]2 (2.21) 21 reaches its minimum. Then, we apply Equation (2.20) to make a normal fuzzy set. Such a suitable function is chosen from standard distributions based on expert's experience or experimental comparison with other classes. The bell-shaped function defined below has been frequently used for such a purpose. where a~ is a location for center, V3/2 defines the inflection points, and -y is height parameter. Learning from Neural Networks The literature dealing with the use of neural networks for constructing membership functions is rapidly growing [9, 52, 2, 3]. Construction of such functions are done by learning patterns from sample data defined in (2.17). Let each input of x be xP its expected output be P' and its actual output be yP. Given suitable hidden layers and activation function, by initializing weights of the network and applying pairs < x'P, tP > of the training set to the neural network, we can calculate the square error. EP, !(yp _ tp)2. (2.23) Then, to minimize the 2,, we update weights according to backpropagation algorithm. At the end of each cycle, a cumulative error defined as E2 = _Z:(yp _ tp) 2 (2.24) 2P=1 is compared against Emax specified by user. A new cycle is initiated until E < Emax. When E < Emax, the desired membership function is obtained. 2.2.3 Fuzzy Controller A fuzzy system is any system whose variables range over states that are fuzzy sets. The most successful application area, of fuzzy system has been the area of fuzzy 22 FUZZY CONTROLLER Action Defuzzification control. ~ ~ ~ ~ Fzz FuzFotolesaeseilzxetsseszntesnethtecy a knowledgeds ersntdb uz inference rulesadnifencegn.Fuz problems henn [10]: enough fonraltimeoprtin showse the gnea architecture of a fuzzy controller knwthis bseionrescusthe deinbrcssoy fuzzy coeec ulsada nrlle Lnet suonsde avrsiefzycontrollers ase shal otliigkowneg iniFigued 2.4. Ihisa figutre, the control m on or o th control variables a hre oineinedasaoerruetenh acta vauemaia oe of the otrllnovrib essan is desre vamle, and be enolutes theirate 23 Disturbances Condition C C Figure 2.4. A simple fuzzy controller NL NM NS AZ PS PM PL (Negative (Negative (Negative (Approximately (Positive (Positive (Positive 0.0 -d -2d/3 -d/3 0 d/3 2d/3 d Figure 2.5. A fuzzy quantization of change in the error. Depending on e and , relevant control actions represented by v are produced. Step 1: Fuzzy Quantization The first step is to obtain fuzzy quantizations by identifying relevant input and output variables and their ranges and selecting appropriate labels (i.e., fuzzy sets) for each variable. The number of labels associated with a variable is generally an odd number between 5 and 9 [10]. For the reason of symmetry, anl odd number is preferred. The number of labels determines the expressiveness and the predictivertess of the fuzzy system [31]. The expressiveness is a measure for the information content that the model provides, while the predictiveness is a measure for its forecasting power. Since these two measures are contradictory, we should compromise. Some observations have been reported in which either three or five labels were about optimal in most 24 1.0 --- - - - - - 0.0 A -d Xcd Figure 2.6. A fuzzified measurement practical applications [8, 53]. Figure 2.5 shows an example of the quantization where triangular fuzzy numbers that are equally spread over the range [-d, d] are used. Step 2: Fuzzification In this step, a fuzzification function f for each input variable is chosen to express measurement uncertainty. For example, by applying an appropriate fuzzification function, a measurement e =xO can be defined by a fuzzy set as shown in Figure 2.6, where E is determined in the context of each particular application. This fuzzy set acts as a fact in the inference process (Step 4). Step 3: Obtain Conceptual Model In this step, a conceptual model is obtained in terms of a set of fuzzy inference rules that describe the action taken on each combination of control variables. Two common ways for obtaining such information are: 1) from human operators or 2) from empirical data by suitable learning methods [31, 51, 38, 50, 49]. The canonical form of the inference rule is IF e is A and is B THEN v is C. Generally, the number of rules required depends on the number of control variables [10]. For example, if a fuzzy controller requires ni control variables and m fuzzy regions for each variable, the system generally requires inrL rules for a total of mn 25 e e V NL NM INS AZ Ps PM PL NL PL PM AZ NM NS {PM Ps AZ AZ PM Ps AZ NS NM PsAZ NS NM - AZ {NM NL Figure 2.7. An example of fuzzy inference rules possible input combinations. Since the number of fuzzy rules grows exponentially with the number of system variables, the search for optimal rules forms one of the main research areas of fuzzy engineering. However, most applications so far have had few inputs and outputs, and this has helped keep the rule explosion manageable [28]. Figure 2.7 shows an example fuzzy rule base in fuzzy sets defined in Figure 2.5. Step 4: Design an Inference Engine Here, to determine the resulting fuzzy set in multiconditional approximate reasoning represented by the form Rule 1: IF e is A, and is B, THEN v is C1 Rule 2: IF e is A2 and is B2 THEN v is C2 Rule n: IF e is A,, and e is B,, THEN v is C,, Fact: e is A' and C' is B' Conclusion: v is C', 26 Ale e B, C1 eeLV e 2 e B2 C2 -- - - - - - - - - - - - - - - - - - - - -- .----<, A CC ee AMvM e=A' e=B' Figure 2.8. An illustration of the min-max composition rule of inference a method called min-max composition rule of inference defined in Definition 2.14~ is commonly applied. That is, through mmn inference, each output membership function is cut off at a height corresponding to the minimum degree of the truth of the rule premise. Then, through max composition, a combined output membership function is constructed by taking pointwise maximums over all of the fuzzy set assigned to the output variable. An illustration of the method for two fuzzy rules is given in Figure 2.8. Step 5: Defuzzification This step performs defuzzification by converting output fuzzy set into a single real number. Two most common methods for defuzzification are centroid and composite maximum. The centroid method takes the center of gravity of the output fuzzy set C' in Figure 2.8. The defuzzified value d is calculated by the formula d - tc, (Vi) x Vi2.5 27 where vi, i =1I... n is an element in fuzzy set C'. In the composite maximum method, a defuzzified value d is defined as the average of the smallest value and the largest value of v for which J-Lc'(v) is the height of C, heightc'. Formally, d=minjvj vi E MI + maxfv, v, E M} 2.6 where M ={vi I IMc'(vi) =height c'} 2.3 Fuzzy Set Theory in Computer Simulation Probability-based methods are useful when most of the uncertainty can be effectively described through the use of large data sets and their associated moments. However, experts often do not think in probability values, but in terms such as much, usually, always, sometimes, etc. In domains where estimation or measurement of probabilities is not amenable, fuzzy set theory offers an alternative [22]. Here, we can use any type of fuzzy number, such as an interval-valued fuzzy number, a triangular fuzzy number, a trapezoidal fuzzy number or a general discrete (or continuous) fuzzy number, depending on the degree of uncertainty. Owing to the extension principle [61] in the fuzzy set theory, nonfuzzy mathematical structures can be made fuzzy. Here is a sample of how this relates to simulation. The following can be made fuzzy [58, 18]: 1) a state variable value including initial conditions, 2) parameter values, 3) inputs and outputs, 4) model structures and 5) algorithmic structures. For example, we can use fuzzy simulation to execute a fuzzy automaton, where its states are characterized by fuzzy sets, and the production of responses and the next states are facilitated by appropriate fuzzy relations [27]. For another example, a fuzzy algorithm, defined as a ordered set of fuzzy instructions, can be used to provide an approximate analysis of systems and decision processes 28 that are too complex for the application of conventional mathematical techniques [58]. Three kinds of fuzzy simulation approaches have been reported in the simulation literature: Qualitative Simulators (i.e., Qua.Si [4]), Fuzzy Qualitative Simulation (i.e., Fusim [46]) and Correlated Uncertainty method [15, 16, 17]. The Qua.Si and the Fusim are useful for qualitative simulations where simulations are performed using fuzzy sets themselves based on fuzzy calculus or fuzzy arithmetic. The third approach takes fuzzy sets from experts and, through deterministic sampling from fuzzy sets, it performs computer simulation quantitatively on discrete event or continuous models using real arithmetic. For such a process, every vertex in the fuzzy number is issued independently to the simulation function, and the outputs of the simulation are mapped into the most closely matched fuzzy linguistic value by a linguistic approximation. Thus, rules can be extracted, and these results can be validated against the expert's domain knowledge. The fuzzy simulation method that we've employed for isolating inconsistency is an extended version of the correlated uncertainty method. The algorithm for fuzzy simulation using the correlated uncertainty method is [17]: 1. Let a fuzzy simulation component such as a parameter p be defined as a triangular fuzzy number F, where F =(a, b, c). Assume the fuzzy number is identified by brackets(i.e., F[2] =b). 2. ForJEl, 2, 3: (a) Let p[j] =F[j].] (b) SIMULATE REAL (c) Vi, obtain xi(t,)[jI, (a) (C) I 29 (b) line size customer satisfaction short happy medium complacent long irritated customer goeysls (d) line customer (e) da satisfaction goeyslsday size satisfaction happy good Saturday short happy Tuesd complacent good morning morni irritated had Figure 2.9. Information presented by an expert where SIMULATE REAL denotes simulation using real arithmetic instead of fuzzy arithmetic, t, = the end time for the simulation, and xi=the state variables of interest. Because of the SIMULATE REAL, the simulation is accomplished by performing multiple simulations; the number of simulations depends on the order of the fuzzy numbers. The outputs of a simulation can be mapped into the most closely matched fuzzy linguistic values by using a distance metric. Let the simulation outputs of mn ordered fuzzy number be defined as F(1), F(2),... F(m), and let the possible n output linguistic values be qi, q2,.. q,~. Then the distance metric is defined by m MinZIF (i) - qgj) (2.27) where j=1, 2, ... , n. Let's take a simple grocery store example [16] to illustrate the concept of the fuzzy simulation. We assume that an employee (cashier) is an expert, and hie or she provides the information represented as in Figure 2.9. Note that the line 5zze day interamrval rate service rate Saturday Morning short fast Saturday Day short medium Saturday Evening medium medium Tuesday Morning long slow service customer rate satisfaction ay slow irritated ng 30 affecting the customer satisfaction is again influenced by both the interarrival rate and the service rate. However, as in this example, the expert may not be able to represent such complex relations precisely. Here, fuzzy simulation comes into play. By replacing those grocery store statistics with the compiled knowledge of queuing models, we can identify the deep knowledge for such complex relations as a hypothesis of the expert's knowledge. A single server queue as a deep model representing a grocery check out line can be associated with the following pseudo-code: schedule an arrival now; while (not end of simulation) do get-next. event; switch on ARRIVAL: schedule REQUEST-SERVER now; schedule ARRIVAL using fuzzy arrival time; REQUEST-SERVER: if server is free then schedule RELEASESERVER using fuzzy service time; else queue customer; RELEASESERVER: release server to next customer; endwhile By executing the fuzzy simulation, we obtained a total of twelve relations between i nterarrival rate, service rate and customer satisfaction. The entire processes are shown in Figure 2.10. Note that, as with any good analysis, the analyst must ensure that the fuzzy number definitions agree as closely as possible with the expert that issued those definitions through the usual knowledge acquisition procedure where the fuzzy knowledge is first elicited. INPUT: interarrival rate shyrt mepiiumn Ipng UI 5.0 10.0 15.0 20.0 Time INPUT: customer satisfaction happy complacent irritated U A 5.0 10.0 15.0 20.0 line size INPUT: service rate f~st meflium slpw sluggish U 5.0 10.0 15.0 20.0 Time OUTPUT: Relation between interarrival/service rate and customer satisfaction interarrival service customer rate rate satisfaction short fast happy short medium happy short slow irritated short sluggish irritated medium fast complacent medium medium complacent medium slow irritated medium sluggish irritated long fast happy long medium happy long slow happy long sluggish complacent Figure 2.10. Applying fuzzy simulation to grocery store example 31 1. 2. Simulation Linguistic Mapping Fuzzy Simulation4 32 By executing the fuzzy simulation in Figure 2.10, we obtain the quantitative plot of line size over time based on each combination of three linguistic values for the interarrival rate and four linguistic values for the service rate. Sampling is based on the three vertices of the fuzzy number being used to define both the interarrival rate and the service rate. From this, we obtain a time-variant description of customer satisfaction over time. This is done by mapping from data for the line size into the fuzzy linguistic values for customer satisfaction using the distance metric. Output of the fuzzy simulation shown in Figure 2.10 forms a more complete rule base as a hypothesis of the expert knowledge. When we compare them to Figure 2.9 (a) and (e), we can note that there is a conflict between the rules concerning Tuesday morning. The queuing model predicted a happy customer, whereas the expert specified an irritated customer. When such conflicts arise, the expert can either reevaluate his original rule as slightly off or a set of parameters can be changed in the queuing model. In this way, the expert must evaluate the new rules created by the fuzzy simulation to see if there is agreement with his expertise. In this chapter, we reviewed the background that is relevant to this research. We discussed the simulation and the fuzzy modeling processes in general and addressed the importance of the expert's role during model validation processes. In the next chapter, we present a new fuzzy simulation approach to bridge the gap between the expert rules and an assumed quantitative model. The fuzzy simulation approach handles, particularly, the possiblistic uncertainty in the expert's rules by directly encoding the uncertainty into the simulation components. CHAPTER 3 A NEW FUZZY SIMULATION APPROACH The fuzzy simulation approach introduced here has extended the original version of correlated uncertainty method discussed in Chapter 2. In this extended approach, by carrying membership degrees of fuzzy sets in the expert's rule premise and issuing them to simulation components, we are able to calculate a confidence factor for each rule, which is then compared against the confidence factor of the expert rule. This quantitative measurement provides us with useful information such as "which is the most inconsistent rule?" and "how consistent are two given rule sets?" This facility plays a basic role for isolating inconsistent knowledge through interactive user control. In the following sections, we assume three types of expert rules as the inputs to the fuzzy simulation. Then, we present how to handle these types of rules differently using the fuzzy simulation approach. 3.1 Expert Rule Format The input of fuzzy simulation is a collection of expert rules. In what follows, we assume that the three following canonical forms of rules are presented by experts. " IF Xis ATHEN Yis B(CF), " IF X is (A, A2) THEN Y is B (CF) and " IF (X is A) *(Y is B) THEN Z is C (CF), where X, Y and Z =fuzzy variables that take real numbers from some universal sets, X, Y and Z, respectively, 33 34 Table 3.1. Notation Notation Usage RULExpert Rules presented by expert RULEfUZZY Rules generated by fuzzy simulation MWexpert Fuzzy Membership Functions presented by an expert M~ UZ Fuzzy Membership Functions generated by fuzzy simulation MFpremise Membership Functions of fuzzy value in rule premise MF0.nseq Membership Functions of fuzzy value in rule consequence OFexpert Confidence Factor presented by an expert C~UZ Confidence Factor calculated by fuzzy simulation A, A, and A2= fuzzy values on X B and C =fuzzy values on Y and Z, respectively, CF = a confidence factor in the rule consequence given that the premise conditions are satisfied, and *and * =arithmetic (+ ,x or --) and logic (or or and) operator, respectively. We call the first type of rule a simplex rule, and the other type a compound rule. The following is an example of the compound rule with the logic operator and: IF (Temperature is High) and (Pressure is Slightly-Low) THEN Heat-Change should be Slightly-Negative (CF =0.8). The premise parts of the last two canonical types of rules can be combined to make a more complex rule such as IF (X is (A, + A2)) or (Y is (B1 + B2)) THEN Z is C. For simplicity, the notation in Table 3.1 will be used in the entire chapter. 3.2 Fuzzy Simulation The fuzzy simulation method introduced here is capable of simulating the expert rules using quantitative models. For each expert rule, this method takes the premise 35 part and its MFpremise, and through simulation, it generates a conclusion with a CFf,,zz. With the intention of comparing this result directly against the expert rule, the fuzzy simulation method is forced to derive a conclusion with the same linguistic value that the expert presented, but with possibly a different C~UZ from the CFexpert. When the expert rule is simplex, fuzzy simulation involves one simulation for each element within a MFpremise. In contrast, when the rule is compound, we first obtain an intermediate fuzzy set by applying the extension principle defined in Definition 2.8 or the rule of conjunctive or disjunctive composition defined in Definition 2.10, depending on whether the operator type in the premise is arithmetic or logic. Fuzzy simulation using the compound rules involves one simulation for each element within the intermediate fuzzy set. Finally, we calculate a C~u,, for that rule using a weighted average method. 3.2.1 Fuzzy Simulation for Simplex Rules Consider a simplex rule of the type IF X is A THEN Y is B, where X, Y =fuzzy variables that take real numbers from universal set X, Y, respectively, and A, B =fuzzy values on X, Y, respectively. Algorithm of Fuzzy Simulation 1. Let a fuzzy simulation component such as a parameter p be defined as a fuzzy set A, where A =IPA(Xl)/xl + ILA(X2)/x2 + - - + IIA(Xn)/X. Assume the element of A is identified by brackets (i.e., A[2] = X2). 36 membership degree membership degree membership function of P membership function of other fuzzy set Figure 3.1. A two-step process of searching membership degree 2. For 3'E1, 2,... , n: (a) Let p[J] =A[j]. (b) SIMULATE REAL. (c) obtain (IIB(Yj)/Yj)(t,). 3. calculate CFf,,,2y, where SIMULATE REAL denotes simulation using real arithmetic, yj, J= 1, . .. , n denotes real values on Y, and te the end time for the simulation. During SIMULATE REAL, the correlated uncertainty method requires that when we replace p with a real number whose membership degree is d, we should replace other fuzzy simulation components with real numbers whose membership degrees are also d. This procedure involves a two-step process of searching for the membership degree of p, and then using this degree to drive the elements of other fuzzy sets. This process is illustrated in Figure 3.1. In what follows, SIMULATE REAL involves this operation. 37 Calculation of CFf,, Just as CFexpert is presented by the expert, we need a way to obtain CFfU,,,, from the fuzzy simulation. By doing this, we benefit from the comparison of the two rules in terms of their CF values. However, since the derivation of the OFxpert involves a subjective opinion, as well as a certain amount of uncertainty, there is no theoretical formulation to calculate the CFf,,,,, whose derivation process is exactly the same as that of the CFxpert. Our solution is to define an equation so its result agrees with a human intuition as much as possible. We used a weighted average method to create such intuition. Given a simplex rule, we define the C~,,, by using the weighted average method: C~UZ - Ej'1(IA(Xj) X ktB(Yj)) (3.1) Enj=1 PA(Xj) where Xj,J 1, 2,. . ,n, denote real values on X in the fuzzy set A, and yj,J 1, 2,. ., n, denote real values on Y obtained from SIMULATE REAL using Xj. The validity of calculating CFfuz2y using the weighted average method is shown as in Figure 3.2. The C~,,, using Equation (3.1) is 1.0 and 0.0 for Figure 3.2 (a) and Figure 3.2 (b), respectively. The results match our intuition. When the CF falls into some range between the above two extreme cases (i-e., 0.0 and 1.0) as shown in Figure 3.3, we can intuitively say that each member in A supports the conclusion B with a higher confidence, the greater CF we get. Using (3.1), the CFu,2y for Figure 3.3 (a) is: C~f UZZY (0.5 x 0.5) + (1.0 x 1.0) + (0.5 x 0.5) 0.5 + 1.0 + 0.5 0.75, 38 I A -, B 1.0 - - - -1.0 - 0.5---------L.I I X y X1 :r2X-3X4 X5 y1y2 Y3Y4 Y5 SIMULATE REAL E A B 1.0 -- - -1.0 0.5----X T-3 X4 X5 Y1 ,Y2 Y3 Y4 Y SIMULATE REAL Figure 3.2. All members or none of members support the conclusion (a) All members of A support the conclusion B with full confidence; (b) None of the members of A support the conclusion B I A X7 2 -T X4 X5 I B 0.5---------L. L A y1 y2 y3 y4 y5 1.0-------0.5---------L x X1X2 ~AX XX SIMULATE REAL I B 1.0 j--0.7 ------ -0.3 0.2 Y1 y2 y3 y4 y5 SIMULATE REAL Figure 3.3. Some members support Conclusion (a) Members of A support the conclusion B with higher confidence compared to the case of (b); (b) Members of A support the conclusion B with less confidence compared to the case of (a) and the C~,,, for Figure 3.3 (b) is: (0. 5 x 0. 3) + (1. 0 x 0. 7) + (0.5 x 0. 2) 0.5 + 1.0 + 0.5 = 0.48. Example Figure 3.4 illustrates how to perform fuzzy simulation using simplex rules, IF X is X THEN Y is B. The results of SIMULATE REAL are artificially made for the purpose of illustration. Applying Equation (3.1), we obtain C~u, by C~fUZY =(1.0 x 0.5) + (0.7 x 0.5) = 0.21. C~f~~~ -0.3 + 0.7 + 1.0 + 0.7 + 0.3 CFf UZZY - 39 A B 0.7 -- - 0.5 - - - - - 0.3 0 1 23 4 56 15 17 26 35 SIMULATE REAL Figure 3.4. Fuzzy simulation using simplex rule 3.2.2 Fuzzy Simulation for Compound Rules with Arithmetic Operations The extension principle [59] defined in Definition 2.8 is a principle for fuzzifying crisp functions. It can be used to generalize crisp mathematical concepts into fuzzy sets. Owing to this principle, models and algorithms involving nonfuzzy variables can be extended to the case of fuzzy variables. By applying Equation (2.8) to a compound rule with arithmetic operation, an intermediate fuzzy set is obtained, and this set is used for fuzzy simulation. Consider a compound rule of the type IF X is (A, *A2) THEN Y is B, where * is one of the four basic arithmetic operators (i.e., , x ) Algorithm of Fuzzy Simulation 1. Apply Equation (2.8) to the rule premise. 2. Let Z be a resulting intermediate fuzzy set, and let a fuzzy simulation component such as a parameter p be defined as a fuzzy set Z, where Z =PZ(zOI/zi + I.'Z(Z2)/Z2 + - - + 1uz(Z.)/Z. Assume the element of Z is identified by brackets (i.e., Z[2] =Z2)- 3. ForJ E1, 2,. .., n: 40 (a) Let p[j] =Z[j]. (b) SIMULATE REAL. (c) obtain (IIB(Y3)/Yj)(t,!). 4. calculate C~,,y Calculation of C~,,, Given a compound rule with arithmetic operations, we define the C~,,, by using the weighted average method ~ Z E1=(AA,(Zj) X IIB(Yj)) (3.2) where zj, j= 1, 2, . . . , n, denote real values on a fuzzy set resulted from the arithmetic operation, A, A2, and Yj, I 1, 2,.. n r, denote real values on Y obtained from SIMULATE REAL using Example Let's assume that we want to perform fuzzy simulation using the following rule: IF Xis (A, +A2) THEN Yis B, where A, and A2 are defined by Figure 3.5 (a) and Figure 3.5 (b). By applying Equation (2.9) defined by IIA, A,(Z) = maxz=A+A2(LtA,(x) A JLA2(W)I we obtain the following set of equation for the intermediate fuzzy set Z. 1uz(2) =(OAO.6)V(0.1AO.3)V(0.3A0) zz0.1, A1 1.0---------0.8 -- - 0.7------0.3 -- - 0. 1- 2 3 4 I (a) 1.0 0.7 0.6 0.3 0.2 0.1 A2 0 1- 2 3 4 I (b) Figure 3.5. Two fuzzy sets for addition ILZ(3) =(0 A 1) V (0.1 A 0.6) V (0.3 A 0.3) V (0.8 A 0) =0.3, A'Z(4) =(0 A 0.2) V (0.1 A 0.) V (0.3 A 1.) V (0.8 A 0.6) V (1 A 0.) V 0.7 0 06 I-tz (5) = (0 A 0.1) V (0.1 A 0.2) V (0.3 A 0.) V (0.8 A 1.) V (1 A 0.6) V (0.7 A 0.) V .6 (0.3 A 0) =0.8, IpZ (7) =(OA0) V(0.3 A0.3) V(0.7 A0.6) V(1Al1) V(0.8 A0.7)V (0.3 A0.2) V (0.1AO0.1) V (OA0) =1, 'z (8) =(0.1 A 0) V (0.3 A 0.1) V (0.8 A 0.2) V (1 A 0.7) V (0.7 A 1) V (0.3 A 0.6) V (0 A 0.3) =0.7, tZ (9) =(0.3 AG0) V (0.8 A 0.1) V (1 A 0.2) V (0.7 A 0.7) V (0.3 A 1) V (0 A 0.6) =0.7, PZ(10) =(0-8 A0) V (IA0.1) V(0.7 A0.2) V(0.3 A0.7) V(0 A1) =0.3, AuZ(11) =(1 A 0) V (0.7 A 0.1) V (0.3 A 0.2) V (0GA 0.7) =0.2, jiz(12) =(0.7 A 0) V (0.3 A 0.1) V (0 A 0.2) =0.1, 41 I.1) 0.6 (0.5 42 1.0 0.8 0.7 0.6 0.3 0.2 0.1 C Z 1 2 3 4 5 6 7 8 9 10(11I 12 13 2 29 32 1 1 r _________ ____ 42 1 51 57 0 so Figure 3.6. Fuzzy simulation of compound (addition) rule Figure 3.6 shows the fuzzy set Z and the result of fuzzy simulation using Z. Again, the results of SIMULATE REAL was created arbitrarily for illustration purposes. Using Equation (3.2), we can calculate C~,,, by C~UZ -(0.1 x 1.0) (0.3 x 0.6) 0.06. 0.1 +0.3 +0.3 +0.6 +0.8 +1.0 +0.7 +0.7 +0.3 +0.2 +0.1 3.2.3 Fuzzy Simulation for Compound Rules with Logic Operations When the expert's rule premise involves logical operators such as and or or, the rule of conjunctive composition and rule of disjunctive composition [61] defined in Definition 2.10 is applied to obtain a possibility distribution. This distribution is used for fuzzy simulation. Consider a compound rule of the type IF (X is A) * (Y is B) THEN Z is C, where * denotes any logical operator. Algorithm of Fuzzy Simulation 1. If * is and operator, y SIMULATE REAL 25 43 then apply the rule of conjunctive composition to the rule premise and calculate a possibility distribution 7w(x, y). If* is or operator, then apply the rule of disjunctive composition to the rule premise and calculate a possibility distribution 7r (X, y).2. Let fuzzy simulation components such as p and q be defined as fuzzy sets A and B, respectively, where A [tPA(X)/X +[tA(X2)/X2 + .+ A(X.)/Xn, B = AB(Y1)/Y1 + IIB(Y2)/Y2 + +... .)Y. Assume the elements of A and B are identified by brackets (i.e., A[2] X2 and B[2] -Y2)3. For i 1,2,...m FordE1,,.n Let p[i] =A [i]. Let q[j] =B[j]. SIMULATE REAL. obtain (/Ic(zij)/zij)(te)4. calculate C~,,y where m and n are the number of elements in A and B, respectively. Notice that in the rule defined above, the universal set of the fuzzy variables A and B are not identical. Otherwise, instead of 7r(x, y), we can get a more simplified fuzzy set as an intermediate set by Definition 2.6 and Definition 2.7 for disjunction and conjunction, respectively. 44 Calculation of CFf,, Given a compound rule with logic operations, C~,,, is defined by using the weighted average method Cf Zy = Zi'= jl(ItA.B(Xi, yj) X ,uC(Zij)) (3.3) where * denotes logical operator. Example Let's assume that we want to perform fuzzy simulation using the following compound rule: IF (X is A) and (Y is B) THEN Z is C, where A and B are defined as A =small 1/1 + 0.6/2 + 0.1/3, B =large 0.1/1 +0.6/2 +1/3. By applying the rule of conjunctive composition, the predicate (X is A) and (Y is B) yields the following possibility distribution: 7r(X,Y) = {[PAandB(X,Y)/(X,Yl)],[IiA.andB(Xl,Y2)/(X1,Y2)], = [PAdB(X1, Y3)/(XI, Y3)], [IL4AandB(X2, Y)/(X2, Y1)], .... [A n B (X3, Y3)/ (X3, Y3)]}I Let's assume that we have the result as shown in Figure 3.7 after performing SIMULATE REAL on this 7r (x, y). Using Equation (3.3), we can calculate C~uz by (1.0 x 0.5) + (0.1 x 0.8) + (0.6 x 0.3) C~f UZZY - 1.0 + 0.1 + 0.6=04. 45 C 1 .0---- - -0.8 1 - - - - - - - - 0.5 --- - - - 0.3 --- - - - - - - 0.6/ (1,2) 1/103) 0.1/(2,1) SM LT 0.6/(2,2) RA 0.6/(2,3) 0.1 /(3,1) 0.1/(3,2)@0 0.1/(3,3) Figure 3.7. Fuzzy simulation of compound (conjunction) rule In this chapter, we presented a new fuzzy simulation approach. We showed that how the fuzzy simulation handles possiblistic uncertainty by means of three approximate reasoning tools and the weighted average method. In the next chapter, we present an environment for isolating inconsistency between the expert rules and an assumed quantitative model. The fuzzy simulation approach forms a basis for building such an environment. This environment handles the linguistic vagueness discussed in Section 2.1.2 and supports interactive user control. CHAPTER 4 A METHOD FOR ISOLATING INCONSISTENCY The purpose of this chapter is to provide an interactive environment to check for consistency and resolve inconsistencies between the qualitative and the quantitative models. Based on the fuzzy simulation approach discussed in the previous chapter, this environment also handles the linguistic vagueness in the expert rules. The en- /7 N no MF available? yes Isolating Inconsistency ConsitencyHuman Adckr visor -- - - - - - - - Intervention Checking consistency o Apply fuzzy simulation directly Change ys Reporting I consistency o Inconsistent RulesoRue eito Expected Rules o MFs no__________ o Simulation ~moet Conducting o Incremental test o Sensitivity test Reporting __________successful Inconsistent rules yes Suggesting o Approximate MF Done -~~~~~~~ ~ ~ ~ ~ - - - - - - - - - -zzy - - - - - - - - - - - - - Figure 4.1. An environment for isolating inconsistency vironment is shown at Figure 4.1. As shown in this figure, the method for isolating 46 Quantitative Models ---------.omponents ---------- 47 inconsistency consists of three major components: Consistency Checker, Advisor and Human Intervention. Consistency Checker performs two major tasks: checking consistency by applying the fuzzy simulation directly, and checking consistency by identifying approximate MFs. Whenever MFxpert is available (i.e., when the expert provides complete fuzzy sets as estimates for linguistic terms), it performs the first task by comparing the rule consequences from the fuzzy simulation against those from the expert in terms of CF. When the amount of inconsistency is out of range, all inconsistent rules and their expected rules from the fuzzy simulations are suggested through Advisor. When there is no MIF,,xprt a priori (i.e., when expert provides central point or interval estimates for linguistic terms), Consistency Checker performs consistencychecking by identifying an approximate M~,,v Using fuzzy simulation, it first focuses on discovering another important knowledge source - linguistic definitions in expert's rules. It tries to produce approximate definitions where the rules from fuzzy simulation maximally match against the rules from the expert. If such definitions can be generated with a fairly good match between the two rule sets, these linguistic definitions are suggested through Advisor, and consistency checking is finished by returning an answer, "consistent." Otherwise, the answer "inconsistent" is returned, and Advisor performs an incremental test or a sensitivity test to find out the source of inconsistency. Using all informations from Advisor, we start to resolve the inconsistency. For this process, Human Intervention is permitted: either the expert rules (including C~expert, MWexpert) or the simulation components can be modified interactively. Every time these modifications occur, Consistency Checker is reinvoked with visual aids, so that the user can easily recognize the effect of the modifications. Therefore, the overall process now involve humans in the loop during the process of checking 48 consistency and resolving inconsistency. We start with this chapter by introducing the quantitative measurements of inconsistency that we have employed. 4.1 Measurements of Inconsistency The consistency between two types of models can be measured by the difference between CF presented by experts and CF calculated from fuzzy simulation on each rule. For each rule, we define its Local Inconsistency, LI, by LI = CfUZ - CFxpert I. (4.1) Global inconsistency is measured by summing up such differences in the entire rule set. Thus, using the LI, we define the Global Inconsistency, GI, in a rule set by GI ZLh, (4.2) where m =total number of rules. Searching for the largest LI enables us to identify the most inconsistent rule (i.e., the worst case rule) between two different knowledge sources. Moreover, calculating the GI in this way allows us to measure the total amount of inconsistency. Note that given two rule sets and their GI, a slightly better GI does not always mean that the rule set leading to this GI is more consistent than the others. This results from possiblistic uncertainty of confidence factors derived by the expert [60] or measurement uncertainty, due to an inability of a measuring instrument to overcome its limiting finite resolution [27]. Using the Equation (4.2), we will say two given qualitative and quantitative models are consistent if GI < e,(4.3) where E is a consistency criterion specified by user. Any inconsistency found may be due to: 49 Reasonng by Human Consequences MI's from expertIdnicaonf Premises No Reasoning by Fuzzy Simulation Consequences Figure 4.2. Identification process of inconsistent rules " an inadequate conceptual model. Specifically, underlying assumptions, model structure, logic, mathematical relations and particular pieces of causal relationship may be inadequate, " an improperly implemented conceptual model, or " improperly designed expert rules. Specifically, improper rule structure, inadequate fuzzy value boundaries and central points or possiblistic uncertainty (confidence factor) may cause the inconsistency. 4.2 Checking Consistency When MF,,,,t is Available By taking MFxpert as an input of fuzzy simulation as introduced in Chapter 3, we obtain each rule consequence associated with a C~,,y Using the CFxpert and C~,,, pair, we get the LI in each rule by Equation (4.1). Then, the GI is obtained from Equation (4.2). This process is shown at Figure 4.2. The most inconsistent rule is considered the rule which has the largest LI. The GI can be used for a performance index. Thus, when any sources of inconsistent components are modified, a comparison between the current GI and the previous GI indicates whether this modification is a good decision or not. Once the inconsistent rules are 50 identified, we must be careful not to eliminate the possibility that the CFxpert was derived in different manner, and that it is not the same as the weighted average method defined in Equation (3.1), Equation (3.2) and Equation (3.3). For reducing such a possibility, Advisor generates rules for all possible consequences associated with the C~,,y whose values are calculated by the weighted average method. 4.3 Checking Consistency When MF,,p,,t is Unavailable The previous section showed how to check for consistency when linguistic terms are defined a priori. Checking for consistency was possible because fuzzy simulations used the expert's precisely predefined linguistic definitions. The algorithm presented here allows us to check for consistency with minimal information provided, such as central point estimates. In this section, we assume that, at the very least, the central point estimates for linguistic values are provided by the expert. For a complete discussion of the various estimate forms that can be covered in this study, see Section 4.3.3. Checking for consistency is possible by generating a set of M~ll, where RULEexpert and RULEf,,,,y maximally match. If such definitions (i.e., MWfu22y) are generated with a GI of less than consistency criterion, E, RULExprt aind RULEfuzy~ are considered to be consistent with such definitions. However, if any set of M~u, does not lead to a GI of less than E, then this means that we cannot find any linguistic definitions that properly fill the gap between two models. This implies that a discrepancy exists between RULExpert and RULE uzzy. In this case, Advisor performs either an incremental test or a sensitivity test to identify inconsistent rules. 4.3.1 Process in General By forcing fuzzy simulation to produce RULEfuzzy and C f ,, which are maximally close to RULEexpert and CFexpert, respectively, we can discover a set of Wfzz 51 Premises I ~ Re11 nin bFuzzy Simulation III Figure 4.3. Identification process of approximate MFfirzzy between these two different models. This optimization process for generating the M~uz is shown in Figure 4.3. The purpose of this process is not to generate finetuned fuzzy membership functions, but to decide whether any set of approximate membership functions exist in which the two models lead to consistent conclusions. This process can be stated as "search for a set of Wfzz which leads to a minimum GI for all rules, and then check if this GI is less than a given consistency criterion, 4.3.2 Heuristic Function and Search Method for Generating Approximate MF To find such a set of MWfuzz2, our algorithm uses the following heuristic function, goal, and search methods in its tuning process: 9 heuristic function: GI. *goal: minimization of GI. nmTh jlhI y, Hua MFs from expert (possibly exist in mind) Consequences Tuning MFs--K \ A\ *1 A - Consequences Goal. Minimize GI t 52 *search method: gradient descent search for finding a minimum GI by always moving in the direction in which the decreasing rate of change is the greatest. Particularly, for each iteration, - pick the most inconsistent rule and then - pick such a subset of the M~,,, in the rule that increasing the spreads of this subset by Ad reduces the GI to the greatest amount. If Ad is too small, then fuzzy membership functions are adjusted very slowly. If a large Ad is chosen, the convergence may be faster, but optimal spreads may be missed. A possible solution is to pick Ad in an adaptive manner. That is, if the GI is decreasing rapidly, take big steps, but if the GI is decreasing slowly, take small steps. Figure 4.4 shows that how the tuning algorithm uses this evaluation function and the search method to generate the MFf,,,,. In this figure, a rule set is assumed to be composed of three rules, R1, R2 and R3. Our purpose in this figure is to find a set of approximate M~,,,, for all linguistic values that leads to the smallest GI. The tuning process begins with the minimal-size fuzzy sets centered on the central point estimates. Then it increases the size of appropriate fuzzy sets by Ad, where the appropriate fuzzy sets are selected using the above strategy. Let us assume that the first fuzzy simulation for each rule provides the information that R2 is the worst case rule. By selecting the R2, the algorithm encounters states, 2, 3, 4, ..., 8 which denote the every possible sets of MFf,,,,,1 after tuning a subset of the MFf',,,y in R2. In this way, a state is defined as a set of MF1UZ2Y. For example, if R2 contains three fuzzy sets A, B and C, the state 2 through state 8 denote each MWfu2zy after tuning f{Al, {B}, {C}, {A, B}, f{B, C1, {A, C1 and {A, B, C1, respectively. Since the fuzzy simulation using state 4 leads to the lowest G1, state 4 (i.e., M~uz after {C} is tuned) is selected for the next tuning process. In this way, more than zero fuzzy sets 53 are tuned at each iteration of the algorithm. When the algorithm reaches a point where such a modification does not reduce the GI compared to the previous GI, this sequence of tuning stops. However, this algorithm may reach either a local minima, LEGEND I selected path initial R 101, (G=8.0) ,. R3 0 selected state iR2 = worst case rule I 2 3 Ipossible subsets for *..tuning MFs 01=759 0=7.7 01=.09in worst case rule ----------------------'3 R2 R3 RI = worst case rule 910 11 possible subsets for Q.21 * *tuning MFs 01=.3 0=597 in worst case rule Figure 4.4. Heuristic function GI and search for the smallest GI a plateau, or a ridge, because it keeps track of only the current states, and does not look ahead beyond the immediate neighbors of that state. Among many possible ways [20, 37, 39, 40] to deal with these problems, we adopt a random-restart gradient descent search [40] by conducting a series of gradient descent searches from randomly generated promising states, running each until it makes no discernible progress. For implementing this strategy, we employ a promising state criterion, 0 defined in terms of GI, which is used for randomly selecting promising states. For example, promising 54 states can be defined as {states IGI < 0,(4.4) where 0 is specified by the user. To improve the search productivity, we always pick a state which doesn't lead to the same modification which has already been conducted. Putting these all together and using the following example of the expert's rules, we illustrate the random-restart gradient descent search in Figure 4.5. RI: IF A THEN C R2: IF B THEN D R3: IF A and B THEN D In this figure, the set representation in a node denotes one subset of MFs being tuned, and the numeric value in a node refers to a GI after this subset is tuned. Notice that state 21 is rejected, even though it has the smallest GI at that point. Because this path leads to the same sequence of modification, D -+ A -> C -4 B, which already has been performed when we visited state 11, we reject state 21. Before presenting a detailed algorithm for identifying the approximate M~UZY we first introduce the expert's estimates that serve as a clue for locating initial positions of MWfU2ZY. 4.3.3 Various Forms of Expert's Estimates on Linguistic Terms The expert's estimates can be one of the following various forms, depending on the uncertainty about his (or her) linguistic terms: * central point estimates, " interval estimates, " approximate fuzzy membership functions, such as triangular or trapezoid fuzzy numbers and * fuzzy membership functions with their complete definitions. 55 REGEND 8.03 OState leading to the same path I already has been R2 visited (B) D) {,D)GI <6, where 06=6.0 7.89- -- - -- - 6.09 ---- 7.7.6 -Picking the most inconsistent rule 5 6 --~ Random restart JA) C AC) 5.21 6.85.97 R2 RI 89 10 232 (A)IC C)C ID) BD t QR2 6114.3494.845 (B) D BD 44.54 Figure 4.5. Random-restart gradient descent search 56 Type Form of estimate Explanation Discretization into fuzzy space confidence A A central point When experts present 0-------------I estimate the center point c of A C confidence Intrvl he epers reen 10A A IneralWhnexers rset.0-- - 2 of the interval [a, b] of A 2 full confidence with a full confidence I________a b confidence Interval When experts present 1.0f 3 of the min-max range [d, eJ A min-max range of A d e confidence When experts present A both the center point c . and the min-max range [d, el of A Approximate d c e 4 fuzzy number confidence when experts present A both the interval [a, bI 1.0 -of A with a full confidenceI and its min-max range I [d4 eJ d a b e confidence A Complete When experts present 1.0----------5 fuzzy number the complete definition Figure 4.6. Five types of estimates about the linguistic terms 57 Figure 4.6 shows the five types of estimates and the way to discretize such an uncertainty into a fuzzy real space. From expert's point of view, the lower on the scale a type is located, the easier it is to estimate. Note that estimating Type 3 is harder than Type 1 and Type 2, since the exact extreme points of that interval are difficult to determine. The algorithm for generating M~UZ has been devised for handling the first two types of estimates (when Type ~4 or Type 5 is presented, tuning M~u~ itself is unnecessary, because the approximate or complete forms of linguistic values are already given). Even though the algorithm presented in the next section deals with the central point estimates, it can be easily extended to cover Type 2 as well. That is, as shown in Figure 4.7, the meaning of the linguistic term of Type 1 can be approximated by a symmetric, triangular fuzzy number. Likewise, the meaning of Type 2 can be approximated by a symmetric, trapezoidal fuzzy number. The difference between those two types is only the shape of the fuzzy numbers. Therefore, these cases can be handled in the same way, if we represent every fuzzy numbers in our algorithm by 4-tuPle symmetric fuzzy numbers [d, a, b, e] as shown in Figure 4.7. For example, the membership degree of two types in this figure can be derived using the same equations defined by 0 if x < d I- 2(-x) if d < x ed(b-a) I-L W Iif a < x < b ed(b-a) i ~~ 0 if X > e. 4.3.4 Algorithm to Generate MFf,,,,,. As shown in Figure 4.8, the algorithm to generate the approximate Wfzz consists of six steps. All steps are explored in detail. 58 I a~b e d a b e (a) Triangular fuzzy number (b) Trapezoidal fuzzy number Figure 4.7. A 4-Tuple Fuzzy Number (a) Triangular fuzzy number; (b) Trapezoidal fuzzy number Step 3 Tuning MFu7Ide o mrv until best saveiGIdenoimrv *[ Find each possible subset of MIFu Step I in the most inconsistent rule fzy Set initial conditions I Step 2 Modiythis sue Fuzzy simulations for all rules o calculate Lis and GI Fuysimulations for all rules Pick one state leading tominimumGl approimat Wfuzy ys current GI < previous GI with"cosistnt" yes I Step 5 Se Pick the minimum GI so far yes ceacuer f n and MFuz leading to this GI random restarts? Figure 4.8. An algorithm for M~, generation 59 Step 1: Set Initial Conditions The following items are initially set by the user. " a tuning size, Ad, of M~,,,,y " a consisency criterion, E, * a promising state criterion, 0 and " number of random restarts to be conducted. Initial sizes of M~,,, can be determined such that each range of M~,,, is 2Ad, based on a central point estimate. The user can initially set the last three items to certain numbers, but it is better determine this later, because the characteristics of these values can be observed after a few iterations of the algorithm. Step 2: Fuzzy Simulations for All Rules For each RULExpert, a fuzzy simulation is executed as discussed in Chapter 3. Sampling is first done at the central point, then in both directions by increasing and decreasing Ad until right before these points exceed the width of the MFpremi,,. The LI for each rule and the GI for a whole-rule set are calculated by Equation (4.1) and Equation (4.2), respectively. Step 3: Tuning MFf,,, until best saved GI does not improve The purpose of this step is to incrementally reduce a GI by selecting the most inconsistent rule and modifying a proper subset of M~,,, among the all subsets of Mf,_ in the most inconsistent rule until we eventually reach the smallest GI. A series of tuning processes stops at this point. During this step, we maintain two lists: a Visitlu~ and a Randomi~t. The Vi'sit wit is a list to save a state which already has been visited. This list is used to avoid revisiting the same states. The Randomis8t is a list which saves a state whose GI is less than given promising state criterion, 0. 60 This list is used when we randomly pick a promising state for random-restarts. As shown in Figure 4.8, this step consists of four substeps. " Step 3.1: Find each possible subset of MFf,,,,, in the most inconsistent rule. That is, first, by picking a rule which has the largest LI, we find the most inconsistent rule. Next, all possible subsets of the M~,,, in this rule are obtained from the power set of the M~,,y Then, for each subset, we execute Step 3.2. " Step 3.2: For each subset of MFf,,,,y, we increase the elements in each subset by Ad and execute fuzzy simulations for all rules to calculate each LI. " Step 3.3: Pick one state leading to a minimum GI. This state is likely to be the best candidate for reducing GI as a whole. Then, update the Visittit and Randomli8t by adding this state to the Visitlijt, and adding the other states whose GIs are less than 0 to the Randomis5t. " Step 3.4 Compare this GI to the previous GI. The purpose of this substep is to make sure that the best candidate obtained from Step 3.3 actually improves the situation. Thus, a stop condition for the tuning process can be written as Current GI > previous GP? If the condition is satisfied, our algorithm proceeds to Step 4. Otherwise, the current GI is saved as a previous GI, and the entire Step 3 is executed until the stop condition is satisfied. Step 4: Random-restarts This step continues to execute Step 3 with random-restarts, each time reinitializing GI to a maximum value, until the algorithm reaches a user specified number of restarts. Note that e before executing Step 3.1, we randomly choose a state in the RandomlSt and immediately deleted it from the list and 61 ein Step 3.3, the state selected should not be in the Vlisitl1t. Step 5: Pick a State Leading to a Minimum GI Among GIs resulting from a series of random-restarts, pick a minimum GI and its MFfuzzy. Step 6: Return Results Depending on the Consistency Criterion. E If the GI above is less than E, then it means that the algorithm was capable of generating the MFf,,,,y in which two models exist in a consistent manner. In this case, the M~uz is returned as approximate linguistic definitions for both the RULEepert and the RULEf,,,2,. Otherwise, the algorithm returns "inconsistent." However, we may still find a source of inconsistency by performing the incremental test or the sensitivity test discussed in the following section. 4.3.5 Identify Inconsistent Rules Even though any set of M~,,z could not be found with a GI of less than E in the previous section, we may still identify inconsistent rules which are responsible for the failure by using two methods. The first method is the incremental test, and the second is the sensitivity test. Advisor performs either or both tests depending on the user's request. In both tests, one should not assume that a newly identified inconsistent rule is the only rule which needs to be analyzed. All existing rules which may cause conflicts with this rule should also be analyzed to resolve the inconsistency. Note that the tests presented here do not work well when a significant portion of a model is inconsistent with the model to which we want to compare it. Incremental Test In this test, we can identify inconsistent rules by observing the rate of changes of the GI as we add rules incrementally and run the MFfUzZy-generation algorithm repeatedly. When the rate of change of the GI is significantly increased by adding rulek, the rulek can be considered as a more inconsistent rule than previously added 62 rules. Once this rulek is added, observing the rate of change of GI is meaningless, since from this point, the inconsistent component (i.e., rulek) also contributes to tuning the M~UZ into an incorrect direction, thereby making the GI less credible. Thus, before the test, an arrangement should be made so that more reliable rules be employed on the test before less reliable rules. For this reason, the incremental test is order-dependent and requires more heuristics than the sensitivity test discussed in the following section. However, when one or more domain experts are available, we can apply a fuzzy individual Or group Preference ordering method [27] for obtaining the ordering within a given compatibility. Having such information available, the advantage of the incremental test over thme sensitivity test is that it requires fewer computation demands. The algorithm for the incremental test can be described as: 1. Let n be the total number of rules for the incremental test. Assume that rules are ordered from the most reliable rule, rule,, to the least reliable rule, rule,,. 2. Divide these rules into two group, ft and U, depending on their reliability such that group ft contains the reliable rules, rule, to rule, and group U contains the unreliable rules, rule+l to rule. 3. Let T be a set of rules for incremental test. Initially T is empty. 4. For i =1 to J (a) Add rule, to T. (b) Run MFfu,,y-generation algorithm on T. (c) Observe the rate of change of GI. (d) If the rate is significant, then report rule, as inconsistent, and stop. 5. For i =]'+I1 to n 63 (a) Add rule, to T. (b) Run MFfu,,y-generation algorithm on T. (c) Observe the rate of change of GI. (d) If the rate is significant, then report rule, be inconsistent, and stop. (e) Delete rules from T. Note that for preventing human experts' possible errors, we inserted Step 4(c) and Step 4(d), even though the rules being added here belong to the reliable group, R. Sensitivity Test A sensitivity test is a validation technique that can be used for both expert systems and simulations [35, 21, 29]. The general idea is to change the system input (i.e., values of variables or parameters) systematically over some range of interest, and study it by observing the effect. In our method, we can employ this idea to identify inconsistent rules by varying the participating rules. The sensitivity test begins with all rules except the first rule and executes the algorithm to generate M~uzy Then, all rules except the second rule are used to generate the M~uzy In this way, given the n rules, the sensitivity test involves a total of n executions of MFfu,,y-generation algorithm, each time with n - 1 rules. When we detect a significant improvement of the GI during this process, we can consider the rule that did not participate at this step of the test as an inconsistent rule, in comparison to other rules already joined. The advantage of this method over the incremental test is that the arrangement of rules based on their reliability is not necessary - thereby eliminating the need for the experts' opinion. However, the computation burden is more severe than the incremental test, since each execution of MFfu~zy-generation algorithm involves n - 1 rules. The general algorithm for sensitivity test can be written as: 64 1. Let n be a total number of rules for sensitivity test. 2. Let T be a set of rules for sensitivity test. Initially T has entire rules. 3. For i =1 to ni (a) Delete rule, from T. (b) Run MFfu,,y-generation algorithm on T. (c) If a significant improvement of GI is detected, then report rule be inconsistent. (d) Insert rule, to T. 4.4 Time Complexity For the analysis of the time complexities of the fuzzy simulation method and M~uz generation algorithm, we consider the following factors: " number of rules: ni, " number of simplex rules: s, " number of sampling points of a fuzzy value: p, " number of input fuzzy variables iii compound rules: mn, " number of possible subsets for modifying MFs in the most inconsistent rule: b, " depth in a search tree: d and * number of random-restarts: r. 4.4.1 Time Complexity for Fuzzy Simulation For a simplex rule, the total p elements of a fuzzy set in the rule premise are issued to each independent fuzzy simulation. Therefore, the time complexity for executing 65 fuzzy simulations for .s simplex rules is 0(SP). (4.5) For a compound rule, an intermediate fuzzy set Z is first obtained from the rule premise, and a fuzzy simulation is executed for each element in Z. This process involves a cartesian product over m - 1 variables (i.e., except for the variable in the rule consequence) in the rule. This leads to the time complexity 0(p'm). Therefore, the time complexity for executing fuzzy simulations for all compound rules is 0(n- s)p"')- (4.6) Combining complexity (4.5) and (4.6), the total time complexity for executing fuzzy simulations for all rules is o (ript). (4.7) 4.4.2 Time Complexity for M~,,, Generation Using the factors, r and d defined above, we can rewrite the overall algorithm for the MFfuzzy generation in Section 4.3.4 as follows: 1. set initial MFs 2. fuzzy simulations for all rules 3. pick the most inconsistent rule and set current GI 4. for i 1 to r while current GI previous GI (a) for j = 1 to b i. modify this subset ii. execute fuzzy simulations for all rules 66 (b) pick one state leading to minimum GI 5. pick best saved GI In the algorithm skeleton above, Step 4(a) determines the branch factor of the search tree. In the worst case, the branch factor is 2' - 1 by ignoring 4'. Therefore, using (4.7), the time complexity involving Step 4(a) to Step 4(b) is 0 (n277tP M). (4.8) The depth d in an arbitrary search tree is determined by the condition (5). Thus, the overall time complexity for generating the M~,,, is 0(rdn2mnpm). (4.9) The overall time complexity shown above demonstrates that, in a worst case, the number of fuzzy variables that can appear in any compound rule (i.e., m) dominates the overall time complexity. Besides, as we pointed out in Section 2.2.3, m also affects the number of fuzzy rules, n, as well [10]. However, most fuzzy applications so far have had few variables and have been in control [28]. This makes the time complexity shown in (4.9) manageable. In other words, the running time of M~,,, generation that we encounter in practical situations are mostly tractable problems. In this chapter, we described an interactive environment for isolating inconsistent knowledge between the expert rules and the quantitative model. To handle the linguistic vagueness in the expert rules, two separate procedures (i.e., applying fuzzy simulation directly and generating MFf UZZY) were presented depending on the types of estimates in the expert rules. In the next two chapters, we illustrate the applications of the presented methodology. CHAPTER 5 FULTON: STEAMSHIP MODELING For a practical application of the method discussed in the previous chapters, we will consider FULTON, a model of a steam-powered ship, as shown in Figure 5.1 [30]. When the fuel valve is open, fuel flows and the furnace heats the sea water in the boiler assembly; when the fuel valve is closed, no fuel flows and the furnace stops heating the water. Heating the sea water produces steam, which is gathered and goes to the turbine, making the steamboat movable. The remaining steam is condensed into liquid in the condenser and is pumped back to the boiler. Among the four components in Figure 5.1, let's assume that we are interested in the boiler assembly, particularly the relation between temperature (T) of the sea water and the amount of steam (A,) gathered in the boiler assembly. In the following two sections, we discuss two models which represent this knowledge quantitatively and qualitatively. Then, we apply our method to isolate any inconsistency between these two models. 5.1 Quantitative Model of Boiler Assembly Consider the boiler assembly in Figure 5.1. The fuel valve is determined to be in one of two states: open or close. Then, depending on the valve position, the behavior of the boiler assembly can be represented by four states of FSA as in Figure 5.2. The low level continuous models for Ml, ..., M4 in that figure are defined as shown below by combining Newton's law with the capacitance low [18]. 1. (MI) COLD: T =ce, A,, =i. A,8 0, 2. (M2) HEATING: 7'kl100 - T), A,, -k2 + Ai,, A, k3, 67 68 BOILER ASSEMBLY v lFe alve O Heating-------------valveOPE - OPEN M4 Cold valve =OPEN MI valve =CLOSE Ta Cooling valve =CLOSE M3 valve = CLOSE Figure 5.2. Four state automanton controller for the boiler assembly 69 3. (M3) COOLING: Tl=k4( - T), A,, -k5 + Aj A, = T * k6 and 4. (M4) BOILING: T =100I 4w -k7 + n As A = k87 where ki, i =1,.. 8 are rate constants, a=the ambient temperature of the water, T =the temperature of sea water, A, the amount of sea water, A, the amount of steam gathered in the boiler, and Ai, the amount of water increased in the boiler by pumping water from the pump assembly. 5.2 Qualitative Model of Boiler Assembly In the expert's point of view, one of the easiest ways to model the physical behaviors of the boiler assembly is to represent that knowledge into natural language. Since the expert is interested in the relationship between the temperature of the sea water and the amount of steam, he or she can form an associative rule-based model by mapping a single input (the amount of time the fuel valve is open) into a single output (the amount of steam) as shown in Figure 5.3. However, as we can see Valve Open Time Amount of Steam CF expert very-very-short very-very-little 0.9 very-short very-little 0.6 short little 0.6 slightly-moderate little 0.6 moderate medium 0.6 slightly-long slightly-much 0.9 long slightly-much 0.5 very-long much -0.9 Figure 5.3. Simnplex rules for the boiler assembly in Figure 5.4 and Figure 5.5, the rate of change in the temperature when the fuel 70 Z 00 0- 70 12 6050 40 30 201 0 2 4 6 8 10 12 14 16 18 20 Time Figure 5.5. Water temperature valve is open is different from the rate of change when the valve is closed. Taking this observation into account, the expert can construct a more complex knowledge base using compound rules to measure the time-dependent dynamic behavior of the system. Figure 5.6 illustrates this process. Putting Figure 5.3 and Figure 5.6 together and using the expert's definition of linguistic terms defined by Figure 5.7, we obtained a complete rule-based model from the expert for the boiler assembly. 5.3 Checking Consistency When MF,,, is Available Once the complete rule-based model is obtained from the expert, the logical step is to run fuzzy simulations on this model, and get RULEf,,,,y and C~UZ to see 71 Valve Open Time Valve Close Time Amount of Steam CFexpent long very-.short much 0.5 slightly-.moderate very-long medium 0.8 moderate short slightly-.much 0.5 short long little 0.5 very-long very-.very.-short much 0.7 slightlylong slightly-moderate much 0.6 very-.short slightly-long little 0.9 very-very-short moderate very-v.eryilittle 0.9 Figure 5.6. Compound rules for the boiler assembly very-.very-.short very-short short slightly-moderate moderate slightly-long long very-long 1 0.03 1 0.07 1 0.091 0#1 1 3 0# 1 01120 0 .0 0.02 0.06 0.08 0.1 0.17 0.19 0.21 0.23 (a) very...very-little very-little little medium slightly-.much much I I I 20.0 25.0 30.0 35.0 45.0 50.0 65.0 80.0 92.0 (b) Figure 5.7. Definitions of the linguistic terms for the boiler assembly (a) MFpremise in the expert's rule; (b) MF,,0,8,9 in the expert's rule 72 if any inconsistency exists. This path is represented as bold arrows in Figure 5.8. Inconsistencies are identified in terms of Us and GI between these two rule sets. We assume that the consistency criterion, e, is set to 0.5 by the user. Figure 5.9 n0 MF expert available? yes Isolating Inconsistency ConsitencyHuman _Ch ck r -- -- -- -- -- -- Advisor -- - - - - - - - Intervention - - - - - - - Change Quantitative Inconsistency 0 Inconsistent RulesoRue eito Expected Rules o MFs% nooSimulation components Conucin o Incremental test o Sensitivity test Reporting "ful'o Inconsistent rules oApproximate ME~y Dn Figure 5.8. Checking consistency when MFxpert is available shows the result of fuzzy simulations. This figure shows that 15th rule is the most inconsistent, and its LI greatly affects the overall GI. To alleviate the human's resolving efforts, Advisor generates the expected rules from the fuzzy simulation by computing the C~,,, for all possible consequences, given the premise part of the 15th rule. Figure 5.10 shows this result. If the user agrees with Advisor by deciding to replace the original rule15 with the 2nd rule in Figure 5.10, he or she gets a fairly small GI of 0.34, as shown in Figure 5.11. Since this GI is less than consistency criterion, 0.5, we conclude that the two models are now consistent. 73 Valve Open Time Vaive Close Time Amount of Steam CFexper CFfUZZY LI very-.very-.short N/A very..yeryjlittle 0.9 0.87 0.03 very-.short N/A very-little 0.6 0.59 0.01 short N/A little 0.6 0.58 0.02 slightly-.moderate N/A little 0.6 0.60 0.00 moderate N/A medium 0.6 0.64 0.04 slightly-long N/A slightly-.much 0.9 0.88 0.02 long N/A slightly-.much 0.5 0.50 0.00 very-long N/A much 0.9 0.94 0.04 long very-.short much 0.5 0.53 0.03 slightly-.moderate very-long medium 0.8 0.76 0.04 moderate short slightly-.much 0.5 0.49 0.01 short long little 0.5 0.51 0.01 veryjlong very-.very-.short much 0.7 0.71 0.01 slightly-long slightly-moderate much 0.6 0.61 0.01 very-short slightly-long little 0.9 0.26 0.64 very-very-short moderate very-.veryjlittle 0.9 0.90 0.00 GI = 0.93 Figure 5.9. The result of fuzzy simulation for the boiler assembly Valve Open Time Valve Close Time Amount of Steam CIFfuzzy very-.short slightly-long very...veryjittle 0.04 very-.short slightly-long very-little 0.46 very-.short slightly-long little 0.26 very-.short slightly-.long medium 0.00 very-.short slightly-long slightly-much 0.00 very-.short slightly-.long much 0.00 Figure 5.10. Rules generated from Advisor for the boiler assembly Valve Open Time Valve Close Time Amount of Steam CFexIt CFfUZ LI very-.very-.short N/A very-.very..jittle 0.9 0.87 0.03 very-short N/A very-.little 0.6 0.59 0.01 short N/A little 0.6 0.58 0.02 slightly-.moderate N/A little 0.6 0.60 0.00 moderate N/A medium 0.6 0.64 0.04 slightly-.long N/A slightly...nuch 0.9 0.88 0.02 long N/A slightly-.much 0.5 0.50 0.00 very-long N/A much 0.9 0.94 0.04 long very-short much 0.5 0.53 0.03 slightly-.moderate very-jong medium 0.8 0.76 0.04 moderate short slightly..much 0.5 0.9 0.01 short long little 0.5 0.51 0.01 very-long very-.very-.short much 0.7 0.71 0.01 slightly-long slightly-.moderate much 0.6 0.61 0.01 very-short slightly .long little 0.5 0.46 0.04 very-.very-short moderate very-.veryjlittle 0.9 0.90 0.00 I GI =0.34 Figure 5.11. Two consistent models 74 5.4 Checking Consistency When MF,,, is Unavailable Adisr ntrvnto yeso Simulation componentsnc Cnitn y e Suumatin Figure5.12.Checking consistency whn iuavlbe The path~~ ~~~~ thatli goi to be expl red intiyeto srpeetda odarw inigura 512Sie Itwondiffeent oucoe (Ineconsistetonosseteed ingon heter ny pprximtset of Expcbenrted withs a GIlesha th ossec rtro rnot aepoudexlsvythog Smten consisency checking procedurehekn illuistrteecaeiytefloig w etos 5.4.1 The Case Who eeif approximate MF,, sScesul eeae The~~~~ ~~ prvossctoohwdnhtrlsdefined in-Figure5.11-are-onsistent Therefore,~~~~~~~~~~~~~ iweueterlsadtecnalpIrnta estaedfndinFgr5.3 less~~~~~~~~~~~~~ thncnssecyciero,0..Wnilsraetisn ress tpb tp 75 Figure 5.13. Central point estimates for MFpemise and MFonseq Set Initial Conditions Figure 5.14 shows the initial M~,.,y The initial size is set to 0.02 for each MFpemtse, and 4.0 for MFonseq. To generate M~,ZY the tuning size, Ad, is set to 0.005 for MFp,,emise, and 2.0 for MFonseq. For random-restarts, the promising state criterion, 0, is set to 5.0. After all tuning processes are done, if a best saved GI is less than consistency criterion of 0.5, then we conclude that we found an approximate set of M~,,, successfully. very-.very-short 0.0 very-very-ittle 20.0 very-.short short slightly-moderate modera I I I A: 0.03 0.06 0.09 very-..little 35.0 0.13 little 50.0 te slightly-long long very-long 0.17 0.2 0.23 Initial Size = 0.02 medium 65.0 slightly-.much much 80.0 92.0 Initial Size = 4.0 Figure 5.14. Initial M~,,, MF rens central point MFose central point preiecne very..very..short 0.0 very_.veryjittle 20.0 very-.short 0.03 very-little 35.0 short 0.06 little 50.0 slightly-.moderate 0.09 medium 65.0 moderate 0.13 slightly-much 80.0 slightly-long 0.17 much 92.0 long 0.2 very-long 0.23 A: 76 Fuzzy Simulations for All Rules The result of fuzzy simulations using the initial M~,,, is shown in Figure 5.15, where rule10 is identified as the most inconsistent rule. Valve Open Time Valve Close Time Amount of Steam CFexpert CFfuZZY LI veryyery-short N/A very-.veryjlittle 0.9 1.00 0.10 very-short N/A very-little 0.6 0.50 0.10 short N/A little 0.6 0.00 0.60 slightly-moderate N/A little 0.6 0.00 0.60 moderate N/A medium 0.6 0.00 0.60 slightly-long N/A slightly-much 0.9 0.50 0.40 long N/A slightly-much 0.5 0.00 0.50 very long N/A much 0.9 1.00 0.10 long very-short much 0.5 0.00 0.50 slightly-moderate very-.long medium 0.8 0.00 0.80 moderate short slightly-much 0.5 0.00 0.50 short long little 0.5 0.00 0.50 very-long very..very..short much 0.7 1.00 0.30 slightly-long slightly-moderate much 0.6 0.00 0.60 very-short slightlyIong very-little 0.5 0.00 0.50 very-very-short moderate very-veryjlittle 0.9 1.00 0.10 I GI 6.80 Figure 5.15. The result of fuzzy simulations using the initial M~",Z Tuning M~,,, Until Best Saved GI Does Not Improve All possible subsets of M~,,, in rule0 are f{slightly..xoderate}, {very-long}, { medium}, {slightly-moderate, veryiong}, Slightly mo derate, medium}, f{very-long, medium} and I{slightly..-moderate, very-long, medium}. After executing fuzzy simulations for each state, the smallest GI of 6.3 was found by modifying a subset {slightly-moderate, medium} as shown in Figure 5.16 and Figure 5.17. Since this GI is less than the previous GI, 6.8, the current GI, 6.3, is saved as the best GI, and Step 3 in Section 4.3.4 is repreated until the GI does not improve. Figure 5.18 shows the actual GUI when the stop condition was met after a series of tuning processes. At this point, Step 4 (random- restarts) begins, very..very-short very-short short slightly-.moderate moderat K.A 0.03 # #09 0.075 0.105 very...very-little very-little 4 20.0 35.0 0.13 little e slightly-long long very-long A A-/ 0.17 0.2 0.23 Tuning Size = 0.005 medium slightly-.much much #4 4 50.0 61.0 65.0 69.0 80.0 92.0 Tuning Size = 2.0 Figure 5.16. Increasing the size of slightly-moderate and medium by Ad Figure 5.17. The result tuned of fuzzy simulations after slightly-moderate and medium are 77 0.0 Valve Open Time Valve Close Time Amount of Steam CFexpert CFfuzzy LI veryvery..short N/A very..veryjlittle 0.9 1.00 0.10 very-short N/A very-little 0.6 0.50 0.10 short N/A little 0.6 0.00 0.60 slightly-.moderate N/A little 0.6 0.00 0.60 moderate N/A medium 0.6 0.25 0.35 slightly-long N/A slightly..much 0.9 0.50 0.40 long N/A slightly-.much 0.5 0.00 0.50 very-long N/A much 0.9 1.00 0.10 long very-.short much 0.5 0.00 0.50 slightly-.moderate veryjlong medium 0.8 0.25 0.50 moderate short slightly-.much 0.5 0.00 0.50 short long little 0.5 0.00 0.50 very-long very..very..short much 0.7 1.00 0.30 slightly-long slightly-.moderate much 0.6 0.00 0.60 very-.short slightly Iona veryitl0. 0.0 .5 very..very-.short moderate very..yery-little 0.9 1.00 0.10 I GI= 6.30 A. A A 78 '1 Ith fcI for' Resdvimg consistency Prbl' uilg Izz zs Siitioll Alair lVindow (;Iluba lnomsiv~*n".' (d4' klet~y M-n~i t MFl. u~s~r n- Valu EX) I ~ I jaJi Al Rs er 1Ur~ tfr15n A 1 _'erv shor ze'o ss -~ v ll esItl 0 0000 0 533300 5 6755 1% A veythl 15ro--'... l 555- 0 0 55l,33. S 5'535 ~~~~ ~ ~ ~ o \ /e e. -) \\e C' C00\\~ 7 y l s v ts . o . S l 500Sf v100 1W OO /~~1 1" , 3 I P ,5 5 0 5 5 1jg~j t\\\ Z-/ It'/t -jj 0 . .5 MONO) 0 S ___________________________________________________ 5f-,j ,& . -f-o 5555Vh555-V555 11,0f) 1015 Sf5JI 0 5000 V 6S4553 111454 itiY ~ 'I ~5 fflfSI3? % '5t-ff) s er - moc _j 7TOWOff 7. Sf5- J rlJV S l 0SSf 0 VOUU j~V ~5f lv Iv 5-3. _ f 's V SO V I05 I CI 0 25 00" 55S \\35 00,1 Sl\\/Ii IQ erj*S Idle I' ',)A- -- 35%0 3 3, 3 ff5-I 5-Sf5 11 55-S f5-IS-e 4-Sf u- ff 5-5-15050j 093)74 f 101 S' I .l b est saved GI M onseq asbe fMfuzzy leading to smallest GI MF premise GI after "medium" was modified Figure 5.18. Satisfying stop condition and Step 3 is reinvoked each time with a promising state whose GI is less than 0, 5.0. Figure 5.19 shows the trend of GIs after seven random-restarts. As we can see in this figure, the first and sixth random-restarts led to the GI of 0.32 and 0.45, respectively, which are less than the consistency criterion, 0.5. Thus, two sets of the MFfzzy leading to these GIs can be considered as good approximation of linguistic definitions, and we conclude that the expert's rules and the quantitative model are consistent if the two sets of M~u,, are used. Figure 5.20 shows the comparison between the approximate set of M~,,z which led to the smallest GI of 0.32, and the MFxpert defined in Fig 5.7. r~1 : ejf c , e i a. ve -hp tzqrclza icp 0 Pi793410 o 55693 o 930270 1 023135 0451647 0,984556 I 4NumWe of Iteation., best saved G1 Figure 5.19. Trend of GI after seven random-restarts very-very-short very-short short slightly-moderate moderate slightlyjlong long very-long 0.0 0.02 0610.071 0.1 0.065 0.0 0.1 10.15 1 0.17i 0.#40.21 0.145 0.16 0.195 0.21 0.18 very-yeryjittle very-little little medium slightly-.much much 2002503003.0 5#150. /65.0 80.0 92.0 36.0 47.0 60.0 64.0 83.0 - -- -- -- MFf y Figure 5.20. The comparison between M~uz and MWexpert Value 79 i i cq so 0.0 0.23 80 Val ue Th, fch&q 3.371 634 33716G34 3079103 50 ,470194 tNumber of 1terattorns best saved 61 Figure 5.21. Trend of GI after seven random-restarts 5.4.2 The Case Where MFf,,,, is Unsuccessfully Generated Section 5.3 showed that expert's rules in Figure 5.9 are inconsistent because of rule15. If we use these rules, we can artificially make the case where M~u, cannot be successfully generated. By executing the MFfu,,y generation algorithm with seven random-restarts, we obtained the best saved GI of 2.98, as shown in Figure 5.21. In this figure, no random-restart lead to a GI less than the given consistency criterion, 0.5. To identify the causes of the inconsistency, we performed the incremental test and the sensitivity test discussed in Section 4.3.5. Incremental Test We assume that compound rules are more prone to inconsistency than simplex rules, because the consequences of all compound rules in the FULTON example require a more complex reasoning process involving two time-related variables. For this reason, we grouped the simplex rules, defined in Figure 5.3, into the reliable group, R, and the complex rules defined in Figure 5.6 into the unreliable group, U. The result of the incremental test is shown at Figure 5.22. Each subfigure shows the result of executing the M~u, generation algorithm (i.e., Step 5(b) of the incremental test procedure in Section 4.3.5) with seven random-restarts right after we added each compound rule to the results obtained from incremental test using 81 the eight simplex rules. By observing the rapid change of GI in Figure 5.22 (g), we can regard the rule15 as the catalyst that caused the inconsistency. Sensitivity Test Figure 5.23 shows the result of the sensitivity test on the rules defined in Figure 5.3 and Figure 5.6. We performed seven random-restarts for each test; each circle in this figure represents the GI we have obtained from each random-restart. A rule i in the x axis represents the rule which did not join for the M~u, generation at that particular test. As shown in this figure, execution of the M~u, generation algorithm (i.e., Step 3(b) of the sensitivity test procedure in Section 4.3.5) without rule15 resulted in a significant improvement of GI, 0.42, compared to the other cases. This indicates that rule15 is the catalyst of the inconsistency. 5.4.3 Human Intervention In Figure 5.24, the path involving human intervention is represented as bold arrows. In Section 5.3, rule15 was identified as an inconsistent rule when the rules defined in Figure 5.3 and Figure 5.6, and the linguistic definitions defined in Figure 5.7, were used. Even in a situation where such complete linguistic definitions were not available, the previous section showed that we were still able to identify rule15 as a catalyst for the inconsistency by conducting the incremental test or the sensitivity test. With this information, the user may change rules (in this case, rule15 or other rules which may be causing the conflict with rule15), central point estimates, the definitions of linguistic terms in these rules or even simulation model components. We built a GUI as shown in Figure 5.25 to interactively visualize the system's responses according to these kinds of user's resolving trials. In this figure, the set of membership functions shown in membership function editor is the best set which led to the minimum GI of 2.98 in the previous section. However, as mentioned, this case fell into the category of unsuccessfully generated M~,,,Y Now, 82 using membership function editor and rule editor, the user can freely change these membership functions, as well as any rules including the CFs. Each time the user makes a modification, this updated input is issued to the fuzzy simulation, and new consistency checking results are provided in terms of LI and GI through evaluation button. In this chapter, we considered FULTON for a practical application of the presented methodology. All algorithms or procedures in Chapter 4 were demonstrated completely. For another application, we consider predator-prey population in the next chapter. In the chapter, we focus on the comparison between the model outputs of the expert rules and the outputs of the quantitative models before and after resolving inconsistency. 83 (b) 1.0 1.6 1.4 1.2 0.( OAF 0.2 0ol 2 3 4 5 6 7 a I (d) 1.8 1.6 1.4 1.2 0.6 04 0.2 1.8 1.4 1.2 0.8 0 0 8 o 1 2 3 4 5 6 7 8 13 (g) 1.8 1.6 1.4 1.2 08 06 0.4 0 0.2 0 8o ' 0 1 2 3 4 5 6 7 8 1 WI.e I .6 .4 .2 i (h) 106 0.1 0.4 o 2 3 4 5 8 7 8 1 WIG, 0 2 3 4 5 6 7 8 I WI.e I 1 2 3 4 5 6 7 8 14 WIGe 1 2 3 4 0 6 7 8 16 role Figure 5.22. Incremental test for the boiler assembly (a) adding ruleg; (b) adding rulelo; (c) adding rule,,; (d) adding rule12; (e) adding rule13; (f) adding rule14; (g) adding rule15; (h) adding rule16 (a) :.8~ 1.2 0.1 0.1 0.. 0.: (C) . 1.6 1.4 1 2 0.8 0.6 0.4 0.2 (e) 1 8 1 6 1.2 - 0.4 0.2 0 4 0 0 84 0 GI obtained from a random-restart 5 4.5 4 3.5- 3 F2.5 2 1.5 1 0.5- 0 2 4 6 8 rule 10 12 14 16 Figure 5.23. Sensitivity test for the boiler assembly no MF expert available? yes Isolating Inconsistency Figure 5.24. Human intervention 01 0 0 8 00 Quantitative Model ConsitencyHuman ,_C eck r . ...I . ... ... ... .. Advisor - -- - - -- - - Intervention - - - - - - - Checking consistency o Apply fuzzy simulation directly inconsistency o Inconsistent Rules no 0Simuation components Checking consistency o Identify approximate MFfz Conducting o Incremental test o Sensitivity test nocesfl Reporting successful Inconsistent rules yes Suggesting o Approximate MFfu2Y Dn ----------------- -------- 85 rule editor ver~hy r ndrto t le o t er-er, litl 0 0onf00 mn zit e4 co cyl t eL2 mor 00000) 5ho ot lag little 0J 5r 000r 0009 lln g Vna c. 007 lightol-ich 0 000000 log 1u03 slightly - zoo Cht 0500300 vet] ion zeroo yc I ;- 0 0 0000 0009 qh (1I lLO~--t LT r ote oo =, ve ' 0-r lote 00000 ver very shr 0vryvr little 20 20 22 Very sort, -f* ver 1,r =5 very litl 35 3 s ho 3 6 1 10 little 20ttle 0 I oe t e[ _-h r 12de t 13 14 slightly m uch07208000 very 10yitl 20 2 very Je3 22232 Reset Roles {RstMFs Exit evaluation button membership function editor Figure 5.25. A GUI for human intervention CHAPTER 6 PREDATOR-PREY POPULATION For another illustration of our method, we considered a predator-prey population. Predator-prey models address the dynamic interaction between a predator species and its prey species. In this chapter, we consider two species: one single predator species P and its single resource 1Zi. 6.1 Qualitative Model In general, there are two kinds of rules which explain the population dynamics in the predator-prey interaction: migration rules and birth/death rules [56, 55]. The migration rules describe the movement of individual P and 7Z. at any interval of time from one location to another. For example, a probability that P will stay in a particular location depends on the configuration of R? and P around that location. The birth/death rates of P and 1Z depend on many factors, such as mortality rate, internal feeding state, predator's encounter rate with prey and so on. These factors are not independent of one another. For example, the mortality rate of P depends on their internal feeding state, and the internal feeding state again depends on the encounter rate with 7Z. Predator-prey dynamics also depends on the natural environment of P and 7Z. When ecologists study the distribution of organisms, they try to discover the physical and biological factors that influence the presence or absence of particular species. In this section, we assume that there are two environmental factors affecting the distribution of 'P and 7?: scale and temperature of the region they live. Given two different environmental factors, an ecologist may predict the population of 7Z in a 86 87 short-term period by considering the causal relation of predator-prey interaction: "In a wide region, the possibility of P's encounter rate with RZ is relatively small, but if the temperature of the territory is warm, then other living food in that region may satisfy the P (which makes the mortality rate low). Therefore, this keeps the population of R from growing too much. On the contrary, in a cold region, the mortality rate of P increases, which makes the population of R crowded." Using the above description, a knowledge engineer can come up with the following two rules. IF P's encounter rate with R? is rare and P's mortality rate is low, THEN the density of the R? is slightly- crowded (CF =0.9), IF P's encounter rate with R? is rare and P's mortality rate is high, THEN the density of the R? is crowded (CF =1.0). After the knowledge acquisition process with the expert in this way, suppose that we obtained 25 rules and membership functions as shown in Figure 6.1 and Figure 6.2, respectively for explaining the population of prey P based on the different combination of the environmental factors. With two inputs and five linguistic values for each of these, there are 5 2 =25 possible rules. In these rules, both triangular and trapezoidal membership functions are used. 6.2 Quantitative Model To describe the dynamics involving growth and decline of the predator-prey population, differential or difference equations are often used. Such mathematical models are designed either for predictive purposes to make accurate short-term forecasts, or to identify generic characteristics and underlying principles. As one of the mathematical models, we consider the Lotka-Volterra predator-prey model [55]: dx(t) _ x(t)(1- x(t)) a x(t) y(t) dt K 1 + ath XMt dy (t) _ acx(t)y(t) M dt l+ ath X(t)-e t) 88 Rue Encounter rate of Mortality rate Destofpy CI predator with prey of predator Dniyo ry C rare rare rare rare rare slightly-rare slightly-rare slightly-rare slightly-are slightly-rare medium medium medium medium medium slightly-frequent slightly-.frequent slightly frequent slig-htly...requent slightly-frequent frequent frequent frequent frequent frequent very-low low moderate slightly-.high high very-low low moderate slightly-high high very-low low moderate slightly-high high very-low low moderate slightly-high high very-low low moderate slightly-.high high slightly-rowded crowded crowded crowded crowded crowded crowded crowded crowded crowded scarce nominal crowded crowded crowded scarce scarce slightly-scarce slightly-crowded crowded scarce scarce slightly-scarce nominal crowded Figure 6. 1. Expert rules for the predator-prey population 0.90 1.00 1.00 1.00 1.00 0.40 0.50 0.90 1.00 1.00 0.60 0.90 0.50 0.90 1.00 0.80 0.80 0.30 0.30 1.00 0.80 0.60 0.30 0.10 0.80 89 Encounter Rate rar asigel slightly feun rar slightly medium frequent feun 1.0 Motlt rate very-low 10,L low moderate slightly-.high high 0.1 0.2 0.3 0.4 0.6 0.8 1.0 Density of prey scarce slightly nmalslightlycrwe L crecrowdedcrwe 1.0 0.0 0.1 0.4 0.5 0.9 1.2 1.3 1.6 1.7 Figure 6.2. Fuzzy membership functions for the expert rules in Figure 6.1 90 where x(t), y(t) =population densities of R? and P as functions of time t,respectively, r = R population's intrinsic rate of increase, K = R's carring capacity, that is the population density the prey population would reach at an equilibrium in the absence of predation, a =P's encounter rate with 7R, c =a conversion rate which maps the P consumption into the R? birth rate, e =P's mortality rate, and th = P's handling time. 1.2 C1. I' 0'8 z 0.6 0.5 0.4 400 50 00 70 00 90 00 1000 2000 3000 Figure 6.3. A time series Time graph for the predator-prey population For example, using initial conditions x(t) =1.05, y(t) = 0.9, r =1.2, K =1.7, a= 1.9, e =0.48, c =0.9 and th =1.0, we obtain a time series graph for the predator-prey population as shown in Fig 6.3, in which the state variables (i.e., density of P and 91 RZ) are graphed against time. To solve the x(t) and y(t), we applied Euler's method [18]. 1.7 1.6 1.45 1.1 0 100 200 300 400 500 600 700 00 900 1000 Ti-~ Figure 6.4. The result of fuzzy simulation on IF rare and low THEN crowded 6.3 Consistency Checking We applied fuzzy simulation to the expert rules in Figure 6.1 using the membership functions defined in Figure 6.2 and initial conditions x(t) = 1.05, y(t) =0.9, r = 1.2, K =1.7, c =0.9 and th =1.0 of the Lotka-Volterra model. For example, Figure 6.4 shows the result of the fuzzy simulation on the expert's second rule, defined in Figure 6.1. To obtain this result, we first applied the rule of conjunctive composition to calculate the possibility distribution w of rare and low. Then each element of the possibility distribution is used for input of the fuzzy simulation, and this gives us a result in terms of membership degrees associated with crowded. Finally, using the weighted average method introduced in Chapter 3, we obtain a CFfU22Y) 1.0. In this way, all 25 expert rules are applied to the fuzzy simulations, and we obtained the results in Figure 6.5. In this figure, rule, is identified as the most inconsistent rule and the rule12 as the second worst case rule. Figure 6.6 shows the suggestion from Advisor. By replacing the original 1st and 12th expert rules 92 Rue Encounter rate of Mortality rate Dniyopry CF 1xet Cfzy L Ru predator with prey of predator DeIt fpe xetL rare rare rare rare rare slightly-.rare slightlyrare slightly-.rare slightly-.rare slightly-.rare medium medium medium medium medium slightly-..frequent slightly-.frequent slightly-.frequent slightly-..frequent slightly-frequent frequent frequent frequent frequent frequent very-low low moderate slightly-.high high very-low low moderate slightly-.high high very-low low moderate slightly-.high high very-.low low moderate slightly-.high high very-.low low moderate slightly-.high high slightly-.crowded crowded crowded crowded crowded crowded crowded crowded crowded crowded scarce nominal crowded crowded crowded scarce scarce slightly-scarce slightly-crowded crowded scarce scarce slightly-.scarce nominal crowded 0.90 1.00 1.00 1.00 1.00 0.40 0.50 0.90 1.00 1.00 0.60 0.90 0.50 0.90 1.00 0.80 0.80 0.30 0.30 1.00 0.80 0.60 0.30 0.10 0.80 Figure 6.5. The result of fuzzy simulation for the predator-prey model with their counterparts from fuzzy simulations, we achieved a GI of 0.38 as shown in Figure 6.7. An idea of how the old and the new rule sets approximate the population of R? differently can be obtained by comparing the response surfaces of these two with one from Lotka-Volterra algebraic formula. The response surface of R.'s density for all 270 combinations of encounter rate =:0.1, 0.2, ..., 2.6 and mortality rate = 0.1, 0.2, ..., 1.0 from the Lotka-Volterra model is shown at Figure 6.8. To create the response surfaces from the fuzzy rule sets defined in Figure 6.1 and Figure 6.7, we used fuzzy logic (discussed in Section 2.2.3) with the min-max composition for inference 0.05 0.85 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 0.37 0.03 0.46 0.04 0.94 0.04 1.00 0.00 1.00 0.00 0.60 0.00 0.09 0.81 0.46 0.04 0.88 0.02 1.00 0.00 0.78 0.02 0.79 0.01 0.29 0.01 0.31 0.01 0.98 0.02 0.77 0.03 0.64 0.04 0.32 0.02 0.15 0.05 0.80 0.00 GI =2.06 93 Figure 6.6. Rules suggested from Advisor for the predator-prey model Rule Encounter rate of Mortality rate Density of prey CFepr CFfuz LI predator with prey of predator epr uz I rare very-low crowded 0.90 0.90 0.00 2 rare low crowded 1.001 1.00 0.00 3 rare moderate crowded 1.00 1.00 0.00 4 rare slightly-.high crowded 1.001 1.00 0.00 5 rare high crowded I .00( 1.00 0.00 6 slightly-rare very-low crowded t0.40 t0.37 0.03 7 slightly-.rare low crowded 0.50 0.46 0.04 8 slightly-rare moderate crowded 0.90 0.94 0.04 9 slightly-..rare slightly-high crowded I 00 1.00 0.00 10 slightly-rare high crowded 1.00 1.00 0.00 11I medium very-low scarce 0.60 0.60 0.00 12 medium tow slightly-.scarce 0.401 0.40 0.00 13 medium moderate crowded 0.50 0.46 0.04 14 medium slightly high crowded 0,90 0.88 0.02 15 medium high crowded 1.00 1.00 0.00 16 slightly-.frequent very-low scarce (1.80 0.78 0.02 17 slightly-frequent low scarce 0.80 0.79 (1.01 18 slightly..frequent moderate slightly-scarce 0.31 0.29 0.01 19 slightly-..frequent slightly-high slightly-crowded 0.30 0.31 0.01 20 slightly-frequent high crowded 1.00 0.98 0.02 21 frequent very-.low scarce 0.80 0.77 0.03 22 frequent low scarce 0.60 0.64 0.04 23 frequent moderate slightly_scarce 0.30 0.32 0.02 24 frequent slightly-high nominal (1.10 0.15 0.05 25 frequent high crowded 0.80 0.80 0.00 GI =0.50 Figure 6.7. Two consistent models for the predator-prey model Encounter rate of Mortality rate Rule predator with prey of predator Density of prey CFfuzzy Rare very-low scarce 0.00 rare very-low slightly-scarce 0.00 rare very-low nominal 0.00 rare very-low slightly_..crowded 0.05 rare very-low crowded 0.95 12 medium low scarce 0.24 medium low slightly-scarce 0.45 medium low nominal 0.09 medium low slightly-.crowded 0.05 medium low crowded 0.21 |

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PAGE 1 MODEL VALIDATION METHODOLOGY FOR ISOLATING INCONSISTENT KNOWLEDGE BETWEEN FUZZY RULE-BASED AND QUANTITATIVE MODELS USING FUZZY SIMULATION By GYOOSEOK KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 PAGE 2 To my parents, wife, Jungsook, my daughter, Karyoon and my son, Seungyup PAGE 3 ACKNOWLEDGEMENTS I wish to express my gratitude to my supervisor and committee chairman, Dr. P. A. Fishwick, for his invaluable guidance, encouragement and patience throughout the past four years of my graduate study. I would like to express my sincere appreciation to Dr. Douglas D. Dankel, Dr. L. M. Fu, Dr. S. Rajasekaran, and Dr. S. X. Bai for serving on my supervisory committee and providing invaluable suggestions for my research. I am grateful to all my colleagues at the Department of Computer and Information Science and Engineering for their help and friendship. I thank Dr. Jinjoo Lee who helped me a lot when I first joined the research group. Special thanks go to Youngsup Kim and Kangsun Lee in my group for their suggestions and companionship during my research. I wish all of them the best of luck in their future endeavors. Last but most importantly, I am deeply indebted to my parents, my wife Jungsook, my daughter Karyoon and my son Seungyup for their encouragement, love and understanding. My greatest gratitude is to them. iii PAGE 4 TABLE OF CONTENTS ACKNOWLEDGEMENTS iii ABSTRACT vi CHAPTERS 1 1 INTRODUCTION 1 1.1 Problem Statement 2 1.2 Purpose of Research 5 1.3 Contribution to Knowledge 7 1.4 Outline 7 2 BACKGROUND AND RELATED WORK 9 2.1 Modeling Process in Simulation and Fuzzy Set Literature 9 2.1.1 Simulation Modeling Process 9 2.1.2 Fuzzy Modeling Process 12 2.2 Fuzzy Set Theory and Its Application 14 2.2.1 Notation, Terminology, and Basic Operations 15 2.2.2 Membership Function Construction 19 2.2.3 Fuzzy Controller 21 2.3 Fuzzy Set Theory in Computer Simulation 27 3 A NEW FUZZY SIMULATION APPROACH 33 3.1 Expert Rule Format 33 3.2 Fuzzy Simulation 34 3.2.1 Fuzzy Simulation for Simplex Rules 35 3.2.2 Fuzzy Simulation for Compound Rules with Arithmetic Operations 39 3.2.3 Fuzzy Simulation for Compound Rules with Logic Operations 42 4 A METHOD FOR ISOLATING INCONSISTENCY 46 4.1 Measurements of Inconsistency 48 4.2 Checking Consistency When MF^xpert is Available 49 4.3 Checking Consistency When MFexpen is Unavailable 50 4.3.1 Process in General 50 4.3.2 Heuristic Function and Search Method for Generating Approximate MFfuzzy 51 4.3.3 Various Forms of Expert's Estimates on Linguistic Terms ... 54 iv PAGE 5 A 4.3.4 Algorithm to generate MFjuzzy ^'^ 4.3.5 Identify Inconsistent Rules 61 4.4 Time Complexity 64 4.4.1 Time Complexity for Fuzzy Simulation 64 4.4.2 Time Complexity for MFjuzzy Generation 65 5 FULTON: STEAMSHIP MODELING 67 5.1 Quantitative Model of Boiler Assembly 67 5.2 Qualitative Model of Boiler Assembly 69 5.3 Checking Consistency When MFgxpert is Available 70 5.4 Checking Consistency When MFexpert is Unavailable 74 5.4.1 The Case Where Approximate MF/Â„2zj, is Successfully Generated 74 5.4.2 The Case Where MFjuzzy is Unsuccessfully Generated .... 80 5.4.3 Human Intervention 81 6 PREDATOR-PREY POPULATION 86 6.1 Qualitative Model 86 6.2 Quantitative Model 87 6.3 Consistency Checking 91 6.4 MFjuzzy Generation 95 7 FUTURE WORK 99 7.1 Limitations and Improvements 99 7.1.1 Resolving Inconsistency 99 7.1.2 Performance Index 99 7.1.3 Local Optimality During MFjuzzy Generation 100 7.2 Application 101 7.2.1 Application in Control Industries 101 7.2.2 Application in MOOSE 102 8 CONCLUSION 106 REFERENCES 108 BIOGRAPHICAL SKETCH 112 V PAGE 6 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A MODEL VALIDATION METHODOLOGY FOR ISOLATING INCONSISTENT KNOWLEDGE BETWEEN FUZZY RULE-BASED AND QUANTITATIVE MODELS USING FUZZY SIMULATION By GYOOSEOK KIM August, 1998 Chairman: Dr. Paul A. Fishwick Major Department: Computer and Information Science and Engineering Model validation is a complicated and multifaceted procedure in which all possible information such as real-world data, other model input-output data, compiled knowledge in the form of mathematical models and expert opinions can be made consistent. Even though validation using real-world data provides the ideal case, in reality, obtaining such data is not always possible. In this dissertation, we assume that expert opinion is presented in the form of fuzzy rules. The fuzzy simulation approach presented here provides a mechanism for directly encoding uncertainty from human reasoning into a computer simulation by mapping the fuzzy linguistic values of the expert's rules into simulation components. To perform a quantitative comparison between the two kinds of models, quantitative measures have been formulated to gauge the sources and the degree of inconsistency. Using the fuzzy simulation approach and the quantitative measures, this method provides an interactive environment for isolating inconsistency between the fuzzy rule-based model and the vi PAGE 7 quantitative model allowing some degree of consistency. This environment also facilitates humans in the loop resolving inconsistency by helping them to identify and revise the most inconsistent component rapidly and then analyze the effectiveness of that modification. Through checking consistency between these two knowledge representations, our approach serves as a method of checks and balances to enhance the complex model validation process. vii PAGE 8 CHAPTER 1 INTRODUCTION While model verification deals with building the model right, model validation deals with building the right model [1]. Validation is concerned with determining whether the model is an accurate representation of the system under study [29]. Model validation is part of the total model development process, and it consists of performing a series of tests and evaluations within the model development process. This validation process is multifaceted and involves the minimal procedure of taking a set of real-system observations and rectifying these observations with an assumed mathematical model or vice versa. This process involves an estimation of parameters of the model that yields a model that best reflects the real-system behavior. In practice, validation is a complicated process, since there may be numerous sources of knowledge comparable with the model output that is under investigation. For many fields, most notably medicine, sociology and economics, there may be a plethora of knowledge and data. This knowledge may be quantitative or qualitative, or both, with experts providing opinions that must somehow be reconciled with quantitative proposed models known to characterize a similar real-world problem. However, currently no algorithm or procedure is available to identify suitable validation techniques [45]. Moreover, one does not always have a complete set of data and a set of models waiting to be identified. The data may be just as incomplete as the model suppositions. The difficulty of achieving the data validity for model validation is discussed in [29]. We have created a methodology that enhances the validation process for such situations. In particular, our method assumes that there is at least one fuzzy rule-based 1 PAGE 9 2 model and one proposed quantitative model. The rule-based model is composed of a set of A'' IF-THEN statements, and the quantitative model is defined as a model whose parameters and state variables take real values. Our method locates inconsistencies between these two representations, and we created an interactive tool for partially resolving inconsistencies. Comparing and contrasting the expert rules with the quantitative model is viewed as being an integral part of an ongoing system validation procedure. 1.1 Problem Statement Knowledge about a given physical system is often obtained from experts in the form of rules. Although the rule-based model is occasionally associative or shallow in nature, this model can easily capture human heuristic and problem-solving knowledge in an efficient way [21, 5, 48]. Fuzzy set theory [57, 58, 59, 27, 14, 61] provides linguistic IF-THEN rules to make use of such experts' knowledge and experience naturally. Through a suitable fuzzy inference scheme, the rule-based system provides a solution without exploiting underlying causal relations on which the solution is based. In some cases, a quantitative model exists which represents all or part of the behaviors of the physical system. This model can be a mathematical model where the system behavior is characterized by one or a series of equations or inequalities. Or this model may be a simulation model where each change in the status of the system is captured over time. This kind of quantitative model provides deeper and more theoretical knowledge when expert system developers need to find solutions for technical problems [5, 16, 48]. To compare and contrast the qualitative model with the quantitative model to serve model validation, we should provide answers for the following fundamental questions: how much do the models differ? and how can one address the difference PAGE 10 3 Figure 1.1. A knowledge acquisition cycle between the two models? One way of handling the inconsistencies between the two different levels of models is to form a knowledge acquisition cycle as in Figure 1.1 [5]. Approaches for creating model bases are discussed within the context of computer simulation [15, 18]. For example, the model base represents compiled knowledge about many domains, such as a mathematical queuing model for waiting line problems. If a match is found during the model base search, then shallow rules (or input-output pairs) are generated by means of deduction process. For such a process, exhaustive simulations are executed based on this deep model. Since the size of the shallow rules in Figure 1.1 resulting from the deduction process is usually too large for a human to study and validate against the original expert's rules, fuzzy induction or fuzzy system identification methods [31, 51, 38, 50, 49] can be employed to obtain a more comprehensible and generalized set of linguistic rules. After completing these processes, the rules obtained as a final product become suitable for a human to study and compare against the expert's original rules. However, the fundamental problems of the knowledge acquisition cycle mentioned above can be described as 1. First and most importantly, forming the above knowledge acquisition cycle to check consistency requires a series of difficult tasks as well as a long process. PAGE 11 4 2. For the direct comparison between the generalized rules in Figure 1.1 and the expert rules, the two rule sets should have the same rule structure, linguistic variables and values. This means that fuzzy induction or identification methods should take the expert's rules into account when they generate the rules. However, most approaches take numerical input and output data to generate fuzzy rules without considering the expert's prior knowledge about the system. Consequently, it is difficult to measure the inconsistency between the two models quantitatively. This could make a decision-making process quite complicated if one wants to resolve inconsistency, especially when a large portion of knowledge components shows inconsistency. 3. Identifying fuzzy rules using only numerical data without properly assumed rule-structure information easily suffers from the curse of dimensionality [32], in which the number of possible rules increases exponentially with the number of possible variables. Moreover, searching for influential variables among all possible variables based only on the numerical data causes the problem of combinatorial optimization. 4. For the deduction process in Figure 1.1, all fuzzy simulation approaches [4, 46, 17] require that proper linguistic definitions (i.e., fuzzy membership functions) be defined a priori before performing fuzzy simulation. Since such definitions of the linguistic terms are difficult to obtain even from experts because of the uncertainty arising from linguistic vagueness, searching for the proper membership functions belongs to the general problem area of knowledge acquisition within the underlying framework of fuzzy set theory [27]. Thus, if such definitions are not available a priori, this additional knowledge acquisition process will clearly become a bottleneck in the cycle shown in Figure 1.1. PAGE 12 5 1.2 Purpose of Research This research is concerned with devising a method for isolating inconsistency between the two different levels of models in an efficient and systematic manner. To achieve this goal, we have developed a knowledge acquisition cycle as shown in Figure 1.2 [26, 24, 25]. Using this approach, we can alleviate the problems discussed in the previous section. Â• First of all, the knowledge acquisition cycle presented here forms a simple process compared to the one in Figure 1.1. The fuzzy simulation approach introduced here directly encodes uncertainty arising from human linguistic vagueness into simulation components and utilizes quantitative models for the deduction process. Since this method accepts the linguistic values in the expert's rule premises as simulation inputs and produces linguistic values as outputs using the same terminologies that the expert used in his (or her) rules, the direct comparison between these two models is possible without an additional induction step. Â• Next, by taking into account the expert's prior knowledge about the system, the major variables affecting the system don't need to be identified. This alleviates major problems such as high computational complexity and local optimality, which usually arise from the structure identification process. Â• Finally, to alleviate the knowledge acquisition bottleneck, this method allows users to employ various levels of estimates depending on the linguistic vagueness in their rules. As shown in Figure 1.2, this method isolates inconsistency through the two phases: 1) consistency checking and 2) interactive user control. In the consistency checking PAGE 13 6 Figure 1.2. A knowledge acquisition cycle using fuzzy simulation phase, depending on the expert's various levels of linguistic vagueness (for example, central points, intervals or fuzzy sets), the result of fuzzy simulations is directly compared against the expert's rules (when fuzzy sets are provided), or fuzzy set definitions, which properly fill in the expert's knowledge, are searched through incremental optimization over fuzzy space (when central points or interval estimates are provided). In both cases, the method suggests the source and the amount of inconsistency to users using quantitative measures we have formulated. If the amount of inconsistency exceeds a reasonable range, human intervention is possible via interactive user control: users may modify either expert rules or simulation model components to reduce the amount of inconsistency. Even at this point, the quantitative measures mentioned above help users identify and revise the most inconsistent component rapidly and analyze the effectiveness of that modification. This humanmachine interaction allows the two models to gradually reach a consensus with a high resolution. O'Keefe [35] pointed out that this kind of visual interaction is one of the most promising validation techniques in expert systems. PAGE 14 1.3 Contribution to Knowledge The primary contribution of this work is that, through checking consistency and resolving inconsistency, our approach serves as a method of checks and balances during the model validation phase of system analysis. During this process, we provide benefits to expert systems from simulation and benefits to simulation modeling from expert knowledge. In particular, when expert system researchers are studying the acquisition of deep knowledge from an expert or validating the expert's knowledge against quantitatively compiled knowledge, the first type of benefits can be obtained from simulation models [16, 15, 17, 54, 34]. The advantage from the reverse process is also obtained when simulation model validations are performed during the simulation modeling process with the aid of the expert knowledge [54, 6, 44, 45]. Additionally, using this method, we can contribute a second benefit from automatically generating fuzzy membership functions where expert rules and quantitative models match maximally. Most methods for constructing fuzzy membership functions [27, 42, 41, 9, 52, 2, 3, 33, 19] rely on a set of sample data (i.e., pairs of an element and its membership degree) from an expert's (or experts') opinions before applying curve-fitting or learning methods. However, gathering such information is not a straightforward matter, even by domain experts, if the system's input, process and response are too complex. If a quantitative model exists for the system, the method presented in this paper allows us to utilize natural rules from the domain experts instead of the restricted sample data to construct approximate fuzzy membership functions. 1.4 Outline In Chapter 2, we discuss some issues arising from the modeling process which appears in simulation and fuzzy set literature. Here, we address the importance of PAGE 15 8 the expert's role for validating simulation models and for constructing fuzzy models. Then, we review the fuzzy set theory that is relevant to this research and its relation to computer simulation. First, we explain the basic definitions of fuzzy set theory and various approaches for constructing fuzzy membership functions. Since one of the most successful application areas of fuzzy systems is the fuzzy controller, we present its overview with its key components. Finally, we briefly discuss the fuzzy simulation approaches which appear in both fuzzy set and computer simulation literature. In Chapter 3, we present a new fuzzy simulation approach. We first assume three forms of experts' rules as the inputs to the fuzzy simulation. Then, we propose three different fuzzy simulation algorithms for directly encoding the fuzziness in the expert rules into computer simulation. In Chapter 4, based on the fuzzy simulation method discussed in Chapter 3, we describe an interactive environment that we've developed for isolating inconsistent knowledge. First, we introduce two measurements of inconsistency that we've employed for quantitative comparison. Then, we provide two consistency checking procedures to handle various forms of experts' estimates on linguistic rules. Finally, we analyze the time complexities of these procedures. In Chapter 5 and Chapter 6, we consider a steam-powered propulsion ship (FULTON) and a predator-prey population, respectively, to illustrate the applications of the methodology. Finally, future work and conclusion are presented in Chapter 7 and Chapter 8, respectively. PAGE 16 CHAPTER 2 BACKGROUND AND RELATED WORK 2.1 Modeling Process in Simulation and Fuzzy Set Literature 2.1.1 Simulation Modeling Process Figure 2.1 [44, 45, 7] shows the general simulation modeling process and its relation to validation and verification. The problem entity is the real or proposed system to be modeled. The conceptual model represents the mathematical, logical or verbal representation of the problem entity, and this model is developed through an analysis or modeling phase. The computerized model represents the conceptual model implemented on a computer, which is developed through a computer programming and implementation phase. Inference about the problem entity is obtained by conducting computer experiments on the computerized model in the experimentation phase. The conceptual model validity determines the validity of the underlying assumptions and theories by using mathematical or statistical methods. Additionally, the process is concerned with whether this specific model's representation of the problem entity being modeled and its structure, logic and mathematical and causal relationships are reasonable for the intended use of the model [44]. For such a process, techniques such as face validation and traces are used. Face validation involves having domain experts evaluate the conceptual model to determine if they believe it is correct and reasonable for its purpose. This usually means examining the flowchart, graphical model or the set of model equations. In the traces the behavior of different types of specific entities in the model are traced through the model to determine if the model's logic is correct and if the necessary accuracy is obtained. 9 PAGE 17 10 Problem Entity Operational / Validity Experimentation Computerized Model Data Validity Analysis Conceptual and Model Modeling Validity Computer Programming and Implementation Conceptual Model Computerized Model ^ Verification Figure 2.1. Modeling process and its relation to validation/verification Computerized model verification determines that the implementation of the conceptual model is correct. The verification techniques used here can be found in the software engineering field, such as formal correctness proof, structured walk-through, top-down and button-up testing [47]. Operational validity is defined as determining that the model output has sufficient accuracy for its intended purpose. Most of the validation efforts take place in this stage. Any discrepancy found may be due to an inadequate conceptual model or an improperly implemented conceptual model. In this case, one set of system data is used for calibration of the simulation model, and another independent set is used for validation. If the simulation output data agree with the system output data, the model can be considered valid. For this reason, data validity is needed for comparing PAGE 18 11 the problem entity's behavior with the model's behavior, as well as for building the conceptual model, for developing theories and for testing the underlying assumption. However, in determining operational validity, it is difficult to use the classical statistical test (t, two-sample chi-square, etc.) between the model output and the corresponding system output data, due to the nature of the data [29]. Specifically, Â• an observation may be nonstationary: the distribution of the successive observations change over time. Â• an observation may be autocorrelated: the observations are correlated with each other. Therefore, techniques commonly used for operational validity are [45] Â• face validation, where experts are asked to make subjective judgements on whether the model has sufficient accuracy, Â• statistical tests for confidence intervals and hypotheses and Â• turing test, where individuals knowledgeable about the system are asked if they can discriminate between system and the model outputs. However, the statistical methods mentioned above also have difficulties in some applications, such as military or manufacturing systems, due to the paucity of real-world data. Because of the difficulty in obtaining data validity, human knowledge about the system takes a relatively important role during the entire validation process, where approaches such as comparison to other models on graphical displays, intuition, opinions or past experience are usefully adopted [6]. The importance of human information and knowledge representation in problem solving tasks are discussed in the field of Systems Engineering [43]. Possibilities for expert aids in model validation are presented in [44]. PAGE 19 12 Physical Sensors Math Algorithm Expert r \ Numerical ^ InputOutput Data Clustering First Cut Fuzzy Rules V J System X Additional Identification Tuning Final Tuned Fuzzy System V / Figure 2.2. Fuzzy engineering 2.1.2 Fuzzy Modeling Process Fuzzy engineering [28] is an emerging terminology which refers to the overall trials for finding the good qualities of the fuzzy system as a function approximator for a given physical systems. The basic unit of the fuzzy function approximation is the IFTHEN rule, and the quality of the approximation depends on how smart the rules are. For example, a few rules give a quick approximation, but are less accurate and which make the tuning process harder thereafter. On the other hand, constructing rules covering all input-output state spaces defined by the Cartesian product of the input pattern and the output pattern easily causes the rule explosion. For these reason, fuzzy engineering mainly concerns itself with developing efficient tuning algorithms or finding optimal set of rules. Figure 2.2 shows a general process for modeling fuzzy systems. The modeling process can start with numerical input-output data. Fuzzy system identification, often called black box modeling, is the most common approach for designing mathematical models of dynamic systems from numerical input-output data. The black box modeling contrasts with the white box modeling, where mathematical models are driven from already known physical laws [32]. The derivation of black box fuzzy models PAGE 20 13 from numerical data employs various clustering methods, neural networks and genetic algorithms to identify fuzzy rules with their membership functions. The recent achievements and comprehensive discussion of the black box modeling can be found in the literature [23, 36]. Their main tasks are structure and parameter estimations. However, the fundamental limitation in using numerical data comes from the curse of dimensionality problem [32]. That is, the number of possible rules, n, increases exponentially with the number of possible variables, r by the formula r n=X[nk, (2.1) where is the number of fuzzy values in the kih. variable. Since all variables are not important to describe the behavior of the underlying system, searching for influential variables among all possible variables causes the problem of combinatorial optimization. Therefore, the modeling process using these techniques suflfers from computational complexity or local optimality [28, 32]. In contrast, the modeling process can also start with a first cut fuzzy system as shown in Figure 2.2. In practice, the first cut fuzzy system can be directly obtained from the expert in the form of a verbal expression. The domain expert forms rules by performing the following three steps [28]: (1) selecting input and output fuzzy variables, (2) selecting fuzzy values of these variables and (3) constructing fuzzy rules by relating the input values to the output values. Rough fuzzy rules also can be obtained from numerical input-output data through unsupervised learning by clustering the data; this gives quick approximation, but is less accurate. The utilization of prior knowledge provided by experts alleviates many problems arising from using traditional system identification, because important variables of the system and their approximate forms of values are provided by the experts. Another advantage of utilizing the expert's prior knowledge is that some system phenomena such as those cannot be revealed by means of collecting the physically observed PAGE 21 14 input-output data can be captured. However, the first potential drawback of using the expert's knowledge is that it is difficult for the human expert to capture all causal system relations, especially in modeling complex systems. Even the expert is often unaware of all of the contextually dependent factors qualifying generalizations [13]. Secondly, obtaining such a good quality of first cut fuzzy rules is not easy in practice. Generally speaking, the gap between the implementation level (i.e., fuzzy rules) and the expert's knowledge is too wide. Specifically, both linguistic imprecision and uncertainty may exist in a fuzzy rule [22]. For example, given a fuzzy rule, IF wind is high THEN the saiUng should be good (0.8), the linguistic imprecision comes from the fuzzy value ''high" and the uncertainty comes from the numeric value (often called confidence factor) "0.8." The linguistic imprecision results from linguistic inexactness, specially, the linguistic vagueness in which the boundary is not clearly defined [22, 62]. Uncertainty can arise because reliable information cannot always be gathered to assign a probablistic uncertainty; moreover, even an expert may be unsure of a particular piece of causal information. Much of the uncertainty in such cases is possiblistic rather then probabilistic in nature [60]. 2.2 Fuzzy Set Theory and Its Application This section presents a review of the relevant aspects of fuzzy set theory which form the basis of our fuzzy simulation. The theory of fuzzy sets can be found in References [57, 58, 59, 27, 14, 61]. Generally speaking, fuzzy sets may be viewed as an attempt to deal with a type of imprecision arising when the boundaries of classes are not sharply defined. A fuzzy set A of a universe of discourse X is characterized by a membership function /x^ : X [0, 1] which associates with each element x of PAGE 22 15 X a number in the interval [0, 1] which represents the grade of membership of X in A. 2.2.1 Notation. Terminology, and Basic Operations Â• Definition 2.1: A fuzzy set A of the universe of discourse X is convex if and only if for all Xi , 0:2 in X //4(Aa;i + (1 \)x2) > Min{^j.A{xi),HA{x2)), where A Â€ [0, 1]. Â• Definition 2.2: A fuzzy set A of the universe of discourse X is called a normal fuzzy set if 3xi e X, fiA{xi) = 1. Â• Definition 2.3: A fuzzy number is a fuzzy set in the universe of discourse X that is both convex and normal. To simplify the representation of fuzzy sets, a finite fuzzy set, A, of X is expressed as A = Ha{Xi)/Xx + ^a{X2)/x2 + . . . + [lA{Xn)/Xn, OV A = "Â£^=1 HA{Xi)/Xi, where + sign denotes the union rather than the arithmetic sum. If the fuzzy set, A, is not finite, A may be represented in the form A = JxfJ'A{x)/x in which the integral sign stands for the union of the fuzzy singletons ixa{x)/x. Â• Definition 2.4: The height of fuzzy set A, height At in the universe of discourse X is the supremum of ^ia{x) over A. Formally, height A = supxexl^A^x). (2.2) Â• Definition 2.5: The complement of A is denoted by A and is defined by (2.3) PAGE 23 16 The operation of complementation corresponds to negation. Definition 2.6: The union of fuzzy sets A and B is denoted by A U B and is defined by AUB= [ {ha{x)^ Mx))/x, (2.4) where V is a maximum operator. Definition 2. 1: The intersection of fuzzy set A and B is denoted hy AnB and is defined by ADB^ [ Mx) A tiB{x))lx, (2.5) where A is a minimum operator. Definition 2.8: If / is an n-ary crisp function which is a mapping from a Cartesian product Xi x Â• Â• Â• x XÂ„ to a space Y, and if A is a fuzzy set in Xiy. Â• Â• y. Xn which is characterized by a membership function //^(xi, . . . , xÂ„), with Xi,i = 1, . . . ,n, denoting a generic point in Xi, then extension principle [59] says /(^) = /(/ fiA{Xi,...,Xn)/{Xi,...,Xn)) JXlX-XXn = / IJ,A{Xi,...,Xn)/f{Xu...,Xn). (2.6) It is assumed that the membership function of A is expressed by fiAiXu . . . , Xn) = llAiiXl) A Ha2{x2) A Â• Â• A fiAÂ„{Xr,), (2.7) where /x^^, i = 1, . . . ,n, is the membership function of Ai. Definition 2.9: Let A and B represent two fuzzy numbers and let denote any of the four basic arithmetic operations. Then, using the extension principle PAGE 24 17 (2.6) under the assumption (2.7), we define fuzzy set, A-kB on 11, where 7^ is a set of all real numbers, as AtA*B('2) = maa;^=x*y(/x^(x) A Hsiy)), (2-8) for all z Â£11. Thus, for example, \i A,B Cll are two fuzzy numbers with respective membership functions /i^(a;) and /iB(2/), then the four basic arithmetic operations, i.e., addition, subtraction, multiplication and division give, for each x,y,z e 1Z, the following results: fiA+Biz) = max^=j,+y{nA{x) A /xb(?/)), (2.9) fiA-B{z) = maX;,=^^y{nA{x) A /^^(y)), (2.10) AiAxB(-2) = max^=xxy{lJ'A{x) A /iB(y)), (2.11) /iA-rB(^) = maX:,=^^y{nA{x) A /iB(y))(2.12) Â• Definition 2.10: Let P be a compound statement of the type {X is A) * (3^ is 5), where A', 3^ = fuzzy variables that take real numbers from some universal set X, F, respectively, A, B = fuzzy values on X, Y, respectively, and * = a conjunction (and) or a disjunction (or). When * is a conjunction, the rule of conjunctive composition [61] states that P can be expressed by a possibility distribution ir{x,y) which is defined by {f^AxBix, y)/(x, y) \ X e X,y eY}, (2.13) where fJ-Axsix^y) = min{fj,A{x), fJ-B{y)), and x = Cartesian product. PAGE 25 18 When * is a disjunction, the rule of disjunctive composition [61] states that P can be expressed by a possibility distribution K{x,y) which is defined by Equation (2.13), where HAxB{x,y) = max(/i^(x), /iB(2/))Â• Definition 2.11: A fuzzy relation [58] R from a set X to a set F is a fuzzy subset of the Cartesian product X xY. R is characterized by a bivariate membership function ij,r{x, y) and is expressed Â• Definition 2.12: A fuzzy conditional statement, W X = A THEN y = B., ox for short, B/m which A and B are fuzzy sets, can be defined as a fuzzy relation Ax B. Â• Definition 2.13: If il is a relation from X to F and 5 is a relation from Y to Z, then the max-min composition [58] of R and 5 is a fuzzy relation denoted hy Ro S and defined by where Vj, is the supremum over the domain of y. Â• Definition 2.14: In traditional logic, one of the main tool for reasoning is modus ponens, that is, {A A {A B)) => B. Then an approximate extension of the modus ponens, called generalized modus ponens can be used for approximate reasoning in the following type of inference Premise: X is A' Implication: IF A' is ^ THEN 3^ is J5 Conclusion: 3^ is B', (2.14) (2.15) PAGE 26 19 where the conclusion B' can be derived by min-max composition rule of inference B' = A'o[AxB]. (2.16) 2.2.2 Membership Function Construction Numerous methods for constructing membership functions have been described in the literature [27, 42, 41, 9, 52, 2, 3, 33, 19]. All these methods can be classified into two approaches: direct and indirect methods. Both methods are further classified depending on whether one expert or multiple experts are involved. In the direct method [27], given a fuzzy set A, an expert is expected to assign a membership degree, fJ-A{x), to each element x, according to his or her opinion. This can be done by defining a complete membership function in terms of mathematical formula, or by exemplifying it by answering a question such as "what is the degree of membership of x in A?" In the indirect method, an expert is required to answer simpler questions which are easier to answer and less sensitive to the various biases of subjective judgement. The most common approach is based on pairwise comparisons [42, 41] of relevant elements, which replaces the direct estimates of membership degree. An example of typical question in this method is "which color, A ov B, has the property of darkness more strongly, and how much more?" For all pairs of elements, comparisons are repeated by giving numerical scale to express the relative strength of the property. The relative weights are represented by a nonsymmetric full matrix. Then the membership degrees are the components of the eigenvector corresponding to the maximum eigenvalue. In this section, we briefly review basic methods for constructing membership functions from sample data gathered from the direct or indirect method. These include the curve-fitting method and learning from neural network. In these methods, PAGE 27 20 we assume that n sample data < xi,ai >,< X2, a2 >, < Xn, an > (2-17) are given, where ajj, i = 1, 2, .., n is an real number and Oj is the membership degree of Xi in a fuzzy set A. Lagrange Interpolation Lagrange interpolation is a curve-fitting method in which the constructed membership function is represented by a polynomial form defined by f{x) = aiLi{x) + a2L2{x) + ... + aÂ„LÂ„(a;), (2.18) where {x ai) ...(x ai_i){x ai+i)...{x an) , . Li{x} = rÂ— rT" (2.19j [Xi ai)...{xi ai_ij(Xj ai+i)...{Xi an). Since values f{x) need not be in [0,1], the following formula is applied to make the fuzzy set A normal: lj,yi{x) = max[0,min[l, f{x)]]. (2.20) Even though the membership function matches the sample data exactly, complexity increases with the number of sample data. Besides, for the values of x outside the given sample range, this method does not work well. This requires that sample data be well distributed over the fuzzy set A. Least-square Curve Fitting Given sample data (2.17) and a suitable function f{x;a,l3,...), where a,/?,... are parameters whose values distinguish function in its class from one another, this method selects a function f{x;ao,f3o,...) from the class for which E = j2[f{xi;aJ,...)-aiY (2.21) 1=1 PAGE 28 21 reaches its minimum. Then, we apply Equation (2.20) to make a normal fuzzy set. Such a suitable function is chosen from standard distributions based on expert's experience or experimental comparison with other classes. The bell-shaped function defined below has been frequently used for such a purpose. /(x;Â«,/3,7) = 7e-(^-"^'/^ (2-22) where a is a location for center, \fPl2 defines the inflection points, and 7 is height parameter. Learning from Neural Networks The literature dealing with the use of neural networks for constructing membership functions is rapidly growing [9, 52, 2, 3]. Construction of such functions are done by learning patterns from sample data defined in (2.17). Let each input of x be xP, its expected output be V and its actual output be y^. Given suitable hidden layers and activation function, by initializing weights of the network and applying pairs < x^, > of the training set to the neural network, we can calculate the square error. E, = \{f-t^f. (2.23) Then, to minimize the Ep, we update weights according to backpropagation algorithm. At the end of each cycle, a cumulative error defined as E=\f:{y'-tn' (2.24) is compared against E^ax specified by user. A new cycle is initiated until E < EmaxWhen E < E^ax, the desired membership function is obtained. 2.2.3 Fuzzy Controller A fuzzy system is any system whose variables range over states that are fuzzy sets. The most successful application area of fuzzy system has been the area of fuzzy PAGE 29 22 FTJZZY CONTROLLER Action Defuzzification Controlled process Fuzzy inference engine Fuzzy rule base Condition Fuzzification Figure 2.3. A general architecture of a fuzzy controller control. Fuzzy controllers are special expert systems in the sense that each has a knowledge base represented by fuzzy inference rules and an inference engine. Fuzzy controllers are capable of utilizing knowledge elicited from human operators in control problems when [10]: Â• one or more of the control variables are continuous, Â• a mathematical model of the process does not exist, or exists but is too difficult to encode, and Â• a mathematical model of the process is too complex to be evaluated quickly enough for real-time operation. In those cases, an imprecise linguistic description consisting of a set of control rules can usually be articulated by the human operators with relative ease. Figure 2.3 shows the architecture of a fuzzy controller. In this section, we discuss the design process of a fuzzy controller. Let us consider a very simple fuzzy controller as shown in Figure 2.4. In this figure, the controller monitors two control variables, e and e, where e is defined as an error between the actual value of the controlling variable v and its desired value, and e denotes the rate PAGE 30 23 Disturbances Action Controlled process Condition Figure 2.4. A simple fuzzy controller -d -im -d/3 0 d/3 2d/3 d Figure 2.5. A fuzzy quantization of change in the error. Depending on e and e, relevant control actions represented by V are produced. Step 1: Fuzzy Quantization The first step is to obtain fuzzy quantizations by identifying relevant input and output variables and their ranges and selecting appropriate labels (i.e., fuzzy sets) for each variable. The number of labels associated with a variable is generally an odd number between 5 and 9 [10]. For the reason of symmetry, an odd number is preferred. The number of labels determines the expressiveness and the predictiveness of the fuzzy system [31]. The expressiveness is a measure for the information content that the model provides, while the predictiveness is a measure for its forecasting power. Since these two measures are contradictory, we should compromise. Some observations have been reported in which either three or five labels were about optimal in most Fuzzy controller PAGE 31 24 0.0 I |_ -d e e Figure 2.6. A fuzzified measurement practical applications [8, 53]. Figure 2.5 shows an example of the quantization where triangular fuzzy numbers that are equally spread over the range [-d, d] are used. Step 2: Fuzzification In this step, a fuzzification function / for each input variable is chosen to express measurement uncertainty. For example, by applying an appropriate fuzzification function, a measurement e = Xq can be defined by a fuzzy set as shown in Figure 2.6, where e is determined in the context of each particular application. This fuzzy set acts as a fact in the inference process (Step 4). Step 3: Obtain Conceptual Model In this step, a conceptual model is obtained in terms of a set of fuzzy inference rules that describe the action taken on each combination of control variables. Two common ways for obtaining such information are: 1) from human operators or 2) from empirical data by suitable learning methods [31, 51, 38, 50, 49]. The canonical form of the inference rule is IF e is /I and e is 5 THEN v is C. Generally, the number of rules required depends on the number of control variables [10]. For example, if a fuzzy controller requires n control variables and m fuzzy regions for each variable, the system generally requires rn" rules for a total of m" PAGE 32 25 e V NL NM NS AZ PS PM PL MI NM PL PM AZ NS PM PS AZ e AZ PM PS AZ NS NM PS AZ NS NM PM AZ NM NL PL Figure 2.7. An example of fuzzy inference rules possible input combinations. Since the number of fuzzy rules grows exponentially with the number of system variables, the search for optimal rules forms one of the main research areas of fuzzy engineering. However, most applications so far have had few inputs and outputs, and this has helped keep the rule explosion manageable [28]. Figure 2.7 shows an example fuzzy rule base in fuzzy sets defined in Figure 2.5. Step 4: Design an Inference Engine Here, to determine the resulting fuzzy set in multiconditional approximate reasoning represented by the form Rule 1: IF e is >li and e is Bi THEN v is Ci Rule 2: IF e is A2 and e is B2 THEN v is C2 Rule n: IF e is An and e is 5Â„ THEN v is CÂ„ Fact: e is A' and e is B' Conclusion: v is C", PAGE 33 26 e = A' e =B' Figure 2.8. An illustration of the min-max composition rule of inference a method called min-max composition rule of inference defined in Definition 2. 14 is commonly applied. That is, through min inference, each output membership function is cut off at a height corresponding to the minimum degree of the truth of the rule premise. Then, through max composition, a combined output membership function is constructed by taking pointwise maximums over all of the fuzzy set assigned to the output variable. An illustration of the method for two fuzzy rules is given in Figure 2.8. Step 5: Defuzzification This step performs defuzzification by converting output fuzzy set into a single real number. Two most common methods for defuzzification are centroid and composite maximum. The centroid method takes the center of gravity of the output fuzzy set C in Figure 2.8. The defuzzified value d is calculated by the formula , _ Er=i/^c'(^^i) X Vi Er=i/^c'(?^z) (2.25) PAGE 34 27 where Vi,i = l...n is a,n element in fuzzy set C. In the composite maximum method, a defuzzified value d is defined as the average of the smallest value and the largest value of v for which fic'iv) is the height of C", heightcFormally, , m.in{vi \vie M} + max{vi \vie M} d = , [Z.zt>) where M = {vi\ Hc'ivi) = heightc}2.3 Fuzzy Set Theory in Computer Simulation Probability-based methods are useful when most of the uncertainty can be effectively described through the use of large data sets and their associated moments. However, experts often do not think in probability values, but in terms such as much, usually, always, sometimes, etc. In domains where estimation or measurement of probabilities is not amenable, fuzzy set theory offers an alternative [22]. Here, we can use any type of fuzzy number, such as an interval-valued fuzzy number, a triangular fuzzy number, a trapezoidal fuzzy number or a general discrete (or continuous) fuzzy number, depending on the degree of uncertainty. Owing to the extension principle [61] in the fuzzy set theory, nonfuzzy mathematical structures can be made fuzzy. Here is a sample of how this relates to simulation. The following can be made fuzzy [58, 18]: 1) a state variable value including initial conditions, 2) parameter values, 3) inputs and outputs, 4) model structures and 5) algorithmic structures. For example, we can use fuzzy simulation to execute a fuzzy automaton, where its states are characterized by fuzzy sets, and the production of responses and the next states are facilitated by appropriate fuzzy relations [27]. For another example, a fuzzy algorithm, defined as a ordered set of fuzzy instructions, can be used to provide an approximate analysis of systems and decision processes PAGE 35 28 that are too complex for the application of conventional mathematical techniques [58]. Three kinds of fuzzy simulation approaches have been reported in the simulation literature: Qualitative Simulators (i.e., Qua.Si [4]), Fuzzy Qualitative Simulation (i.e., Fusim [46]) and Correlated Uncertainty method [15, 16, 17]. The Qua.Si and the Fusim are useful for qualitative simulations where simulations are performed using fuzzy sets themselves based on fuzzy calculus or fuzzy arithmetic. The third approach takes fuzzy sets from experts and, through deterministic sampling from fuzzy sets, it performs computer simulation quantitatively on discrete event or continuous models using real arithmetic. For such a process, every vertex in the fuzzy number is issued independently to the simulation function, and the outputs of the simulation are mapped into the most closely matched fuzzy linguistic value by a linguistic approximation. Thus, rules can be extracted, and these results can be validated against the expert's domain knowledge. The fuzzy simulation method that we've employed for isolating inconsistency is an extended version of the correlated uncertainty method. The algorithm for fuzzy simulation using the correlated uncertainty method is [17]: 1. Let a fuzzy simulation component such as a parameter p be defined as a triangular fuzzy number F, where F = {a,b,c). Assume the fuzzy number is identified by brackets(i.e., F[2] = b). 2. For j G 1,2,3: (a) Let p[j] = F[j]. (b) SIMULATE REAL (c) Vi, obtain Xi{te)[j], PAGE 36 29 (a) day interarrival rate service rate (b) line aiz.c i^itctompr cstiQfaption Saturday Morning short fast short happy Saturday Day short medium Saturday Evening medium medium medium complacent Tuesday Morning long slow long irritated (c) customer satisfaction grocery sales (d) day line size customer satisfaction (e) day service rate customer satisfaction happy complacent good good Saturday morning short happy Tuesday morning slow irritated irritated bad Figure 2.9. Information presented by an expert where SIMULATE REAL denotes simulation using real arithmetic instead of fuzzy arithmetic, tg Â— the end time for the simulation, and Xi = the state variables of interest. Because of the SIMULATE REAL, the simulation is accomplished by performing multiple simulations; the number of simulations depends on the order of the fuzzy numbers. The outputs of a simulation can be mapped into the most closely matched fuzzy linguistic values by using a distance metric. Let the simulation outputs of m ordered fuzzy number be defined as F(1),F(2), . . . F{m), and let the possible n output linguistic values be gi, 92, Â• , 9nThen the distance metric is defined by m Mm^|F(z) -9,(2)1, (2.27) where j = 1, 2, . . . , n. Let's take a simple grocery store example [16] to illustrate the concept of the fuzzy simulation. We assume that an employee (cashier) is an expert, and he or she provides the information represented as in Figure 2.9. Note that the line size PAGE 37 30 affecting the customer satisfaction is again influenced by both the interarrival rate and the service rate. However, as in this example, the expert may not be able to represent such complex relations precisely. Here, fuzzy simulation comes into play. By replacing those grocery store statistics with the compiled knowledge of queuing models, we can identify the deep knowledge for such complex relations as a hypothesis of the expert's knowledge. A single server queue as a deep model representing a grocery check out line can be associated with the following pseudo-code: schedule am arrival now; while (not end of simulation) do get_next_event ; switch on PAGE 38 31 INPUT: interarrival rate sh^rt mejdium Ipng INPUT: service rate f^st mefiium slpw sluggish t y 1. Simulation 2. Linguistic Mapping Fuzzy Simulation INPUT: customer satisfaction happy complacent irptated 5.0 10.0 15.0 20.0 line size 5.0 10.0 15.0 20.0 Time OUTPUT: Relation between interarrival/service rate and customer satisfaction interarrival rate service rate customer satisfaction short fast happy short medium happy short slow irritated short sluggish irritated medium fast complacent medium medium complacent medium slow irritated medium sluggish irritated long fast happy long medium happy long slow happy long sluggish complacent Figure 2.10. Applying fuzzy simulation to grocery store example PAGE 39 32 By executing the fuzzy simulation in Figure 2.10, we obtain the quantitative plot of line size over time based on each combination of three linguistic values for the interarrival rate and four linguistic values for the service rate. Sampling is based on the three vertices of the fuzzy number being used to define both the interarrival rate and the service rate. From this, we obtain a time-variant description of customer satisfaction over time. This is done by mapping from data for the line size into the fuzzy linguistic values for customer satisfaction using the distance metric. Output of the fuzzy simulation shown in Figure 2.10 forms a more complete rule base as a hypothesis of the expert knowledge. When we compare them to Figure 2.9 (a) and (e), we can note that there is a conflict between the rules concerning Tuesday morning. The queuing model predicted a happy customer, whereas the expert specified an irritated customer. When such conflicts arise, the expert can either reevaluate his original rule as slightly off or a set of parameters can be changed in the queuing model. In this way, the expert must evaluate the new rules created by the fuzzy simulation to see if there is agreement with his expertise. In this chapter, we reviewed the background that is relevant to this research. We discussed the simulation and the fuzzy modeling processes in general and addressed the importance of the expert's role during model validation processes. In the next chapter, we present a new fuzzy simulation approach to bridge the gap between the expert rules and an assumed quantitative model. The fuzzy simulation approach handles, particularly, the possiblistic uncertainty in the expert's rules by directly encoding the uncertainty into the simulation components. PAGE 40 CHAPTER 3 A NEW FUZZY SIMULATION APPROACH The fuzzy simulation approach introduced here has extended the original version of correlated uncertainty method discussed in Chapter 2. In this extended approach, by carrying membership degrees of fuzzy sets in the expert's rule premise and issuing them to simulation components, we are able to calculate a confidence factor for each rule, which is then compared against the confidence factor of the expert rule. This quantitative measurement provides us with useful information such as "which is the most inconsistent rule?" and "how consistent are two given rule sets?" This facility plays a basic role for isolating inconsistent knowledge through interactive user control. In the following sections, we assume three types of expert rules as the inputs to the fuzzy simulation. Then, we present how to handle these types of rules differently using the fuzzy simulation approach. 3.1 Expert Rule Format The input of fuzzy simulation is a collection of expert rules. In what follows, we assume that the three following canonical forms of rules are presented by experts. Â• IF A" is A THEN yisB [CF), Â• IF A' is {Ai A2) THEN y is B {CF) and Â• IF {X is A) * [y is B) THEN Z is C (CF), where X, y and Z = fuzzy variables that take real numbers from some universal sets, X, Y and Z, respectively, 33 PAGE 41 34 Table 3.1. Notation Notation Usage RU LEgxpert RU LE fuzzy F expert MFfuzzy M Fpj-ejjiise M F ^onseq C Fgxpert CF fuzzy Rules presented by expert Rules generated by fuzzy simulation Fuzzy Membership Functions presented by an expert Fuzzy Membership Functions generated by fuzzy simulation Membership Functions of fuzzy value in rule premise Membership Functions of fuzzy value in rule consequence Confidence Factor presented by an expert Confidence Factor calculated by fuzzy simulation A, Ai and A2 = fuzzy values on X, B and C = fuzzy values on Y and Z, respectively, CF Â— a confidence factor in the rule consequence given that the premise conditions are satisfied, and and * = arithmetic (+, -, x or ^) and logic (or or and) operator, respectively. We call the first type of rule a simplex rule, and the other type a compound rule. The following is an example of the compound rule with the logic operator and: IF (Temperature is High) and (Pressure is Slightly _Low) THEN Heat.Change should be Slightly J^egative (CF = 0.8). The premise parts of the last two canonical types of rules can be combined to make a more complex rule such as IF {X is {Ai + ^2)) or {y is {Bi + B2)) THEN Z is C. For simplicity, the notation in Table 3.1 will be used in the entire chapter. 3.2 Fuzzy Simulation The fuzzy simulation method introduced here is capable of simulating the expert rules using quantitative models. For each expert rule, this method takes the premise PAGE 42 35 part and its MFpremise, and through simulation, it generates a conclusion with a CFfuzzy With the intention of comparing this result directly against the expert rule, the fuzzy simulation method is forced to derive a conclusion with the same linguistic value that the expert presented, but with possibly a different CFfuzzy from th6 C F expert' When the expert rule is simplex, fuzzy simulation involves one simulation for each element within a MFpremiseIn contrast, when the rule is compound, we first obtain an intermediate fuzzy set by applying the extension principle defined in Definition 2.8 or the rule of conjunctive or disjunctive composition defined in Definition 2.10, depending on whether the operator type in the premise is arithmetic or logic. Fuzzy simulation using the compound rules involves one simulation for each element within the intermediate fuzzy set. Finally, we calculate a CFfuzzy for that rule using a weighted average method. 3.2.1 Fuzzy Simulation for Simplex Rules Consider a simplex rule of the type IF A" is THEN y is B, where X,y = fuzzy variables that take real numbers from universal set X, Y , respectively, and A, B = fuzzy values on X, Y , respectively. Algorithm of Fuzzy Simulation 1. Let a fuzzy simulation component such as a parameter p be defined as a fuzzy set A, where A = [Ia{xi)Ixi + tiA{x2)/x2 + . . . + flAM/XnAssume the element of A is identified by brackets (i.e., ^[2] = X2). PAGE 43 36 membership degree membership degree A V 0 membership function of P membership function of other fuzzy set Figure 3.1. A two-step process of searching membership degree 2. For j Â€ l,2,...,n: (a) Let p[j] = A[j]. (b) SIMULATE REAL. (c) obtain {fj.B{yj)/yj){te)3. calculate CFf^zzy, SIMULATE REAL denotes simulation using real arithmetic, yj,j = 1, . . . , n denotes real values on Y, and te = the end time for the simulation. During SIMULATE REAL, the correlated uncertainty method requires that when we replace p with a real number whose membership degree is d, we should replace other fuzzy simulation components with real numbers whose membership degrees are also d. This procedure involves a two-step process of searching for the membership degree of p, and then using this degree to drive the elements of other fuzzy sets. This process is illustrated in Figure 3.L In what follows, SIMULATE REAL involves this operation. where PAGE 44 37 Calculation of CFf.,Â„Â„j Just as CFexpert IS presented by the expert, we need a way to obtain CFj^zzy from the fuzzy simulation. By doing this, we benefit from the comparison of the two rules in terms of their CF values. However, since the derivation of the CFexpert involves a subjective opinion, as well as a certain amount of uncertainty, there is no theoretical formulation to calculate the CFj^zzy whose derivation process is exactly the same as that of the CFexpertOur solution is to define an equation so its result agrees with a human intuition as much as possible. We used a weighted average method to create such intuition. Given a simplex rule, we define the CFf^zzy by using the weighted average method: fuzzy Â— ^ ~~r7~\ ' W--*-;) where Xj,j = 1, 2, . . . , n, denote real values on X in the fuzzy set A, and yj,j = 1, 2, . . . , n, denote real values on Y obtained from SIMULATE REAL using The validity of calculating CFj^zzy using the weighted average method is shown as in Figure 3.2. The CF fuzzy using Equation (3.1) is 1.0 and 0.0 for Figure 3.2 (a) and Figure 3.2 (b), respectively. The results match our intuition. When the CF falls into some range between the above two extreme cases (i.e., 0.0 and 1.0) as shown in Figure 3.3, we can intuitively say that each member in A supports the conclusion B with a higher confidence, the greater CF we get. Using (3.1), the CF fuzzy for Figure 3.3 (a) is : (0.5 X 0.5) + (1.0 X 1.0) -I(0.5 X 0.5) CF. fuzzy Â— 0.5 + 1.0 + 0.5 -0.75, PAGE 45 38 Xj Xi 3^ Xg yj jtj ^3 Jj; X^XjjX^ Xg ^1 ^2 ^3 ^4 ^5 SIMULATE REAL SIMULATE REAL Figure 3.2. All members or none of members support the conclusion (a) All members of A support the conclusion B with full confidence; (b) None of the members of A support the conclusion B Figure 3.3. Some members support Conclusion (a) Members of A support the conclusion B with higher confidence compared to the case of (b); (b) Members of A support the conclusion B with less confidence compared to the case of (a) and the CFjuzzy for Figure 3.3 (b) is: (0.5 X 0.3) + (1.0 X 0.7) + (0.5 x 0.2) fuzzy Â— 0.5 + 1.0 + 0.5 = 0.48. Example Figure 3.4 illustrates how to perform fuzzy simulation using simplex rules, IF X is X THEN y is B. The results of SIMULATE REAL are artificially made for the purpose of illustration. Applying Equation (3.1), we obtain CFjuzzy by (1.0 X 0.5) + (0.7 x 0.5) fuzzy Â— 0.3 + 0.7 + 1.0 + 0.7 + 0.3 0.21. PAGE 46 39 A B 1.0 0.7 0.5 0.3 0 1 2 3 4 5 6 15 17 26 35 Li SIMULATE REAL Â• Â• Â• Figure 3.4. Fuzzy simulation using simplex rule 3.2.2 Fuzzy Simulation for Compound Rules with Arithmetic Operations The extension principle [59] defined in Definition 2.8 is a principle for fuzzifying crisp functions. It can be used to generalize crisp mathematical concepts into fuzzy sets. Owing to this principle, models and algorithms involving nonfuzzy variables can be extended to the case of fuzzy variables. By applying Equation (2.8) to a compound rule with arithmetic operation, an intermediate fuzzy set is obtained, and this set is used for fuzzy simulation. Consider a compound rule of the type where is one of the four basic arithmetic operators (i.e., +, -, x, -^). Algorithm of Fuzzv Simulation 1. Apply Equation (2.8) to the rule premise. 2. Let Z be a resulting intermediate fuzzy set, and let a fuzzy simulation component such as a parameter p be defined as a fuzzy set Z, where IF X is {Ai A2) THEN y is B, Assume the element of Z is identified by brackets (i.e., Z[2] = Z2). 3. For j e 1,2, ...,n: PAGE 47 40 (a) Let p[;] = Z\j]. (b) SIMULATE REAL. (c) obtain {nB{yj)/yj){te), 4. calculate CFjuzzy Calculation of CFfÂ„,,,j Given a compound rule with arithmetic operations, we define the CFjuzzy by using the weighted average method f-^^ fuzzy 7-^; ) W-^) where Zj,j = 1,2, ...,n, denote real values on a fuzzy set resulted from the arithmetic operation, A^-k A2, and yj,j = 1, 2, . . . , n, denote real values on Y obtained from SIMULATE REAL using Zj. Example Let's assume that we want to perform fuzzy simulation using the following rule: IF X is {Ai + A2) THEN y is B, where Ai and A2 are defined by Figure 3.5 (a) and Figure 3.5 (b). By applying Equation (2.9) defined by fJ-Ai+Aiiz) = maX;,^Ai+A2{fJ'Ai{x) AHAiix)), we obtain the following set of equation for the intermediate fuzzy set Z. Atz(l) = (0A0.3) V(OAO.l) ==0, //z(2) = (0A0.6) V(0.1 A0.3) V(0.3A0) = 0.1, PAGE 48 012 3 4567 01234567 (a) (b) Figure 3.5. Two fuzzy sets for addition ^z{3) = (0 A 1) V (0.1 A 0.6) V (0.3 A 0.3) V (0.8 AO) = 0.3, = (0A0.7)V(0.1 Al) V(0.3A0.6) V(0.8A0.3) V(l AO) = 0.3, fj,z{5) = (0 A 0.2) V (0.1 A 0.7) V (0.3 A 1) V (0.8 A 0.6) V (1 A 0.3) V (0.7 A 0) = 0.6, /iz(6) = (OAO.l) V(0.1A0.2) V(0.3A0.7) V(0.8A1) V(l A0.6) V(0.7A0.3)V (0.3 AO) = 0.8, fiz{7) = (0A0)V(0.3A0.3) V(0.7A0.6) V(l Al) V(0.8A0.7)V(0.3A0.2) V (0.1 A 0.1) V (0 AO) = 1, ^z{8) = (0.1 AO) V (0.3 A 0.1) V (0.8 A 0.2) V(l A 0.7) V (0.7 A 1)V (0.3 A 0.6) V (OA 0.3) = 0.7, l_iz{9) = (0.3 A 0) V (0.8 A 0.1) V (1 A 0.2) V (0.7 A 0.7) V (0.3 A 1) V (0 A 0.6) = 0.7, /i^(lO) = (0.8 AO) V (1 A 0.1) V (0.7 A 0.2) V (0.3 A 0.7) V (OA 1) = 0.3, ^lz{n) = (1 A 0) V (0.7 A 0.1) V (0.3 A 0.2) V (OA 0.7) = 0.2, ^z(12) = (0.7 A 0) V (0.3 A 0.1) V (0 A 0.2) = 0.1, /xz(13) = (0.3 AO) V (OAO.l) = 0. PAGE 49 42 1.0 o.g 0.7 0.6 0.3 0.2 0.1 I 1 Â— r Â— I 1 2 3 4 5 6 7 8 9 10 11 12 13 25 29 32 42 51 57 SIMULATE REAL Figure 3.6. Fuzzy simulation of compound (addition) rule Figure 3.6 shows the fuzzy set Z and the result of fuzzy simulation using Z. Again, the results of SIMULATE REAL was created arbitrarily for illustration purposes. Using Equation (3.2), we can calculate CFjuzzy by CP ^ (0.1 X 1.0) + (0.3 X 0.6) ^ ^""''^ 0.1 + 0.3 + 0.3 + 0.6 + 0.8 + 1.0 + 0.7 + 0.7 + 0.3 + 0.2 + 0.1 ' ' 3.2.3 Fuzzy Simulation for Compound Rules with Logic Operations When the expert's rule premise involves logical operators such as and or or, the rule of conjunctive composition and rule of disjunctive composition [61] defined in Definition 2.10 is applied to obtain a possibility distribution. This distribution is used for fuzzy simulation. Consider a compound rule of the type IF {X is A) * {y is B) THEN Z is C, where * denotes any logical operator. Algorithm of Fuzzy Simulation 1. If * is and operator, PAGE 50 43 then apply the rule of conjunctive composition to the rule premise and calculate a possibility distribution 7c{x,y). If * is or operator, then apply the rule of disjunctive composition to the rule premise and calculate a possibility distribution K{x,y). 2. Let fuzzy simulation components such as p and q be defined as fuzzy sets A and B, respectively, where B = HB{y\)lyi + tJ'B{y2)/y2^ + iJ-B{yn)lynAssume the elements of A and B are identified by brackets (i.e., A[l] = X2 and B\2] = 2/2). 3. For z e 1, 2, . . . , m For j e 1,2, . . . ,n Let p[i] = A[i]. Let q[j] = B[j]. SIMULATE REAL, obtain {iJ.c{zij)/zij){te). 4. calculate CF fuzzy, where m and n are the number of elements in A and B, respectively. Notice that in the rule defined above, the universal set of the fuzzy variables A and B are not identical. Otherwise, instead of i[{x, y), we can get a more simplified fuzzy set as an intermediate set by Definition 2.6 and Definition 2.7 for disjunction and conjunction, respectively. PAGE 51 44 Calculation of CFf,Â„,y Given a compound rule with logic operations, CFjuzzy is defined by using the weighted average method fuzzyÂ— v-n ( \ ' where * denotes logical operator. Example Let's assume that we want to perform fuzzy simulation using the following compound rule: IF {X is A) and {y is B) THEN Z is C, where A and are defined as A = sma// = 1/1 + 0.6/2 + 0.1/3, B = larger OA/1 +0.6/2 + 1/3. By applying the rule of conjunctive composition, the predicate {X is A) and {y is B) yields the following possibility distribution: T^{x,y) = {[tJ'AandBixi,yi)/{xi,yi)],[tlAandB{xuy2)/{xuy2)], = {fJ'AandB{Xl,y3)/{Xi,y3)],[^AandB{x2,yi)/{x2,yi)], [/^/i and 0(2:3, 2/3)/ (a^a, ya)]} = {[0.1/(1, 1)], [0.6/(1, 2)], [1/(1, 3)], [0.1/(2, 1)], [0.6/(2, 2)], [0.6/(2, 3)], [0.1/(3,1)], [0.1/(3, 2)], [0.1/(3, 3)]}. Let's assume that we have the result as shown in Figure 3.7 after performing SIMULATE REAL on this 'K{x,y). Using Equation (3.3), we can calculate CFjuzzy by (1.0 X 0.5) + (0.1 X 0.8) + (0.6 X 0.3) ^^fuzzv Â— ~ Â— = 0.44. ^ ^ 1.0 + 0.1 + 0.6 PAGE 52 45 0.1/(1,1) 0.6/(1,2) 1/(1,3) 0.1/(2,1) 0.6/(2,2) 0.6 / (2,3) 0.1/(3,1) 0.1/(3,2) 0.1/(3,3) Â• Â• Â• Figure 3.7. Fuzzy simulation of compound (conjunction) rule In this chapter, we presented a new fuzzy simulation approach. We showed that how the fuzzy simulation handles possiblistic uncertainty by means of three approximate reasoning tools and the weighted average method. In the next chapter, we present an environment for isolating inconsistency between the expert rules and an assumed quantitative model. The fuzzy simulation approach forms a basis for building such an environment. This environment handles the linguistic vagueness discussed in Section 2.1.2 and supports interactive user control. PAGE 53 CHAPTER 4 A METHOD FOR ISOLATING INCONSISTENCY The purpose of this chapter is to provide an interactive environment to check for consistency and resolve inconsistencies between the qualitative and the quantitative models. Based on the fuzzy simulation approach discussed in the previous chapter, this environment also handles the linguistic vagueness in the expert rules. The enConsistency Checker Isolating Inconsistency Advisor Human Intervention Checking consistency 0 Apply fuzzy simulation directly Reporting 0 Inconsistent Rules o Expected Rules Change 0 Rules o MFs 0 Simulation components Checking consistency 0 Identify approximate MF^^^^^ Conducting o Incremental test 0 Sensitivity test Reporting o Inconsistent rules yes Suggesting 0 Approximate ME fuzzy Figure 4.1. An environment for isolating inconsistency vironment is shown at Figure 4.1. As shown in this figure, the method for isolating 46 PAGE 54 47 inconsistency consists of three major components: Consistency Checker, Advisor and Human Intervention. Consistency Checker performs two major tasks: checking consistency by applying the fuzzy simulation directly, and checking consistency by identifying approximate MFs. Whenever MFexpert is available (i.e., when the expert provides complete fuzzy sets as estimates for linguistic terms), it performs the first task by comparing the rule consequences from the fuzzy simulation against those from the expert in terms of CF. When the amount of inconsistency is out of range, all inconsistent rules and their expected rules from the fuzzy simulations are suggested through Advisor. When there is no MFg^pert a priori (i.e., when expert provides central point or interval estimates for linguistic terms). Consistency Checker performs consistencychecking by identifying an approximate MFjuzzy Using fuzzy simulation, it first focuses on discovering another important knowledge source linguistic definitions in expert's rules. It tries to produce approximate definitions where the rules from fuzzy simulation maximally match against the rules from the expert. If such definitions can be generated with a fairly good match between the two rule sets, these linguistic definitions are suggested through Advisor, and consistency checking is finished by returning an answer, "'consistent.'''' Otherwise, the answer 'Hnconsistenf is returned, and Advisor performs an incremental test or a sensitivity test to find out the source of inconsistency. Using all informations from Advisor, we start to resolve the inconsistency. For this process. Human Intervention is permitted: either the expert rules (including CFexpert, MFexpert) Or the simulation components can be modified interactively. Every time these modifications occur, Consistency Checker is reinvoked with visual aids, so that the user can easily recognize the effect of the modifications. Therefore, the overall process now involve humans in the loop during the process of checking PAGE 55 48 consistency and resolving inconsistency. We start with this chapter by introducing the quantitative measurements of inconsistency that we have employed. 4.1 Measurements of Inconsistencv The consistency between two types of models can be measured by the difference between CF presented by experts and CF calculated from fuzzy simulation on each rule. For each rule, we define its Local Inconsistency, LI, by LI Â— \CFfuzzy Â— CFg^pertl(4.1) Global inconsistency is measured by summing up such differences in the entire rule set. Thus, using the LI, we define the Global Inconsistency, GI, in a rule set by m GI = J2LI^, (4.2) i=l where m = total number of rules. Searching for the largest LI enables us to identify the most inconsistent rule (i.e., the worst case rule) between two different knowledge sources. Moreover, calculating the GI in this way allows us to measure the total amount of inconsistency. Note that given two rule sets and their GI, a slightly better GI does not always mean that the rule set leading to this GI is more consistent than the others. This results from possiblistic uncertainty of confidence factors derived by the expert [60] or measurement uncertainty, due to an inability of a measuring instrument to overcome its limiting finite resolution [27]. Using the Equation (4.2), we will say two given qualitative and quantitative models are consistent if GI < e, (4.3) where e is a consistency criterion specified by user. Any inconsistency found may be due to: PAGE 56 49 Reasoning by Human Consequences > Premises MFs from expert Identification of inconsistent rules Reasoning by Fuzzy Simulation Consequences Figure 4.2. Identification process of inconsistent rules Â• an inadequate conceptual model. Specifically, underlying assumptions, model structure, logic, mathematical relations and particular pieces of causal relationship may be inadequate, Â• an improperly implemented conceptual model, or Â• improperly designed expert rules. Specifically, improper rule structure, inadequate fuzzy value boundaries and central points or possiblistic uncertainty (confidence factor) may cause the inconsistency. 4.2 Checking Consistency When MK-r^.^,-/ is Available By taking MF^xpert as an input of fuzzy simulation as introduced in Chapter 3, we obtain each rule consequence associated with a CFjuzzy Using the CFexpert and CF fuzzy pair, we get the LI in each rule by Equation (4.1). Then, the GI is obtained from Equation (4.2). This process is shown at Figure 4.2. The most inconsistent rule is considered the rule which has the largest LI. The GI can be used for a performance index. Thus, when any sources of inconsistent components are modified, a comparison between the current GI and the previous GI indicates whether this modification is a good decision or not. Once the inconsistent rules are PAGE 57 50 identified, we must be careful not to eliminate the possibility that the CF^xpert was derived in different manner, and that it is not the same as the weighted average method defined in Equation (3.1), Equation (3.2) and Equation (3.3). For reducing such a possibility, Advisor generates rules for all possible consequences associated with the CF fuzzy, whose values are calculated by the weighted average method. 4.3 Checking Consistency When MF^n-p^rt is Unavailable The previous section showed how to check for consistency when linguistic terms are defined o priori. Checking for consistency was possible because fuzzy simulations used the expert's precisely predefined linguistic definitions. The algorithm presented here allows us to check for consistency with minimal information provided, such as central point estimates. In this section, we assume that, at the very least, the central point estimates for linguistic values are provided by the expert. For a complete discussion of the various estimate forms that can be covered in this study, see Section 4.3.3. Checking for consistency is possible by generating a set of MFj^zzy where RU LEexpert and RULEfuzzy maximally match. If such definitions (i.e., MFf^zzy) are generated with a GI of less than consistency criterion, e, RU LEexpert and RULE fuzzy are considered to be consistent with such definitions. However, if any set of MFfuzzy does not lead to a GI of less than e, then this means that we cannot find any linguistic definitions that properly fill the gap between two models. This implies that a discrepancy exists between RU LEexpert and RULE fuzzy In this case. Advisor performs either an incremental test or a sensitivity test to identify inconsistent rules. 4.3.1 Process in General By forcing fuzzy simulation to produce RULE fuzzy and CF fuzzy which are maximally close to RU LEexpert and CFexpert, respectively, we can discover a set of MFfuzzy PAGE 58 51 Reasoning by Human MFs from expert (possibly exist in mind) Consequences Reasoning by Fuzzy Simulation Figure 4.3. Identification process of approximate MFfuzzy between these two different models. This optimization process for generating the MFfuzzy is shown in Figure 4.3. The purpose of this process is not to generate finetuned fuzzy membership functions, but to decide whether any set of approximate membership functions exist in which the two models lead to consistent conclusions. This process can be stated as "search for a set of MFfuzzy which leads to a minimum GI for all rules, and then check if this GI is less than a given consistency criterion, e." 4.3.2 Heuristic Function and Search Method for Generating Approximate MFj,Â„,y To find such a set of MFfuzzy, our algorithm uses the following heuristic function, goal, and search methods in its tuning process: Â• heuristic function: GI. Â• goal: minimization of GI. PAGE 59 52 Â• search method: gradient descent search for finding a minimum GI by always moving in the direction in which the decreasing rate of change is the greatest. Particularly, for each iteration, pick the most inconsistent rule and then pick such a subset of the MF fuzzy in the rule that increasing the spreads of this subset by Ad reduces the GI to the greatest amount. If Ad is too small, then fuzzy membership functions are adjusted very slowly. If a large Ad is chosen, the convergence may be faster, but optimal spreads may be missed. A possible solution is to pick Ad in an adaptive manner. That is, if the GI is decreasing rapidly, take big steps, but if the GI is decreasing slowly, take small steps. Figure 4.4 shows that how the tuning algorithm uses this evaluation function and the search method to generate the MF fuzzy In this figure, a rule set is assumed to be composed of three rules, R\, R2 and R3. Our purpose in this figure is to find a set of approximate MFfuzzy for all linguistic values that leads to the smallest GI. The tuning process begins with the minimal-size fuzzy sets centered on the central point estimates. Then it increases the size of appropriate fuzzy sets by Ad, where the appropriate fuzzy sets are selected using the above strategy. Let us assume that the first fuzzy simulation for each rule provides the information that R2 is the worst case rule. By selecting the R2, the algorithm encounters states, 2, 3, 4, 8 which denote the every possible sets of MFfuzzy after tuning a subset of the MFfuzzy in R2. In this way, a state is defined as a set of MFfuzzy For example, if R2 contains three fuzzy sets A, B and C, the state 2 through state 8 denote each MFfuzzy after tuning {A}, {B}, {C}, {A,B}, {5,C}, {A,C} and {A,B,C}, respectively. Since the fuzzy simulation using state 4 leads to the lowest GI, state 4 (i.e., MFfuzzy after {C} is tuned) is selected for the next tuning process. In this way, more than zero fuzzy sets PAGE 60 53 are tuned at each iteration of the algorithm. When the algorithm reaches a point where such a modification does not reduce the GI compared to the previous G/, this sequence of tuning stops. However, this algorithm may reach either a local minima, Rl R3 R2 = worst case rule LEGEND o selected path selected state f ^ \ Â• Â• Â• ! Vg1=6.09 J ^^^^^^^^^^ possible subsets for tuning MPs in worst case rule / R2 / Rl = worst case rule R3 R3 possible subsets for Â• Â• Â• tuning MPs V 01=6.38 J Y Gl=5.97 J in worst case rule Figure 4.4. Heuristic function GI and search for the smallest Gl a plateau, or a ridge, because it keeps track of only the current states, and does not look ahead beyond the immediate neighbors of that state. Among many possible ways [20, 37, 39, 40] to deal with these problems, we adopt a random-restart gradient descent search [40] by conducting a series of gradient descent searches from randomly generated promising states, running each until it makes no discernible progress. For implementing this strategy, we employ a promising state criterion, 6 defined in terms of GI, which is used for randomly selecting promising states. For example, promising PAGE 61 54 states can be defined as {states \GI PAGE 62 55 Figure 4.5. Random-restart gradient descent search PAGE 63 Type Form of estimate Explanation Discretization into fuzzy space 1 central point estimate When experts present the center point c of A conf 1.0dence / c L 2 Interval of full confidence When experts present the interval /a, 67 of A with a full confidence conf 1.0 dence A a b 3 Interval of min-max range When experts present the min-max range [d, e] of A conf 1.0 dence A d e 4 Approximate fuzzy number When experts present both the center point c and the min-max range [d, e] of A conf i.o' dence d c L e when experts present both the interval [a, b] of Awith a full confidence and its min-max range [d,e] conf 1.0 dence A da be 5 Complete fuzzy number When experts present the complete definition of A confi i.o' dence i A 5 Figure 4.6. Five types of estimates about the linguistic terms PAGE 64 57 Figure 4.6 shows the five types of estimates and the way to discretize such an uncertainty into a fuzzy real space. From expert's point of view, the lower on the scale a type is located, the easier it is to estimate. Note that estimating Type 3 is harder than Type 1 and Type 2, since the exact extreme points of that interval are difficult to determine. The algorithm for generating MFjuzzy has been devised for handling the first two types of estimates (when Type 4 or Type 5 is presented, tuning MF fuzzy itself is unnecessary, because the approximate or complete forms of linguistic values are already given). Even though the algorithm presented in the next section deals with the central point estimates, it can be easily extended to cover Type 2 as well. That is, as shown in Figure 4.7, the meaning of the linguistic term of Type 1 can be approximated by a symmetric, triangular fuzzy number. Likewise, the meaning of Type 2 can be approximated by a symmetric, trapezoidal fuzzy number. The difference between those two types is only the shape of the fuzzy numbers. Therefore, these cases can be handled in the same way, if we represent every fuzzy numbers in our algorithm by .^-iup/e symmetric fuzzy numbers [d, a, 6, e] as shown in Figure 4.7. For example, the membership degree of two types in this figure can be derived using the same equations defined by li X < d ii d < X < a ii a < X < b ii b < X < e if X > e. 4.3.4 Algorithm to Generate MFfÂ„^,y As shown in Figure 4.8, the algorithm to generate the approximate MFjuzzy consists of six steps. All steps are explored in detail. r 0 1 1 2(a-i) eÂ— dÂ— (6Â— a) 1 2(x-6) eÂ—dÂ—{bÂ—a) 0 PAGE 65 (a) Triangular fuzzy number (b) Trapezoidal fuzzy number Figure 4.7. A 4Tuple Fuzzy Number (a) Triangular fuzzy number; (b) Trapezoidal fuzzy number Stepl Set initial conditions Step 2 Fuzzy simulations for all rules o calculate Lis and GI Advisor Return ^approximate MF^^^y with "consistent Step 6 Step 5 Step 3 Tuning MFfÂ„ until best saved GI does not improve Find each possible subset of MJ^^^^y in the most inconsistent rule ^ Modify this subset Fuzzy simulations for all rules ^ Pick one stat e leading to minimum Gl yes Pick the minimum Gl so far and MFj^^y leading to this Gl yes Figure 4.8. An algorithm for MFfuzzy generation PAGE 66 59 Step 1: Set Initial Conditions The following items are initially set by the user. Â• a tuning size, Ad, of MFjuzzy, Â• a consistency criterion, e, Â• a promising state criterion, 6 and Â• number of random restarts to be conducted. Initial sizes of MFjuzzy can be determined such that each range of MFfuzzy is 2Ad, based on a central point estimate. The user can initially set the last three items to certain numbers, but it is better determine this later, because the characteristics of these values can be observed after a few iterations of the algorithm. Step 2: Fuzzy Simulations for All Rules For each RULEgxpert-, a fuzzy simulation is executed as discussed in Chapter 3. Sampling is first done at the central point, then in both directions by increasing and decreasing Ad until right before these points exceed the width of the MFpremiseThe LI for each rule and the GI for a whole-rule set are calculated by Equation (4.1) and Equation (4.2), respectively. Step 3: Tuning MFj^^^y until best saved GI does not improve The purpose of this step is to incrementally reduce a GI by selecting the most inconsistent rule and modifying a proper subset of MFfuzzy among the all subsets of MFfuzzy in the most inconsistent rule until we eventually reach the smallest GI. A series of tuning processes stops at this point. During this step, we maintain two lists: a Visitiist and a RandomnstThe Visitust is a list to save a state which already has been visited. This list is used to avoid revisiting the same states. The Randomnst is a list which saves a state whose GI is less than given promising state criterion, 9. PAGE 67 60 This list is used when we randomly pick a promising state for random-restarts. As shown in Figure 4.8, this step consists of four substeps. Â• Step 3.1: Find each possible subset of MF fuzzy in the most inconsistent rule. That is, first, by picking a rule which has the largest LI, we find the most inconsistent rule. Next, all possible subsets of the MFjuzzy in this rule are obtained from the power set of the MFjuzzy Then, for each subset, we execute Step 3.2. Â• Step 3.2: For each subset of MF fuzzy, we increase the elements in each subset by Ad and execute fuzzy simulations for all rules to calculate each LI. Â• Step 3.3: Pick one state leading to a minimum GI. This state is likely to be the best candidate for reducing GI as a whole. Then, update the Visitust and Randomiist by adding this state to the Visitust, and adding the other states whose GIs are less than 9 to the RandomustÂ• Step 3.4: Compare this GI to the previous GI. The purpose of this substep is to make sure that the best candidate obtained from Step 3.3 actually improves the situation. Thus, a stop condition for the tuning process can be written as Current GI > previous GI? If the condition is satisfied, our algorithm proceeds to Step 4. Otherwise, the current GI is saved as a previous GI, and the entire Step 3 is executed until the stop condition is satisfied. Step 4: Random-restarts This step continues to execute Step 3 with random-restarts, each time reinitializing GI to a maximum value, until the algorithm reaches a user specified number of restarts. Note that Â• before executing Step 3.1, we randomly choose a state in the Randomust and immediately deleted it from the list and PAGE 68 61 Â• in Step 3.3, the state selected should not be in the Fzsi^iisf. Step 5: Pick a State Leading to a Minimum GI Among GIs resulting from a series of random-restarts, pick a minimum GI and its M F fuzzy step 6: Return Results Depending on the Consistency Criterion, e If the GI above is less than e, then it means that the algorithm was capable of generating the MFjuzzy in which two models exist in a consistent manner. In this case, the MFfuzzy is returned as approximate linguistic definitions for both the RULEexpert and the RULE fuzzy Otherwise, the algorithm returns ''inconsistent." However, we may still find a source of inconsistency by performing the incremental test or the sensitivity test discussed in the following section. 4.3.5 Identify Inconsistent Rules Even though any set of MFfuzzy could not be found with a GI of less than e in the previous section, we may still identify inconsistent rules which are responsible for the failure by using two methods. The first method is the incremental test, and the second is the sensitivity test. Advisor performs either or both tests depending on the user's request. In both tests, one should not assume that a newly identified inconsistent rule is the only rule which needs to be analyzed. All existing rules which may cause conflicts with this rule should also be analyzed to resolve the inconsistency. Note that the tests presented here do not work well when a significant portion of a model is inconsistent with the model to which we want to compare it. Incremental Test In this test, we can identify inconsistent rules by observing the rate of changes of the GI as we add rules incrementally and run the MF/Â„22j,-generation algorithm repeatedly. When the rate of change of the GI is significantly increased by adding ru/cfc, the rulck can be considered as a more inconsistent rule than previously added PAGE 69 62 rules. Once this rulek is added, observing the rate of change of GI is meaningless, since from this point, the inconsistent component (i.e., rulek) also contributes to tuning the MF/uzzy into an incorrect direction, thereby making the GI less credible. Thus, before the test, an arrangement should be made so that more reliable rules be employed on the test before less reliable rules. For this reason, the incremental test is order-dependent and requires more heuristics than the sensitivity test discussed in the following section. However, when one or more domain experts are available, we can apply a fuzzy individual or group preference ordering method [27] for obtaining the ordering within a given compatibility. Having such information available, the advantage of the incremental test over the sensitivity test is that it requires fewer computation demands. The algorithm for the incremental test can be described as: 1. Let n be the total number of rules for the incremental test. Assume that rules are ordered from the most reliable rule, rulei, to the least reliable rule, rulcn2. Divide these rules into two group, R and U, depending on their reliability such that group R contains the reliable rules, rulei to rulcj, and group U contains the unreliable rules, rulsj+i to rulen3. Let T be a set of rules for incremental test. Initially T is empty. 4. For i = 1 to j (a) Add rulci to T. (b) Run MF/u22j^-generation algorithm on T. (c) Observe the rate of change of GI. (d) If the rate is significant, then report rulci as inconsistent, and stop. 5. For i = j + 1 to n PAGE 70 63 (a) Add rulci to T. (b) Run MF/uz2j/-generation algorithm on T. (c) Observe the rate of change of GI. (d) If the rate is significant, then report rulci be inconsistent, and stop. (e) Delete ru/cj from T. Note that for preventing human experts' possible errors, we inserted Step 4(c) and Step 4(d), even though the rules being added here belong to the reliable group, R. Sensitivity Test A sensitivity test is a validation technique that can be used for both expert systems and simulations [35, 21, 29]. The general idea is to change the system input (i.e., values of variables or parameters) systematically over some range of interest, and study it by observing the effect. In our method, we can employ this idea to identify inconsistent rules by varying the participating rules. The sensitivity test begins with all rules except the first rule and executes the algorithm to generate MFfuzzy Then, all rules except the second rule are used to generate the MFfuzzy In this way, given the n rules, the sensitivity test involves a total of n executions of MF/u22j,-generation algorithm, each time with n Â— 1 rules. When we detect a significant improvement of the GI during this process, we can consider the rule that did not participate at this step of the test as an inconsistent rule, in comparison to other rules already joined. The advantage of this method over the incremental test is that the arrangement of rules based on their reliability is not necessary thereby eliminating the need for the experts' opinion. However, the computation burden is more severe than the incremental test, since each execution of MF/u^^y-generation algorithm involves n 1 rules. The general algorithm for sensitivity test can be written as: PAGE 71 64 1. Let n be a total number of rules for sensitivity test. 2. Let r be a set of rules for sensitivity test. Initially T has entire rules. 3. For i = 1 to n (a) Delete rulci from T. (b) Run MF/uzzy-generation algorithm on T. (c) If a significant improvement of GI is detected, then report rulci be inconsistent. (d) Insert rulei to T. 4.4 Time Complexity For the analysis of the time complexities of the fuzzy simulation method and MFfuzzy generation algorithm, we consider the following factors: Â• number of rules: n, Â• number of simplex rules: s, Â• number of sampling points of a fuzzy value: p, Â• number of input fuzzy variables in compound rules: m, Â• number of possible subsets for modifying MFs in the most inconsistent rule: b, Â• depth in a search tree: d and Â• number of random-restarts: r. 4.4.1 Time Complexity for Fuzzy Simulation For a simplex rule, the total p elements of a fuzzy set in the rule premise are issued to each independent fuzzy simulation. Therefore, the time complexity for executing PAGE 72 65 fuzzy simulations for s simplex rules is 0{sp). (4.5) For a compound rule, an intermediate fuzzy set Z is first obtained from the rule premise, and a fuzzy simulation is executed for each element in Z. This process involves a cartesian product over m Â— 1 variables (i.e., except for the variable in the rule consequence) in the rule. This leads to the time complexity 0(p"*). Therefore, the time complexity for executing fuzzy simulations for all compound rules is Combining complexity (4.5) and (4.6), the total time complexity for executing fuzzy simulations for all rules is 4.4.2 Time Complexity for MFf.,Â„,y Generation Using the factors, r and d defined above, we can rewrite the overall algorithm for the MFfuzzy generation in Section 4.3.4 as follows: 1. set initial MFs 2. fuzzy simulations for all rules 3. pick the most inconsistent rule and set current GI 4. for i = 1 to r 0{{n-s)p"'). (4.6) 0(np'"). (4.7) while current GI j previous GI (a) for j = 1 to b i. modify this subset ii. execute fuzzy simulations for all rules PAGE 73 66 (b) pick one state leading to minimum GI 5. pick best saved GI In the algorithm skeleton above, Step 4(a) determines the branch factor of the search tree. In the worst case, the branch factor is 21 by ignoring 0. Therefore, using (4.7), the time complexity involving Step 4(a) to Step 4(b) is 0(n2>'"). (4-8) The depth d in an arbitrary search tree is determined by the condition (5). Thus, the overall time complexity for generating the MF fuzzy is 0(rrfn2>"'). (4.9) The overall time complexity shown above demonstrates that, in a worst case, the number of fuzzy variables that can appear in any compound rule (i.e., m) dominates the overall time complexity. Besides, as we pointed out in Section 2.2.3, m also aflFects the number of fuzzy rules, n, as well [10]. However, most fuzzy applications so far have had few variables and have been in control [28]. This makes the time complexity shown in (4.9) manageable. In other words, the running time of MFjuzzy generation that we encounter in practical situations are mostly tractable problems. In this chapter, we described an interactive environment for isolating inconsistent knowledge between the expert rules and the quantitative model. To handle the linguistic vagueness in the expert rules, two separate procedures (i.e., applying fuzzy simulation directly and generating MF fuzzy) were presented depending on the types of estimates in the expert rules. In the next two chapters, we illustrate the applications of the presented methodology. PAGE 74 CHAPTER 5 FULTON: STEAMSHIP MODELING For a practical application of the method discussed in the previous chapters, we will consider FULTON, a model of a steam-powered ship, as shown in Figure 5.1 [30]. When the fuel valve is open, fuel flows and the furnace heats the sea water in the boiler assembly; when the fuel valve is closed, no fuel flows and the furnace stops heating the water. Heating the sea water produces steam, which is gathered and goes to the turbine, making the steamboat movable. The remaining steam is condensed into liquid in the condenser and is pumped back to the boiler. Among the four components in Figure 5.1, let's assume that we are interested in the boiler assembly, particularly the relation between temperature (T) of the sea water and the amount of steam [As) gathered in the boiler assembly. In the following two sections, we discuss two models which represent this knowledge quantitatively and qualitatively. Then, we apply our method to isolate any inconsistency between these two models. 5.1 Quantitative Model of Boiler Assembly Consider the boiler assembly in Figure 5.1. The fuel valve is determined to be in one of two states: open or close. Then, depending on the valve position, the behavior of the boiler assembly can be represented by four states of FSA as in Figure 5.2. The low level continuous models for Ml, M4 in that figure are defined as shown below by combining Newton's law with the capacitance low [18]. 1. (Ml) COLD: T = a,A^ = AÂ„, A, = 0, 2. (M2) HEATING: t = k,{100 T), = -k^ + = T * ^3, 67 PAGE 75 68 BOILER ASSEMBLY CONDENSER Figure 5.1. a model of a steam-powered ship valve = CLOSE Figure 5.2. Four state automanton controller for the boiler assembly PAGE 76 69 3. (M3) COOLING: t = hia T), = -h + An, As = f*ke and 4. (M4) BOILING: T = 100, = -kj + iiÂ„, = A;8, where ki,i = 1, . . . , 8 are rate constants, a = the ambient temperature of the water, T = the temperature of sea water, Ayj = the amount of sea water, As = the amount of steam gathered in the boiler, and Ain = the amount of water increased in the boiler by pumping water from the pump assembly. 5.2 Qualitative Model of Boiler Assembly In the expert's point of view, one of the easiest ways to model the physical behaviors of the boiler assembly is to represent that knowledge into natural language. Since the expert is interested in the relationship between the temperature of the sea water and the amount of steam, he or she can form an associative rule-based model by mapping a single input (the amount of time the fuel valve is open) into a single output (the amount of steam) as shown in Figure 5.3. However, as we can see Valve Open Time Amount of Steam CF ^ expert very_very_short very_very_little 0.9 very_short very_little 0.6 short little 0.6 slightly_moderate little 0.6 moderate medium 0.6 slightly_long slightly_much 0.9 long slightly _much 0.5 veryjong much 0.9 Figure 5.3. Simplex rules for the boiler assembly in Figure 5.4 and Figure 5.5, the rate of change in the temperature when the fuel PAGE 77 70 z u Ol. o o u 10.0 14.0 100 90 80 _ 70 t S feo E ID ^ 5040 30 20 Figure 5.4. Fuel valve input 7^ 10 Time Figure 5.5. Water temperature 12 14 16 18 20 valve is open is different from the rate of change when the valve is closed. Taking this observation into account, the expert can construct a more complex knowledge base using compound rules to measure the time-dependent dynamic behavior of the system. Figure 5.6 illustrates this process. Putting Figure 5.3 and Figure 5.6 together and using the expert's definition of linguistic terms defined by Figure 5.7, we obtained a complete rule-based model from the expert for the boiler assembly. 5.3 Checking Consistencv When MF^^^^^t is Available Once the complete rule-based model is obtained from the expert, the logical step is to run fuzzy simulations on this model, and get RULEfuzzy and CFjuzzy to see PAGE 78 Valve Open Time Valve Close Time Amount of Steam CF ^ expert long very_short much 0.5 slightly_modetate veryjong medium 0.8 moderate short slightly_much 0.5 short long little 0.5 very_long very_very_short much 0.7 slightlyjong slightly_moderate much 0.6 very_short slightlyjong little 0.9 very_very_short moderate very_very_little 0.9 Figure 5.6. Compound rules for the boiler assembly (a) very_very_little veryjittle little medium slightly_much much I I I I 20.0 25.0 30.0 35.0 45.0 50.0 65.0 80.0 92.0 (b) Figure 5.7. Definitions of the linguistic terms for the boiler assembly (a) MFpremise in the expert's rule; (b) MFconseq in the expert's rule PAGE 79 72 if any inconsistency exists. This path is represented as bold arrows in Figure 5.8. Inconsistencies are identified in terms of Lis and GI between these two rule sets. We assume that the consistency criterion, e, is set to 0.5 by the user. Figure 5.9 Isolating Inconsistency Advisor Human Intervention Checking consistency o Apply fuzzy simulation directly Reporting o Inconsistent Rules o Expected Rules Change 0 Rules 0 MFs 0 Simulation components Conducting o Incremental test o Sensitivity test Reporting o Inconsistent rules Suggesting o Approximate MI^^ -(Done ) Figure 5.8. Checking consistency when MFgxpert is available shows the result of fuzzy simulations. This figure shows that 15th rule is the most inconsistent, and its LI greatly affects the overall GI. To alleviate the human's resolving efforts, Advisor generates the expected rules from the fuzzy simulation by computing the CFf^^^y for all possible consequences, given the premise part of the 15th rule. Figure 5.10 shows this result. If the user agrees with Advisor by deciding to replace the original ruleir, with the 2nd rule in Figure 5.10, he or she gets a fairly small GI of 0.34, as shown in Figure 5.11. Since this GI is less than consistency criterion, 0.5, we conclude that the two models are now consistent. PAGE 80 Valve Open Time Valve Close Time Amount of Steam CF expert LI very_very_short N/A very_very_little 0.9 0.87 0.03 very_short N/A very_little 0.6 0.59 0.01 short N/A little 0.6 0.58 0.02 slightly_moderate N/A little 0.6 0.60 0.00 moderate N/A medium 0.6 0.64 0.04 slightly_long N/A slightly_much 0.9 0.88 0.02 long N/A slightly_much 0.5 0.50 0.00 very_long N/A much 0.9 0.94 0.04 long very_short much 0.5 0.53 0.03 slightly_moderate very_long medium 0.8 0.76 0.04 moderate short slightly_much 0.5 0.49 0.01 short long little 0.5 0.51 0.01 very_long very_very_short much 0.7 0.71 0.01 slightly_long slightly_moderate much 0.6 0.61 0.01 very_short slightly Jong litUe 0.9 0.26 0.64 very_very_short moderate very_very_little 0.9 0.90 0.00 Gl = 0.93 Figure 5.9. The result of fuzzy simulation for the boiler assembly Valve Open Time Valve Close Time Amount of Steam fuzzy very_short slightlyjong very_very_little 0.04 very_short slightly_long veryjittle 0.46 very_short slightly_long little 0.26 very_short slightlyjong medium 0.00 very_short slightlyjong slightly_much 0.00 very_short slightlyjong much 0.00 Figure 5.10. Rules generated from Advisor for the boiler assembly Valve Open Time Valve Close Time Amount of Steam CF expert CF fuzzy LI very_very_short N/A very_very_little 0.9 0.87 0.03 very_short N/A very_little 0.6 0.59 0.01 short N/A little 0.6 0.58 0.02 slightly_moderate N/A little 0.6 0.60 0.00 moderate N/A medium 0.6 0.64 0.04 slightly_long N/A slightly_much 0.9 0.88 0.02 long N/A slightly_much 0.5 0.50 0.00 very Jong N/A much 0.9 0.94 0.04 long very_short much 0.5 0.53 0.03 slightly_moderate very_long medium 0.8 0.76 0.04 moderate short slightly_much 0.5 0.49 0.01 short long little 0.5 0.51 0.01 very_long very_very_short much 0.7 0.71 0.01 slightly_long slightly_moderate much 0.6 0.61 0.01 very_short slightly_long little 0.5 0.46 0.04 very_very_short moderate very_very_little 0.9 0.90 0.00 GI = 0.34 Figure 5.11. Two consistent models PAGE 81 74 5.4 Checking Consistency When MF^^p^rf is Unavailable Isolating Inconsistency Advisor Human Intervention Checking consistency 0 Apply fuzzy simulation directly Reporting 0 Inconsistent Rules 0 Expected Rules Change 0 Rules 0 MFs 0 Simulation components Conducting o Incremental test o Sensitivity test Reporting o Inconsistent rules Suggesting 0 Approximate MIj^^ Figure 5.12. Checking consistency when MF^xpert is unavailable The path that is going to be explored in this section is represented as bold arrows in Figure 5.12. Since two different outcomes (i.e., consistent ox inconsistent, depending on whether any approximate set of MFf^zzy can be generated with a GI less than the consistency criterion or not) are produced exclusively through the consistency checking procedure, we illustrate each case in the following two sections. 5.4.1 The Case Where Approximate MFfÂ„,,y is Successfully Generated The previous section showed that rules defined in Figure 5.11 are consistent. Therefore, if we use the rules and the central point estimates defined in Figure 5.13, then we can illustrate that the approximate set of MFf^zzy can be found with a GI less than consistency criterion, 0.5. We illustrate this process step by step. PAGE 82 75 MF premise central point MF conseq central point very_very_short 0.0 very_very_Iittle 20.0 very short 0.03 very little 35.0 short 0.06 little 50.0 slightly_moderate 0.09 medium 65.0 moderate 0.13 slightly_much 80.0 slightlyjong 0.17 much 92.0 long 0.2 veryjong 0.23 Figure 5.13. Central point estimates for MFpremise and MFconseq Set Initial Conditions Figure 5.14 shows the initial MFfuzzy The initial size is set to 0.02 for each MFpremise, ^ud 4.0 for MFconseqTo generate MFfuzzy, the tuning size, Ad, is set to 0.005 for MFpremise, ^ud 2.0 for MFconseqFor random-restarts, the promising state criterion^ 9, is set to 5.0. After all tuning processes are done, if a best saved GI is less than consistency criterion of 0.5, then we conclude that we found an approximate set of MFfuzzy successfully. very_very_short very_short short slight!y_moderate moderate slightlyjong long veryjong i \ \/ \l \/ / ^ 1 + 1 f f 1 0.0 0.03 0.06 0.09 0.13 0.17 0.2 0.23 Initial Size = 0.02 very_very_litt!e veryjittle little medium slightly_much much I I I 20.0 35.0 50.0 65.0 80.0 92.0 Initial Size = 4.0 Figure 5.14. Initial MFfuzzy PAGE 83 76 Fuzzy Simulations for All Rules The result of fuzzy simulations using the initial MFjuzzy is shown in Figure 5.15, where ruleio is identified as the most inconsistent rule. Valve Open Time Valve Close Time Amount of Steam CF expert ^^fuzzy I T L.1 very_very_short IN /A very_veryjittle 1 on yj. L\j vprv Qhnrt V vi y aiivji I N/A very little 0.6 0.50 0.10 short N/A little 0.6 0.00 0.60 sIightly_moderate N/A little 0.6 0.00 0.60 moderate N/A medium 0.6 0.00 0.60 slightlyjong N/A slightly_much 0.9 0.50 0.40 long N/A slightly_much 0.5 0.00 0.50 veryjong N/A much 0.9 1.00 0.10 long very_short much 0.5 0.00 0.50 slightly _moderate veryjong medium 0.8 0.00 0.80 moderate short slightly_much 0.5 0.00 0.50 short long little 0.5 0.00 0.50 veryjong very_very_short much 0.7 1.00 0.30 slightly_long slightly_moderate much 0.6 0.00 0.60 very_short slightly_long very_little 0.5 0.00 0.50 very_very_short moderate very_very_little 0.9 1.00 0.10 GI = 6.80 Figure 5.15. The result of fuzzy simulations using the initial MFj. Tuning MFfÂ„yyy Until Best Saved GI Does Not Improve All possible subsets of MFjuzzy in ruleio are {slightly_moderate}, {very Jong}, {medium}, {slightly .moderate, very Jong}, {slightly jnoderate, medium}, {very Jong, medium} and {slightly jnoderate, very Jong, medium}. After executing fuzzy simulations for each state, the smallest GI of 6.3 was found by modifying a subset {slightly .moderate, medium} as shown in Figure 5.16 and Figure 5.17. Since this GI is less than the previous GI, 6.8, the current GI, 6.3, is saved as the best GI, and Step 3 in Section 4.3.4 is repreated until the GI does not improve. Figure 5.18 shows the actual GUI when the stop condition was met after a series of tuning processes. At this point, Step 4 (random-restarts) begins, PAGE 84 77 very_very_short very_short short slightly_moderate moderate slightlyjong long veryjong il / / / + + + f 1 f 1 + t + 0.0 0.03 0.06 0.09 0.13 0.17 0.2 0.23 very_very_litt!e veryjittle little medium slightly_much much 20.0 35.0 50.0 61.0 65.0 69.0 80.0 92.0 Tuning Size = 2.0 Figure 5.16. Increasing the size of slightly -moderate and medium by Ad Valve Open Time Valve Close Time Amount of Steam CF expert ^'^fuzzy LI very_very_short N/A very_very_litt!e 0.9 1.00 0.10 very_short N/A veryjittle 0.6 0.50 0.10 short N/A Uttle 0.6 0.00 0.60 slightly_moderate N/A little 0.6 0.00 0.60 moderate N/A medium 0.6 0.25 0.35 slightlyjong N/A slightly_much 0.9 0.50 0.40 long N/A slightly_much 0.5 0.00 0.50 very_long N/A much 0.9 1.00 0.10 long very_short much 0.5 0.00 0.50 sIightly_moderate veryjong medium 0.8 0.25 0.50 moderate short slightly_much 0.5 0.00 0.50 short long little 0.5 0.00 0.50 veryjong very_very_short much 0.7 1.00 0.30 slightlyjong slightly_moderate much 0.6 0.00 0.60 very_short slightIy_long very_little 0.5 0.00 0.50 very_very_short moderate very_very_little 0.9 1.00 0.10 GI = 6.30 Figure 5.17. The result of fuzzy simulations after slightly ^moderate and medium are tuned PAGE 85 78 !Ui.d iot Re Main Wimiow Wenlify biconsislenl Rules i Idttntify Consistent vavh ( 3 873410 ) PAGE 86 best saved GI Figure 5.19. Trend of G/ after seven randomrestarts very_very_short very_shoi1 short .slightly_moderate moderate slightlyjong long veryjong f f 1 \ ^ t 0.0 0.02 0.06 0.07 0.1 0.15 0.17 1 0.19 1 0.2 1 0.23 0.065 0.08 0.11 0.145 0.16 0.195 0.21 0.18 PAGE 87 80 i34 3 371634 3.0791 03 10,0 5.0 3,470192 3,516656 3,476064 Number of iterations best saved GI Figure 5.21. Trend of GI after seven random-restarts 5.4.2 The Case Where MFfÂ„,.,y is Unsuccessfully Generated Section 5.3 showed that expert's rules in Figure 5.9 are inconsistent because of rulei5. If we use these rules, we can artificially make the case where MF fuzzy cannot be successfully generated. By executing the MFj^zzy generation algorithm with seven random-restarts, we obtained the best saved GI of 2.98, as shown in Figure 5.21. In this figure, no random-restart lead to a GI less than the given consistency criterion, 0.5. To identify the causes of the inconsistency, we performed the incremental test and the sensitivity test discussed in Section 4.3.5. Incremental Test We assume that compound rules are more prone to inconsistency than simplex rules, because the consequences of all compound rules in the FULTON example require a more complex reasoning process involving two time-related variables. For this reason, we grouped the simplex rules, defined in Figure 5.3, into the reliable group, R, and the complex rules defined in Figure 5.6 into the unreliable group, U. The result of the incremental test is shown at Figure 5.22. Each subfigure shows the result of executing the MFf^zzy generation algorithm (i.e.. Step 5(b) of the incremental test procedure in Section 4.3.5) with seven random-restarts right after we added each compound rule to the results obtained from incremental test using PAGE 88 81 the eight simplex rules. By observing the rapid change of GI in Figure 5.22 (g), we can regard the rulei^ as the catalyst that caused the inconsistency. Sensitivity Test Figure 5.23 shows the result of the sensitivity test on the rules defined in Figure 5.3 and Figure 5.6. We performed seven random-restarts for each test; each circle in this figure represents the GI we have obtained from each random-restart. A rule i in the X axis represents the rule which did not join for the MFf^zzy generation at that particular test. As shown in this figure, execution of the MFfmzy generation algorithm (i.e., Step 3(b) of the sensitivity test procedure in Section 4.3.5) without rulei5 resulted in a significant improvement of GI, 0.42, compared to the other cases. This indicates that rulei^ is the catalyst of the inconsistency. 5.4.3 Human Intervention In Figure 5.24, the path involving human intervention is represented as bold arrows. In Section 5.3, rulcx^ was identified as an inconsistent rule when the rules defined in Figure 5.3 and Figure 5.6, and the linguistic definitions defined in Figure 5.7, were used. Even in a situation where such complete linguistic definitions were not available, the previous section showed that we were still able to identify rulei^ as a catalyst for the inconsistency by conducting the incremental test or the sensitivity test. With this information, the user may change rules (in this case, ru/eis or other rules which may be causing the conflict with ru/eis), central point estimates, the definitions of linguistic terms in these rules or even simulation model components. We built a GUI as shown in Figure 5.25 to interactively visualize the system's responses according to these kinds of user's resolving trials. In this figure, the set of membership functions shown in membership function editor is the best set which led to the minimum GI of 2.98 in the previous section. However, as mentioned, this case fell into the category of unsuccessfully generated MFjuzzy Now, PAGE 89 82 using membership function editor and rule editor, the user can freely change these membership functions, as well as any rules including the CFs. Each time the user makes a modification, this updated input is issued to the fuzzy simulation, and new consistency checking results are provided in terms of LI and GI through evaluation button. In this chapter, we considered FULTON for a practical application of the presented methodology. All algorithms or procedures in Chapter 4 were demonstrated completely. For another application, we consider predator-prey population in the next chapter. In the chapter, we focus on the comparison between the model outputs of the expert rules and the outputs of the quantitative models before and after resolving inconsistency. PAGE 90 83 Figure 5.22. Incremental test for the boiler assembly (a) adding ruleg; (b) adding ruleio] (c) adding rulen] (d) adding ruleu] (e) adding ruleis; (f) adding ruleu, (g) adding rulei^; (h) adding ruleie PAGE 91 O GI obtained from a random-restart rule i Figure 5.23. Sensitivity test for the boiler assembly Isolating Inconsistency Checking consistency 0 Apply fuzzy simulation directly Checking consistency 0 Identify approximate MF^^^ Advisor Reporting o Inconsistent Rules o Expected Rules Conducting o Incremental test o Sensitivity test Reporting 0 Inconsistent rules Suggesting o Approximate MF^^^^ Human Intervention Change 0 Rules 0 MPs 0 Simulation components Figure 5.24. Human intervention PAGE 92 85 very_v8ry_Â»lBrt tvce *> very very little 0 VBt^Tslioifc zero Â•> very little 0 bOOCOO Â«eEO Â«> little 5,600008 slj^tifjiiiderate iÂ«r& Â•> little 0 SOOOOD m^tattT 2Â»to Â•> Â»e4Â»Â« 0.688000 *3i(pÂ»tly laagr slieitely such 0 900000 iMi$ s?w> aucli ff.SOODOO TOty 1Â«J zero Â»> Â»iicE 0. 908600 lung" my jtott Â»> Â«>sb S.S86000 sligfetl? KoierBte wty long Â»> 0. wfoite" sbsrt Â•> sli^Stlj 8-SOSOOe ^tt iMlg Â•> little D.SSI088 sli^tly Ion; ili#ttly nsdecate <=> Mtdi 8. very short alightly_lon} Â»> little 0. 30(B00 imrCwiT-'tet Mderate -> Â»Â«y_ PAGE 93 CHAPTER 6 PREDATOR-PREY POPULATION For another illustration of our method, we considered a predator-prey population. Predator-prey models address the dynamic interaction between a predator species and its prey species. In this chapter, we consider two species: one single predator species V and its single resource TZ. 6.1 Qualitative Model In general, there are two kinds of rules which explain the population dynamics in the predator-prey interaction: migration rules and birth/death rules [56, 55]. The migration rules describe the movement of individual V and 71 at any interval of time from one location to another. For example, a probability that V will stay in a particular location depends on the configuration of TZ and V around that location. The birth/death rates of V and TZ depend on many factors, such as mortality rate, internal feeding state, predator's encounter rate with prey and so on. These factors are not independent of one another. For example, the mortality rate of V depends on their internal feeding state, and the internal feeding state again depends on the encounter rate with TZ. Predator-prey dynamics also depends on the natural environment of V and TZ. When ecologists study the distribution of organisms, they try to discover the physical and biological factors that influence the presence or absence of particular species. In this section, we assume that there are two environmental factors affecting the distribution of V and TZ: scale and temperature of the region they live. Given two different environmental factors, an ecologist may predict the population of 7?. in a 86 PAGE 94 87 short-term period by considering the causal relation of predator-prey interaction: "In a wide region, the possibility of P's encounter rate with 11 is relatively small, but if the temperature of the territory is warm, then other living food in that region may satisfy the V (which makes the mortality rate low). Therefore, this keeps the population of Tl from growing too much. On the contrary, in a cold region, the mortality rate of V increases, which makes the population of Tl crowded." Using the above description, a knowledge engineer can come up with the following two rules. IF "P's encounter rate with TZ is rare and "P's mortality rate is low, THEN the density of the TZ is slightly-crowded (CF = 0.9), IF "P's encounter rate with 71 is rare and P's mortality rate is high, THEN the density of the 71 is crowded (CF = 1.0). After the knowledge acquisition process with the expert in this way, suppose that we obtained 25 rules and membership functions as shown in Figure 6.1 and Figure 6.2, respectively for explaining the population of prey V based on the different combination of the environmental factors. With two inputs and five linguistic values for each of these, there are 5^ = 25 possible rules. In these rules, both triangular and trapezoidal membership functions are used. 6.2 Quantitative Model To describe the dynamics involving growth and decline of the predator-prey population, differential or difference equations are often used. Such mathematical models are designed either for predictive purposes to make accurate short-term forecasts, or to identify generic characteristics and underlying principles. As one of the mathematical models, we consider the Lotka-Volterra predator-prey model [55]: dt ~ "^^^'^^^ K > l + ahx{ty dy{t) acx{t)y{t) dt l + ath.x{t) -ey{t), PAGE 95 Rule Encounter rate of predator with prey Mortality rate of predator Density of prey CF 1 rare very_low shghtly_crowded 0.90 2 rare low crowded 1.00 3 rare moderate crowded 1.00 4 rare slightly_high crowded 1.00 5 rare high crowded 1.00 6 slightly_rare very_low crowded 0.40 7 slightly_rare low crowded 0.50 8 slightly_rare moderate crowded 0.90 9 slightly_rare slightly_high crowded 1.00 10 slightly_rare high crowded 1.00 11 medium veryjow scarce 0.60 12 medium low nominal 0.90 13 medium moderate crowded 0.50 14 medium slightly_high crowded 0.90 15 medium high crowded 1.00 16 slightly_frequent very_low scarce 0.80 17 slightly_frequent low scarce 0.80 18 slightly_frequent moderate slightly_scarce 0.30 19 slightly_frequent slightly_high slightly_crowded 0.30 20 slightly_frequent high crowded 1.00 21 frequent veryjow scarce 0.80 22 frequent low scarce 0.60 23 frequent moderate slightly_scarce 0.30 24 frequent slightly_high nominal 0.10 25 frequent high crowded 0.80 Figure 6.1. Expert rules for the predator-prey population PAGE 96 89 Encounter Rate rare ^''g'i"y rare 0.0 0.3 Mortality rate very_low 0.1 0.2 Density of prey medium slightly frequent 0.6 0.7 0.8 1.2 1.6 moderate 0.3 0.4 slightly scarce 0.6 frequent 2.0 2.1 0.8 nominal slightly crowded 2.6 1.0 crowded 0.0 0.1 0.4 0.5 0.9 1.2 1.3 1.6 1.7 Figure 6.2. Fuzzy membership functions for the expert rules in Figure 6.1 PAGE 97 90 where x{t), y{t) = population densities of 71 and V as functions of time t, respectively, r = TZ population's intrinsic rate of increase, K = Tl's earring capacity, that is the population density the prey population would reach at an equilibrium in the absence of predation, a = "P's encounter rate with TZ, c = a conversion rate which maps the V consumption into the K birth rate, e = "P's mortality rate, and th = V^s handling time. 1.3 1 1 1 1 1 \ 1 1 1 r 1.2 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure 6.3. A time series graph for the predator-prey population For example, using initial conditions x{t) = 1.05, y{t) Â— 0.9, r = 1.2, K = 1.7, a = 1.9, e = 0.48, c = 0.9 and t^ = 1.0, we obtain a time series graph for the predator-prey population as shown in Fig 6.3, in which the state variables (i.e., density of V and PAGE 98 91 11) are graphed against time. To solve the x{t) and y{t), we applied Euler's method [18]. 1 I 1 . r r 1 1 1 1 1 r O 100 200 300 400 500 600 70O 80O QOO 1000 Tim* Figure 6.4. The result of fuzzy simulation on IF rare and low THEN crowded 6.3 Consistency Checking We applied fuzzy simulation to the expert rules in Figure 6.1 using the membership functions defined in Figure 6.2 and initial conditions x{t) = 1.05, y{t) = 0.9, r Â— 1.2, K = 1.7, c = 0.9 and t/j = 1.0 of the Lotka-Volterra model. For example. Figure 6.4 shows the result of the fuzzy simulation on the expert's second rule, defined in Figure 6.1. To obtain this result, we first applied the rule of conjunctive composition to calculate the possibility distribution tt of rare and low. Then each element of the possibility distribution is used for input of the fuzzy simulation, and this gives us a result in terms of membership degrees associated with crowded. Finally, using the weighted average method introduced in Chapter 3, we obtain a CFfuzzy, 1-0. In this way, all 25 expert rules are applied to the fuzzy simulations, and we obtained the results in Figure 6.5. In this figure, rulei is identified as the most inconsistent rule and the rulsu as the second worst case rule. Figure 6.6 shows the suggestion from Advisor. By replacing the original 1st and 12th expert rules PAGE 99 92 Rule Encounter rate of predator with prey Mortality rate of predator CF Density of prey expert ^Â•^fuzzy LI 1 rare very_Iow slightly _crowded 0.90 0.05 0.85 2 rare low crowded 1.00 1 00 0 00 3 rare moderate crowded 1.00 1 00 0.00 4 rare slightly_high crowded 1.00 1.00 0.00 5 rare high crowded 1.00 1.00 0.00 6 slightly_rare very_low crowded 0.40 U. J / 7 slightly_rare low crowded 0.50 U.*tO ft ftA U.U4 8 slightly_rare moderate crowded 0.90 n 04 u.y* 0 04 9 slightly_rare slightly_high crowded 1.00 1 00 0 00 10 slightly_rare high crowded 1.00 1 00 0 00 1 1 medium very low scarce 0.60 0.60 0.00 medium low nominal 0.90 0.09 0.81 medium moderate crowded 0.50 0.46 0.04 14 medium slightly high crowded 0.90 0.88 0.02 1 e medium high crowded 1.00 1.00 0.00 lo slightly_frec)uent very_low scarce 0.80 0 78 0.02 1 1 1 1 slightly_frequent low scarce 0.80 0 79 0 01 1 c lo slightly_frequent moderate slightly_scarce 0.30 0 29 0 01 ly slightly_frequent slightly_high slightly_crowded 0.30 ft "^1 0 01 slightly_frequent high crowded 1.00 n OR ft ft? 21 frequent very_low scarce 0.80 0.77 0.03 22 frequent low scarce 0.60 0.64 0.04 23 frequent moderate slightly_scarce 0.30 0.32 0.02 24 frequent slightly_high nominal 0.10 0.15 0.05 25 frequent high crowded 0.80 0.80 0.00 Gl = 2.06 Figure 6.5. The result of fuzzy simulation for the predator-prey model with their counterparts from fuzzy simulations, we achieved a GI of 0.38 as shown in Figure 6.7. An idea of how the old and the new rule sets approximate the population of TZ differently can be obtained by comparing the response surfaces of these two with one from Lotka-Volterra algebraic formula. The response surface of 7^'s density for all 270 combinations of encounter rate = 0.1, 0.2, 2.6 and mortality rate Â— 0.1, 0.2, 1.0 from the Lotka-Volterra model is shown at Figure 6.8. To create the response surfaces from the fuzzy rule sets defined in Figure 6.1 and Figure 6.7, we used fuzzy logic (discussed in Section 2.2.3) with the min-max composition for inference PAGE 100 Rule Encounter rate of predator with prey Mortality rale of predator Density of prey Â•^f^fuzzy 1 rare very_low scarce 0.00 rare very_low slight]y_scarce 0.00 rare very_low nominal 0.00 rare very_low slightly_crowded 0.05 rare very_low crowded 0.95 12 medium low scarce 0.24 medium low slightly_scarce 0.45 medium low nominal 0.09 medium low slightly_crowded 0.05 medium low crowded 0.21 Figure 6.6. Rules suggested from Advisor for the predator-prey model Rule Encounter rate of predator with prey Mortality rate of predator Density of prey CF^^^ ^''fuzzy LI rare very_low crowded 0.90 0.90 0.00 2 rare low crowded 1.00 1.00 0.00 3 rare moderate crowded 1.00 1.00 0.00 4 rare slighdy_high crowded 1.00 1.00 0.00 5 rare high crowded 1.00 1.00 ooo 6 .slightly_rare very_low crowded 0.40 0.37 0.03 7 slightly_rare low crowded 0.50 046 0.04 8 slightly_rare moderate crowded 0.90 0.94 0.04 9 .slightly_rare )ilightly_high crowded 1. 00 1.00 OOO 10 slightly_rare high crowded 1.00 1.00 0.00 11 medium very_Iow scarce 0.60 0.60 0.00 12 medium low slightly_scarce 0.40 0.40 0.00 13 medium moderate crowded 0.50 0.46 O04 14 medium slightly_high crowded 0.90 0.88 0.02 15 medium high crowded 1.00 1.00 OOO 16 s!ightly_frequen! veryjow scarce 0.80 0.78 0.02 17 slightly_frequenl low scarce 0.80 0.79 0.01 18 slightly_frequenl moderate slightly_scarce 0.30 0.29 0.01 19 slightiy_frequent .slightly_high slightly_crowded 0.30 0.31 0.01 20 slightly_frequent high crowded 1.00 0.98 0.02 21 frequent veryjow scarce 0.80 0.77 0.03 22 frequent low scarce 0.60 0.64 0.04 23 frequent moderate slightly_.scarce 0.30 0.32 0.02 24 frequent slighdy_high nominal 0.10 015 0.05 25 frequent high crowded 0.80 0.80 0.00 GI = 0.50 Figure 6.7. Two consistent models for the predator-prey model PAGE 101 94 and the centroid method for defuzzification. First, in the min inference, each CF was considered to calculate the membership degree of the consequence membership function. That is, given a encounter rate a and mortality rate e and a Rulcj, the truth value Tvj of the output membership function Dj can be calculated by Tvj = ifiEjia) A i^Mjie)) x CFj, (6.1) where Ej and Mj are two fuzzy values defined in the premise of the rulcj, and A is the minimum operator. Figure 6.8. Population of prey obtained from the Lotka-Volterra model Through the min-inference, the output membership function Dj is cut off at a height corresponding to the Tvj. Then, we obtain a combined output membership function D' through the max composition by taking the pointwise maximum over all of the fuzzy sets obtained from the min-inference for all RulcjJ = 1, .., 25. Finally, PAGE 102 95 we obtain the density d by the defuzzification formula, , _ T:^=ld^^lD'{d^) where d^, i = 1, ...n is an element defined in the fuzzy set D'. Figure 6.9 and Figure 6.10 shows the population of Tl obtained in this way for the rule sets defined in Figure 6.1 and Figure 6.7, respectively. As we can see in these figures, by replacing the original 1st and 12th rules with the rules suggested from the fuzzy simulation, we can obtain a better approximation for the population of 72.. An explanation may be possible as to why the two models show inconsistency meshexpertl.epsmeabout the particular two causal relations defined in the 1st rule and 12th rule. First, when P's encounter rate with TZ is rare, the mortality rate of the V almost doesn't affect the density of the (by observing the 1st rule to the 5th rule in Figure 6.7). The interval between the two boundaries of rare defined by the expert (Figure 6.2) seems so narrow, that every combinations of rare and any value of the mortality rate result in the approximately same population of IZ. Second, by observing the 11th rule to the 13th rule in that figure, when Vs encounter rate with IZ is medium, the increase of P's mortality rate from veryJow to low results in a lesser change of the 7^'s population than when it increases from low to moderate. 6.4 MFf.Â„,Â„j Generation In this section, we illustrate how to obtain an approximate set of MF fuzzy using the rules, defined in Figure 6.7, with only central points or intervals of full confidence, shown at Figure 6.2. For the MF/Â„2^j^-generation process, we set the tuning size Arf = 0.1 for the encounter rate, 0.05 for the mortality rate and 0.1 for the density of Tl. The promising state criterion, 9, and the consistency criterion, e, are set to 2.4 and 0.7, respectively. Under these settings, we conducted a total of 15 random-restarts during the MF/uz2j,-generation process and obtained the trend of GI, as shown at PAGE 103 96 1st rule 12the rule Figure 6.9. Population of prey obtained from the original expert rules defined in Figure 6.1 Figure 6.10. Population of prey obtained from the consistent rules defined in Figure 6.7 PAGE 104 97 O random-restart 3 I 1 Â— 1 1 1 ' Promising state criterion Consistency criterion the smallest GI = 0.56 Figure 6.11. Trend of GI during 15 random-restarts Figure 6.11. As we can see in that figure, a total of 36 sets of MF fuzzy led to a GI of less than the consistency criterion, 0.7. From this, we conclude that the expert rules defined in Figure 6.7 and the Lotka-Volterra model are consistent with any of these MFfuzzy Figure 6.12 shows the comparison between the MFjuzzy leading to the smallest GI of 0.56 against the original MF^xpert defined in Figure 6.2. The response surface generated from this set of MFjuzzy and the consistent rules is shown at Figure 6.13. In this chapter, we considered another example, predator-prey population to illustrate the application of the presented methodology. Given the expert rules and the quantitative model, we demonstrated that how we made these two models more consistent by employing the methodology to isolate inconsistency and resolve inconsistency. PAGE 105 Encounter rate slightly rare rare medium slightly frequent frequent Mortality rate very low moderate slightly high 0.3 0.4 Density Of prey ,j j^^, scarce nominal slightly crowded crowded Figure 6.13. Population of prey obtained from MF fuzzy PAGE 106 CHAPTER 7 FUTURE WORK In this chapter, we first mention the limitations of the presented methodology and suggest ways improving or reducing such limitations in the future. Then, we suggest two promising application areas: application in control industries and application in MOOSE. 7.1 Limitations and Improvements 7.1.1 Resolving Inconsistency In this study, the main concern is to isolate the particular piece of causal relations which show inconsistency between the expert's rules and an assumed quantitative model. However, after isolating these inconsistent rules, most efforts to resolve the problem rely on Human Intervention. We can add the following capabilities to Advisor to support the human efforts to resolve the inconsistencies: Â• suggesting the ideal rule structure by determining the optimal number of fuzzy variables and values, Â• locating the expert's missing rules if they exist and Â• locating the expert rules upon which we should elaborate. 7.1.2 Performance Index In this research, Global Inconsistency (GI) serves as a performance index to search for better linguistic definitions where expert rules and quantitative model match maximally. Since the value of GI depends on the number of rules, this value may differ from an application to an application. To make this value independent on 99 PAGE 107 100 y (a) (b) Figure 7.1. Local optimality using the weighted average method (a) Fuzzy simulation with the initial size of fuzzy sets A and B; (b) Fuzzy simulation with the desirable size of fuzzy sets A and B any particular application, a normalizing procedure (such as dividing by the number of rules) is necessary. After such a standard representation of GI is obtained, the goodness of GI can be more strictly studied in the future. 7.1.3 Local Optimality During MFfÂ„r,y Generation The weighted average method employed in this study may cause local optimality during MFjuzzy generation when stand-alone rules exist in expert rules. Here, we define a stand-alone rule as a rule whose fuzzy values are never used in other rules. To illustrate this problem, we consider a stand-alone expert rule, IF Ais A THEN y isB (0.7). Suppose that we obtain CF fuzzy of 0.75 by executing the fuzzy simulation with the initial size of fuzzy sets A and B, as shown in Figure 7.1 (a). This is certainly a problem if the membership function defined in Figure 7.1 (b) is the goal for which we are looking. The stand-alone rule will never be picked as the worst case rule by the MFjuzzy generation algorithm because CFexpert ~ CFf^zzy at its initial sizes of A and B. We can consider the following approaches to reduce such local optimality problems: PAGE 108 101 Â• Always allow the best case rule to be picked at the end of the tuning process (i.e., Step 3 of the algorithm described in Section 4.3.4). Continue this process to see if the GI eventually decreases and reaches its previous minimal point again. Â• Detect all stand-alone rules and report them to the user. If the user is able to provide more rules using the fuzzy sets A and B, then the rule is not a stand-alone rule any more. 7.2 Application 7.2.1 Application in Control Industries As one of the promising applications of the presented methodology, we can consider the application involving Proportional Integral Derivative (PID) controller. To illustrate this kind of applications, we particularly consider subway control system, because, in this area, relevant quantitative models are known precisely and the expert model may exist at the same time. Train operation is broadly classified into two control modes: (1) train speed regulation control and (2) train stopping control. For some systems, the conventional PID automatic train operation control hardwares were already being installed. The PID controller takes the error between the goal speed and the actual train speed to control the train's motor and brake. This requires a linerized system model, a desired state and an error criterion. However, human can still control the train because they can evaluate the system objectives on the basis of experience. In the subway control system, temperature and pressure are state variables, decrease is a control action, and fuel is controlled. An expert (or operator) may provide a rule like "IF Temperature is High and Pressure is Slightly Jiigh THEN decrease fuel." In this way, he (or she) evaluates the system state variables and decides on the control action which acts on the fuel. Moreover, PAGE 109 102 subjective performance indices such as riding comport and accurate stopping can also be incorporated in the rule. A fuzzy controller based on such rules has already applied to the Sendai system in Japan in 1987. Given these the expert rules and the PID models mentioned before, we may apply our methodology to check for consistency between these two models. When we find inconsistency, we may make slight alterations on the rules and (or) the membership functions in the fuzzy rules. Since most applications employing PID models usually can be replaced with experienced human operators and rules, this area is one of the most promising application areas to apply the presented methodology. 7.2.2 Application in MOOSE For various application-oriented examples, we are particularly interested in the application within MOOSE (Multimodal Object Oriented Simulation Environment) [12, 11]. MOOSE is an enabling environment under development at University of Florida for modeling and simulation based on 00PM. 00PM extends object-oriented program design with visualization and reinforces the relation of "model" to "program." This permits a tight coupling between a model author and the modeling and simulation process through an interactive HCI (Human Computer Interface). MOOSE consists of four major components [12]: Modeler, Translator, Engine and Scenario. Modeler interacts with a model author via a GUI in a way which helps the author make a valid conceptual model of the system. Translator is a bridge between a model design and a model execution. It reads the output from Modeler and automatically builds the corresponding structures of the conceptual model with C+-Icode, therefore it ensures that the program is a valid representation of the conceptual model. Engine is a C+-Iprogram, composed of Translator output plus runtime support, compiled and linked once, then repeatedly activated for the model execution. Scenario is a visualization-enabling GUI which interacts PAGE 110 103 Problem Entity / N / / Conceptual and Experimentation Physical Modeling and visualization Conceptual Model Classes and relations Attributes Methods Physical Model CODE, FSM, FBM, RBM, EQN Figure 7.2. Modeling process in MOOSE with Engine and displays Engine's output in a meaningful way, so that the output of MOOSE can be validated against the author's expertise. Even though this process reinforces the relation of the "model" to the "program" in a natural way, any adequate validation technique for the modeling process has not yet been developed. Therefore, by incorporating the method discussed in this research into MOOSE, we can obtain two types of benefits: one from validating the expert's rules against the simulation models, and another from validating the simulation models against the expert's knowledge. To make this point more understandable, we consider the modeling process in MOOSE as shown in Figure 7.2. MOOSE supports many different types of models [12] including CODE, FSM (Finite State Machine), FBM (Functional Block Model), RBM (Rule Based Model) and EQN (EQuatioNal Constraint model) for the physical modeling process. Then, by translating the conceptual model into C++ code, it constructs the computerized model. MOOSE does not yet employ validation or verification techniques. Therefore, using the methodology introduced in this dissertation, the /ace validation process Computerized Model (C++) Model Translation PAGE 111 104 Problem Entity 7 Experimentation ' f ^ and visualization | Expert's /' V Rules Conceptual and Physical Modeling /' Consistency Checking Conceptual Model Classes and relations Attributes Methods Computerized Model (C++) Model Translation Physical Model CODE, FSM, FBM, RBM, EQN Figure 7.3. Consistency checking in MOOSE with an expert can be automated, thereby contributing to validate the computerized models in MOOSE. A prerequisite for this process is that an expert's rule set for the system of interest must exist. Given that this condition is satisfied, the fuzzy simulation method can perform a consistency c/iec/s between the expert's rules and the computerized model shown in Figure 7.3. Any inconsistency found can be considered to be due to an inadequate conceptual or physical model of MOOSE or an improperly translated physical model on the computer. To make this validation available, the following development steps are recommended: 1. make fuzzy simulation method available in MOOSE so that all quantitative models represented in CODE, FSA, FBM, EQN and RBM can be simulated using fuzzy sets. 2. develop a user interface in MOOSE for accepting the expert's fuzzy rules. 3. make a consistency-checking facility available between the expert rules and the computerized model by using the fuzzy simulation. PAGE 112 105 : Temperature Fiizzy Set'^w^Water.depth | Start PolrttO ' "Ctkt ft*rtioo'^ Shape of MF # Mangle .^JraiJCzraiti Fuzzy Values Medrum Deep Ittsert Mniiily Diileie Save & Quit Shallow Medium Dt>ep Veryjleep Cajicel : Left Point WiWle Point niyhl Point a 0 30 a 65 30 C5 101 f,f> UJ Figure 7.4. GUI for fuzzy simulation in MOOSE 4. develop a user interface via human intervention for resolving inconsistency. Currently, fuzzy simulation is available in MOOSE, as shown in Figure 7.4. To proceed the remaining steps, especially step 3 and step 4, the approaches we discussed in Chapter 4 can be used. The consistency between two types of models (expert's rules and computerized models in MOOSE) can be measured by the difference between the CF presented by an expert and the CF calculated from the fuzzy simulation on each rule. This gives MOOSE's model author useful information, such as which components of the model should be further investigated. Consequently, by incorporating our method into MOOSE, we can obtain a benefit from validating the simulation models against the expert's knowledge. PAGE 113 CHAPTER 8 CONCLUSION The motivation for this work lies with the problem of isolating the difference between qualitative and quantitative forms of models about physical systems. Comparing and contrasting the qualitative model, especially expert rules, with the quantitative model is viewed as being an integral part of an ongoing system validation procedure. The primary contribution of this work is that by presenting an interactive tool for checking for consistency and resolving inconsistency, we provide a methodology to serve as a check and balance to enhance the complex model validation process. For this purpose we introduced a fuzzy simulation method that bridges the gap between the two different levels of models and showed how we can directly compare and maintain these two models in a systematic manner using the fuzzy simulation method. Since the uncertainty arising from the human reasoning process is easily represented by rules associated with confidence factors, we devised a way for replicating such processes by showing how the fuzzy simulation can derive confidence factors from quantitative models. To handle the possibilistic uncertainty arising from the human reasoning process, we've employed the extension principle in fuzzy set literature, various approximate reasoning tools such as the rule of conjunctive or disjunctive composition and the weighted average method. Moreover, because the expert's qualitatively described rules naturally contain linguistic vagueness, we have devised our method to handle the expert's various levels of estimates, depending on the confidence level on his (or her) linguistic terms. When the expert's complete linguistic definitions are available a priori, a direct application 106 PAGE 114 107 of fuzzy simulations gives us useful information, such as which knowledge components show inconsistency. Even without exact linguistic definitions, the method presented here determines consistency by searching for approximate fuzzy membership functions where two different levels of models maximally match. 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PAGE 119 BIOGRAPHICAL SKETCH Gyooseok Kim received the B.S. degree in Electronics from Korean Air Force Academy in 1984 and the M.S. degree in Computer Science from Korean National Defense College in 1989. In earlier days, he served with Korean Air Force as a chief programmer of Korean Air Defense System. In 1998, he received a doctoral degree in the Computer and Information Science and Engineering department at the University of Florida. His major research area is modeling for computer simulation. His minor interests are fuzzy simulation, and knowledge acquisition and validation from qualitative and quantitative simulation. 112 PAGE 120 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pau^A. FishwicK, Chairman Associate Professor of Computer and Information Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ci-Min Fu Associate Professor of Computer and Information Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy, Douglas D. Dankel II Assistant Professor of Computer and Information Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a disserUtion for the degree of Doctor of Philosophy. Sangulhevar Rajasekaran Associate Professor of Computer and Information Science and Engineering PAGE 121 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sherman X. Bai Assistant Professor of Industrial and Systems Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1998 Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School |