# New methodologies in the design of a general purpose fuzzy expert system: Applications with AI based precipitation retrieval designed for satellite microwave measurements

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O rder N um ber 8906233 N ew m ethodologies in the design of a general purpose fuzzy expert system : A pplications w ith A I based precipitation retrieval designed for satellite microwave m easurem ents Oh, Kyung Whan, Ph.D. The Florida State University, 1988 UMI 300N.ZeebRd. Ann Aibor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES NEW METHODOLOGIES IN THE DESIGN OF A GENERAL PURPOSE FUZZY EXPERT SYSTEM: APPLICATIONS WITH AI BASED PRECIPITATION RETRIEVAL DESIGNED FOR SATELLITE MICROWAVE MEASUREMENTS by KYUNG WHAN OH A Dissertation submitted to the Department of Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy Approved : Fall Semester, 1988 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. NEW METHODOLOGIES IN THE DESIGN OF A GENERAL PURPOSE FUZZY EXPERT SYSTEM: APPLICATIONS WITH AI BASED PRECIPITATION RETRIEVAL DESIGNED FOR SATELLITE MICROWAVE MEASUREMENTS (Publication No. ______ ) Kyung Whan O h , P h .D . THe Florida State University, 1988 Major Professor: Abraham Kandel, Ph.D. In the design of expert systems, management of uncertainty is related to a computational analysis premises to the conclusion. In of uncertainty this work from the we propose a new reasoning method with the equivalence operator instead of the implication operator in modus ponens. It demonstrates a fuzzy- logic-based computational framework. Based on this method, we introduce process. the concept Through the of coimplication coimplication representation scheme and concept, an inference engine Although several ad hoc models with vagueness, in there has been have been the inference a knowledge are designed. developed to deal a strong need for a globally applicable method of dealing with vagueness in expert systems. The General Purpose Fuzzy Expert System (G P F E S ) is an attempt to model uncertainty in the general domain of expert systems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Precipitation retrieval by means of remote sensing long standing problem in radiometry from space has Satellite great Meteorology. potential for is a Microwave estimating rainfall because upwelling radiation over a cloud is directly responsive to precipitation microphysics. simulation provide studies an various been carried out in an attempt interpretation measurements of remotely sensed to microwave in precipitating atmospheres and to explore the feasibility retrieval. have Therefore, of passive multi-channel microwave precipitation Numerical retrieval is studies have shown that precipitation an ill-conditioned problem from a mathematical perspective because of problems with non-unique relationships between the radiation signals From be and a theoretical perspective, overcome by There remains, simply however, analytical framework. adopting to intensity. some of these problems, can a multispectral strategy. a number of gaps within a purely We propose to close these gaps with an expert system approach which coimplication precipitation handle uses our new technique uncertainty called in general domains of expert systems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To Miran, Minhu and my mother Kuemsoon iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I am deeply indebted to my major professor, for his encouragement, Dr. A. Kandel, guidance and constructive criticism. His effort and thought to my professional development shall not be forgotten. My gratitute is also due to the other members of my supervisory committee: Drs. H. Levitz, L. Hawkes, and E.A. Smith for their assistance. I am especially grateful to Dr. E.A. Smith for the construction of the precipitation retrieval knowledge base. My gratitute is also extended to my friendly colleagues Dr. H. Cooper, M. Smith, and A. Mehta for their helpful collaboration, to Dr. Frank Wentz for providing the SSM/I data set, and to The National Science Foundation for their assistance in obtaining the INSAT data. This dissertation is dedicated to my wife, Miran, my son, Minhu, and my mother, Keumsoon. It is no exaggeration to say that without Miran's support, encouragement and love over the past life in Tallahassee, this work would not have been accomplished. This research has been supported by NASA grant NAGW-991 and DOE grant DE-FC05-85ER250000. A portion of the computational support has been provided by the Supercomputer Computations Research Institute (S C R I ), at the Florida State University, under the above DOE Contract. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents 1. 2. 3. Introduction 1 1.1 Problem Statement ----------------------------------- 1 1.2 Literature Survey ----------------------------------- 5 Background 8 2.1 Fuzzy S e t s ------------------------------------------- 8 2.2 Possibility Theory ---------------------------------- 17 2.3 Fuzzy logic and Approximate Reasoning -------------- 20 Classical Expert Systems 33 3.1 Knowledge Representation --------------------------- 37 3.2 Inference Engine ------------------------------------ 41 3.3 Reasoning with uncertainty in expert systems ------ 43 3.3.1 Problem definition --------------------------- 43 3.3.2 Bayesian M e t h o d ------------------------------- 46 3.3.3 Certainty Factor ----------------------------- 49 3.3.4 Dempster-Shafer Theory of Evidence ----------- 53 3.4 Development of Expert Systems ---------------------- 4. Building a General Purpose Fuzzy Expert SystemtGPFES) 57 67 4.1 Development toward fuzzy expert systems ---------- 67 4.2 Development of the c o i m p l i c a t i o n ------------------- 74 4.3 Knowledge representation in GPFES ----------------- 82 4.4 Coimplication and Resolution procedure ------------ 91 4.5 GPFES inference engine ----------------------------- 93 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5. Application of GPFES to microwave precipitation retrieval technigue 107 5.1 I n t r o d u c t i o n ----------------------------------------- 107 5.2 The use of satellite remote s e n s i n g ---------------- 111 5.3 Review of Passive Microwave methods to retrieve r a i n f a l l --------------------------------------------- 117 5.4 Estimation of rainfall rate from multiple microwave frequencies using GPFES ---------------------------- 121 5.5 Data set for a case study with SSM/I measurements 125 5.6 Knowledge Base to retrieve Precipitation --------- 138 5 .7 R e s u l t s ---------------------------------------------- 155 6. Conclusions 157 References 161 Appendices A. System Configuration for Image Processing to retrieve precipitation from Satellite Microwave Measurements B. Interactive sample sessions vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 185 List of Tables 4.1 Fuzzy implication operators -------------------------- 72 4.2 Knowledge schema --------------------------------------- 88 5.1 Temporal and spatial resolution of SSM/I channels — 127 5.2 Rules based on coimplication in the second tier ---- 146 5.3 Asymmetrical difference for each r u l e --------------- 147 5.4 Membership functions used in the second tier ------- 147 5.5 Rain rate i n d e x ---------------------------------------- 149 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 3.1 Major components of an expert system ---------------- 34 4.1 Knowledge database design through schemas ----------- 87 4.2 Diagram for G P F E S ---------------------------------- 4.3 diagram of GPFES inference engine -------------------- 4.4 A detailed flow diagram of fuzzy inference engine in G P F E S ----------------------------------------------- 104 5.1 87 96 Relationship between brightness temperature and rainfall rate over w a t e r ------------------------------------ 123 5.2 Footprint geometry for the four SSM/I frequencies — 129 5.3 SSM/I orbit and scan g e o m e t r y ---------------------- 131 5.4 An example of SSM/I orbit and scan geometry --------- 133 5.5 INSAT-IR i m a g e -------------------------------------- 5.6 Composite images of 136 a. vertically polarized brightness temperature at 19 GHz ---------------------------------------------------------- 136 b. vertically polarized brightness temperature at 37 GHz ---------------------------------------------------------- 137 c. vertically polarized brightness temperature at 85 GHz ---------------------------------------------------------- 137 5.7 A schematic example of coimplication rule 1 --------- 154 5.8 The image, representing rainfall rate, obtained from G P F E S ----------------------------------------------- 156 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1 Introduction 1.1 Problem Statement The analysis of expert systems consists mainly of developing a measure of uncertainty and establishing mechanisms of infe rence. Because much of the information in the knowledge base of a typical expert system is imprecise and vague, management of uncertainty is an important issue. expert systems use two-valued However, logic Many of the existing and probability theory. it is widely recognized that such methods have some shortcomings. Zadeh[Z183a] points out that conventional approaches to the management of uncertainty in expert systems are intrinsic ally inadequate because the uncertainty they fail to consider that much of in such systems is possibilistic rather than pobabilistic in nature. Expert systems are the fastest growing and the most vis ible field of Artificial Intelligence [BF82,Gw82,Md79]. Al though several ad hoc models have been developed to deal with vagueness, there is a strong need for a globally applicable Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 method of dealing with models developed vagueness thus far have in expert systems. The no theoretical base and have been inadequately analyzed. The objective of of an expert in expert systems is to get the knowledge a specific problem an appropriate structure, domain, and distribute in the same problem domain. represent it in it to other To accomplish this, users management of uncertainty is needed together with knowledge acquisition, knowledge representation, an inference engine, some control strategies and a user interface. Little development has made thus far towards the representation been and propagation of uncertainty. Management of uncertainty is related to a analysis of computational uncertainty from the premises to the conclusion [Z183a]. Uncertainties in the premises will be transmitted to the conclusion. In other words, the premise and the conclusion We attempt to establish there is a relation between in an inference. such a computational framework based on fuzzy logic to deal with uncertainty in fuzzy expert systems. In a fuzzy conditional inference in fuzzy expert systems the problem is that if we are given values for A -> B and A', we modus ponens. have to find In the real a consistent value for B ’ through fuzzy logic system we propose to use the equivalence relation for modus ponens of the infer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 ence on fuzzy expert systems instead of implication, survey a property in equivalence roximate reasoning and introduce a new approach to app using this new concept Based on the equivalence method, of dealing with we have of modus ponens. developed a theory vagueness in expert systems which is called coimplication. Approximate reasoning is implemented using a "generalized modus ponens" logics, modus ponens allows {Y is B} to be derived from {X is A} and {(X is A) -> (Y is B)}. [Z177c]. In classical If A and B are described by fuzzy values, we have to define the fuzzy implication match the meaning of any given and we need a mechanism to situation with the premise of the rule. The generalized modus ponens gives a definition for such a mechanism. The method we propose will also give such a mechanism. The purpose of this research is to provide gui dance for modeling general domains. The goals of this research are : 1. to develop the concept of coimplication based on the equivalence operator. 2. to use coimplication to develop the structures, facts and rules which represent vague information and propaga tion of uncertainty in expert systems, and to design an inference engine for handling them. 3. to design a general purpose fuzzy expert system (GPFES) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 based on the concept of the coimplication in a fuzzy en vironment and to use the concept of the schema-like knowledge representation for the management of the know ledge data structure in the implementation of GPFES. 4. to apply GPFES, based on the construction the coimplication method, to of a knowledge base for passive micro wave based precipitation retrieval [BT85,Md79]. Precipitation retrieval by means long standing problem in Meteorology. of remote sensing is a It is an ill conditioned problem from a mathematical perspective because of non-unique relationships between the radiation signals and tation intensities. Mugnai [SM88] Mugnai and Smith [MS88] have developed a theoretical the precipi and Smith and foundation for avoiding some of the problems based on a multi-spectral pass ive microwave approach. theoretical However, calculations that channel framework, it is even in evident from their the case of a multi there remain ambiguities in the relation ships between microwave brightness temperatures. We have developed a knowledge base for testing the co implication technique specifically for precipitation retrie val from multi-channel passive microwave radiometer ments obtained from the Air Force SSM/I satellite. plication theorem is designed to remove measure The coim the ambiguities in the functional relationships in order to assign specific rain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 rate intensities temperatures. to a 4 channel The test is combination conducted of brightness over the Indian ocean within the Southwest-East Asian monsoon domain. 1.2 Literature Survey There are four main reasons for the presence of vagueness in an expert system[BT85]. 1) The reliability of information : the factual knowledge is a result of ill-defined concepts in the observation, or in accurate and poor instruments used, and the result of weak implication when the expert or ineer is unable to rule set is a knowledge eng establish a strong correlation between premise and conclusion. 2) The inherent imprecision of language : If rules are the rule not expressed in a formal language, their meaning cannot be interpreted exactly. matching is no longer adequate to compare with the premise. representation This requires a Thus, a lexical subsets "semantic" of facts matching com paring the approximate meaning of facts and premise. 3) Incomplete information : In this case we need to par tially match facts and premise. Approaches to deal with this kind of uncertainty range from default reasoning with consis tent assumptions[Rr80] to analogical reasoning[Wp80]. 4) The aggregation of rules from different knowledge Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 sources or different experts : There are four possible errors that can occur in knowledge represented [SSS81]; conflicting! redundant, as production rules subsuming, and missing rules, i) conflicting rules : They succeed under the same circum stances, but make contradictory conclusions and increase the level of uncertainty by creating inconsistencies, ii) redundant rules : They succeed under the same circum stances, make the same conclusion and may create an inflated assessment of the certainty of the conclusion, iii) a subsumed rule : The premise subset of the second of and creates an the first rule is a over-estimate of the certainty of the common conclusion, iv) missing rules : They fail sion under the right to provide a needed conclu circumstances and create incomplete information. Expert systems have modelled uncertainty and imprecision in various ways. Many systems, including the PROSPECTOR mine ral exploration system[DHN78], have used the Bayesian approach for the relative likelihood ratio to quantify the strength of a given rule. Shortliffe devised a scheme based on what he called certainty factors (CFs) for measuring that could be placed conclusion as a result of in any given the evidence so far[Se76,SB75]. A certainty the confidence factor is the difference between two component measures; CF[h:e] = MB[h:e]- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 MD[h:e], where CF[h:e] is certainty of the hypothesis h given evidence e, MD[h:e] MB[h:e] is a measure is a measure of function [Sg76] has of belief in h given e and disbelief in h been given e. developed within The belief the framework of Dempster’s work on upper and lower probabilities induced by a multivalued mapping[Da67]. These models have several problems when applied to general domains[BT85]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 Chapter 2 Background 2.1 Fuzzy sets In this section some basic concepts discussed. in fuzzy set theory The theory of fuzzy sets, 1965[Z165], introduced by Zadeh in is a generalization of abstract set attempted to generalize the classical notion of proposition sense. are to accomodate uncertainty in the In classical set theory an item either theory. a set and a nonstochastic belongs to set or it does not. It can not partially belong to a set. the concept of the set. fuzzy sets, Fuzzy set theory founded basis for is used in an item may provides the modeling He a In partially belong to a systematic and well- of imprecision. Imprecision the sense of vagueness rather than lack of know ledge about the value of a parameter as in tolerance analysis. Fuzzy in set theory which provides vague conceptual rigorously studied. a strict phenomena mathematical framework can be precisely and It can also be considered as a modeling language well suited for situations in which fuzzy relations, criteria, and phenomena exist. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 Let us consider person is heavy. the problem Someone who sidered heavy by some people. of determining when a is 180 pounds may not be con However he is certainly not light. In classical set theory one might say he was almost heavy and allow him a grade of membership of 0.8 in the set of heavy people. This seems to be clearer. Let us now turn to the formal framework of fuzzy set theory. Definition 2.1: Let U be a collection of objects denoted generically by x. Then a fuzzy set F in U is a set of ordered pairs: F = X {(x,X (x)) ! x is a member of U } . F (x) is called the membership function or grade of memberF ship of x in F which maps U to the closed interval [0,1]. The grade-of-membership value X (x) of an object x in F F can be interpreted as the degree of compatibility of the pre dicate associated with F and the object x. ble to interpret It is also possi X (x) as the degree of possibility that x is F the value of a parameter fuzzily restricted by F. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Example 2.1: F = "Real numbers considerably larger than 20" F = {(xfpf(x)) F ! x is a member of R) , where 0 if x < 20 -2 -1 if x > 20 1+(x - 2 0 ) The assignment of the membership function of a fuzzy set is subjective in nature and, in general, reflects the context in which the problem is viewed. Definition 2.2: The support of which % a fuzzy set F is the set of all x 6 U at (x) is positive. F Example 2.2: Let the universe as temperature. be the interval [0,120], with x interpreted A fuzzy subset F of U labeled WARM may be defined by a grade of membership function such as 0 if x < 60 if 60 < x < 70 if 70 < x <.80 if 80 < x i 110 /x - 60 \ V X 20 / 2 <*> = F tx -80^ ' 20 1 - 2 / Then, the support of WARM is the interval (60,110]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Definition 2.3 The height of a fuzzy set F is the supremum of X (x) over the F universe U. hgt(F) = sup [X (x)} x feU F In the Example 2.2, the height of old is effectively 1. Definition 2.4: The crossover point of a fuzzy set F is the point in the universe U whose grade of membership in F is 0.5. Definition 2.5: A fuzzy set F is said to be normal if its height is unity, that is, if In sup xeU Example 2.2, X = F the crossover point of WARM is 70 fuzzy set WARM is normal. fuzzy sets, To and simplify the representation of it is convenient to use the following notation: a finite fuzzy set F on U is expressed as n F = the % (xl)/xl + .... F + /C (xn)/xn F X = j=l (xj)/xj. F when U is not finite, we can use the notation f = £ r p <x>/x. u Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 Definition 2.6: A fuzzy set F is convex iff ^xlfeU, ^ x 2 6 U , % F When we belongs [Axl + (1 - X )x2] > mint# (xl) F want to exhibit to a fuzzy set F, an element X.6[0,l], (x2) ] . F x6U that typically we may demand that its membership value be greater than some threshold. Definition 2.7: The o^-cut Fot, of F is the ordinary set of elements their membership values are greater than or such equal that to some threshold o(£(0,l]. F* = {x <£, U, X F Similarly, <x > > o O • ~ the strong o(-cut is defined by F~ = {x <= U, X <x > ><*> • F The membership function of a fuzzy set F can be expressed terms of the characteristic functions of its to the formula % (x) = F where X -cuts according sup min[o< , (x) ] , *€[0,1] Hx (X) = F<* 1 if x 6 F ^ 0 otherwise, s in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 The o^-cut sets are used for setting decision thresholds in an expert system environment. At some point a yes/no decision is needed in all expert systems. We will now define unions and intersections of fuzzy sets. Definition 2.8: Let A and B be A and fuzzy sets of U. Then the union of fuzzy sets B is denoted by A U B and is defined A U B = \ J (X (x) V % A by (x)) / x , B U where V is the symbol for maximum. Definition 2.9: Let A and B be fuzzy sets of U. fuzzy sets A and B is denoted by A A AB f\ B the intersection of and is defined by (x))/x, where A is the symbol for minimum. = \ J U Then A B Definition 2.10: The complement of a fuzzy set A is denoted by A and is defined by A = 1 - PC (x) )/x. U Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Definition 2.11: A linguistic hedge or a modifier is an operation, which modifies the meaning of a term, or more generally, of a fuzzy set. If F is a fuzzy set then the modifier m generates the composite term G = m(F). Mathematical models used for modifiers are : :X Concentration (x) = KX Con(F) Dilation : Z <x) = ( Die(F) 2(X Contrast intensification F (x)) F X 1/2 (x) ) F if# <*>> 1 - 2(1 N (x) e [0,0.5] F X (x)) otherwise F Generally the following linguistic hedges are associated with the above mentioned mathematical operators. If F is a fuzzy set then very F = Con(F) more or less F = Dil(F). Definition 2.12: The product of A and B is denoted by AB and is defined by AB = \ * ( x ) A U X (x) /x. B Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 Definition 2.13: If Fl,..,Fk are Cartesian defined fuzzy subsets of Ul,..,Uk, product as a fuzzy respectively, the of Fl,..,Fk is denoted by Flx..xFk and is subset of U l x . .xUk whose membership function is expressed by y (xl,..,xk) = Fix..xFk X (xl)A**A FI X (xk). Fk Example 2.3: Let A be the set of weights in pounds of five people. A = {(Anna,140),(H a r r y ,175),(Na n c y ,165),(B i l l ,150), (Eric,190)}. Let JC (x), the membership function of the fuzzy set Heavy Heavy, be defined on the following expression: Heavy = 0.2/Anna + 0.8/Harry + 0.6/Nancy + 0.4/Bill + l./Eric. Let B be the set of heights in inches of people. B = {(Anna,57),(Harry,73),(Nancy,67),(Bill,62),(Eric,80)). Let Tall, fC. (x), the membership function of the fuzzy set Tall be defined on the following expression: Tall s 0.3/Anna + 0.78/Harry + 0.7/Nancy + 0.45/Bill + l./Eric. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Then Heavy U Tall = 0.3/Anna + 0.8/Harry + 0.7/Nancy + 0.45/Bill + l./Eric Heavy Tall = 0.2/Anna + 0.78/Harry + 0.6/Nancy + 0.4/Bill + 1./Eric Heavy = 0.8/Anna + 0.2/Harry + 0.4/Nancy + 0.6/Bill + O./Eric Heavy Tall = 0.06/Anna + 0.62/Harry + 0.42/Nancy + 0.18/Bill + l./Eric 2 Heavy = 0.04/Anna + 0.64/Harry + 0.36/Nancy + 0.16/Bill + l./Eric 0.5 Heavy = 0.1/Anna + 0.4/Harry + 0.3/Nancy + 0.2/Bill + 0.5/Eric Con(Tall) = 0.09/Anna + 0.61/Harry + 0.49/Nancy + 0.2/Bill + l./Eric Dil(Tall) = 0.56/Anna + 0.88/Harry + 0.84/Nancy + 0.67/Bill + l./Eric Example 2.4: Let XI = X2 = {2,4,6} and let A1 = 0.5/2 + 1/4 + 0.6/6 and A2 = 1/2 + 0.6/4. Then AlxA2 = 0.5/(2,2) + 1/(4,2) + 0.6/(6,2) + 0.5/(2,4) + 0.6/(4,4) + 0.6/(6,4). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 2.2 Possibility theory The subject of possibility has been introduced by Zadeh[Z178]. Let X be a variable taking values in U. Then a possibility distribution, |T , associated with X may be viewed as a fuzzy X constraint on the values that may be assigned to X. Such a distribution is characterized by a possibility distribution function TT :U -> [0,1] which associates with each xfi U the X degree of the possibility that X may take x as a value. One of the central concepts of possibility distribution. distribution, possibility theory is In order to define it is convenient to introduce the that of a a possibility notion of a fuzzy restriction. Definition 2.14: Let X be a variable taking values in U characterized by a membership function ^ variable (u). F is a fuzzy restriction on the F X if it acts as an elastic constraint on the values that may be assigned to X in the sense that the assignment of value u to X has the form: X = U : X <u > • F Whether a fuzzy set can be considered as a fuzzy restric tion or not obviously depends is only the case if it acts on its interpretation. This as a constraint on the values of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 a variable, which might take the form of linguistic term or a classical variable. Let R(X) be a fuzzy restriction associated with X as de fined in definition 2.14. assignment equation Then R(X)=F is called a relational which assigns the fuzzy set fuzzy restriction R ( X ) . For instance, F to the A(X)=age of Jack and F is the fuzzy set "young". The proposition "Jack is young" can be expressed as R(A(X)) = F. Definition 2.15: Let F be a fuzzy set in an universe of discourse U which is characterized by its membership function X (u), which in F turn is interpreted as the compatibility of u £ U with F. Let X be a variable taking values in U and F act as a fuzzy rest riction, R(X), associated with X. F", which translates distribution,JT Then the proposition "X is into R(X) = F associates a possibility i which is postulated to be equal.to R(X). X The possibility distribution function, f|" (u), characterX izing the possibility distribution || is defined to be nuX merically equal to the membership function (u) of F, i.e., F = X xF . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Definition 2.16: Let F be a fuzzy set in the universe U and ]"[" a possibility X distribution associated with a variable X which takes values in U. The possibility measure, X IC (A) = sup u 6A natural U is defined by (u) F Zadeh[Z177b] proposes, meaning in TTL(A), of A C in studying the representation of language, the concept of a possibility m easure. Example 2.5: Let us consider the possibility distribution which is induced by the proposition "X is a small integer" . jT — {(1,1),(2,1),(3,.8),(4,.6),(5,.4),(6,.2)} and X Then the possibility measure IT(A) of A is A—{3,4,5}• 1C (A ) = max (.8,.6,.4) = .8 Similar to probability theory there exist also conditio nal possibilities. Such a conditional possibility distribut ion can be defined [Z181]. Definition 2.17: Let X and Y be variables in the universes U and V, respec tively. The conditional possibility distribution of Y, X, is then induced by a proposition of given the form "If X is F Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 then Y is G" and is denoted by TT(Y/X) (v/u). If we let A and B be fuzzy subsets of U, then the possi bility measure of the union is defined as 1C(B) and the possibility and B is related means maximum and by measure 1C(A U B) = TC(A) V of the TC(A A B) < tt:(A) intersection of A /\ TC(B), where V means minimum. 2.3 Fuzzy logic and approximate reasoning In Boolean logic truth values can be 0(false) or l(true) and by means of these truth values the operators are defined via truth tables. Zadeh Fuzzy logic [Z175b] has been proposed by and is an extension of set theoretic multivalued in which the discussion truth values are linguistic variables. logic In this we limit consideration to possibilistic interpre tations of linguistic variables and will discuss mainly the role of fuzzy logic and approximate reasoning for the manage ment of uncertainty in expert system. Fuzzy logic is essentially considered as an application of possibility theory to logic. For numerical truth values v(A) and v(B), the logical operators are defined as v(A and B) = min(v(A),v ( B ) ) v(A or B) = max(v(A),v ( B )) v(A -> B) = min(1,l-v(A)+v(B)) v(not A) s 1 - v(A). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 The employment of fuzzy ' logic as a framework for the management of uncertainty in expert systems makes it possible to consider a number of issues effectively or correctly For example: which cannot be dealt with by conventional techniques [Z183a]. 1) If x is small then Y is large with CF = 0.8, in which the antecedent "X is small" and the consequent "Y is large" are fuzzy predicates propositions because the denotations of the small and large are fuzzy subsets SMALL and LARGE of the real line. 2) Since the number of rules in an expert system is usually relatively small (i.e. of hundred), there such as "X is A ’" the order of two are likely to be many cases in which a fact may not match exactly the antecedent of any rule of the form "If X is A then Y is B with CF = a". The conventional treat it rule-based systems usually avoid this issue or in an ad hoc manner because partial matching does not lend itself to analysis within the confines of two-valued logic. fuzzy By contrast, the gradation of truth and membership in logic provides a natural way of dealing with partial matching through the use of the compositional rule ence and interpolation. of infer 3) In many cases, the antecedent and /or the consequent of a rule contain fuzzy quantifiers such as most, many, implicit or explicit few, many more, usually, much of, etc. Consider the disposition "Student is young", which may Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 be interpreted which in turn as the proposition "Most students are you n g " , may be as the conditional expressed as a rule, or, equivalently, proposition "If x is a student then it is likely that x is young.". Here a disposition is a proposition with implicit fuzzy quantifiers typical expert system are, The loping in effect, disposition[Z183b]. approximate measures of uncertainty and rules of inference, i.e., bilistic expert In statisti semantic evaluations are in the form of proba logic relations establishing systems consisting of axioms, and deductions based recursively on them. systems, ried the rules in a analysis of expert systems consists mainly of deve theories cal and many of for which supplies the calculus the of basic operators probability. In and existing systems, the computation of certainty factors is car out through a combination of methods which are based on two-valued nized logic and probability theory. that such methods have It is widely recog serious shortcomings and, for the most part, are hard to rationalize. Zadeh [Z183a] shows to the management of intrinsically inadequte that the conventional uncertainty because in they expert approaches systems are fail to come to grips with the fact that much of the uncertainty in such systems is possibilistic rather than probabilistic suggests in nature. He also a fuzzy-logic-based computational framework be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. em- 23 ployed to deal with both possibilistic and probabilistic un certainty within a single conceptual system. Fuzzy logic provides a natural framework for the manage ment of uncertainty in expert systems because its main pur pose is to provide a systematic basis for representing and in ferring from imprecise knowledge ledge. The greater rather than precise expressive power of from the fact that it contains tional two-valued as well as know fuzzy logic derives as special multi-valued cases the logics. tradi The main features of fuzzy logic which are of relevance to the manage ment of uncertainty in expert systems are the following[Z183a]. (a) In fuzzy logic, the truth values are allowed to range over the fuzzy subsets of T. For example, if T is the unit inter val, then a truth-value in fuzzy logic, e.g., very true, be interpreted as a fuzzy defines the possibility subset of the unit interval distribution associated may which with the truth-value in question. A fuzzy truth-value may be viewed as an imprecise charaterization of an intermediate truth-value. (b) In fuzzy logic, the predicates may be crisp, e.g., mortal, even, father of, etc. or, more generally, tired, large, tall, much heavier, etc. as multi-valued logics allow some. However, fuzzy,e.g., ill, (c) Two-valued as well only two quantifiers, all and fuzzy logic allows the use of fuzzy quantifiers exemplified by most, many, several, few, much of, frequently, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 occasionally, about, etc. Such quantifiers may be interpreted as fuzzy numbers which provide an imprecise characterization of the cardinality of one or more fuzzy or nonfuzzy sets, (d) Fuzzy logic provides a method for representing the meaning of both nonfuzzy and fuzzy predicate-modifiers exemplified by not, very, more or less, extremely, slightly, much, a little, etc. This leads to a system for variables[Z175a], i.e., computing variables with linguistic whose values are words or sentences in a natural or synthetic language. For example, Age is a linguistic variable when its values are assumed to be: young, old, very young, not very old, etc., with each value interpreted as a possibility distribution over the real line, (e) In two-valued logical qualified, principally systems, a proposition, p, may be by associating with p a truth-value, true or false : a modal operator such as possible or neces sary: and an intensional operator such as know, believe, etc. In fuzzy logic, the three are : truth qualification, principal modes of qualification expressed as p is 7* , in which is a fuzzy truth-value ; probability qualification, as p is X » i-n which \ y expressed is a fuzzy probability ; and possi bility qualification, expressed fuzzy possibility, e.g., quite as p is 1C , in which possible, K. is a almost impossible, etc. Since the inference processes in existing expert Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 systems are based on two-valued logic and/or probability theory, the principal tools for inference are the modus ponens and/or its probability analog-Bayes’ rule and its variations, the validity of these inference processes is open to question since most of the information in the knowledge base of a typ ical expert system consists of a collection of fuzzy rather than nonfuzzy propositions. Among viewed as the traditional rules of inference - which may be special cases of the compositional rule of infer ence - is the modus ponens. derive To establish this fact, we shall from the compositional rule of inference a more gene ral version of the modus ponens which in fuzzy logic is ref erred to as the generalized modus ponens[MG81,Z177c]. Consider a pair of propositions {P1,P2} of the form PI s If X is F then Y is G, P2 = X is F ’, in which F, F ’ and G are fuzzy sets (or fuzzy predicates). PI and P2 may be expressed as PI -> TT = H (Y|X) P2 -> where TT (YlX) tion of Y given X, X iu,v) H s1 A |T = F ’ , X denotes the conditional (1 H is ->£(u) F defined in +X possibility terms of distribuF and G by (v)), u and v are generic values G Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 of X and Y respectively and A is the min operator. pC :UxV -> [0,1] is the membership function of H. H By applying the compositional rule and P2, the possibility distribution of of inference Y to PI is found to be given by TTY = HoF* where the right-hand side represents the composition of H and F ’with respect to X. More concretely, X Y (v) = sup<;t (u ,v ) /\ u H = s u p < 2 (u) A F’ x F’ (1 - X <u) + X F (v))). G This conclusion may be stated in the form of the syllogism If X is F then Y is G X is F* Y is H o F ’ where HoF* is defined by X , (v )* This syllogism expresses the Y generalized modus ponens. The generalized modus ponens differs from its classical version in two respects. First, F ’ is not required to be identical classical case. And second, the with F, predicates as IX is in the F, G and F ’ are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 not required to be crisp. and the predicates It can be verified that when F s F ’ are crisp, HoF' deduces G and the above syllogism becomes If X is F then Y is G X is F Y is G Standard formulations both an of inference processes involve implication and an inference rule, like modus ponens or modus tollens. In the modelization of approximate reasoning by fuzzy sets, most researchers have basically focussed their attention on the study of implication. some difficulties identifying Generally, the statement "If A there are then B " , even when dealing with classical logic. One of the reasons is that the statement is understood i.e. the whole as B will occur if A does, modus ponens schema is implicitely assumed [TV85]. Trillas and Valverde [TV85] are devoted to the modus ponens, study of especially in the context of multivalued logic used to modelize approximate reasoning by means of fuzzy logic. After an overview of the concept and uses modus ponens in some logics, it is pointed out that procedure implied by modus ponens may be in many cases the described by means of a function. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 We now discuss the concept and meaning of logical rules. A rule can be a directive which expresses the assertion of a formula [Rh80]. a permission as to Consequently, requirement for a rule of inference should be that from true statements to true statements." an inference is not a statement [Yr85]. the only "It leads Therefore, like an implication, but a procedure which uses implication to get true statements. Such a procedure can be described only in a rule and expressed in a schema. Thus, symbolically in the case of modus ponens, the schema is a -> b a b that is, if a Since a rule is is true and a -> b is a procedure and not true then b is true. a formula, it requires a justification which follows from the truth table defining the implication connective. antecedent "a" is Except both the truth value of the false, by applying this procedure to the whole table we can determine quent "b" given when the the truth value of the conse truth values of "a" and "a -> b " . Modus ponens is identified with the the probability of the sentence "b", given both the probabil ities sentences and "a ->b", according to the of the "a" procedure to determine following schema Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 I P(a) £ P(a->b) 1 7j P(b) > m(£,lj) where the function m depends on the If P(a->b) , definition is defined as P(b a) , then be checked by applying the = of €.Yj P(a->b). , as it can law of total probabilities, i.e., P(b) = P(b|a) P ( a ) + P(b|la) P(Ta) 2.P(b|a) P(a). If P(a->b) = P(Ta V b ) , then we will have P(a->b) = P(1a V b) = 1 - P (a ) + P(b) - P(ia f\ b) 1 1 - P(a) + P ( b ) i.e. £+7j - = max(0, From v(a->b) m should be variable. also A ± 1). v(a->b*) if v(b) < v|b'), non-decreasing function m it follows that with respect to the second satisfying MP1 through MP4, where MP1 through MP4 are defined in the end of this section, will be called the modus ponens generating function. The nature of the domain of a modus ponens generating function may vary from the one system to another. Thus in Boolean, probabilistic and multivalued set of truth logics, this domain will be VxV, V being the values. In fuzzy logic m is a function from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 U UxV V [0,1] X [0,1] into [0,1]. We digress to the analysis of the existence of modus ponens generating functions for some of the implication func tions of the multivalued logic system introduced in the framework of fuzzy sets theory. Fuzzy implication functions are functions I from [0,1]X [0,1] into [0,1], defined by I(x,y) = G(n(x),y), continuous t-conorm and n a strong negation G being a function. Among strong implications, the most known and used are IL(x,y) = min(l-x+y,l) generated by G(x,y) = min(x+y,l) and N(x) = 1 - x, IM(x,y) = max(l-x,y) generated by G(x,y) = max(x,y) and N, and IP(x,y) = 1 - x + xy generated by G(x,y) = x +y - xy and N. Definition 2.18: Triangular or T-Norms[SS63] are functions used tional conjunction with domain for proposi- in the unit interval of the real line satisfying the axioms: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 (i) T(0,0) = 0, T(a,1) = T(l,a) = a (ii) T(a,b) <. T(c,d) whenever a <_c , b < d (iii) T (a ,b ) = T(b,a) (iv) T(T(a,b),c) Similarly, = T(a,T(b,c)) formulas for disjunction are triangular cono rms i.e. T-conorms. T-Norms with T-conorms also the exception of the satisfy first the axioms axiom, which of is modified to be (i) T ( 1,1) = 1, T(a,0) = T(0,a) = a. In many cases, the procedure modus ponens may be descri bed by a function m from [0,1]X[0,1] among others, the two basic into [0,1] which satisfy, conditions m(x,x->y) <_ y and m ( 1,1)=1. According to the previous discussion of a modus ponens generating function, the function m satisfies MP1 through MP4, in such a way that m(v(a,v(a->b))) is a lower bound for the validity of "b", i.e. MP1: m(v(a),v(a->b)) < v ( b ) ; MP2: ■(1,1) = 1, which guaranties the preservation of classical modus ponens; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 MP3: m(0,y) = 0, which may be rephrased as "from a false statement, anything can be inferred", and; MP4: If x <L x ’ then m(x,y) <. m(x' ,y), which enables us to consider the chaining of inferences i.e. m(m(v(a),v(a->b)),v(b->c)) < m ( v ( b ),v(b->c)) £ v ( c ) . Modus ponens has been treated as a be represented procedure which may by means of functions to get valid statements from valid conditional statements. Modus ponens generating functions for some fuzzy implication functions of multivalued logic systems have been characterized. In some way, the repr esentation of a conditional statement by means of an implica tion connective may be viewed as a sort of formal construc tion which is justified precisely because of their participa tion in a rule of inference which works. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Chapter 3 A Review of Expert Systems Computer scientists have tried that perform remarkable to develop computer functions, in a way that would considered intelligent if carried out by research effort, processing and humans. including robotic devices, expert systems, programs be The entire natural language is usually called artificial intelligence(AI). A collection of AI techniques, which enables computers to assist people in analyzing decisions, problems and making is called expert systems. An expert system is one that handles real-world, complex problems requiring an expert’s interpretation and solves problems using a computer model of expert human reasoning, reaching the same conclusion that the human expert would reach. The most expert systems is that they systems that can perform significant thing about are highly successful. These are skilled medical diagnosis, mass- spectrogram interpretation, predict crop disease, prospect for minerals to give just a few examples. based on an extensive body of An knowledge problem area. This knowledge is organized rules which allow the system to draw expert system is about a specific as a collection of conclusions from given Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 data or premises. There are in fact four essential fledged expert system : knowledge components of a base, inference knowledge-acquisition module and explanatory ful- engine, interface. The structure of an expert system is shown in Figure 3.1. Figure 3.1 Major components of an expert system users^------------- explanatory interface ^------ inference engine domain----------- ^knowledge < experts acquisition module The knowledge base 1 ) knowledge base is the program’s store of facts and associations it "knows" about a subject area (such as medi cine). A critical design decision is how such knowledge is to be represented within the program. There are many choices. For MYCIN, they chose to represent knowledge as conditional statements, or rules, of the following form : If A and B, then C. The inference mechanism can take many forms. We often speak of the control structure to reflect the fact that there are different controlling strategies for the system. A set of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 rules may be chained as follows: If A then B ( rule 1 ) If B then C ( rule 2 ) A C This is input ( conclusion called forward ) chaining, or data-directed inference, because the data that are known drive the infer ences from left to right in rules, with rules chaining together to deduce a conclusion C. In MYCIN, a backward goal-directed a system control strategy is used. chaining or a In backward chaining starts with a statement of the goal to achieve and works "backward" through inference rules i.e. from right left, to find the data that establish that goal Find out about C (goal If B Then C ( rule 1 ) IF A then B ( rule 2 ) If A then C ( implicit to : ) rule ) Question : Is A true ? ( data ) Since there are many rule chains and many pieces ofdata about Which the system needs to inquire, we say that MYCIN is an evidence gathering program. Knowledge acquisition is a bottleneck in expert system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 development. Typically with an expert, the knowledge engineer must sit down in many long sessions, and extract the human expert's domain knowledge for use in the expert knowledge expert’s engineer area will often not must have some system. understanding the so that he can talk to the expert. The expert be able to provide general problem solving heuristics. Example problems and the heuristics in cess of the expert solving them must be presented. ledge gathered of The tends to be incomplete the pro Some know and often not quite correct. In order to develop an expert system, a domain expert for the knowledge initial stages with the expert we have acquisition to find process. The will be an intense process of finding the relevant information and how to get it. The know ledge engineer is responsible for weeding out the inconsis tencies in the knowledge base. Some tools have been developed to aid the knowledge acquisition process, [DL82] which aids knowledge the expert system acquisition such as TEIRESIAS and building EMYCIN [Vw79]. refinement for SEEK[PZM81] gives advice about rule refinement during the development of a diag nostic-type expert system. A computer program that functions like an expert in a given domain is more likely to be accepted by experts in that domain, and by nonexperts seeking its advice, if the system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 can explain its actions. need not be a A consultative psychological model, rule-based imitating system a human’s reasoning process. The important point is that the system and a human expert arrive at the use same the same knowledge about the domain to conclusions knowledge base contains the expert as well as consideration. facts to given conditions. domain-specific knowledge about a particular problem The of an under When a rule is used, its actions make changes to the internal data base, which contains the system’s decis ions. Explanations require displays of how the rules use the information provided by the user to make various intermediate deductions and information finally to arrive at the answer. If the contained in these rules adequately shows why an action was taken, an explanation can simply entail displaying each rule or a free-text translation. The purpose of the explanation capacity is to give the user access to as much of the system’s knowledge as possible. It should be easy for a user to get a complete, understand able answer to any sort of question about the system’s know ledge and operation. 3.1 Knowledge representation There are several methods of any of which can be used knowledge alone or in representation, conjunction with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 others to build expert systems. Each method provides the program with certain benefits, such as making it more effici ent, more easily understand, or more easily modified. A thor ough description of them is given in "The Handbook of Arti ficial Inteligence" [BF82]. The three main knowledge repre sentation schemes are rules, semantic networks and frames. Rule-based premise THEN knowledge representation uses the form "IF conclusion". When the current problem situation matches the condition part it is performed. This of a rule, the conclusion part of conclusion part may direct program control or may instruct the system to reach a real conclusion, i.e. add a new fact to the database, and so on. Rules provide a natural way and changing for describing processes driven by a complex environment. A set of rules can specify how the program should react to the detailed advance knowledge changing about the data without requiring flow of control. The term rule has a much narrower meaning than it does in ordinary language. It refers to the most popular type of knowledge representation technique[WH78]. Rules provide tions, directives, a formal way of representing recommenda or strategies. The matching of a r u l e ’s con dition to the facts can produce what are called inference chains. The inference chain indicates how the system used the rules to infer the identity of a final conclusion. An expert Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 system's inference chains can be displayed to the user to help explain how the system reaches its conclusions. Semantic networks were originally as psychological models of human memory developed for use but are a standard representation method for AI and expert systems[Br79]. CASNET [WKS78] is an important example of using the semantic network method. The semantic network consists of points called nodes, and arcs. The latter connect those nodes which are related by some semantic relation. The nodes in stand for objects, concepts, or events. representing hierarchies include a semantic network Common arcs used for isa and has-part. Given a semantic network, because we know about the properties of the relations linking the nodes, we can infer a conclusion from the network as in the rule-based method. The isa relation and the has-part relation establish a property inheritance hierar chy in the network. Items lower in the network can inherit properties from items higher up in the network. This can save space be since information about similar nodes d o e s n ’t have to repeated at each node. Semantic networks are a useful way to represent knowledge in domains that use taxonomies to simp lify problem solving. Also, the semantic network representa tion is useful because it provides a standard way of analyzing the meaning of a sentence and have been used in natural lang uage research to represent complex English sentences [NR75]. However, semantic networks can suffer from combinatorial Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 explosion as well as over generalization!for X isa bird, birds have-property flies, thus X flies; but X is an ostrich.) Frame-based knowledge representation uses a network of nodes connected by relations, and organized into a hierarchy. Each node represents a concept that may be attributes and values associated with the the hierarchy inherit properties of term frame refers to network. Frames A frame is node. Nodes low in higher-level nodes. are organized The the method employed by much like a semantic In a frame system each node is defined by a collec tion of attributes(e.g. size, attributes(e .g . large, tall ed by a special way of representing common concepts and situations. CENTAUR[Aj83]. described height), and values of those ), where the attributes are call slots. Each slot can have procedures attached to it which are executed when the Each slot can have information in the slot any number of procedures Three useful types of procedures attached to is changed. attached to it. slots are as follows: i) if-added : when new information is placed in the slot ii) if-removed : when information is deleted from the slot iii) if-needed : when information is needed from the slot, but the slot is empty. Frame systems expectations about are the useful for problem domains where form and content of the data play an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 important role in problem solving such as interpreting visual scenes or understanding speech. 3.2 Inference engine The inference knowledge base to engine solve is an interpreter that uses the the problem at hand. The main point of interest in inference engine is that structure information must be kept in the all of the control inference engine. The explicit division between a knowledge base and an infere nce engine offers a degree of domain-independence. Knowledge base content is strongly influenced by the control strategy used in the inference engine. The inference engine and knowl edge base are not cases. necessarily totally independent in many It is responsible for determining what piece of know ledge to inference use next and for scheduling necessary actions. The engine also is responsible for determining when to ask the user a question and when to search the knowledge base for the information. It must deal with imprecise and uncer tain information. Inference engines operate in chaining, backward chaining one of or both. two ways: Forward forward chaining begins with some data and continues along the inference chain until it reaches a final conclusion. Backward chaining begins with a final conclusion and continues backward until it finds Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 a complete path of evidence for one of the satisfactory states. EMYCIN[Vw79] is an example of a backward chaing method. [Mj82a] is a forward chaining expert system. systems which use a combination of chaining methods; e.g. SPERILL[OFY84]. use meta-knowledge, which is forward XCON There are some and backward The inference knowledge about engine the system’s knowledge i.e. how best to utilize various knowledges, which pieces of knowledge to use first, whether a piece of knowledge should be inferred and when knowledge may be in the same to stop processing. The meta knowledge base or in a separate knowledge base. EMYCIN[Vw79], MYCIN[Se76], basically a domain-independent version of is an appropriate skeletal system for developing a consultation program that can request data about a case and provide an interpretation or analysis. EMYCIN MYCIN inference engine to be applied to a new whose problem-specific MYCIN rule language, knowledge providing can all allows the problem domain be represented in the of M Y C I N ’s features as well as a human-engineered system-building environment, which greatly facilitates entering and debugging the knowledge base. In EMYCIN, the initial goal determines the value of the level goal attribute. top- EMYCIN works on the goal of establish ing the value of the attribute of some object. To do this, retrieves a it precomputed list of rules whose consequents are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 known to bear on that goal, and rules until it either establishes certainty or exhausts the deduced it attempting truth resorts to of its to rule it attempts to apply the the value list. asking the If with complete no value can be user for the value. In apply a rule, EMYCIN must first establish the antecedent, certainty of each of its which requires conditions. that the To do this, the system typically has to establish the values of objects. This sets up subgoals determining other attributes of are addressed by using the same mechanism recursively. In KAS[Rr81], At any given time there is no formal top-level hypotheses. KAS is either trying to identify the best top-level hypothesis to pursue or trying to identify the best question to ask the user to first mode will be called establish that hypothesis. The the goal-selection mode and the second, the question-asking mode. Goal selection is guided by information the user supplies during the session. 3.3 Reasoning with uncertainty in expert systems 3.3.1 Problem definition Uncertainty just in data can be compounded when aggregat ing uncertain data in the premise certainty measures to the conclusion. and when propagating Suppose that "X is A" Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 has a degree of confidence xl and there is a rule "(X is A) -> (Y is B) with a degree of confidence rl". Then we should be able to decide yl, the degree of confidence related to the conclusion "Y is B", yl must be a function of xl and rl. Bonissone and Tong[BT85] have stated the questions related to uncertainty management for reasoning as follows: How can we elicit from the expert consistent values of rl ? How can we compute or otherwise provide the value of xl ? What are the forms by which rl and xl should be repres ented : a scalar, an interval, a distribution, a lingui stic expression ? Are they defined on an absolute scale or on a relative one ? In a more of multiple general case, clauses, how when the premise is can we aggregate composed the degree of certainty of the facts comprising the clauses of the premise ! i.e. what is the function T(xl,,,xn) that determines xp, the degree of certainty of that conclusion ? Prade[Ph77] has modeled this kind of uncertainty using a fuzzy extension of modal logic, necessity and possibility. the degree of certainty based on Zad e h ’s concepts of It has also been is rather than a scalar[Bp83]. suggested that represented by a fuzzy interval The degree of necessity[Ph77] is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 represented by the lower bound of the fuzzy interval degree of possibility by its upper bound. and the Necessity and pos sibility respectively represent the amount of support for the hypothesis, and the amount of failure to refute the hypothes is provided by the evidence. The value of the necessity of an hypothesis is always smaller than or equal to the value of its possibility. Violation of this constraint during aggrega tion of conclusions from inconsistencies among the different experts knowledge suggests sources. rule Our method to handle uncertainty reasoning will be discussed in Chapter 4. In the numerical expert system, sed in representation of uncertainty in the there are several methods and will be discus this section. The Bayesian approach, employed in PROSPECTOR(D 7 ), uses an effective likelihood ratio to quanti fy the strength measures the of a given rule. sufficiency This likelihood ratio of a piece of evidence to prove a given hypothesis. The certainty factor (CF) in MYCIN[Se76]. belief and a The CF is the difference between a degree of degree of disbelief in a given hypothesis after supporting evidence is found. preted as approach [SB75] has been used the relative The CF increase was originally or inter decrease of a ratio of probability. The belief function, proposed by Shafer [Sg76], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was 46 developed within and lower the framework probabilities of Dempster’s work on upper induced by a multivalued mapping [Da67]. In this context, the lower probability was associated with a degree of belief degree of plausibility. this approach. and the upper probability with a However, there are two problems with A computational problem has been encountered, since the evaluation of the degree of belief and plausibility requires the exponential in the cardinality of the hypothesis set. The second problem results from a normalization process present in both Dempster’s work and Shafer’s. Zadeh[Z179] has argued that this normalization process can lead to incorrect and counter-intuitive results. 3.3.2 Bayesian Method Many systems, including the PPOSPECTOR mineral explora tion system[DHN78], have used Bayes’s theorem as the thread for tying together information from disparate sources. Bay e s ’s rule provides for computation of relative likelihoods between competing hypotheses on the strength of the evidence. It depends on the formula L R (H :E ) = P(E:H) / P(E:H’), where the of the event likelihood ratio LR is defined as the probability or evidence E given a particular hypothesis H Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 divided by the probability of the evidence given the falsity of that hypothesis (H’). Thus if we know the the evidence probability of given the hypothesis, and its negation, we can work out the likelihood of the hypothesis in the light of the evidence. For given instance, chicken-pox chicken-pox if we know the probability of spots and the patients, probability of spots among non we can compute the probability of chicken-pox given spots. This piece of likelihood evidence ratio to measures prove the a given hypothesis. Pednault, Zucker and Muresan [PZM81] concluded that conditional independence hypotheses and inconsistent under of the the negation could be assumption of both under the of the hypotheses were with the other assumptions of an exhaustive and theories rest on the belief that for unlikely the evidence mutually exclusive space of hypotheses. how sufficiency of a it true. is, there no matter probability about some hypo thesis there must be some evidence our views(beliefs) on the matter. there remaining forever unchanged. everything, is a prior probability that it Given a prior process would stop right Essentially, Ba y e s ’s we can call on to adjust If there were not, then the with the prior probability Given relevant evidence, we can modify this prior probability to produce a posteriori proba bility of the same hypothesis given some new evidence. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Let P(H) be the prior probability of some hypothesis and P(H:E) be the posterior probability of the same hypothesis given some item of relevant evidence E. By definition, P(H and E) P(H:E) and P(E and H) = P(E:H) = P(E) Thus, we get P(H) P(E:H)P(H) •P(H:E) = P(E) One it assumes important criticism independence of the variables being used. This is quite a serious point. symptoms. on the Bayesian method is that The temperature". first For example, suppose that we have two was "fever" There would be no and the second was "high point in having both items because they mean the same thing. They are exactly correlated. But, in some cases, twice and therefore probability of we would have included the same evidence either incremented, or decremented, the influenza more than we should have done and the final posterior probabilities would be wrong. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 3.3.3 Certainty Factor Shortliffe devised a scheme based on what he called a certainty factor (CF) for measuring the confidence that could be placed in any given conclusion as a result of the evidence so far [Se76]. A certainty factor is the difference between two component measures: CF[h:e] where CF[H:e] is the certainty of the hypothesis h given evi dence e, MB[h:e] MD[h:e] = M B [h :e ] - MD[h:e], is a measure of belief in h is a measure of disbelief in h given e. from -1 (false) to 1 (true) meaning ignorance. with CF can range values and 0 MB and MD can range between 0 and 1. CF reflects a simple balancing The above fractional given e and formula does of evidence for not permit conflict of evidence as opposed to The and against. any distinction between lack of evidence i.e. MB and MD both high, or MB and MD both low, respectively. MB and MD obey some of the axioms of probability but are not derived from a population sample of any kind and therefore cannnot be given a statistical interpretation. CF is an alternative to conditional probability that has several advantages in M Y C I N ’s domain. CF is in the statement of decision rules themselves. In corresponds to the conclusions in this the case the evidence E premise of the rule. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 x Thus A and B and C -> D is a representation of the statement CF[D: A and B and C] = x. In accordance with subjective probability theory, it may be argued that the expert’s personal probability P(h) reflects his or her belief in h at any given time. Thus 1 - P(h) can be viewed as an estimate of the expert’s the truth of h. tion If P(h|e) disbelief regarding is greater than P(h), the observa of e increase the expert’s belief in h while decreasing disbelief regarding the truth of h. In fact, the proportionate decrease in disbelief is given by the following ratio: P ( h !e ) - P(h) 1 - P(h) This ratio is called the measure of increased belief in h resulting from the observation of e, i.e. MB[h:e]. Suppose, on the other hand, observation while of that P(h|e) e would were less than P(h). Then the decrease the expert’s belief in h increasing his or her disbelief regarding the truth of h. The proportionate decrease in belief in this case is given by the following ratio: ( P(h)-P(h!e))/P(h). We call this ratio the measure of increased disbelief in h resulting from the observation of e, i.e., MD[h:e]. To sum Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 marize these results, we consider the measure of increased belief, MB[h:e] to be the proportionate decrease in disbelief regarding the hypothesis h that results from the e. observation Similarly, the measure of increased disbelief, MD[h:e] is the proportionate decrease in belief regarding the hypothesis h that results from the observation e, where belief is estim ated by P(h) at any time and disbelief is estimated by 1 - P(h) . The above definition may be specified formally in terms of conditional and a priori probabilities : if P(h) = 1 M B [h :e ] m a x [P (h | e ),P(h)] - P(h) o . w. m a x [1,0] - P (h ) if P ( h ) M D [h :e ] = ' min[P(h|e),P(h)] - P(h) o.w min[l,0] - P(h) * And we find a measure termed a certainty factor (CF), that combines the MB and MD : CF[h:e] = M B [h :e ] - MD[h:e]. The certainty factor is an artifact for combining degree of belief and disbelief into a single number. Such a number Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 is needed in order to facilitate comparisons of the eviden tial strength of competing hypotheses. A property of CF is CF[h:e] + CF[ not H:e] = 1, confirms h. That is, although where e evidence may support a hypo thesis with degree x, it does not support the negation to the hypothesis with degree 1 - x. CF[not h:e] = MB[not h:e] - MD[not h:e] P(not h|e) - P(not h) = 0 - - P(not h) [1 - P ( h | e )] - [1 - P(h)] 1 - P(h) P(h) - P ( h | e ) 1 - P(h) CF[h:e] = M B [h :e ] - MD[h:e] P(h|e) - P(h) = - 0 1 - P(h) Thus CF[h:e] + CF[not h:e] = 0. This result occurs because for any h and any e MB[h:e] = MD[not h:e]. This is intuitive ly appealing since it states that evidence that supports a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 hypothesis disfavors the negation of the hypothesis to an equal extent. The CF model is equivalent in part to the simple proba bility model. The empirical success stands in spite of theoretical of CF objections. model in MYCIN The fact that in trying to create an alternative to probability theory or rea soning, Shortliffe and Buchanan duplicated the use of stan dard theory demonstrates the difficulty of creating a useful and internally consistent system that is not isomorphic to a portion of probability theory [Aj76]. 3.3.4 Dempster-Shafer Theory of Evidence The theory was first developed by and extended by Glenn Shafer. Dempster in the 1960s He published "A mathematical theory of Evidence"[Sg76]. Recently researchers have begun to investigate applications of the theory to expert systems[Bj81, F 1 8 1 ,G LF81]. The advantage lity to model the of the Dempster-Shafer theory is its abi narrowing of accumulation of evidence and diagnostic reasoning medicine in a the hypothesis set with the process and that characterizes expert reasoning in general. An expert uses evidence that, instead of bearing on a single hypothesis bears on a larger in the original hypothesis set, subset of this set. The often functions and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 combining rule of the Dempster-Shafer theory are well suited to represent this type of evidence and its aggregation. The generality of the Dempster-Shafer belief functions is an avoidance of the Bayesian restriction that the ment of belief to a hypothesis impies commitment of commit the re maining belief to its negation i.e., that P(h) = 1 - P(not H). As in the CF model, the beliefs original set need not sum allotted to subsets of to 1: the in each hypothesis some of the original in the belief can be hypothesis set. The Dempster-Shafer model includes many of the features of the CF model but is based on a firm mathematical foundation. This is a clear advantage over the ad hoc nature of CF m o d e l . The Dempster-Shafer [0,1] to indicate evidence. belief This number supports the hypothesis. Shafer model theory in a is the Unlike uses a number in the range hypothesis given a piece of degree to which the evidence the CF model, the Dempster- avoids the use of negative numbers. The impact of each distinct piece of evidence on the subsets of 0 , which is a frame of discernment (hypothesis s e t ) , represented by a function called a is a generalization basic probability assignment (bpa). of the traditional A bpa probability density e function. Using 2, the enlarged domain of all subsets of 0, a bpa denoted m assigns a number in [0,1] to every subset such that the numbers sum to 1. The of @ quantity m(A) is exactly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 A, where A is an element of 2 & and the total belief is 1. This portion of belief cannot be further subdivided among the subsets of A and does not include portions of belief commit ted to subsets of A. Since entails belief in subsets belief in containing a subset certainly that subsets, be useful to define a function that computes a of belief in A. This quantity would include not only it would total amount belief committed exactly to A but belief committed to all subsets of A. Such a function, called a belief function denoted Bel, corresponding to a specific bpa, m, assigns to every subset A of 0 the sum of A by m. of the beliefs committed exactly to every Thus, Bel(A) subset is a measure of the total amount of belief in A and not of the amount committed precisely to A by the evidence giving rise to m. The Dempster combination rule differs from the MYCIN combining function in the pooling of evidence supporting mut ually exclusive hypotheses. Let Bell and Bel2 and ml and m2 denote two belief func tions and their respective b p a ’s. new bpa, denoted ml ® m2, Dempster’s rule computes a which represents the combined effect of ml and m2. The corresponding belief function, deno ted Bel ® Bel2, is easily computed from ml ® m2 by the defin ition of a belief function. If we sum all products of the form ml(x)m2(Y), where X and Y run over all subsets of 0, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 56 result is 1 by elementary algebra and the definition of a bpa: ^Eml(X)m2(Y) = ^ m l (X )2lm2 (Y ) = 1 x 1 The = 1. bpa representing the combination of ml and m2 apportions this number 1, the total amount of belief, among the subsets of 0 by assigning ml(X)m2(Y) There are typically several to the intersection of X and Y. different subsets of 0 whose intersection equals that of X and Y. Thus, for every subset A of 0, Dempster’s rule defines ml@m2(A) to be the sum of all products of the form ml(X)m2(Y), where X and Y run over all subsets whose intersection is A. In the most general situation, a given piece of evidence supports many of the subsets of 0, each to varying degrees. The simplest situation is that in which the evidence supports one subset to a certain degree and the assigned to d. Because of the modular way remaining belief is in which knowledge is captured and encoded in MYCIN, this latter situation appl ies in the case of MYCIN rules. If the premises confirm the conclusion of a rule with degree s, where s is above thres hold value, then the r u l e ’s of © effect can be represented by a bpa. singleton corresponding on belief in the subsets This bpa assigns s to the to the hypothesis in the conclusion of the rule, call it A, and assigns 1 - s to 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 The Dempster-Shafer theory is appealing in its potential for handling evidence bearing on categories of diseases as well as on specific disease entities. It facilitates the ag gregation of evidence gathered at varying levels of detail or specificity. Thus collaborating experts that refer to semantic concepts at could specify rules whatever level in the domain hierarchy is most natural and appropriate. The theory is probability based and it has been specu lated to be useful in building expert systems. A problem with this theory is that rules of evidence combination may create a too large certainty measure of the combined evidence about a fact if a normalization is used to eliminate or hide a con tradiction [Z184a,Z184b]. 3.4 Development of Expert Systems The DENDRAL [BF78] program is the first AI program to emphasize the power of specialized knowledge over generalized problem-solving methods. It was started in the mid-1960s by Lederberg and Feigenbaum as an investigation of the use of AI techniques for hypothesis formation. tions of empirical data It constructed explana in organic chemistry, specifically, explanations of analytic data about the molecular structure of an unknown organic chemical-compound. By the mid-1970s, there were several large programs, collectively called DENDRAL, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 which interacted to help organic chemists elucidate molecular structures. The programs are knowledge-intensive, require very specialized knowledge of chemistry in i.e., they order to produce plausible explanations of the data. Thus a major con cern in research on DENDRAL was how to represent specialized knowledge of a domain like chemistry so that a computer pro gram could use it to solve complex problems. MACSYMA[MF71] auspices of the MAC has been developed project. at M.I.T. under the It is able to provide symbolic solutions to calculus problems and various algebraic problems. Some will argue that it is makes use of algorithms it has achieved high not an expert system since it to symbolically solve problems. But competence in the symbolic computations associated with applied analysis. Many mathematicians seek its assistance in algebraic computation. The roots of MACSYMA can be traced.back to early algebraic manipulation programs for general mathematical simplications[Ht61,Kk65,W d 6 3 ]. MYCIN[SD75] is one of the first medical expert systems. It gives consultative advice on diagnosis and theraphy for infections diseases that compares favorably with advice given by experts in infections disease. MYCIN's is represented in terms of production tainty ing. rules medical knowledge involving cer factors, which help accommodate probabilistic reason The rules are invoked using a backward-chaining control Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 strategy since that it effectively works backward makes from MYCIN hypothesis-driven, its conclusions. factors are loosely based on probability theory. Certainty They are an ad hoc method of uncertainty handling and have not proved to be uniformly applicable. in the sense MYCIN is an outgrowth of DENDRAL that many of the lessons learned in the struction of DENDRAL were used in the design tion of MYCIN. erberg and applied to and implementa The senior members of the DENDRAL team, Feigenbaum, Buchanan that con AI ideas had convinced that made themselves and Bruce DENDRAL a problem of medical import. Led- work could be MYCIN has developed the idea of making many uses of one data base. Using backwardchaining of rules, MYCIN can give the reasons for its sions in terms of its rules and can carry on asking the user for information needed ward chaining. deci a dialogue by to continue the back MYCIN has led to a more general framework for expert systems such as EMYCIN [VSB84], where EMYCIN is called Essential MYCIN or often is called Empty MYCIN. used as an expert system building tool if you fill its knowledge base with the EMYCIN is under the theory that appropriate knowledge it will then be an expert in that domain. domain It is ess entially an expert system shell. When you fill the shell with the appropriate knowledge a full fledged expert system is born. But EMYCIN has been successful for limited problem types. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 TEIRESIAS[Dr77] facilitates automatic acquisition of new knowledge in the MYCIN system. of meta-level knowledge domain. It is concerned with the use that is not specific to MYCIN or its TEIRESIAS does assume that rule-like units ledge are used and of know that a chain of such rules suffices as an explanation of system behavior. Metaknowledge is knowledge about how MYCIN knowledge is represented and used. TEIRESIAS has rule models and rules about the structure of such meta-rule dictates that rules rules. One mentioning the culture site of an organism should also mention the organism’s portal of entry. Such a rule enables TEIRESIAS to detect faulty rules as they enter the system. Using the context of the dia logue and its own knowledge of what a rule should look like. TEIRESIAS can fill TEIRESIAS was Randy D avis’dissertation research original in much of a new rule for an expert. and its two goals were the development of an intelligent assis tant, and to develop a set of tools for knowledge base truction and maintenance in order to abstract cons from them a methodology applicable to a range of systems. Rl [Mj82b] is one of the first commercially sucessful expert systems developed by John McDermott at Carnegie-Mellon University. It configures tal Equipment Corporation. much uncertainty and is VAX 11/780 computers for the Digi It does able not have to reason with to effectively use a match, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 backtrack search method on the knowledge base. It has evolved into the more versatile expert system called XCON[Mj82a]. It gave rise to the expert system building language 0PS5 [Fc81]. Several speech-understanding systems have been developed during the 1970s. HEARSAY-II[EHL80] is one of them and used a global working memory ("blackboard"), types and levels of uniform structure. information in which different were all integrated into a HEARSAY-II provided ways of focussing, so the system could shift its attention appropriately area of the interpretation problem to another. from one This idea of independent knowledge modules has been used frequently in ex pert systems and contributes to the ability to modify and ac quire knowledge. University. [REF73]. It has been HEARSAY-II is Unfortunately, developed a higher at Carnegie-Mellon version of HEARSAY-I it only obtained the expertise of a 10 year old in some very narrow domains. With several differ ent knowledge sources, which works via a blackboard, be thought of as several small expert systems, to form a larger system. cessors, thereby laying it may loosely coupled It was implemented on separate pro the ground work for distributed expert systems. When a blackboard pattern matched the domain’s schema, the system triggers a knowledge source. HEARSAY has attacked a very large domain, unlike most expert systems. Earlier we mentioned EMYCIN which is an expert system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 building tool. EMYCIN was used to build PUFF [AKS83], an example of an expert system obtained from filling out a shell. PUFF is a rule-based any medical expert expert system and is as successful system to date. It has been used as as a teaching system at a San Francisco Hospital. CASNET(Casual Associational Network program) was devel oped during the early and primary use has in been midseventies [WKS77, WKS78]. the glaucoma, and ophthalmologists close to expert. handling, diagnosis theraphy of have rated its performance as It introduces its own method of uncertainty which has a resemblance to both fuzzy methods. and Its probabilistic and This expert system shows that LISP is not the only possible language for development. They developed CASNET in FORTRAN. The program provides modeling diseases instead of program was one of attempts expert development of EXPERT [WK79], applied to glaucoma. model theumatology diseases. determines the system Within this effects of systems, a general and The CASNET program framework for merely modeling glaucoma. the first framework for building a general This to provide a general and it led tool that endocrinology as to the has been well as to introduced a casual network to network the program reasons to a theraphy or diease. represents the disease as a dynamic Thus process the about which it reasons and can explain to the user. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 SOPHIE[BB75] acts as an electronics laboratory instructor that interacts with a student who attempts to debug a piece of malfunctioning equipment. Responding quickly in restricted English, it is expert enough for actual instruction. employs a simulation model of electronics. It SOPHIE also has de clarative knowledge of a particular circuit, encoded in a se mantic network, and routines that reason with this form of knowledge. PROSPECTOR[DGH79], structured similarly to MYCIN, expert advice Networks are on finding ore deposits from geological data. used both judgmental to express the knowledge in PROSPECTOR, knowledge which is expressed static knowledge about domain objects. The a knowledge acquisition system(KAS) that as rules and program contains facilitates the ac quisition of all types of knowledge in PROSPECTOR. tinually gives prompts the user until structure are filled in. KAS con all missing parts of a new This process is driven by an exter nal grammar that can be changed without difficulty, making it easy to modify KAS as PROSPECTOR evolves. a network PROSPECTOR editor and that understands The core of KAS is various mechanisms in gives the user a limited ability to edit new knowledge in terms of content rather than form. AGE[ABN81] for attempt is an expert system building tool. It stands to generalize. It is an ambitious attempt to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 develop a tool that is capable of being used expert system for any desired application. by the user for specific applications. to build an It may be modified In addition its infer ence engine can be configured in various ways. CENTAUR[Aj83], which has been developed by Aikens, example of a second generation expert system. offspring of PUFF. is an CENTAUR is an It also diagnoses pulmonary disfunctions. Its questioning is more directed than P U F F ’s and it is a bet ter diagnostician. The improvements are accomplished by plicating the control knowledge. The knowledge representation technique use ex of frames as its allows the control know ledge to be put in one of the frame slots. Hence it combines rules and frames, showing them to be a powerful combination. CADUCEUS[PW84], consultation system domain of which is called INTERNIST, is a medical that attempts to make a diagnosis in the internal medicine. The program displays expert performance in about 85 percent of internal medicine, so its knowledge base is one of the largest The diagnosis problem is in any expert system. complicated because a have more than one disease, which makes patient can the number of pos sible combinations enormous. Like CASNET, CADUCEUS represents its medical knowledge in a structure, the disease tree, about which the program reasons dynamically. data-drivened and hypothesis-directed CADUCEUS combines reasoning in the same Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 framework. The patient data are used first theses, and these tions that must to predict hypo are then used to predict other manifesta either be confirmed or used to change the hypotheses. Kohout and Bandler[BK80b] propose methods to represent and manipulate knowledge in fuzzy expert systems, brief introduction to the concept of and give a a fuzzy expert system [KB82]. Yager[Yr83] fuzzy sets, introduces a robot planning methodology using which may be seen as a first step in developing an expert system for robot planning. SPERILL-II[OFY84] is an expert system designed to perform damage assessment of existing structures. They use fuzzy sets to represent imprecise data. Dempster and Shafer’s theory for combining fuzzy sets with certainty factors is used in doing inexact inference. FINDEX[WS85] appropriate products. is an interactive expert system and suggests techniques It is a for forecasting linguistic sales of commercial fuzzy production rule system. They have made some extensions to the wenstop language [Wf79] of linguistic variables and incorporated this into the FESS[H186], which has been developed by HALL, is a re reasoning process of an expert system. usable fuzzy expert system. It provides a methodology Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 66 develop a multi-knowledge porates fuzzy reasoning source expert system which incor techniques. In FESS, a blackboard concept is used for communication and the developed methodology of it allows the construction of an expert system which has no domain knowledge built into it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Chapter 4 Building a General Purpose Fuzzy Expert System (GPFES) 4.1 Development toward fuzzy expert systems In a domain with little imprecision an expert system can likely be developed. But, the development of an expert system is very difficult in a domain with a large amount of impreci sion. There are several sources of imprecision and uncertain ty in expert system areas and when a solution is imprecise it must be presented to the system user in a manner which cates the uncertainty in it. ally lead to an answer system must be able to indi An imprecise question will usu which is also phrase imprecise. questions in captures their somewhat vague meanings An expert an manner which when such arise. The process of acquring knowledge is quite imprecise. It is like ly that the knowledge acquired doesn't expert's since the expert is often exactly capture the unaware of all the tools used in the reasoning process. Fuzzy reasoning techniques can provide the basis for representing in an expert’s knowledge. the imprecision inherent Incomplete information gives another uncertainty to the expert system. Common sense reasoning in- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 eludes many inferences from incomplete information. It has been attempted with fuzzy reasoning according to Zadeh[Z184c]. Uncertainty also have weak occurs in implications. accept and use the weak the knowledge The expert base where we may system must implications be able to together with other sources of information to come to a conclusion about the prob lem. Uncertainty also arises in knowledge sources i.e. experts and references, etc. from There may be flicting, redundant, subsuming or missing knowledge. be noted of different con It should that some of the applications with the least amount imprecision attempted. in expert systems have been successfully But those with large amounts of imprecision not been as successfully solved. have Imprecision and uncertainty must be handled theoretically. There are several different ways that imprecision may be handled in etical methods an expert system. basis of i.e. handling In a sense they have no theor have been handled imprecision in ad hoc ways. Most are based on probability. Experts tend to think in terms of much, usually, always, some times. They d o n ’t use probabilistic values. introduced several methods those methods are effective in to handle specific guaranteed to be effective in extreme ral domains. In Chapter 3, we uncertainty. cases, While they are not situations or to gene Fuzzy set theory gives us a theory-based method Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 of dealing with uncertainty. By fuzzy expert system we mean an expert system which is based on fuzzy sets and fuzzy logic in handling imprecision and uncertainty for the inference process and knowledge represenation scheme. The theory of fuzzy sets and fuzzy logic is well-founded and strong. The theoretical basis behind fuzzy techniques will allow us to deal with uncertainty in a manner that is well-supported. SPERILL-II[OFY84] is an expert system for damage assess ment of existing structures imprecise data. and uses fuzzy sets to represent Knowledge may be represented in the proposi- tional calculus but it is not good enough for general applica tions. Predicate calculus is a more useful one. But those are very precise representation formalisms and have been used in in most expert systems. The fuzzy logic is good for imprecise applications pervasive form. That but it is still somewhat cumbersome. The most form of knowledge representation is the rule-based is because it is a natural form and easy to understand. In the design of expert systems recognition of the need for graded use of degree of implication, there is increasing production rules i.e. the leading to degree of certainty or possibility attached to the conclusions. This accords with imprecise, incomplete and faulty input information. We Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. have 70 surveyed probabilistic methods in the previous chapter. In this section, we survey a method of fuzzy implication for re presenting a relation between premise and conclusion in a rule. Bonissonne [BB85] represent and proposes using linguistic qualifiers or conclusions of a rule-based fuzzy numbers to variables in the premise knowledge representation scheme. A t-norm and t-conorm pair is chosen and they provide guidence on the choice Conclusions of an appropriate implication scheme. are given not with single-valued certainties but as ranges, or, more likely, qualified as truth by some appro priate linguistic qualifier. REVEAL [Jp84] is uses fuzzy reasoning. a general decision support tool which In REVEAL, fuzzy sets take the role of adjectives or qualifiers in natural languages. production rules for performing inference. tors It uses fuzzy Functional opera such as very and fairly are available both with default values and those values defined by the user. FLOPS[BST86] built in is viewed as a fuzzy expert system. the expert system building language OPS5. FLOPS has been applied to be classification of regions in grams. It makes use of fuzzy numbers as well as confidence values linguistic It is for descriptors system conclusions. echocardio multi-valued It can translate into fuzzy sets and back. The system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 uses fuzzy production rules for knowledge represen tation . ARIES [AR85] provides mechanisms for the representation and manipulation of multiple as defined by several degrees of propositional truth, criteria of truth, belief, or likeli hood. It has been implemented as a general purpopse approxi mate inference subsystem capable of being easily incorporated into a wide variety cf information systems. Degrees of truth, which may be interpreted in ARIES as either several ways, are represented in classical or interval of the real line. fuzzy intervals These intervals of the [0,1] represent con straints on the possible truth values of facts or conditional propositions (rules). Implication relations are used for inference chaining in the expert system. The implication premise and a conclusion. relation consists of a The premise is made up of a number of clauses. Each clause has the form "X is Y " , where X and Y represent variable linguistic statements. Y may contain fuzzy qualifiers such as "almost" or "very". The premise consists of one to n claues. The clauses are connected by the conjunc tion and disjunction operators/\, V. Any t-norm, t-conorm pair may be used ; see Definition 2.18. The conclusion is a set of one or more clauses connected by the linguistic connective and, which is used as an uniform Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 1. S # Standard Sharp a —♦> b — 1 iff a 1 or 6 = 1 otherwise 0 2. S Standard Strict a -*j 1 if.fa<b 0 otherwise 1 iff a < 6 otherwise 6 —1 3. S* Standard Star a. — b — 6 4. G43 Gaines 43 4*. G43* Modified Gaines 43 a — b = m in(l, -— -) a l— o 5. L Lukasiewicz a —»s 6 = m tn(l, 1 —a + b) 5.5 KDL Kleene-Dienes-Lukasiewicz a -♦ 5 , 5 6 6 = 1 ~ c + ob . KD Kieene-Dienes a —>8 6 = (1 —o) V 6 7. EZ Early Zadeh a —* 7 5 = (a A b) V (1 —a) Table 4.1 Fuzzy im plication operators Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 conjunctive operator. That means that each clause in the con clusion holds with the same strength as the overall conclusion. Each relation is given some a priori strength which we call a certainty. This with terms, certainty may be represented linguistically such as usually, sometimes, or occasionally. It may be represented with discrete numerical values in the real -valued interval [0,1] into which the linguistic terms are currently translated. There are which may be at least nine different implication operators used in fuzzy implication relations which have been classified by Bandler and Kohout [BK80b]. They are shown in Table 4.1. Oh and Bandler[OB87] survey these fuzzy implica tion operators. given i.e. In fuzzy expert systems, the problem is that, x->y and x, we must find a value y which is consistent to find the value of the conclusion of an implication relation by the use of fuzzy modus ponens[BK84]. What is done with the conclusion when it has been determ ined with some truth value ? It is used in the continuance of the reasoning process. system has come The reasoning process halts when the to a conclusion which provides the informa tion that the user of the expert system seeks or there are no more relations available to determine an answer for the user. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 4.2 Development of coimplication Fuzzy logic has been proposed by Zadeh [Z175b] and is based on the concepts of fuzzy sets[Cc88,Cs71,Gj67,Mp69,Z165]. Based on [Z175b] we have investigated an equivalence property of fuzzy logic and applied it to the new approach to approxi mate reasoning using a fuzzy logic which uses the infinite multivalued logic of Lukasiewicz[Rn69]. Oh and Bandler [OB87] have defined the difference between the truth degrees Q and Q -> P i.e. d = v(P->Q)-v(Q->P), of P -> and have investigated some properties regarding the difference in the fuzzy implica tion operators classified by Bandler and Kohout[BK80a]. Definition 4.1 shows the other cases of this difference between two propositions P and Q with operator 5 in Bandler and Kohout [BK80a]; (see Table 4.1). Definition 4.1: 1) symmetrical difference: v(P | Q) = |v(P -> Q) - v(Q -> P)j = Ip - q \ 2) asymmetrical difference: i) v(P ii) v(P |— Q ) = v(P -> Q) - v(Q -> P) = q - p -\ Q) = v(Q -> P) - v(P -> Q) = p - q The concept of symmetrical difference of fuzzy sets is discussed in Dubois and Prade[DP80], and Kandel[Ka86]. Theorem Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 4.1 shows a relation between the equivalence and the symmetri cal difference of P and Q. Theorem 4.1 : v(P <-> Q) = 1 - | p - q| = 1 - v(P | Q) = v(not(P | Q ) ) pro o f : v(P <->Q)=v(P -> Q and Q -> P) = min(min(l - p + q,l), min(1 - q + p , 1)) i) p > q : v(P <-> Q) = min(l - p + q fl) = 1 - p + q = i - Ip - q| ii) p < q : v(P <-> Q) = min(l|l - q + p ) = l - q + p = 1 - | p - q| Consider a method applying these operators to an infer ence through modus ponens. The classical and plausible modus ponens is the formula "IF P -> Q AND P THEN Q" which is a tautology. The new approach is which isa similarity relation expert systems i.e. the to use the equivalence operator, [Ka86], in formula modus ponens "IF P <-> Q AND P THEN Q" which is still a tautology. Theorem 4.2: 1) 2) in (P <-> Q ) A P -> Q implies ( P -> Q ) /\ P -> Q P <-> Q implies P -> Q Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 proof trivial. The following is a typical type of the fuzzy conditional inf erence : Ant 1 : If x is A then y is B Ant 2 : x is A ’ Cons y is B 1 Let us consider a specific example for the fuzzy conditional inference. Ant 1 : If tomato is red (0.7) then tomato is ripe Ant 2 : tomato is red (0.8) tomato is ripe When we use truth (0.8) Lukasiewicz’s fuzzy ( ? ) implication operator, the degree of Ant 1 is 1; thus, given x -> y and x, then y is determined. In this case, the truth degree of "tomato is ripe" is greater than or equal to 0.8. Consider the above example by the equivalence operator : Ant 1 : tomato is red (0.7) <-> tomato is ripe (0.8) Ant 2 : tomato is red (0.8) tomato is ripe ( ? ) If we use the equivalence operator then the truth degree of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 "tomato is method ripe" is 0.9. is more specific If the truth degree of Therefore, than 0.8 in the implication method. "tomato is red" truth degree of"tomato is ripe", same results. 0.9 in the equivalence then is greater than the both methods have the Based on this equivalence method, we develop the coimplication concept. The definition of inference from a set of premise[Yr85] can be improved in a fuzzy real environ ment system using the equivalence method. Definition 4.2 : In multivalued logic, we say that (Pl,,,Pn) I— atoms in Pl,,,Pn implies P, denoted P i f , for every consistent interpretation of the the propositions Pl,P2,,Pn and P, the relationship | (PlA. «APn) 1 <_ | P I is true. Note that this definition subsumes the inference rule in binary logic which requires that ((PlA..APn) -> P) is a taut ology. Definition 4.3 : A fuzzy statement form which is always greater than or to x, x £ (0,1], as a truth degree, values of its equal no matter what the truth fuzzy statement letters may be, is called x-tautology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a 78 Definition 4.4 : P denoted (PI,..,Pn) H P, Pl,P2,...Pn fuzzily coimplies P, if for every fuzzy consistent interpretation of the atoms in the propositions Pl,,,Pn and P, p=sup(|v(Q) - v(P)|) and p < where p is Q and P, and 0.5, PiQ “ a maximum possible symmetrical difference between Q = PlA. .APn. In binary logic, the inference rule requires that Pn)->P tion (PlA..A is a tautology, but we use <-> instead of ->. Defini 4.4 subsumes the coinference rule in binary logic which requires that ((PlA. .APn)<->P) is a tautology. Theorem 4.3: In binary logic, 0 (Pl,..,Pn) H P if (P1A.A P n ) < - > P is a tautology. Proof: Assume (PIA. «APn)<->P is a tautology. Under the assumption that all the P i ’s have truth value one, i.e. v(PlA. .APn) = 1. From the assumption, v((PlA. .APn)<->P) = l, and then v(P)=l by the equivalence property. Therefore p ^ 0 . 5 . Under the assump tion that at least one of the P i ’s have truth value zero then PlA. .APn have truth value 0. From the assumption, P has a truth value 0. Therefore p <,0.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 Theorem 4.4: In fuzzy logic, 1-x (Pl,..,Pn) | | P if (PlA. .APn)<->P is a x-tautology. Pr o o f : Assume (PlA. A P n )<->P is a x-tautology, where x>. 0.5. From the property of <->, this requires that l-\v(PlA. .APn)-v( P)| i.e. |v (P1A«-APn) - v(P)l >_ x, < 1 - x. Thus p = sup(\ v (P1A« *APn)~ Pi ,P v(P) | ) = 1—x <_ 0. 5 . Theorem 4.5: 0 Let P be a proposition. Then (P) 1 (P proof s u p (Iv(P) - v ( P ) I ) = 0 P Theorem 4.6 (Syllogistic Reasoning): Let P, Q and R be propositions. pi p2 If (P) |--- 1 Q, (Q) i 1 R and pi + p2 < 0.5 then (P) pl+p2 1--\ R. Pro o f : Let p - v ( P ) , q = v (Q ) and r = v(R). Then p - q < pi and |q - r | 1 p2 . -pi <_ p - q <_ pl and -p2 < | P ” r I £. pl + p 2 . q - r < p2. Therefore pl+p2 Thus (P ) I— -*— 1R Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 Theorem 4.7: pl p2 )— 1 P 2 , (P2) 1---\P3,, p then (Pl) — 1Pn If (Pl) pn-1 n-1 (Pn-1) I--- 1 Pn and p = 2 p i i= 1 10.5 Theorem 4.8: Assume (Pl,,,Pn) is a set of premises and P is some conclusion. Let (Pa,,,Pk) be any subset of the original premises (Pl,,,Pn). P P+d If (Pa,..,Pk) V— H P then (Pl,..,Pn) )■--- 1 P and p + d £ 0.5, where d = min(Pa,,,Pk) - min(Pl,,,Pn) . proof - ( p + d ) l ~ ( p + min(Pa,..,Pk) - min(Pl,..,P n )) 1 m i n ( P l ,..,Pn) - v(P) < p + min(Pl,..,Pn) - min(Pa,..,Pk) <_ p + d. Theorem 4.9: Assume P and Q are any two propositions. Ip-ql i) (P,Q) I 1P ip-ql ii) (P) | 1 P v Q, where p = v(P) and q = v(Q). proof I min(P,Q) - p I = \min(0,q-p)| 0 i) q < P : (PiQ) M P p-q ii) q < p : (P,Q) |— | P Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 The knowledge representation method of uncertainty, based on the coimplication in fuzzy expert systems, will be discuss ed in the next section. fuzzy coimplication, but We have discussed the concept of for an actual application in fuzzy expert systems we are using the following definition. Definition 4.5: Pl,,Pn fuzzily coimplies P approximately with an asymmetrical (d) difference d, denoted (Pl,..,Pn) -[P, if for every fuzzy consistent interpretaion of the atoms in the propositions Pl,, Pn and P, pl = inf(v(P) - v(Q)) and p2 = sup(v(P) - v ( Q ) ), and p2 - pl <^0.5 and d = (pl + p2) / 2, where Q = PlA. *APn and d = v(P) - v(Q). Theorem 4.10: (dl) If (A) V 1 C, (d2) (B) |--- 1 C (min(dl,d2)) then (A V B) I------------ 1 C Proof: Let a = v(A), b = v(B) and c = v(C). v(C) - v(A) a: dl and v(C) - v(B) fir d2. Then v(C) - m a x (v (A ),v (B )) fi: min(dl,d2). Using this concept, the four modes of inference in fuzzy expert systems surveyed by Bandler and Kohout [BK84] can be considered. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 Remarks: 1. Ponens a -> b and a : b (d) a b— 1 b and a b cs a + d 2. Denial a -> b and not a : not b (d) a \ -- 1 b and not a not b s 1 - a + d 3. Confirmation a -> b and b : a (d) a I— | b and b a CS b - d 4. Tollens a -> b and not b : not a 4.3 (d) a H — \ b and not b not a is d + 1 - b Knowledge Representation in GPFES One of the main purposes in this research is to develop the structure of facts and rules to represent vague informat ion and handle propagation of uncertainty in expert systems using the coimplication. Knowledge representation is one parts in the expert system design. of the most important Syntactical structures of knowledge representation were discussed in Chapter 3. But we emphasize the semantical structure in this section for GPFES. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 In syntactical types of knowledge representation, there are semantic networks, production systems and frame (schema)-like representation methods. A knowledge base knowledge about a problem domain. facts about objects, instructions) and contains facts and Examples of knowledge are events ( or actions ), performance ( or meta-knowledge ( e.g. the reliability of certain information ). The combination powerful tool of of rules and schemas can provide a the knowledge representation scheme. More over* the schema concept contains the concept of the semantic network. Since network, all network rules three can be easily described in a semantic combinations of the rule, the semantic and the schema (frame)-like knowledge representation would appear true. When people are faced with a new large amounts of highly structured experience; [Bf32] used "schema" word they use knowledge which they have acquired from previous the situation, Barlett [Bf32]. Barlett to refer to this situation. Marvin Minsky[Mm75] used the concept of a frame as a fundamen tal structure representing common concepts and frame is organized much like system (or schema system) organized in a hierarchy, general a semantic situations. A network. A frame is a network of nodes and relations where the topmost nodes represent concepts and the lower nodes more specific instances Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 of those concepts. Davis and Buchanan [DB77] introduce the concept of a data structure schema, a device that provides a framework in which representations can be specified. The most important point in the notion specification of many kinds of tions. In the top level, together, of schema is the detailed information about representa a schema system links every schema indicating which categories of data structure exist in the system and the relationships among them. level of organization, there are individual In the next schemas, the basic units around which the information about representations is organized. Each schema indicates the structure and inter relationships of a single type of data structure. In the last level, there are slots, which are the attributes (e.g., name, c o l o r , size). Use of the schemas in knowledge acquisition process relies on several ideas: i) Information creating in a new the schema is viewed as a guide to instance of the representation it describes. ii) That guidance is supplied by a) the structure description information, in is the form of a prototype and b) the relations information, as which pointers to a number which is interpreted of structures that may Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 require updating to ensure that necessary data structure interrelations are maintained, iii) The interrelation process drives the knowledge trans fer dialogue. iv) The advice present in the schema adds a level of so phistication to the dialogue. The main knowledge base, purpose in using the schema is to maintain the base systematically. for example, entation, in the knowledge adding a new instance of a known repres will not violate necessary data structures. base, One change Besides the relationships between maintenance of the knowledge Davis and Buchanan[DB77] offer a convenient mechanism for organizing and implementing data structure access and storage function. The main idea for the schemas is to utilize them as points around which knowledge is organized. facts are represented with represented a truth In slots, interval and rules are with the asymmetrical difference value for co implication. In individual schemas, the name of a coimplication ation is contained. of the coimplication It contains the value for each relation rel attribute and the asymmetrical differ ence value for the coimplication. The known designing set of facts we will call the data base. a knowledge database using the schema-like In repre Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 sentation, the a communication scheme knowledge knowledge base has been considered. representation the expert between the database and The schemas in the may be a conceptual description of system for some application area. The term "know ledge schema" is used instead of the word "schema" knowledge representation. Figure 4.1 and Figure 4.2 in show the a knowledge database design scheme through schemas, and repres ents a diagram for GPFES interface among with the communication knowledge schema, database and scheme or knowledge database, respectively. Knowledge the conclusion indicate what schemas in any are used to store information about reasoning process. Knowledge schemas schemas give evidence about the conclusion and what schemas use the conclusion. A coimplication relation may have an a priori determin ed asymmetrical difference representing interval [-0.5,0.5], which is given a certainty by the Coimplication relations are the primary the which inference can be process. They are relations An the domain expert. very flexible used for any application. on used for relations example of the coimplication relation is as follows: (-0 .01) Thunderstorm is strong)------- )There is a lightning. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Figure 4.1 Knowledge database design through schemas schema hierachy I---------non-fuzzy schemas fuzzy schemas slots with attached procedures slots with fuzzy predicates and attached procedures Figure 4.2 Diagram for GPFES Domain Expert Conceptual knowledge schema 4' .1 Knowledge Engineer GPFES , rt I i l Knowledge database^I » -I management system I < I I I Inference engine^. ^database schema I ->•knowledge database — I 'I users FS: fuzzy schema NF: non-fuzzy schema Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 This is read as "If the thunderstorm is strong then there a lightning" as in (-0.01) indicates the that normal this is implication relation. The coimplication relation usually holds. Table 4.2 shows an example of the knowledge schema. Table 4.2 Knowledge schema type coimplication name lightning clause-1 in premise Thunderstorm conclusion there is lightning a priori asymmetrical difference (-0.01) : This is assumed to be a partitioned coimplication on a subinterval which will be discussed premise value [0.85,0.9] For representation of facts the closed interval [0,1] of truth degree is divided into n disjoint intervals; I2=[b2,B2],,,In=[bn,Bn], Il=[bl,Bl], where Ii and Ij are disjoint (i=j), and bl=0,Bl=b2,,,Bn-l=bn,Bn=l. Let {fi} be the membership values in the frequencies interval of items Ii (i=l,n) fuzzy subset F. The basic form of an uncertain cise fact is (predicate,F,{(Ii,fi,ui)},b,B), are the taking fuzzy related to a and/or impre where b and lower bound and B the upper bound of the truth values fi of the fact, respectively, and ui= (xk)/fi, (x k ) ; ui k= 1 F F 'Z.X X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 is an usual value of the interval Xi; see Zadeh[Z186]. Suppose that a universe of discourse U has 100 elements, i.e. xl,. . ,xl00. A fuzzy subset F in U is characterized by a membership function uF which takes the value in the interval [0,1], i.e. X :u -> [0,1]. Because U is discrete, F is represented as F = F ^< xl)/xl Let +. . + Ii X <xl00)/xl00. and Ij be the intervals containing b and B, j respectively, where i < j, and M = 2 ^ * Then the certainty = k=l degree p is approximately defined as p = 1 - P(UIk) +1/100 = 1 4 k 1 - M/100 + 1/100. imprecise truth transform them fi upward and If b=B=l value and fn=l i then p = 1. J Given an r and its certainty degree p, we can into b and B. The basic idea is to count for downward, alternately, from a partitioned interval la containing r as 101 - lOOp. In the real world, implies the conclusion we are not sure that the antecedent in every point of truth degree, i.e. in many cases, given an uncertain truth degree of the antece dent, it is possible that the conclusion has several different truth degrees. plying the For solving asymmetrical this kind of problem, we are ap coimplication to the rules on the partitioned intervals. A rule conclusion, is and composed of two parts, called antecedent and usually is denoted clusion". A relation is suppoed by "Antecedent -> Con to hold between them. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The 90 approach to represent rules is to use the coimplication (d) (Pl,,Pn) |-- 1 P and, instead of finding a coimplication on the closed interval [0,1], coimplication of the goal is to find the pieces of the predicate on partitioned intervals Ii, i=l,n. The interval into [Ii}i=l,n. [0,1] of the As in facts, asymmetrical difference between antecedent will be divided we wish to find every possible Ii and some values in the closed interval [0,1] of the conclusion. Definition 4.18: Let Di s [d:d=ci - a i , a i & I i truth degree of conclusion approximate asymmetrical and from ai}. Then di is called an difference on Ii di = (inf(Di) + sup(Di))/2 if sup(Di) Let SDi s [sd:sd= ci - ai possible truth degree called a = sup(SDi) of ci is every possible , - inf(Di) ^ 0 . 5 . aifeli conclusion and is defined by and ci is from ai}. Then every sdi is symmetrical difference on Ii and is defined by sdi if sdi <^0.5. The structure of a rule in the knowledge base is con structed by (A , C , [{(Ii,di,s d i )}i = s ,t ] ,b,B). In the next sec tion, we will discuss how to infer the conclusion and to find the truth interval of it on the partitioned coimplication. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 4.4 Coimplication and resolution procedure In Section 4.3, we have discussed the knowledge sentation scheme of vagueness through repre the partitioned coim plication. From those structures, we have to decide the truth interval of the conclusion which is a resolvent of rules and facts (or user input) on the concept of coimplication. Before we truth interval, discuss how to resolve a conclusion with the le t ’s the disjunction of the consider the conjunction operation and propositions on the antecedent and splitting scheme of the truth interval into partitioned truth intervals. Corollary 4.1: Let truth Pl,P2,..,Pn be propositions. Let [bl,B 1 ],..,[bn,Bn] be intervals of Pl,..,Pn, respectively. interval [b,B] of the conjunction b=inf(bi) and B=inf(Bi), and the Then the truth P1AP2A. .APn is defined by truth interval [b,B] of the disjunction P1VP2V..VPn is defined by b=sup(bi) and B=sup(Bi). In the resolution process, the truth first of all, we have to find interval of the antecedent and find a subpartition covering it. Algorithm 4.1 shows the splitting procedure of the truth interval of the antecedent la, calculated by Corol lary 4.1, into a subpartition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 Algorithm 4.1: Splitting 1. 2. Find a minimum subpartition {Ii}i=s,t covering the truth t s-1 interval la of the antecedent such that la £ ( J Ii, U Ii f\ n i=s i=l la = and U Iif\Ia. = h. i = t+l Set the left bound of Is to the left bound of la and the right bound of It to the right bound of la i.e. a sub partition {Ii’}i=s»t From {Ii*} i=s,t, difference between the and the property antecedent of asymmetrical and the conclusion we can infer that conclusion has the possible interval Ic=[bc,Bc] of the truth degree of the conclusion as in Algorithm 4.2. Algorithm 4.2: Resolution 1. From {Ii’}i=s,t, find {Ii"}i=s,t by Ii" = I i ’ + di 2. Find the truth degree [be,Be] of the conclusion by be = inf(U Ii") and Bc=sup(U Ii"). i=s,t i=s,t 3. Find the symmetrical difference SD by SD = sup(sdi). i = s ,t In expert systems, variable may be several conclusions concerning a obtained from different rules. The structure for facts and rules defined in the previous section will be applied to compute a final truth degree of the conclusion and combine several different pieces of information. Basically, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 (dl) (d2) (dn) — \ C, A2 t- —( C, . . ,An |—— t C are combined <d) V. .V An |-- 1 C, where d = inf(d i )i = l,n . AI into AI V A2 Algorithm 4.3 : Combination (d) A V B h 1C ) ( Case of 2 rules i.e. 1. Go to Algorithm 4.1 with the disjunction interval [b,B] and find {Ii’}i=s,t. 2. Find di[A]i=s,t and di[B]i=s,t, and if di does not exist for some i, i=s,t then set di to 0. 3. Calculate di[A V B] by min(di[A ] ,d i [ B ] ) for i=s,t. 4. Go to Algorithm 4.2 for the combined truth interval of C. 4.5 Fuzzy inference engine The fuzzy coimplication relation is knowledge representation knowledge formalism. inference chaining consists of a premise relation. which in GPFES. The are used for coimplication up of variable linguistic statements. such as most, or less. clauses connected by any relation and a conclusion like the implication a type "0 is F " , more used to model Coimplication relations The premise is made Each clause has can be a new approach to The a number where 0 and F F can contain conclusion the connective A * of clauses. Each represent a qualifier is a set of coimplication Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 relation is given some a priori asymmetrical difference. strength which we call an It is represented with a numerical value in [-0.5, 0.5]. Classical modus ponens has the follow ing form: IF x->y and x then y We use a method based on the coimplication to determine that a conclusion holds with some true value in the interval [0,1]. The d denote problem may be stated in the following, letting the asymmetrical difference relation, x denote the truth value of the coimplication of the premise X and _ y denote the truth value of the conclusion Y: d s? v(y) <d) or (X) |-- \ Y. - v(x) Given values for x and d, we must find a consistent value for b. That Then we may find conclusion. be calculated easily by y = m i n ( 1 ,max<x+d,0)). the truth degree of Y as a confidence in the In Table 4.3, modus ponens for the fuzzy co implication method is shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 x 0 . 1 .2 .3 .4 .5 .6 .7 .8 .9 1. -.5 -.4 -.3 -.2 -. 1 .0 .1 .2 .3 .4 .5 .0 .0 .0 .0 .0 .0 .1 .2 .3 .4 .5 .0 .0 .0 .0 .0 .1 .2 .3 .4 .5 .6 .0 .0 .0 .0 .1 .2 .3 .4 .5 .6 .7 .0 .0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1. .1 .2 .3 .4 .5 .6 .7 .8 .9 1. 1. .2 .3 .4 .5 .6 .7 .8 .9 1. 1. 1. .3 .4 .5 .6 .7 .8 .9 1. 1. 1. 1. .4 .5 .6 .7 .8 .9 1. 1. 1. 1. 1. .5 .6 .7 .8 .9 1. 1. 1. 1. 1. 1. y = min(1,max(x+d,0)) Table 4.3 Based on this simple method, given an a priori asymmetr ical difference as a certainty for a coimplication relation of -0.1 and a certainty in the premise which lies anywhere in the range [0.8,0.9], we find that the certainty value conclusion is in the range [0.7,0.8]. The of the reasoning process seems to be reasonable. An inference engine Executing rules is also is needed referred to to execute the rules. as firing rules. The inference engine must determine which rules are relevant to a given data memory and choose one to apply. This selection or control strategy is called conflict resolution. The inference engine can be described as a finite-state machine with a cycle consisting of three action states : match-rules, select- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 rules, and fire-rules. In the state of match-rules, the infe rence engine finds all of the rules that are satisfied by the current contents of working memory according to some compari son algorithms. candidates for conflict set. The rules execution. Algorithm 4.4 that are found are all potential They are shows referred to as the the match procedure. the second state, select-rules, the machine determines In which rules will be actually be executed and is showed in Algorithm 4.5. the It then transfers to the state of fire-rules, and fires rules selected. this execution first state Algorithm 4.6 or firing, and the explains machine is ready to start it. cycles over again. Following back to the Figure 4.3 shows the architecture of the GPFES inference engine model. Figure 4.3 The diagram of GPFES inference engine $ data knowledge base match-rules ■^working memory select-rules ^ conclusion!s) fire-rules no conclusions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 The inputs to the inference engine are the user input data and the knowledge base. The output is the conclusions of rules that have been executed. Working memory stores during the problem-solving the current process by state of knowledge holding symbols that 4k represent facts assumption about the and goals. domain and the problem solver’s The elements in working memory may be created, modified, or removed for various reasons. is too old to be of intermediate A variety useful in interest computation of or a fact was If a fact the result of then facts may be removed. attributes of working memory elements are determining which of the rule matches are most relevant and should be selected for firing. Algorithm 4.4: Match Let CL be a list of matched rules i.e. the conflict set. Let m be the number of rules in the knowledge base. Let [blj,Blj] be a truth interval of the element in working memory. Let [b2j,B2j] be a pre-defined a priori truth interval of elements in knowledge base. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 Let [b3jfB3j] be a truth interval produced by the match process. ( Pattern or Literal match ) 1. Set i = 1 and match = 0 2. If rule i is already fired then i = i + 1; if i £ m then go to 2 endif; endif; if i > m then go to Algorithm 4.5 endif; 3. Get every clause of the premise in rule i 4. Compare the compiled clauses to the elements in working memory or data base of facts. 5. If a pattern or literal match is made then match = 1; goto 6; else i - i + 1; if i > m and match = 0 then go to Algorithm 4.7 endif; goto 2 endif Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 ( Truth interval match ) Let n be the number of clauses to be compared. 6. for j = l ,n If [blj,B1j] £ [ b 2 j , B 2 j ] then set [b3j,B3j]=[blj,B1j]; MDj=l; endif; If [bl j ,B1 j ] Otl>2 j ,B2 j ] then set [b3j,B 3 j ]=[b2jfB 2 j ]; MDj=(B2j-b2j)/(Blj-blj); e nd i f ; If [blj,Blj] f\[b2j,B2j] / 0 and not case 1 and not case 2 and blj <b2j then set [b3j,B3j] = [b2j,Blj]; MDj=(Blj-b2j)/(Blj-blj); endif; If [bl j ,B1 j ] (\ [b2 j ,B2 j ] ^ <f) and not case 1 and not case 2 and b2j < blj then set [b3j,B3j] = [blj,B2j]; MDj = (B2j-blj)/(Blj-blj) ; endif; If [bl j ,B1 j ] A [ b 2 j ,B2 j ] = f then MDj=0; endif; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 7. Find MD of the total match degree of the premise by MDs V MDt = max(MDs,MDt) and MDs /\ MDt = min(MDs,MDt). 8. Find the truth interval of the premise i.e. [b3,B3] using Corollary 4.1. 9. Put the rule i into CL with the match degree MD and [b3,B3]. 10. i=i+l; goto 2. Algorithm 4.5: Selection Let SL be the selected conflict set. 1. If CL is empty then ask the user to start again. 2. Find the rules with the match degree > 0.5 and put the rules into SL. 3. If SL is empty then find the rule with the highest match degree and put it into SL. 4. If MD in SL is equal to 0 then ask the user to start again; else goto Algorithm 4.6; endif Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 Algorithm 4.6: Firing Let FL be a list of the fired rules. For each rule in SL, 1. Get the conclusion of the rule. 2. Find the truth interval of the conclusion and the trical difference based on symme the coimplication relation by the method descibed in Algorithm 4.2. 3. Place the fired conclusion with the rule number on working memory. 4. Put the fired rule into FL. 5. If the rule is a conclusion then put the fired rule into a conclusion list. Algorithm 4.7: Combination 1. Find the conclusions. 2. If some conclusions are same then find a new truth inter val using Algorithm 4.3. The match process key words, which means the same semantically. match degree. working memory In order matches has two stages. Steps 1-5 match the identical keywords or that they are Steps 6-10 do some evaluation for the to the see how premise well the elements in with a priori defined truth interval they evaluate the match degree. The reasoning Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 process continues until the system has come to a conclusion which provides the information for which the user of the expert system is searching, or there are no more coimplication relat ions to determine an answer for the user. GPFES will be imple mented in a forward chaining mode, but it will not be difficult to implement in a backward chaing mode or in both modes later. In the forward chaining or data-driven style, the sys tem begins by choosing an initial coimplication relation for the inference. Given a certainty value of the premises, the conclusion will have a certainty degree through the new modus ponens method. If the conclusion is satisfactorily, determined to be certain it is used to direct the choice of the next coimplication relation to be done. This conclusion will be in the premise evaluated as of the the continues until next best the coimplication one final for relation processing. conclusion is This which is process reached with the certainty value above a threshold or the intermediate conclu sion is determined to have a certainty value below a prede fined threshold. There can be several paths to reach the same conclusion Suppose that and then we have to choose the best one of those. there are n paths {Pn} to reach a conclusion. Then, for the coimplication relations of each path, they have {sdij} 1<_ j<_ li, 11 i<_ n i.e. the symmetrical difference as a certainty degree, where li is the path length of path i. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For 103 determining the best one of many possible paths, we will take ii li one with a smallest value of ( ^ tsdi.il } 1^ i<_ n w h e r e ^ L [sdijj j= l — ~ j= l is assumed to be less than or equal to 0.5. If we have a path with the match degrees MDij, j=l,li for some i then we would say that the conclusion have the match degree MDil X MDi2 X . .. X M D i l i . Figure 4.4 is a detailed diagram of the fuzzy inference engine.to show relationships among algorithms in this section. The forward chaining scheduler in Figure 4.4 performs a com munication interface between data base of facts and knowledge base. Although GPFES is not implemented in the backward chain ing mode, let us consider it. first coimplication tains a goal or In backward chaining, relation chosen will be one final conclusion. relation whose conclusion causes That which the con is a coimplication the system to halt when it has a satisfactory certainty value. If t*he system finds a co implication relation, determine then the premise the uncertainty value of the clauses must be evaluated to the conclusion. in the premise will have some certainty Some of values supplied by the user, while other clauses will be the conclu sions of some other coimplication relations. several coimplication relations taining the clauses which we to want. There might be reach a conclusion con This backward chaining Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 F ig u re 4 .4 A d e t a i l e d flo w diagram o f fu zzy i n f e r e n c e e n g in e i n GPFES START s c h e d u lin g queue; i fo rw ard c h a in in g { f a c t u a l queries.. d a ta b ase ( fa c ts ) J d e c id e a n e x t f a c t u a l query • t o be answ ered from u s e r i I i a sk u s e r to answ er w ith J a c e rta in ty (tru th in te rv a l) , J i i p u t a com piled answ er o f f a c t u a l J query from a u s e r i n t o th e w orking LnSSPSDL w ith_a J :r u tJ i_ in te r v a l r knowledge base ( c o im p lic a tio n based r u l e s ) p a t t e r n m atch working memory match (A lg o rith m 4 .4 ) MATCH tr u th in te rv a l m atch s th e re m atch s th e re i n a l con [sto p or ■continue ■ "error" due to ^ Jp o o r knowledge ■base o r m issin g [in fo rm a tio n s ■from u s e r ■ ■ ■ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Jput m atched r u l e s w ith m atch J 'd e g re e s i n t o a c o n f l i c t s e t i i i (c o n f l i c t set selection(Algorithm 4.5) J. J s e l e c t re a s o n a b le r u l e s w ith match d eg ree g r e a t e r th a n or 'e q u a l t o 0 .5 and p u t them i n t o th e s e le c te d c o n f l i c t s e t [ selected conflict set j f ir in g ( A lg o r ith m 4 .6 ) J. J p u t c o n c lu s io n s o f th e f i r e d r u le s i i n to a c o n c lu s io n l i s t T c o n c lu s io n l i s t I combination(Algorithm 4.7) I ip u t c o n c lu s io n s o f th e combined r u l e s Jw ith t r u t h i n t e r v a l and sy m m e tric a l 'd i f f e r e n c e i n t o w orking memory in te r m e d ia te o n c lu s io n working • memory J jgo to forw ard c h a in in g s c h e d u le r to | a sk a n e x t f a c t u a l query « Igo to th e m atch s ta g e ” f o r " a "conJfirma"tiqn 'a s f i n a l c o n c lu s io n s MATCH START Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 process will continue until a coimplication relation, which has every clause supplied by the user or the facts, is found. Then certainty sions and they are propagated paths to give us which is values are shown in along the a certainty value our goal. the intermediate conclu forward for the final conclusion As in the forward chaining mode, choose a best path of many possible paths to goal. The essential building structures reach of reasoning scheme for GPFES have been described. mentations are simple and flexible. chaining we can the same an approximate Their imple Those concepts will give a general guide to manage uncertainty in every domain in expert system fields. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 107 Chapter 5 Application of GPFES to microwave precipitation retrieval technique 5.1 Introduction Precipitation is a vital exchange process within the hydrological cycle, and represents the net result of heating from condensation in the atmosphere. The importance of prec ipitation monitoring is growing within the current trends in environmental research. Rainfall studies are growing in sig nificance because of the steep upward curve in the demand of human society for water supplies as standards and expectations have risen populations, in living many parts of the world [BM81]. Precipitation varies duration, respect to its frequency, intensity and spatial pattern, not to mention its propensity to sleet or snow. zero, with fall in various forms such as rain, Rainrate intensities range from to above 100 mm/h. practically Intensity, to some extent, inverse function of duration. Patterns hail, is an of rainfall organi zation and distribution prompted by the overriding influence of the general circulation of the atmosphere are strongly Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 influenced by global geography. The sizes, surface configu ration, and dispositions of the land masses influence their own precipitation climatologies. The factors the motion properties, governing the occurrence of rainfall are of cloud parcels and their condensation for these determine the concentration, size distribution, and nature of cloud particles. cloud has formed, condensation and aggregation of evolve a raindrop spectrum. is left uninterrupted, initial Once a droplets precipitation process these microphysical processes steadily increase the average radius Whether particles grow If the nuclei to determined by air motion, and the range precipitation of drop size. size is largely through its control of macroscale properties such as cloud-dimensions, water content, and life time. These latter conditions govern the roles of microscale processes and and therefore the length of time over which they operate, the maximum size which a cloud drop can attain. In view of the wide range of uses of rainfall data, mon itoring rainfall with the desired often difficult. We first detail and summarize accuracy is problems arising from measurement of rainfall by conventional means. "Conventional" refers to those means which rely upon well-established types of instruments: not only the traditional in situ devices Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 (i.e. the raingauge), but also ground-based remote sensing systems (principally weather radar). Measurement of rainfall by gauges interrelated factors of topography, design. site, aspect, cation in relation to hills and ridges. raingauges tropics are not located likely on to give rainfall over the ocean in their scale, wind, and gauge The gauge catch may be representative of a small or large area depending on slope, example, is affected by the elevation, and lo Taking extreme volcanic an islands in the readings representative of vicinity. nature of the local surface and the presence of nearby objects and struc tures. the gauge catch is influenced by the On a very small Wind is the single factor contributing most signifi cantly to errors in gauge measurements. is well sited and exposed, Even if a raingauge owing to deflection of raindrops in the disturbed airstream around the orifice of the gauge, monthly average winds as light as 5 m/s may result in errors of the underestimation monthly as large as rainfall Raingauges designs are comes 10 % in regimes where half at rates less than 100 mm/h. intended to reduce wind effects. those problems must be added a plethora organizational difficulties, To of practical and/or some of which are more signi ficant in certain types of regions than others: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 (i) accessibility of desired raingauge locations, (ii) The availability of suitable personnel for reading and servicing the gauges, (iii) a suitable power supply for some types of continuous recording raingauges, (iv) preserving the rainfall catch in accumulating rain- gauges especially when read infrequently, (v) security of the rainfall station from vandalism and other da m a g e , (vi) access to a suitable data can be sent communication link so that the quickly to a central facility for processing and archiving, and (vii) transcription and transmission errors, which tend to introduce positive errors in reported rainfall. In case of weather radar for rainfall monitoring there is a comparable range of problems which circumscribe the use of such a system. It is sufficient for present purposes to say that there are difficult problems as yet not completely solved relating to the proper relationship of microwave energy the to drop size spectrum, backscattered partial filling of radar beam, attenuation of the radar beam by intervening drops, absorption and reflection propagation), and practice calibration the by the ground (anomalous signal calibration [B173,HAP79,Mb71]. In of radars used for measurement of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill precipitation includes an adjustment to match gauge measure ments of rainfall within the radar scan. Radar ding in has the singular advantage over gauges of provi a spatialy continuous view. support nations and of raingauge of North America, the requisite support It is employed extensively networks in some more advanced Europe and Asia. However its cost sophisticated technical and have limited the operational use engineering of radar for rain fall monitoring in many parts of the world. Despite the enor mous local value of radars, on rainfall 5.2 most of the w o r l d ’s information continues to come from raingauges. The use of satellite remote sensing As born a scientific approach remote sensing was effectively in the photography. mid-nineteenth As a tool in century with the invention of rainfall monitoring it began with the development of meteorological radar after The term "remote sensing" was popularized after the birth of space exploration by satellites which prompted the new dis cipline to take off in the late rapidly expanding World War II. 1960's as one of the fields of scientific technological most endea vours . There namely are two types of Earth observation satellites, "Earth resource" and "environmental and weather" sat Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 ellites. In general terms, the differences between the two types arise from design capabilities for the channel frequen cies, and the ground spatial resolutions optics. For precipitation studies, have relatively an infrequent repeat cycle. for rainfall resource of equipped environmental is so highly satellites employ On the other hand, although no dedicated hydrological satellite has number the channel Earth resource satellites little to offer, variable through time and Earth of and yet to be launched, weather satellites have a been with sensors which yield data of value for rainfall monitoring. Environmental occupied two types and weather satellites have of orbits and may be grouped commonly accordingly into: (a) polar-orbiting sun-synchronous satellites, and (b) geostationary satellites. The polar orbiting satellites occupy level orbits (usually between 500-1500 km of Earth), orbit Poles. relatively above crossing the equator at high angles takes such a satellite close to the Earth rotates on its polar axis within the surface so that each North Sun-synchronous orbits are organized so and South that as the the orbital ellipses, each new orbit results in the presentation of a new the global target to the satellite in low- such a way strip of that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 113 relationship between Earth surface and sun angle is kept rela tively constant (i.e. satellite passage is invariant in local time). Characteristics orbital periods are 100 min, orbits being required for each satellite to entire globe under daylight conditions course, since such satellites every investigate 24 spend one-half of over the nighttime side of the globe, 14-15 hours. the Of each orbit each unit area on the surface is thus viewed twice every 24 hours, once in daylight and once at night. Geostationary satellites are placed into orbit at approx imately 35,400 km above the earth surface, in the plane of the equator, and advance in the same direction as the rotation of the Earth. This type of orbit is geosynchronous, satellite keeps polar axis, pace with i.e. the the rotation of the earth on its and geostationary in that it appears to be fixed or stationary above a given Such a satellite point on the E a r t h ’s surface. is able to record the same geographic field of view very frequently throughout the diurnal cycle, commonly at intervals of 30 min. The earlist geostationary satellites were the experimen tal Applications Technology Satellite(A T S ) of the early 1970s. In 1979, five geostationary satellites provided circumglobal coverage between latitudes of the equator during a complete of 80 degree N and S the Global Weather Experiment(G W E ). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 These included three American Geostationary Operational E n vironmental Satellites(GOES-E and W and GOES-1), one covering the Americas and the western Atlantic, Americas and ocean. eastern Pacific, and a second the western a third over the Indian The European Space Agency (ESA) satellite, Meteosat, was located over Europe-Africa; The Japanese Satellite(GMS-l) was located over the western Pacific. Since the GWE, the Indian Space Agency has launched three INSAT satellites which are three axis stabilized geostationary satellites. Satellites have contributed science of enormously to the related meteorology and climatology through the last two decades. The attributes of satellite systems which have made application to atmospheric science problems possible are as follows: (a) Satellite coverage of data, systems can provide a complete global thereby greatly extending our appreciation of the atmosphere environment particularly in data-remote regions; (b) Satellite imaging systems yield spatially continu ous data, contrasting strongly with those obtained from the irregular networks (c) Satellites of surface weather systems; can selected meteorological investigate parameters the distributions of much more than in situ observing networks which deploy consistently large numbers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 of instrument packages; (d) Geostationary satellites can give a much higher temporal frequency of information than commonly obtained from surface and upper air weather stations; (e) Satellites provide a unique view of the atmosphere, observing it from above, rather than from within; (f) Satellite data can be obtained for broad areas in near real time. Satellite data are studies because they and time. problems of great complement utility in atmospheric conventional data in space On the other hand, satellite data have a number of and associated difficulties. (a) there are These include: problems converting satellite parameters into calibrated units; (b) Elaborate convert parameters transformation procedures are required to satellite-observed which can be radiances integrated into with meteorological data provided by conventional networks; (c) The user of from the data satellite source; processing operation data important is generally procedures of removed the data may remain obscure to him. Further information on the role of satellites as remote sensing systems for studies of the Earth’s atmosphere can be found in [AV73,Be74,S S A70]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 Satellite on data provided sensors: meters retrieval by of rainfall has a small number depended largely of different types of cross-track scanning radiometers, spin-scan radio and vertical profile radiometers. Cross-track ning radiometers are flown on polar-orbiting satellites. These monitor radiation a revolving mirror satellite track. which scans revolution adjacent to the one before. the operational target via the target across the sub As the satellite each new from scan advances along its orbit of the mirror yields a new scan line The radiation collected by the mirror is passed through a beam splitter and spectral filters to give the desired wavelength separation. The wavelengths usually used in rainfall studies are visible (VIS: radiation with the wavelength infrared radiation, with the wavelength 0.5-0.7 /am), infrared (IR: thermal escaping through the atmospheric window of 10.5-12. 5 ;um) , and microwave naturally emitted radiation at radar wavelengths from 1.55 cm). Microwave (MW: 0.33- instruments employ antennas or antenna reflectors to achieve their scan pattern. Examples of crosstrack scanning radiometers are the four channel Advanced Very High Resolution Radiometer (AVHRR) on TIROS-N type NOAA satellites and the Scanning Multispectral meter (SMMR) on Nimbus-7, Microwave observing microwave Radio radiation five different wavelengths. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. at 117 Spin-scan radiometers are flown on geostationary satel lites using spin stabilization of approximately 100 RPM. coming radiation is received In by a fixed mirror angled at 45 s degree to the optical axis of the satellite system, which is aligned parallel to the axis about which the satellite itself spins. The spinning motion of each satellite provides a westto-east scan motion across the target when the spin parallel to to West). stepping each of axis is the polar axis of the Earth (METEOSAT spins East Latitudinal coverage is the scanning mirror achieved north by sequentially to south at the end of rotation(METEOSAT steps South to North). Thus an image the entire disc of the earth is built up over a period of about 20 minutes. The INSAT satellite is three axis stabilized and scans in a back and forth fashion. 5.3 Review of Passive Microwave methods to retrieve rainfall There are several satellite rainfall monitoring methods; cloud-indexing methods [Be70], GWB76,S077a, S 0 77b], bi-spectral life-history methods [SMS79, and cloud [Lm67,DV73,Ga73,Wd79], rainfall monitoring model methods from visible and infrared images[MS72,S077a,SMS79,WS71], and passive microwave methods[Sd69]. In this section, we discuss passive microwave to retrieve rainfall from remotely sensed data. methods COSPAR, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 118 Committee on Space Research, assessed possibilities for mea suring rain from satellites, and in 1967 endorsed observa tions of passive microwave radiation as potentially the most viable approach. first In 1968 the meteorological ment [SH73]. USSR launched Cosmos 243, the satellite to carry a microwave instru The radiometer on Cosmos 243 Although the operating lifetime had four channels. of each instrument was less than 2 weeks, the observations they transmitted were suffici ent to construct latitudinal profiles of liquid water content in the atmosphere[GD70,S m 7 2 ,SH73]. Nimbus-5 microwave (NEMS) instruments and (ESMR-5). in 1972 an carried : a Nimbus-E Electrically NEMS was of the first American orbiting Microwave Scanning Spectrometer Microwave Radiometer little use in measuring rainfall b e cause of its 180km resolution and limited coverage, the ESMR acheived some success in oceanic rainfall however applicat ions. ESMR-5 measured passive microwave radiation at 1.55 cm. The Nimbus-6, next satellite in this series, carried a conical scanning version of the ESMR at a shorter wavelength i.e. 0.81 cm. The Scanning Multichannel Microwave Radiometer (SMMR) on Nimbus-7 and Seasat, both launched in 1978, employed a new antenna design and provided brightness temperature information at 5 microwave frequencies. Resolution was improved to 20 km at the smallest wavelength. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 The first objective attempt to microwave radiances to rain intensity relate simple-channel was made by Wilheit [WCR77], who developed a technique for quantitatively mapping precipitation over the oceans by means data at 1.55 cm. This study of satellite radiance has guided much subsequent re search on the methodology of associating microwave brightness temperatures to precipitation rates. Kidder and Vonder Harr [KV77] used data from the Nimbus 5 ESMR-5 in- conjunction with Wilheit's freezing level approach to calculate seasonal preci pitation frequencies for the tropical oceans. Rao et al.[RAT76] used all the available ESMR-5 data for 1973-1974 to the oceans. assemble a global atlas of rainfall rates over Their maps are qualitatively correct in many areas, however, there are many instances in which the computed precipitation amounts are quite different from actual tological values. While discussing clima- the possible sources of error for the rainfall atlas, Rao[Rm84] pointed out the over simplified treatment of [WCR77] model and on freezing level. to the rain-cloud in the Wilheit et al. the unrealistic dependence of the results The latter problem led Rao et al.[RAT76] apply 'ad hoc* corrections to the climatological freezing levels in order to produce realistic results for cold pheres. atmos Rodgers and Adler [RA81] later utilized measurements from ESMR-5 to estimate latent heat release within tropical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 storm systems, however, to avoid the uncertainties associated with the Wilheit et al.[WCR77] freezing level approach, utilized an empirical relationship based on they radar derived rainrates. Wilheit et al.[WCK82] have compared simultaneous obser vations of a precipitating storm from five microwave radiome ters aboard the NASA CV-990 aircraft. Rodgers and Siddallin- gaiah[RS83] have shown how the 10 and 18 GHz channels on SMMR can be used to improve the rain.versus wet ground discrimina tion at 35 GHz in regards to the earlier Rodgers et al[RCW79] study. Spencer and Santek[SS85] have examined the use of two frequencies (18 and 35 GHz) to better events over a land background. The tendency studies is to use extra channels select precipitation in these latter to help select rain events but to quantify the rain-rate with an individual channel. Mugnai and Smith[MS88], and Smith and Mugnai[SM88] recently extended the Wilheit et al.[WCR77] include a time-dependent, have investigation vertically-inhomogeneous to cloud microphysical component in conjunction with the variable rain layer. To dependent, do so they have obtained the results from a timetwo-dimensional numerical cumulus model developed by Hall[Hw80], which provides detailed water drop spectra the horizontal and vertical directions, hydrometeor-size domain. in as well as in the The model has been integrated up to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 the time at which precipitation has just begun to fall through the cloud base. The radiative calculations are based on a ver tically and angularly detailed plane-parallel radiative-transfer model for unpolarized microwave radiation. A sequence of seven microwave frequencies, which undergo both absorption and scattering in a precipitation medium, have been considered. They have thus explored the feasibility of using multichannel passive microwave techniques for estimating rainfall. 5.4 Estimation of rainfall rate from multiple microwave fre quencies using GPFES Precipitation retrieval by conditioned problem remote sensing .is an ill from a mathematical perspective because of the non-unique relationships which exist between the radia tion signals and precipitation intensity. Mugnai and Smith [MS88], and Smith and Mugnai[SM88] have developed a theoreti cal foundation for avoiding some of application of a multispectral a purely analytical these gaps with a GPFES, Figure 5.1 shows problems by the approach using passive micro wave measurements. There remains, within these however, operation. a number of gaps We propose to close i.e. an expert system approach. theoretical relationships between brightness temperatures and rainfall rates, developed by Smith and Mugnai[SM88] over ocean. A selection of eight rain layer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 thicknesses[Flwc=0.0, 0.25, 0.50, 0.75, 0.85, 0.90, 0.95, 1.0] are inserted into a model cloud environment at three different evolution times during the cloud lifetime [Tc=1000, 1900 seconds]. The heights of the rain layers are defined in conjunction with an integral property water; more details for 1500, and of the cloud liquid see Smith and Mugnai[SM88]. rainrate ranges marked with black arrows are The deemed to be the most probable rainrate intervals associated with a given time step. The three time steps are generally related to light, medium and heavy rainfall steps. (BT) shown at each frequency (i.e. (V+H)/2, temperature and The brightness temperatures correspond to unpolarized B T ’s where V is a vertical polarization brightness H is a horizontal polarization brightness temperature). When considering passive remote sensing algorithms, cloud development time and rain layer thickness are generally not known "a priori". That is, through brightness tempera tures we have to infer a rainrate, cloud development time and rain layer thickness. fall by remote Thus, the problem of estimating sensing is a far more complex process suggested by studies which are based on rain than single parameter, invariant cloud microphysics. At first glance, approaches are highly Figure 5.1 suggests ambiguous since that one channel the functional rela- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 tc»IQOOsec , tca 1900sec A .v = 19.35 GHz i i H>0 1 ^ 1 1 1 i 1 1 A .-i.i*” • Lv = 22.235 GHz . 290' ! 260 w 3 I 1 1 kS— iJ 1 1 1 230 <____ «» Q. 200 E at 290 t ,V = 37.0 GHz , et ! ^ in in 260 c JC g» u 03 i i 230 . 200' 290 ' r i !^ = 89.5 GHz , « '260 t _ _ 1 230 i 1 1 « ------------> 1 1 1 200 1 , „ 10 <— >1 i - 100 1 10 100 10 100 Rainfall rate (mm/hour) Figure 5.1 Relationship between brightness temperature and ra in fa ll ra te over water Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 tionships between brightness generally not temperatures single valued. approach is incorporated, However, and rainrates are when a multi-channel the ambiguities can be reduced. When considering the detection and measurement of preci pitation over water backgrounds, it microwave signatures induced by contrasted to the background. microwave frequencies, is essential that the the precipitating clouds are In general, at standard passive rain layers appear "hot" against the low emission "cold" ocean. When considering the detection and measurement of preci pitation over high contrast arguments detected, emissivity apply. depending on land backgrounds, In general, frequency, the same precipitation can be if the background is obscured to the extent that rain clouds appear "cool" against a "warm" background. This land effect was first pointed out by Savage and Weinman [SW75] at the 37 GHz frequency. In order to unambiguously microwave measurements backgrounds, which it is important surface variation, of emission and interpret frequency dependent precipitating to clouds understand reflection, contribute to the total the column surface emissivity variation, eliminate i.e. degree to and its potential brightness peratures. Multi-channel differential techniques for this problem because they over land the are tem useful dependence on they isolate the relative Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 effect of the precipitating cloud regardless of the absolute backgroud. Spencer and Santek[SS85] and Spencer et al.[SHS87] have discussed the use of 18-37 from SMMR from effectively measurements discrimination in warmer threshold GHz temperature differences screening intense precipitation land backgrounds. of 20 degrees They utilized a which was obtained empirically, and which they maintained was somewhat arbitrary and could only be used to isolate intense precipitation. 5.5 Data set for a case study with SSM/I measurements In order to test our knowledge base on passive microwave measurements, we have used a test case from a new SSM/I data set aided measurements by satellite (INSAT). from from the Indian geosynchronous INSAT satellite measurements are obtained the 22 km VHRR special data set prepared under the aus pices of the Indo-U.S. Science Smith et al [SOS88]. and Technology On June 19, 1987, the program, see Special Sensor Microwave/Imager(S S M / I ) was launched aboard the Block 5D-2 F8 spacecraft, a polar orbiting satellite from the Air Force Defense Meteorological Satellite Program!DMSP). This satellite radiometer detects 19.35, 22.235, ation on microwave radiation 37.0 and 85.5 GHz. a variety of at four frequencies These data provide inform environmental parameters, atmospheric water, wind speed, and sea ice. including The data used in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 this investigation have been produced by Remote Sensing Systems [Wf88]. The data set is a compacted, chronologicallyordered version of the Temperature Data Records (TDR’s) being produced by the Fleet Numerical Oceanographic Center. The Compacted SSM/I antenna temperature(TA) data are available on a series of 6250 bpi magnetic tapes (8 tapes per month of d a t a ). The DMSP orbit is circular, polar, sun-synchronous, and near- with an altitude of 860 km and an inclination of 98.8 degrees. The orbital period is 102 minutes; for the ascending equatorial crossing is A.M.. SSM/I actually radiometers, each incorporates simultaneously 7 the local time approximately separate measuring 6:00 total-power the microwave emission coming from the Earth and the intervening atmosphere. Table 5.1 gives the frequencies, and spatial polarization, resplution of the 7 channels. measurements are taken vertical polarization is at 19.35, measured 37.0 at and temporal Dual-polarization and the 85.5 GHz; only 22.235 GHz water vapor channel. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 Table 5.1 Temporal and spatial resolution of SSM/I channels Frequency (GHz) Polarization Integration Period 3 dB Footprint Size Along--track Cross--trs 19.35 vertical 7.95 ms 69 km 43 km 19.35 horizontal 7.95 ms 69 km 43 km 22.235 vertical 7.95 ms 50 km 40 km 37.0 vertical 7.95 ms 37 km 29 km 37.0 horizontal 7.95 ms 37 km 29 km 85.5 vertical 3.89 ms 15 km 13 km 85.5 horizontal 3.89 ms 15 km 13 km To facilitate multi-spectral processing of the 4 individ ual frequencies, the 22, 37, and 85 GHz channels are voluted to match the 19 GHz diagram given footprint size. The decon- schematic in Figure 5.2 describes the raw antenna foot print geometry[GH78] for the 4 indivisual SSM/I frequencies. The deconvolution scheme is based on the use of discrete, nonrecursive filters whose weights correspond to the percentage overlaps of the higher resolution/higher frequency footprints with respect to the lowest resolution 19 GHz footprint. The following four equations are used to produce spatially consis tent B T ’s for each (i,j) grid point in the 19 GHz domain, where i is a down-track index and j is a cross-track index: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BT(19)ltJ - BT(19)1(j f . u u u BT(22)ij a [6«BT(22)ltJ + BT(22)1_1 j + BT(22)1+1j ] / 8 F _ U U BT<37)i.j » [4*BT(37)1(J + B T O T J j . i j U ♦ BT(37)I+1>j U U + BT(37)ltJ_, + BT{37)1(J+1]/8 BT<85h , j - 1|BT{85)i j + BT(85)i_ltJ ♦ BT(85)i+1j U U + BTfBSJij.j + BT(85)IiJ+1 U U + BT(85)|_2,j ♦ BT(85)I+2(j 0 U + BT(85)!-x,j-j ♦ BT(85)i+1,j-i + BT (8S >Y+1,J—1 ♦ BT(85)i+ 1 J + 1 } U U + 0.5 |BT(85)1_2(j _1 ♦ BT(85)I_2,j+1 U U + BT(85)|+2> j_j + BT(85)i+2,j+i) U U + 0.25 (BT(85)|tj_2 + BT(85)i,j+2}]/ll.5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 FREQUENCY >85 >19 >22 >37 GHz GHz GHz GHz = ->SCAN B -->SCAN A = ->SCAN B > SCAN A Figure 5.2 Footprint geometry for the four SEM/I frequencies Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 The antenna portion of the SSM/I consists of a parabolic reflector of dimensions 61 by 66 cm, which focusses the micro wave radiation into a corrugated, broad-band, feedhorn. 7-port antenna The radiometer and antenna spin as a unit about an axis parallel to nadir. The rotation period is 1.9 s. A cold calibration reflector and a hot reference load are attached to the spin axis and do not rotate. The rotating antenna feedhorn observes the fixed cold reflector and hot load once each scan. In this way scan. Earth observations are segment of the calibration rotation in the aft direction, observations are taken every taken cycle as is shown during when a 102.4 degree the SSM/I is looking in Figure 5.3. The 102.4 degree arc is centered on the spacecraft subtrack and corres ponds to a 1394 km wide swath on the Ear t h ’s surface. During each scan, the 85 GHz channels are sampled 128 times over the 102.4 degree arc. The integration period for a single sample is 3.89 ms. This sampling scheme results in 128 V - p o l (vertical ly polarized) footprints and 128 H-pol(horizontally polarized) footprints having an effective 3-dB resolution of about 15 km. Observations at the lower three frequencies are only taken every other scan. Scans during which the lower channels are sampled are called ’A-scans'; called ’B-scans’, i.e. 85 GHz only. arc of an A-scan, 64 samples the During of each of other scans are the 102.4 degree the lower channels Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 6.58 KM /SEC V E LO C IT Y GROUND TRACK 1 0 2 .° A CTIVE SCAN ANGLE 19 A N D J 1394 KM SWATH W ID TH SL1^ 85?”*'- SCAN A SCANS 1 2 5 hm > - 1 .9 0 SECT SCAN A SC A N S Figure 5.3 S3-1/I orbit and scan geometry ( f rom [H LS87 ] ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 are taken, with the integration period being 7.95 ms. 5.2 also shows Figure the spatial resolutions of the samples taken on the ’A-scans’ and 'B-scans' [HLS87]. Figure 5.4 shows an example of successive SSM/I orbits and their scan routes. The predecessor to SSM/I was the Scanning Multichannel Microwave Radiometer(S M M R ); different versions of this radio meter were flown aboard Seasat and Nimbus-7, both launched in 1978. The SSM/I sensor design has several advantages compared to the SMMR design. First the parabolic reflector and feedhorn spin as a unit. For SMMR, the feedhorn was stationary, while the parabolic reflector scanned back and forth. As the reflec tor scanned above the fixed feedhorn, the orientation of the reflected Ear t h ’s vertical and horizontal polarization vectors were rotated relative to Thus the mixture of V-pol varied during the scan. further complicated the and feedhorn polarization vectors. H-pol radiation continuously The polarization rotation effect was by spacecraft attitude variations. DMSP spacecraft attitude is also much more The stable than that for SeaSat or Nimbus-7. The second ibration of SSM/I. ports that go to advantage is the simplified, The SSM/I 7 separate external cal has a single feedhorn radiometers. required and all channels observe the same sources. In the SMMR design, No two ferrite switches with 7 switching is calibration were used to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 figure 5.4 An e x a m p l e of SSM /I orb it and scan g e o m e t r y wmmm W:iM iiiss M •• .y .: m k \ xrnmm i'!h .»«v\'v ' „■ . i » y “ -39 L 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 switch back and forth between the cold horn and the earth- viewing horn introducing an extra transmission gate. The spatial than that for the and temporal Seasat and coverage of SSM/I is better Nimbus-7 S M M R ’s. The 1394 km wide swath of SSM/I is about twice that of the SMMR. Further more, in contrast to operations schedule, ously. the Nimbus-7 S M M R ’s the SSM/I instrument The periodic turning on and off calibration problems; when the SMMR every other day operates continu of the SMMR warms up after causes turn-on, calibration drift occurs. The SSM/I antenna temperature data reside on a series of 6250 b p i , 2400-ft magnetic tapes. Each data file corresponds to a single SSM/I orbit. The beginning of an orbit is defined as the ascending equator crossing of the spacecraft (i.e., south-to-north crossing). The data files (i.e., orbits) chronologically The ordered. orbit period is are 102 minutes, and thus there are 14.1 orbits per day. A tape contains three or four days of antenna temperatures. For computation of calibrated brightness temperatures, we have used the algorithm outlined in [Wf88]. For the case study used in this inves tigation of our research, we have choosen a 5 day period from July 16 to July 20 during 1987. INSAT satellite data indicate a major monsoon surge during this period. The area under study is bounded by (40 S,40 N) and (35 E,115 E). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 Figure 5.5 shows a composite period under study; INSAT-IR image for the the composite involves data from July 16, 18, and 20 and corresponds to SSM/I orbit swaths necessary to generate full coverage. One SSM/I orbit pass is missing on July 20; therefore there is a missing data sector in the com posite imagery. Figure 5.6 shows composite images of SSM/I during the same period for 3 different channels of vertically polarized brightness temperature. brightness temperature data at The vertically 22 GHz have because they show virtually the same pattern as images and have been Graphics prepared Enhancement been polarized excluded 19 GHz. The on the Micro-based Image Display Tool (MIDGET) [SSA87,SOS88]; Appendix A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. see 136 Figure Figure 5.5 5.6.a INSAT-IR SSM/I-19 GHz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 Figure 5.6.b SSM/1-37 GHz Figure 5.6.C SSM/l-85 GHz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 5.6 Knowledge base to retrieve precipitation Numerical studies have shown that precipitation retrieval is an ill-conditioned problem from a mathematical perspective because of problems with non-unique relationships between the radiation signals and precipitation intensity. reduce some of these problems, base In order to we have developed a knowledge for precipitation retrieval incorporating multi-channel passive microwave radiometer measurements from the Air Force SSM/I instrument for application within the Southwest-East Asian Monsoon domain. To aid the selection and classification of appropriate targets, we also incorporate INSAT imagery. The knowledge system. The base first select cloud targets lexes, is developed tier is as a three tier logic "rule-based" and is designed to (isolated clouds, meso-scale cloud comp or deep organized convective depressions) as possible candidates for precipitation and teristics. The VIS-IR channel to describe general charac rule base structure information from utilizes high frequency INSAT along with the SSM/I brightness temperatures to locate potential targets, cribe their morphological structures, ocean, to characterize the residual to to des discriminate land- polarization, and to identify large ice particles(graupel). This rule base is also used to assign most probable rainfall categories for light and heavy rainfall(for both water and land background), which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 are assigned as the final solution if the tier 2 cannot provide a unique rainrate solution. In and 3 logic the jargon of fuzzy logic these would be referred to as possibility indices; they are useful for situations in which paradoxical conditions in a set of microwave brightness temperatures prevent an exact solution. All facts and rules are based on the behavior of the SSM/I passive microwave measurements [19 (HV), 22 (V), 37 (HV) and 85 (HV) GHz] satellite. along with VIS/IR images In the tier 1 rule definitions, H is ly polarized microwave channel; microwave channel; from the INSAT a horizontal V is a vertically polarized U = (H + V)/2 is an unpolarized estimate; D = (V - H) is a degree of polarization; F means a fuzziness to represent the proximity of values which d o e s n ’t have fixed boundaries or thresholds. In the first tier of the knowledge base, ions are made concerning the classification of several decis precipitating clouds. The following tests are carried out on the MIDGET(see Appendix A for detailed procedures): 1. The first decision is to determine whether a cloud is pre sent based on a visible(VIS) channel image and/or an infrared (IR) channel image. 2. The second decision concerns: 2.1 the cloud radiometric properties; i.e., whether Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 the cloud temperature is (cold, medium, warm) and whether the cloud albedo is (high, medium, 2.2 the cloud morphological properties; or appearence of cloud, specifically low) i.e., the shape whether it is cumuliform (convective) or stratiform(layered). In this discrimination test, we use the basic signatures of cumuliform and stratiform clouds: cumuliform: unstable, large vertical versus horizontal development, short evolutionary growth (minutes) and cotton-like appearence, stratiform: stable(F), small vertical versus horizontal development, long evolutionary growth(hou r s ), homogene ous over large area. 3. The land third decision or water. concerns whether the cloud is over There are four cases: cumuliform over land, cumuliform over water, stratiform over land and stratiform over water. 4. The fourth decision concerns the residual polarization due to the land background; IF the 37 GHz signal is highly polar ized i.e., 37D brightness temperature exceeds about 18 degrees (F) THEN the polarized signature of the and rain is unlikely. work for water; The value of surface is dominant "about 18 degrees" will it is different for land. However we have not yet determined an optimum value. This rule can be applied to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 both cumuliform or stratiform clouds. 5. The fifth decision concerns the likely rainfall category for over water situations; IF 19H 20 degrees (F) THEN decide heavy, - 37H does not exceed about light or no rain: 5.1 IF the cloud is over water and 19U exceeds about 240K (F) THEN there is heavy rain; 5.2 IF the cloud is over water about and 19U does not exceed 240K (F) but is greater than or equal to about 195K (F) THEN there is light rain; 5.3 IF the cloud is over water and 19U 195K (F) THEN it is not raining. 6. The sixth decision concerns ice particles the identification (graupel) over land; 37H exceeds about 20 degrees less than about 255K (F)", is less than about of Let A be a clause large "19H - (F)" and B be a clause "85PCT is where 85PCT=1.82*85V - .82*85H: 6.1 IF the cloud is over land and A and B THEN large ice particles are present; 6.2 IF the cloud is over land, and (A and not B) or (not A and B) THEN go to a special care branch(under development); 6.3 IF the cloud is over land and stratiform is the cloud type and 6.1 is true THEN the situation is impossible, and thus return to step 2.2 to reevaluate the appearence of the cloud. 7. The seventh decision concerns the likely rainfall category Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 for over land situations: 7.1 IF 6.1 is true THEN there is heavy rain; 7.2 IF 19H - 37H does not exceed about 20 degrees(F), 85PCT is less than about 275K (F) and 37U is less than [.9(F)*ST - 10(F)] THEN there is light rain, where ST is a surface temperature (determined from INSAT IR temperature); 7.3 IF The neither 7.1 or 7.2 are true THEN it is not raining. second tier of the inference network to remove knowledge base incorporates an the ill-conditioned aspects of the problem based on the behaviour of the 4 microwave signals at 19, 22, 37 and 85 GHz. The rules in the second tier are derived from Figure 5.1 over water. This portion of the know ledge base stresses the coimplication relation which is posed as a form of fuzzy logic. We are currently developing co implication based rules over land to extend the application of the technique. An important feature of the second tier is that brightness temperature boundaries or thresholds need not be "fixed" as done in conventional rainfall algorithms. Instead the methods of "fuzzy logic" are used which effective ly alter implications within the proximity of boundaries. The elimination of ambiguities is the main advantage of the coimplication method in an ill conditioned invoking problem such as rainfall retrieval. The coimplication based rules over water in tier 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are 143 classified in terms of 3 categories; probable and low probable with for the conclusion high probable, a reliability medium of rain class according to the development time-step of the cloud. A coimplication based rule structure the antecedent part, consists of the conclusion part and the asymmetrical differnces representing the coimplication simplification of rule representation, relation. For the we have used several notations: 1) General structure of Coimplication; The "general structures of coimplication" notation is of the form (PI),,(Pn) I— | Q( j) , where Pi is a clause in the antecedent, and j is a number from an asymmetrical Q is a conclusion difference array representing the coimplication relation of the rule. Table 5.2 provides rules which describe describe the the syntax the behavior behavior of for nine coimplication of Figure 5.1. the non-linear The rules relationships in Figure 5.1 interms of linear segments in which discrete ranges of the B T ’s are given high, conjunction with heavy their rainfall intervals. medium and low probabilities in relationship to light, In the rule definitions, medium and the asym metrical difference transforms are given in Table 5.3. differences represent partial These mapping between the antecedent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 and the conclusion in their truth values. 2) clause in the antecedent; The "clause in antecedent" notation is of the form (l,u,C,V(i)), where 1 is a lower bound of brightness temperature, u is an upper bound of brightness temperature, frequency and V(i) is a fuzzy subset C is a channel which is used to evaluate the "truth degree" representing the characteristic of the interval [l,u] which has membership function i. The degrees 5.4; We membership in functions used for evaluating the truth the nine coimplication have used rules are given in Table linear relationships as the form of the membership functions. The individual clauses used in the nine rules are given explicitly in Table 5.2. 3) conclusion; The "conclusion" notation is of the form X(l,u,t) , where X is H (high probable) / M (medium probable) L (low probable), / 1 is a lower bound of rain index, u is an upper bound of rain index and t is the development time step. Table 5.2 indicates the conclusions for the nine rules; The relationship between a rain index and a rainrate is given in Table 5.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 To provide an example, a condition (267,281,85,high(H14)) means that a brightness temperature within the interval [267,281] H14. The high denotes that the BT interval is over a high H14 ranges from 0.7 to 1.0. temperatures 0.7, referred as Therefore, The outside that interval would be zero. e^-cut has been discussed where The in Section 2.1. if a brightness temperature is 267 degrees, then most obtained truth degree for code The truth degree for any bright the truth degree of this condition is function, the 85 GHz channel has a membership function code range of brightness temperatures. ness at brightness temperature using a 0.7 o(-cut; 0.7 has been fixed 0.7. see by In a membership intervals have been Definition 2.7 for o<-cut, trial-and-error while finding asymmetrical differences of the coimplication relation. Asym metrical differences are expressed by Definition 4.18. value 0.7 could be changed later. highly index probable that there is light rain interval [1,6] seconds.". An H ( 1,6,1000) example at the of development means with step The "it is a rain rate time 1000 a complete coimplication rule is (204,251,19,l o w ( H U ) ) , (246,268,22, medium low(H12)), (240,273,37,medium low(H13)), W h In words, (267,281,85,h i g h (H 1 4 )) (1,6,1000)(1). it is interpreted as "IF BT(brightness temperature) at 19 GHz is low and BT at 22 GHz is medium low and BT at 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 GHz is medium low and BT at 85 GHz is high THEN light rain is highly probable in conjunction with asymmetrical difference in rule 1. Table 5.2 Rules based on coimplication in the second tier RuleOl: (204,251,19,low(Hll)), (246,268,22,medium low(H12)), (240,273,37,medium low(H13)), (267,281,85,high(H14)) W H ( 1,6,1000)(1) Rule02: (255,273,19,high medium(H21)), (265,274,22,high medium(H22)), (253,273,37,high(H23)), (252,266,85,medium(H24)) M H(4,8,1500)(2) Ru le03: (228,257,19,low(H31)), (227,255,22.medium low(H32)), (225,251,37,low(H33)), (234,250,85,low(H34)) M H(9,17,1900)(3 ) Ru le04: (215,267,19,low medium(M41)), (252,273,22,high medium(M42)), (265,276,37,high medium(M43)), (264,281,85,high(M44)) M M ( 5,8,1000)(4 ) R u le05: (254,258,19,high medium(M51)), (267,269,22,high medium(M52)), (270,272,37,high(M53)), (261,266,85,high(M54)) W M<1,4 , 1500)(5) R u le06: (242,277,19,not high medium(M61)), (243,277,22,not high medium(M62)), (236,273,37,not medium high(M63)), (248,266,85,medium low(M64)) M (M(9,17,1500)(6) R u le07: (252,257,19.medium low(M71)), (249,255,22.medium low(M72)), (239,251,37,medium low(M73)), (237,250,85,medium low(M74)) M M(5,8 , 1900)(7) R u le08: (254,276,19,medium(L81)), (257,277,22,high(L82)), (243,277,37,high(L83)), (259,281,85,high(L84)) M L(9,17,1000)(8) R u le09: (257,257,19,medium(L91)), (255,255,22,m e dium(L 9 2 )), (250,251,37,medium(L93)), (246,250,85.medium high(L94)) L(l,4 , 1 9 0 0 X 9 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 Table 5.3 Asymmetrical Difference for each rule itemxlOO rule No. 1 2 3 4. 5. 6. 7. 8. 9. (70,75,3)(76,80,4)(81,85,0)(86,91,3)(92,95,0)(96,100,1) (70,80,5)(81,85,6)(86,90,1)(91,95,0)(96,100,-3) (70,75,14)(76,85,-5)(86,95,-2)(96,100,1) (70,75,3)(76,85,8)(86,90,-2)(91,95,1)(96,100,-4) (90,91,0)(92,92,1)(93,94,-1)(95,96,1)(97,100,-2) (70,76,13)(77,91,-2)(92,94,-1)(95,100,1) (70,75,14)(76,90,-1)(91,95,-3)(96,100,0) (70,72,7)(73,75,10)(76,80,0)(81,84,2)(85,87,-1)(88,100,-5) (90,99,2)(100,100,-5) Table 5.4 Membership functions used in the second tier In code type XNF, X means reliability class (High-H; Medium-M; Low-L), N is a rule number in Tier-2, and F means channel(1=19 GHz, 2=22 GHz, 3=37 GHz, 4=85 GHz) or C=conclusion. In the function definition x is a brightness temperature or rain index, and y is a possibility truth degree. Code HI 1 H12 H13 H14 H1C H21 H22 H23 H24 H2C Function y=-3/470*(x-204)+0.7 y=-3/220*(x-246 )+0 .7 y=-3/330*(x -240)+0.7 y= 3/140*(x-267)+0.7 y=-3/50 *(x-l) + 1. y=-3/180*(x-255)+l. y= 3/40 *(x-265)+0.7 -3/50 * (x-269)+l. y= 3/200*(x - 2 5 3 )+0.7 y= 3/140*(x-252)+0.7 y=-l/20 *(x - 4 ) +0.95 X X X X X X X X X X X in in in in in in in in in in in [204, 251] [246, 268] [240, 273] [267, 281] [1,6] and [255, 273] [265, 269] [269, 274] [253, 273] [252, 266] [4,8] and and X is BT and X is BT and X is BT and X is BT x is rain index and X is BT and X is BT and X is BT and X is BT x is rain index Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 Table 5.4: Continued H31 H32 H33 H34 H3C M41 M42 M43 M44 M4C M51 M52 M53 M54 M5C M61 M62 M63 M64 M6C M71 M72 M73 M74 M7C L81 L82 L83 L84 L8C L91 L92 L93 L94 L9C y=-3/290*(x-228)+l. y=-3/280*(x-227)+1. y=-3/260*(x-225)+l. y=-3/160*(x-234)+l. y= 3/80 *(x - 9 ) +0.7 y=-3/520*(x-215)+l. y=-3/210*(x-252)+l. y=-3/110*(x-265)+l. y= 3/170*(x-264)+0.7 y=-23/300*(x - 5 )+0.93 y=-l/40 *(x - 2 5 4 )+1. y=-l/40 *(x - 2 6 7 )+1. y=l/40 * (x-270)+0.95 y=l/50 *(x- 2 6 1 )+0.9 y=-l/30*(x - 1 ) + 1. y=-3/280*(x-242)+l. 3/170*(x-270)+0.7 y=-3/260*(x-243)+l. 3/80 * (x-269)+0.7 y=-3/240*(x-236)+1. 3/130*(x-260)+0.7 y =-1/60 *(x- 2 4 8 )+ 1 . y= 3/80 *(x - 9 ) + 0.7 y=-3/50 *(x - 2 5 2 )+1. y=-3/60 *(x - 2 4 9 )+1. y=-3/120*(x-239)+l. y=-3/130*(x-237)+l. y= 1/10 *(x - 5 ) + 0.7 y=-15/140*(x-254)+l. = 0.85 y= 3/200*(x-257)+0.7 y= 3/340*(x-243)+0.7 y= 3/220*(x- 2 5 9 )+0.7 y=-3/80 * (x-9) + 1. y= 1 y= l y= l y= 1/40 *(x - 2 4 6 )+0.9 y=-l/30 *(x-l) + 1. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in in [228,257] and X is BT [227,255] and X is BT [225,251] and X is BT [234,250] and X is BT [9,17] and x is rain index [215,267] and X is BT [252,273] and X is BT [265,276] and X is BT [264,281] and X is BT [5,8] and x is rain index [254,258] and X is BT [267,269] and X is BT [270,272] and X is BT [261,266] and X is BT [1,4] and x is rain index [242,270] [270,277] and X is BT [243,269] [269,277] and X is BT [236,260] [260,273] and X is BT [248,266] and X is BT [9,17] and x is rain index [252,257] and X is BT [249,255] and X is BT [239,251] and X is BT [237,250] and X is BT [5,8] and x is rain index [254,268] [268,276] and X is BT [257,277] and X is BT [243,277] and X is BT [259,281] and X is BT [9,17] and x is rain index [257,257] and X is BT [255,255] and X is BT [250,251] and X is BT [246,250] and X is BT [1,4] and x is rain index Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 Table 5.5 Rain rate index 1 Index 2 3 4 5 6 7 8 R a t e (mm/hour) 0.2 0.4 0.8 1.0 2.0 5.0 8.0 Index 11 13 14 R a t e (mm/hour) 20. 30. 12 15 16 9 10 10. 12. 16. 17 40. 60. 80. 100 200 The final tier of the knowledge base uses Table 5.4 Table 5.5 as a "frame system" to uniquely tation intensity to any combination of and assign a precipi microwave brightness temperatures through the "Possibility Distribution Functions" which are the inverses of the membership functions. ventional logic notation» these would be referred In con to as the "Probability Distribution Functions" directly associated with the nine individual coimplication relationships. In the ins tances that tier 2 cannot arrive at a unique solution, we say that paradoxical conditions exist and the GPFES reverts most to a probable rainfall category involving no, light or heavy rain based on the tier 1 conclusions. Some sample interactive runs to show relationships between tier 1 and tier 3 are given in Appendix B. We now formulate a retrieval algorithm based on how tier Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 2 and tier 3 generate a unique rainfall measurement. Let i (i = l, 4) be a channel number representing frequencies at GHz, 22 GHz, 37 GHz and 85 GHz, respectively. Let j(j=l,9) be a coimplication rule number in tier 2. degree of a membership function 19 Let P(i,j) be a truth given brightness temperatures BT(i) as input. in Let Table 5.4 with C(j) be a truth degree evaluation from a combination of 4 individual frequency membership function calculations based on a conjunction oper ation. Let T(j) be the truth degree of a possible rain class obtained from the asymmetrical difference adjusment. Let MT be the maximum truth degree from the set of nine truth degrees associated with the nine coimplication rules(only a subset of these will have non-zero values): In summary: 1. input BT(i) (i=l,4) 2. find P(i,j) (i=l,4 and j=l,9) 3. calculate by conjunction C (j ) = min[P(l,j),P(2,j),P(3,j),P(4,j)] (j=l,9) 4. find asymmetrical difference d(j) associated with the subinterval containing C(j) (j=l,9) in Table 5.3 5. calculate T(j) = C(j) + d(j) (j=l,9) 6. find MT = max[T(j),j = l ,,9] and N s j of 7. if (MT is greater than 0.0) then MT i) find a rain index from MT and rule number N using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 the membership function with conclusion given in Table 5.4 ii) find a rain rate from a rain index using Table 5.5 8. paradoxical conditions: if MT=0 then use tier 1 decision on none, light or heavy rain as the most probable rainfall category. An example to retrieve a rain rate based on the algorithm is as follows: Let BT1 = 210, BT2 = 250, BT3 = 248 and BT4 = 277. Then Pll = 0.96 ( H U , Table 5.2), P21 = 0.95 P31 = 0.93 (H13, Table 5.2), P41 = 0.91 (H 1 4 , P12 = o. , P22 = 0. P13 = o. , P23 = 0.75, P33 = 0.73, P43 = 0. P14 = 0. , P24 = 0. , P34 = 0. , P44 = 0.93 P15 = o. , P25 = 0. , P35 = 0. , P45 = 0. P16 = 0. , P26 = 0.91, P36 = 0.94, P46 = 0. P17 = o. , P27 = 0.95, P37 = 0.78, P47 = 0. P18 = 0. , P28 = 0 . , P38 = 0.74, P48 = 0.95 P19 , P29 = 0 . , P39 = 0 . 0. , P32 = 0. (H12, , P42 = 0. , P49 = 0. Therefore, Cl = 0.91, C2 = C3 =...= C8 = C9 = 0. But Cl is in [0.86,0.91] with a rule number 1 in Table 5.2 and dl = 0.03. Thus T1 = Cl + dl, T2 = T3 =...= T8 = T9 = 0 . Then MT = T1 = max(Tl,,T9) and N = 1. Using H1C in Table 5.4, we find a rain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 index 2 (0.4 mm/h; see Table 5.5). Thus if a set of B T ’s pass all the way through Tier 1 Tier 3, rate. then to the retrieval system can assign a unique rain- Since there is no intersection point among brightness temperature intervals at 85 GHz associated with coimplication rules 1-3 in Table 5.4, = i.e. [267,281]A [252,266] ft[234,250] we find at most only one coimplication rule with a truth degree greater than different possible or equal to 0.7 as a combination of 4 brightness temperatures associated with a high reliable conclusion of rain class. Similarly for 85 GHz, the same argument is applied to the case conclusion of rain class because brightness temperature maximum of [259 ,251 ] intervals a low reliable [246 ,250] = for rules 8 and 9. in If a truth value in the conclusion associated with a high reliable class has the same value as the conclusion associated with a low reliable class, then we take the case with a high reliable class to avoid a multivalued solution. rules 4 and 7, the same [264, 281 ] C\[237 ,250] = at 85 GHz. argument <f> for can be applied it is not unique conclusion examination, however, class involves immediate obvious with a non-zero note because brightness temperature intervals But, the relationship among rules 4, the medium reliable conclusion Therefore For cases of that 5 and 6 of intersection. that there truth degree. exists a On futhur the intersection interval Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 [264,281]fl[261,266]fl[248,266] of brightness temperatures with rules 4, 5 and 6 at 85 GHz, [264,266] respectively, we find a possible truth rule number 4; [0.96,1.0] for rule number 6. The is [264,266]. From interval [0.7,0.74] for rule number 5; and for [0.7,0.73] value 0.72 is the only common inter section point of truth degrees(at brightness temperature 265) for rules 4 and 6, and since the membership function at 85 GHz of rule 4 is an increasing function whereas the membership function at 85 GHz of rule 6 is a decreasing function, is only one intersection point at 0.72. Therefore, there there is a guaranteed maximum certainty associated with the coimplication rules of tier 2. As a final illustration of the tier 2 and tier 3 logic, Figure 5.7 schematically presents the behaviour of the member ship functions of step by “««8tep Hll, H12, H13, H14, and H1C along with the calculations brightness temperatures of the algorithm which satisfy The rainrate solution assumes for a set of coimplication rule 1. the truth degrees associated with coimplication rules 2-9 are all zero. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 Figure 5.7: A schematic example of coimplication rule 1 A rain-rate is assigned under the assumption that the truth degrees of rules 2-9 are 0.0. Rain 19 GHz 0.7Low r ■5 204 3 4-1 251 277 [204,251]->[1.0,0.7] y=-3/470*(x-204)+l 22 GHz Light 1 6 17 [1,6]->[1.0,0.7] y=-3/50*(x-l)+l Asymmetrical Difference Transform Medium (0.7,0.75,0.03) (0.76,0.8,0.04) (0.81,0.85,0.0) (0.86,0.91,0.03) (0.92,0.95,0.0) (0.96,1.0,0.01) Example £ 246 268 277 g [246,268]->[1.0,0.7] ** y=-3/220*(x-246)+l 37 GHz Medium •f? 225 240 273 277 [240,273]->[l,0.7] y=-3/330*(x-240)+l BT1=210 BT2=250 BT3=248 BT4=277 Then we find fuzzy truth degrees. Plla0.96 P21=0.95 P31=0.93 P41=0.91 Using conjunction operation, Cl-0.91 Asymmetrical Difference Adjustment: Cl=0.91 [0.86,0.91]->0.03 0.91+0.03=0.94 Optimal Rule Selection: MAX[0.94,0.,0.,0.,0.,0.,0.,0.,0.] Rainrate Conclusion: v(Light Rain in [1,6])=0.94 Therefore, x=50/3*(l-y)+l =2->0.4 mm/hr 85 GHz High 234 3 Ih 4J r 267 281 [267,281]->[0.7,1] y=3/140*(x-267)+0.7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 5.7 Results Figure 5.8 demonstrates the results for the test case showing precipitation areas rain", that have been "light rain", background. Areas classified and "no rain" categories over land into over "heavy a water which qualify as possible rain areas have also been identified. The figure indicates that there are infrequent instances when tier 2 identifies paradox ical conditions in which case the algorithm tier 1 most probable rainfall category. reverts to the That is, if there is no coimplication rule which a set of multi-channel brightness temperatures satisfy, no rain, then we have assigned to those points light rain or heavy rain index based on tier 1 con clusions. We are currently developing a tier-2 system for land based precipitation. Figure 5.8 shows good correlation to the temperature pat tern in the INSAT IR image previously shown in Figure 5.5. In Figure 5.8, we also show a small inset which is the cloud category map prepared on MIDGET during the classification the cloud type (convective or stratiform); see Appendix A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.8 The image, representing rainfall rate, obtained from GPFES blue:light rain, red:heavy rain, magen:most probable heavy rain cyan:most probable light rain, green:possible rain areas over land, other:no rain 157 Chapter 6 Conclusions The concept of a general purpose fuzzy expert system which makes use of fuzzy reasoning techniques based on coimplication has been discussed and incorporated into a General Purpose Fuzzy Expert System (GPFES) together with an application to passive microwave satellite precipitation retrieval. The GPFES is one in which no specific domain knowledge is incorporated in the inference engine. vagueness in various ways fuzzy techniques. implication method. Many expert systems operate although they do with not make use of We have modelled vagueness through the co The models developed thus far have no theoretical base and have been inadequately analyzed. We have attempted to provide guidance for modelling general domains. Once GPFES was implemented, its performance was necessary. the GPFES performance, some method of evaluating In order to provide a test of we have applied our technique to the problem of passive microwave precipitation retrieval. A knowledge base to retrieve precipitation from satel lite microwave measurements has been developed and in GPFES. The evaluated investigation has been primarily attempted to test the usefulness of coimplication on space-based measure- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 merits of passive microwave signals, specifically as they pertain to the retrieval of precipitation over water and land backgrounds when the BT-RR functions exhibit problems with non-uniqueness. Any attempt to develop a quantitative formulation between microwave brightness temperatures sities at one considerations and or more frequencies somewhat beyond precipitation inten is adversely effected by the control of contemporary techniques. Mugnai and Smith[MS88], and Smith and Mugnai[SM88] have developed a theoretical foundation for avoiding one of the problems of "non-unique relationships between the radiat ion signals and precipitation intensity". They do this by application of a multi-spectral approach using passive micro wave measurements. There remains, with a purely analytical close these gaps operation. however, a number of gaps We have attempted to with an expert system approach based on the coimplication method. A well-founded method of reasoning under uncertainty been developed and implemented in GPFES. mental system. has GPFES is an experi In GPFES, we h a v e n ’t considered natural lang uage facilities which are needed in real applications. Future research will involve implementing natural language facilites in GPFES. In the application to precipitation retrieval, this feature was not needed because GPFES with receives answers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 respect to queries from a numerical data set and transforms results into imprecise truth values. The most important problem facing the expert system builders is to acquire knowledge from a domain expert. a time consuming and ledge acquisition difficult process. would enable knowledge interactively. It is Automation of know the domain expert to supply The research and ideas described in this dissertation are not the final solution to approximate reasoning in expert system areas. New methodologies are needed if we are to eventually develop a general purpose fuzzy expert system which involves principles for in general domains. However, modelling uncertainties the results of this dissertation clearly demonstrate that 1) a new reasoning method, based on the equivalence operator instead of the implication oprator in modus ponens, provides for a fuzzy logic-based computational framework 2) the concept of coimplication in the inference process is a potential solution to model uncertainty in the general domain of expert 3) an expert method, system systems approach, based on the coimplication is also a potential solution to handle an ill conditioned problem in precipitation retrieval from remotely sensed microwave measurements. We note also some open problems in expert systems area related Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 to this dissertation research as follow: 1) GPFES is a shell-like expert system tool. It has been developed with a single application as a target. Future work will involve having GPFES handle multiple knowledge sources and support a communication interface among different knowledge sources 2) The most important problem currently facing the builders of expert systems is that of knowledge acquistion. It is a time consuming and difficult process. There is a great need to automate this process. In GPFES, the knowledge representation scheme uses asymmetrical differences between antecedent and conclusion, in the partitioned truth interval. To find asymmetrical differences makes the knowledge acquisition process more difficult 3) Common sense knowledge is embodied in most human reasoning and is very broad and hard to define. It has not been captured to any extent in expert system knowledge bases. Zadeh[Z183b] has done some work on this topic. In GPFES, the knowledge structure to handle common sense knowledge must be described as poor 4) A great difficulty in expert systems development is that of non-monotonic reasoning. Exceptions can be represented in a knowledge base but when there are a large number, space problems happen. We plan to solve this non-monotonic reasoning problem using coimplication theorems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 References [Aj76] Adams, J. B. (1976), A probability model of medical reasoning and the MYCIN model, Mathematical Biosciences 32: 177-186. [ABN81] Aiello, N., Bock, C., Nii, H.P. and White, W.C. (1981), Joy of AGEing: An introduction to the AGE-1 system, Report HPP-81-23, Computer Science Dept., Stanford Univ. [Aj83] Aikens, J.S. (1983), Prototypical Knowledge for Expert Systems, Artificial Intelligence, Vol. 20, [AKS83] Aikens, J.S., K u n z , J.C. and 163-210. Shortliffe, E.H. (1983), PUFF : An Expert System for Interpretation of pulmonary Function Data, Computers and Biomedical Research, Vol. [AV73] 16, 199-208. Anderson, R.K. and Veltischev, N.F. (1973), The use of satellite pictures in weather analysis and forecasting, WMO technical note No. 124, WMO, Geneva, pp. 275. [AR85] Appelbaum, L. and Ruspini E. H. (1985), ARIES: An Approximate Reasoning Inference Engine, Approximate Reasoning in Expert Systems, M.M. Gupta; A. K a n d e l ; W. Bandler and J.B. Kiszka (e d s .), Elsevier Science Publishers B.V. (North-Holland). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 [BK80a] Bandler, W. and Kohout, L. J. (1980), Fuzzy power sets and fuzzy implication operators, Fuzzy sets and systems 4(1), July, 13-80. [BK80b] Bandler, W. and Kohout, L. J. (1980), Semantics of Implication Operators and Fuzzy Relational Products, Int. J. Man-Machine Studies, [BK84] 12, 89-116. Bandler, W. and Kohout, L. J. (1984), The Four Modes of Inference in Fuzzy Expert Systems, Cybernetics and Systems Research 2, R. Trappl(ed.), North-Holland. [Bf32] Barlett, F.C. (1932), Remembering: A study in Experimental and Social Psychology, Cambridge, U.K.: Cambridge Univ. Press. [Bj81] Barnett, J. A. (1981), Computational methods for a mathematical theory of evidence, 7th Int. Joint Conference on A.I. In proceedings of the (Vancouver, B. C.), pp. 868-875. [BF82] Barr, A. and Feigenbaum, E.A. (1982), The handbook of Artificial Intelligence, Los Altos, CA; William Kaufmann, [Be70] Inc.. Barrett, E.C. (1970), The estimation of monthly rainfall from satellite data, Mon. Wea. Rev. 98, 322-327. [Be71] Barrett, E.C. (1971), The tropical Far East: ESSA satellite evaluations of high season climatic patterns. Geog. J. 137, 535-555. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 [Be7 4] Barrett, E.C. (1974), Climatology from Satellites, Methuen, London, pp418. [BM81] Barrett, E.C. and Martin, D.W. (1981), The use of satellite data in rainfall monitoring, Academic Press Inc. [B173] Ill fifth Av. NY, NY 10003. Battan, L.J. (1973), Radar Observation of the Atmosphere, The Univ. of Chicago Press, Chicago and London, [Bp83] 324 pp. Bonissone, Piero P. (1983), Coping with uncertainty in expert systems: a comparative study, Proceedings of the American Control Conference(ACC), pp 12301232, San Francisco, California. [BB85] Bonissone, P. P. and Brown Jr., A. L. (1985), Expanding the Horizons of Expert Systems, Proc. of the Conference on Expert Systems and Knowledge Engineering, Gottlieb Duttwailer Institut, Zurich, Switzerland, April. [BT85] Bonissone, P.P. and Tong, R.M. (1985), Editorial: Reasoning with uncertainty in expert systems, Int. J. Man-Machine Studies, 22, 241-250. [Br79] Brachman, R.J. (1979), On the epistemological status of semantic networks. In N. V. Findler(ed.) Associative Networks: Representation and use of Knowledge by Computers, Academic Press. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 [BB75] Brown, J.S. and Burton, B.C. (1975), Multiple representations of knowledge for tutoring reasoning, in D.G. Bobrow and A. Collins, eds., Representation and understanding: Studies in cognitive science, NY, Academic Press, pp. 311-349. [BF78] Buchanan, B.G. and Feigenbaum, E.A. (1978), DENDRAL and Meta-DENDRAL: Their Applications Dimension, A.I., V o l . 11. [BST86] Buckley, J. J.; Siler, W. and Tucker, D. (1986), FLOPS, A Fuzzy Expert Systems: Applications and Perspectives, In C.V. Negoita and H. Prade(EDs.) Fuzzy logics in Knowledge Engineering, Verlag TUV Rheinland, Gm. [Cc88] Chang, C.L. (1988), Fuzzy topological spaces, J. Math. Anal. A p p l . 24, 1, 182-190. [Cs71] Chang, S.K. (1971), Fuzzy programs, theory and applications, P r o c . Brooklyn Polytechnical Institute Symp. on Computers and Automata, vol. XXI. [DL82] Davis, R. and Lenat, D.B. (1982), Knowledge-Based Systems in A.I., McGraw-Hill Inc., N.Y., N.Y. [Dr77] Davis, R. (1977), Interactive transfer of expertise: Acquisition of new inference rules, in IJCAI 5:321-328. [DB77] Davis, R. and Buchanan, B.G. knowledge: (1977), Meta-level Overview and applications, in proceedings Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 of the 5th Int. Joint Conference on A . I .,Cambridge,M A , 920-927. [Da67] Dempster, A. P. (1967), Upper and lower probabilities induced by a multivalued mapping, The Annals of Mathematical Statistics, [DV73] 38(2), 325-339. Dittberner, G.J. and Vonder Haar, T.H. (1973), Large scale precipitation estimates using satellite data; application to the Indian Monsoon, Arch. Met. Geoph. Biokl. Ser.B., 21, 317-334. [DP80] Dubois, D. and P r a d e , H. (1980), Fuzzy sets and Systems: Theory and Applications, Academic press, NY. [DGH79] Duda, R.O., Gaschnig, J.G. and Hart, P.E. (1979), Model design in the PROSPECTOR consultant system for mineral exploration, in D. Michie, e d . , Expert systems in the micro-electronic age, Edinburgh, Edinburgh Univ. Press, pp. [DHN78] Duda, R. 0. 153-167. ; Hart, P. E. and Nilsson, N. J. (1978), Semantic network representation in rule-based inference systems, in waterman, D. A. and Hayes-Roth, F. e d s . Pattern Directed Inference Systems, pp. 203- 221, Acad. Press. [EHL80] Erman, L.D., Hayes-Roth, F., Lesser, V. and Reddy, D. (1980), The HEARSAY-II speech-understanding system : Integrating knowledge to resolve uncertainty, Computing Surveys 12, no.2, 213-253. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 [Fc81] Forgy, C.L. (1981), 0PS5 u s e r ’s manual, Report CMU-CS- 81-135, Computer-Science-Dept., Carnegie-Mellon Univ., Pittsburgh, Pa. [F181] Friedman, L. (1981), Extended plausible inference, In proceedings of the 7th International Joint Conference on Artificial Intelligence(Vancouver, B . C . ), pp. 487-495. [GLF81] Garvey, T.D.; Lowrence, J.D. and Fischler, M. A. (1981), An inference technique for integrating knowledge from disparate sources, In proceedings of the 7th Int. Joint Conference on A.I. (Vancouver, B. C . ), pp. 319-325 . [Gw82] Gevarter, W.B. (1982), An overview of expert systems, NBSIR 82-2505, National Bureau of Standards, Washington, D . C . . [GH78] Gloersen, P. and Hardis, L. (1978), The scanning multichannel microwave radiometer(SMMR) experiment. The Nimbus 7 u s e r ’s guide, GSFC, Greenbelt, MD, 213-245. [Gj67] Goguen, J.A. (1967), L-Fuzzy Sets in: JMAA 18(1967) pp. 145-174. [GWB76] Griffith, C . G . , Woodley, W . L . , Browner, S., Teijeiro, J., Maier, M . , Martin, D . W . , Stout, J. and Sikdar, D. N. (1976), Rainfall Estimation from Geosynchronous Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 Satellite Imagery during Daylight Hours, NOAA Tech. ERL 356-WMPO 7, Boulder, Col., [Ga73] Gruber, A. 106 pp. (1973), Estimating rainfall in regions of active convection, J. A p p l . M e t eorol. 12, 110-118. [GD70] Gurvich, A.S. and Demin, V.V. (1970), Determination of the total moisture content in the atmosphere from measurements on the Cosmos 243 satellite. Atmos. Oceanic Phys. 6, 453-457. [H186] Hall, L.O. (1986), A Methodological Approach to a Re-Usable Fuzzy Expert System, Ph.D Dissertation, Dep. of Comp. Sci., FI. St. Univ., Tallahassee, [Hw80] Hall, W.D. FL. (1980), A detailed microphysical model within a two-dimensional dynamic framework, description and preliminary results, model J. Atmos. Sci., 37, 2486-2507. [Ht61] Hart, T. (1961), Simplify Memo 27, A.I. Group, Project MAC, MIT, Cambridge, Mass. [HLS87] Hollinger, (1987), J.; L o , G. Poe; Savage, R. and Peirce, J. Special Sensor Microwave/Imager U s e r ’s Guide, Naval Research Lab., Washington DC, 177 pp. [HAP79] Hudlow, M . D . ; Arkell, R . ; Patterson, V.;Pytlowany, P.J.; Richards,F. and Geotis, S . (1979), Calibration and Inter-comparison of the GATE C-Band Radars, NOAA Technical Report EDIS-31, Center for Environmental Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 Assessment Services, NOAA, Washington, D.C., [Jp84] Jones, P.L.K. 132 pp. (1984), REVEAL: An expert systems support environment, In R. Forsyth(ed.)Expert systems principles and case studies, Chapman and Hall, N.Y., 131-150. [Ka86] Kandel, A. (1986), Fuzzy Mathematical Technics with Applications, Addison-Wesley Pub. Co.. [KV77] Kidder, S.Q. and Vonder Haar, T.H. (1977), Seasonal oceanic precipitation frequencies from Nimbus 5 microwave data, J. Geophys. Res., 82, 2083-2086. [KB82] Kohout, L.J. and Bandler, W. (1982), Fuzzy Expert Systems, Proceedings Second Technical Conference of the British Computer Society’s Expert System Specialist Group, Brunei Uni., Runnymede, England. [Kk65] Korsvold, K (1965), An on-line algebraic simplification program, A.I. Project, Memo 37, Stanford Univ., Stanford, CA. [Lm67] Lethbridge, M. (1967), Precipitation probability and satellite radiation data, Mon. Wea. Rev., 95, 487-490. [MG81] Mamdani, E. H. and Gaines, B. R. (1981), Fuzzy reasoning and its Applications, Academic Press, L o ndon. [Mp69] Marinos, P.N. (1969), Fuzzy logic and its application to switching systems, IEEE Trans. 018,4,343-348. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 [MS72] Martin, D.W. and Suomi, V.E. (1972), A satellite study of cloud clusters over the tropical north Atlantic Ocean, Bull. Amer. Meteorol. Soc. 53, 135-156. [MF71] Martin, W.A. and Fateman, R.J. (1971), The Macsyma System, Proceedings of the 2nd Symposium on Symbolic and Algebraic Manipulation, Los Angeles, [Mb71] Mason, B.J. 59-75. (1971), The Physics of Clouds, Oxford Univ. P r e s s (Charendon), London and New York, 671 pp. [Mj82a] McDermott, J. (1982), Domain Knowledge and the Design Process, Design Studies, Vol. 3, No. [Mj82b] McDermott, J. 1. (1982), R 1 : A Rule-based Configurer of Computer Systems, artificial Intelligence, [Md79] Michie, D. 19, 39-88. (1979), Expert Systems in the Micro electronic Age, Edinburgh University Press, Edinburgh, UK. [Mm75] Minsky, Marvin (1975), A framework for representing knowledge, In P. Winston (eds.), The Psychology of Computer Vision, McGraw-Hill. IMS88] Mugnai, A. and Smith, E.A. (1988), Radiation transfer to space through a precipitating cloud at multiple microwave frequencies, Part I: Model Description, J. Clim. Appl. Meteor., Sep. 1055-1073, Vol. 27, [Nc85] Negoita, C.V. (1985), Expert Systems and Fuzzy Systems, Benjamin-Cummings Co., Reading, Ma. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 [NR75] Norman, D . A . , and Rumelhart, D.E. (eds.) (1975), Explorations in Cognition, San Francisco: W.H. Freeman and Company. [OFY84a]Ogawa, H . ; Fu, K.S. and Yao, J.T.P. (1984), An Expert System for Damage Assessment of Existing Structures, Proceedings of the First Conference on Artificial Intelligence Apllications, IEEE Computer Society. [OFY84b]Ogawa, H.; Fu, K.S. and Yao, J.T.P. (1984), Knowledge Representation, and Inference Control of SPERILL-II, ACM Conference Proceedings, Novemver. [OB87] Oh, Kyung-Whan and Bandler, W. fuzzy implication operators, Approximate Reasoning, [PZM81] Pednault, E.P.D., (1987), Properties of International Journal of 1(3), July, 273-286. Zucker, S.W. and Muresan, L.V. (1981), On the independence assumption underlying subjective Bayesian updating, Artificial Intelligence, 16,213-222. [PW84] Politakis, P. and Weiss, S.M. (1984), Using Empirical Analysis to Refine Expert System Knowledge Bases, A . I . , Vol. 22, 23-48. [Ph77] Pople, H . E . , Jr. (1977), The formation of composite hypotheses in diagnostic problem solving: An exercise in synthetic reasoning, [Ph81] Prade, H. in IJCAI 5, p p . •1030-1037. (1981), Modal semantics and fuzzy set Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 theory, In Yager, R. R. Ed. Fuzzy set and Possibility: Recent Development, pp. 232-246, Pergamon Press. [Rm84] Rao, M.S.V. (1984), Retrieval of worldwide precipitation and allied parameters from satellite microwave observations. Advances in Geophys. 26, Academic Press, 237-336. [RAT76] Rao, M . S . V . , Abbott III, W.V. and Theon, J.S. (1976), Satellite-derived global oceanic rainfall atlas (1973 and 1974), NASA SP-410, NASA/GSFC, Greenbelt, MD, 31 p . , 5 appen. [Rr81] Reboh, R. (1981), KAS: Knowledge engineering techniques and tools in the PROSPECTOR environment. SRI technical note 243, SRI, 333 Ravenswood A v ., Menlo Park, CA, June. [REF73] Reddy, D . R . , Erman, L . D . , Fennel, R . D . , and Neely, R.B. (1973), The HEARSAY speech understanding system: An Example of the Recognition Process, proc.IJCAI 3, 185-193. [Rh80] Reichenbach, H. (1980), Elements of Symbolic Logic, The Mcmillan Co., 1947, also in Dover Publications I n c .. [Rr80] Reiter, R. A.I., [Rn69] (1980), A logic for default reasoning, 13, 81-132. Rescher, N. (1969), Many-valued logic, McGraw-Hill, NY. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 [RA81] Rodgersi E.B. and Adler, R.F. (1981), Tropical cyclone rainfall characteristics as determined from a satellite passive microwave radiometer, Mon. Wea. Rev., 109, 506-521. [RCW79] Rodgers, E.B., Chang, A.T.C. and Wilheit, T. (1979), A statistical technique for determining rainfall over land employing Nimbus 6 measurements, Meteor., [RS83] J. A p p l . 18, 978-991. Rodgers, E.B. and Siddalingaiah, H. (1983), The utilization of Nimbus 7 SMMR measurements to delineate rainfall over land. J. Clim. Appl. Meteor., 22, 1753-1763. [SW75] Savage, R.C. and Weinman, J.A. (1975), Preliminary calculations of the upwelling radiance from rain clouds at 37. and 19.35 GHz, Bull. Amer. Meteor. Soc., 56, 1272-1274. [SS63] Schweizer, B. and Sklar, A. (1963), Associative Functions and Abstract Semigroups, P u b l . Math. Debrecen 10, 69-81. [S077] Scofield, R.A. and Oliver, V.J. (1977), A Scheme for Estimating Convective Rainfall from Satellite Imagery, NOAA Technical Memorandum NESS86, Washington, D.C., 47pp. [S077] Scofield, R.A. and Oliver, V.J. (1977), Using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 satellite imagery to estimate rainfall from two types of convective systems, Papers, Eleventh Technical Conference on Hurricanes and Tropical Meterology, (13-16 D e c • 1977, Miami Beach), Meteorological Society, [Sg76] Shafer, G. American Boston, 204-211. (1976), A Mathematical Theory of Evidence, Princeton, N J ; Princeton Univ. Press. [Se76] Shortliffe, E. H. (1976), Computer-based Medical Consultations: MYCIN, New York; American Elsevier Pub. Co. [SB75] Shortliffe, E. H. and Buchanan, B. G. (1975), A model of inexact reasoning in medicine, Mathematical Biosciences 23, 351-379. [SD75] Shortliffe, E. H. and Davis, R. et.al. (1975), Computer-based Consultations in Clinical Therapeutics: Explanations and Rule Acquisition Capabilities of The MYCIN System, Computers and Biomedical Research, Vol.8, 303-320. [Sm72] Shtern, M.I. (1972), Investigations of the Upper Atmosphere and Outer Space Conducted in 1970 in the USSR. NASA TT-F-666, Washington, D.C. [SSA70] Sikdar, D.N.; Suomi, V.E. and Anderson, C.E. (1970), Convective transport of mass and energy in severe storms over the United States-an estimate from a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 geostationary altitude, Tellus 22, 521-532. [SM88] Smith, E.A. and M u gnai, A. (1988), Radiation transfer to space through a precipitating cloud at multiple microwave frequencies, Part II: Results and Analysis, J. Clim. Appl. Meteor., Sep. 1074-1091, Vol. 27. [SOS88] Smith, E , A . , Oh, K.W. and Smith, M.R. (1988), A PC- based interactive imaging system designed for INSAT data analysis and monsoon studies, Submitted to Bull. Amer. Meteor. Soc. for publication. [SSA87] Smith, M . R . , Smith, K.W. E.A., Ahlquist, J.E. and Oh, (1987), Interactive graphics demonstration of "MIDGET" using data from the southwest-east asian monsoon, 3rd International Conference on Interactive Information and Processing Systems for Meteorology, Oceanography, and Hydrology., Jan. 12-16, New Orleans, LA. [SHS87] Spencer, R.W., Howland, M.R. and Santek, D.A. (1987), Severe storm identification with satellite microwave radiometry, an initial investigation with Nimbus-7 SMMR data. J. Clim. Appl. Meteor., [SS85] Spencer, R.W. and Santek, D.A. 26, 749-754. (1985), Measuring the global distribution of intense convection over land with passive microwave radiometry, J. Clim. Appl. M e t e o r .,24, 860-864. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 [Sd69] Staelin, D. H. (1969), Passive remote sensing at microwave wavelengths, Proc. IEEE 57, 427-439. [SH73] Stoldt, N.W. and Hav a n a c , P.J. (1973), Compendium of Meteorological Satellites and Instrumentation, NSSDC 73-02, NASA G S F C : Greenbelt, M D . , 455 pp. plus appendix and indices. [SMS79] Stout, J.E.; Martin, D.W. and Sikdar, D.N. (1979), Estimating GATE rainfall with geosynchronous satellite images, Mon. Wea. Rev. 107, 585-598. [SSS81] Suwa, M . ; Scott, A.C. and Shortliffe, E.H. (1981), An approach to verifying completeness and consistency in a rule-based expert system, HPP81-5, Heuristic Programming Project, Stanford Univ., Stanford, C A . . [TV85] Trillas, E. and Valverde, L. (1985), On mode and implication in approximate reasoning, Approximate reasoning in Expert Systems, Madan M. Gupta e d s .), North Holland. [Vw79] Van Melle, W. (1979), A domain-independent Production- rule System for Consultation Programs,Proceedings IJCAI-79, 923-925. [VSB84] Van Melle, W . , Shortliffe, E.H. and Buchanan, B.G. (1984), EMYCIN : A Knowledge Engineers’s tool for constructing rule-based expert systems, In rule-based expert systems, B. Buchanan and E. Shortliffe (eds.), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 Addison-Wesley, NY 302-328. [WH78] Waterman, D.A. and Hayes-Roth, F.(eds.) (1978), Pattern-directed inference systems, New York: Academic Press. [WK79] Weiss, S.M., Kulikowski, C.A. (1979), EXPERT: A system for developing consultation models, in IJCAI 6, pp. 942-947. [WKS77] Weiss, S.M., Kulikowski, C.A., and Safir, A. (1977), A model-based consultation system for the long-term management of glaucoma, in IJCAI 5. pp. 826-832. [WKS78] Weiss, S.M., Kulikowski, C.A., and Safir, A. (1978), A model-based method of computer-aided medical decision-making, Artificial Intelligence, [Wf79] Wenstop, F. 11, 145-172. (1979), Exploring linguistic Consequences of Assertions in Social Sciences, In M. Gupta, R. Ragade, and R. Yager (eds.) Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam. [Wf88] Wentz, Frank J. (1988), U s e r ’s manual, SSM/I antenna temperature tapes, Remote Sensing Systems, 1101 college a v e . , suite 200, santa rosa, CA 95404. [WS85] Whalen, T. and Schott, B. (1985), Goal-directed approximate Reasoning in a Fuzzy Production System, In approximate Reasoning in expert Systems, Gupta et. al (eds.), North-Holland, N.Y. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 [WCK82] Wilheit, T . T . , Chang, A.T.C., King, J.L., Rodgers, E.B., Nieman, R . A . , and Krupp, B . M . , Milman, A.S., Stratigos, J.S. and Siddallingaiah, H.(1982), Microwave radiometric observations near 19.35, 92 and 183 GHz of precipitation in tropical storm core, J. Appl. Meteor., [WCR77] 21. 1137-1145. Wilheit, T . T . , Chang, A.T.C., Rao,M.S.V., E.B. and Theon, J.S. Rodgers, (1977), A satellite technique for quantitatively mapping rainfall rates over the oceans. J. Appl. Meteor., [Wp80] Winston, P.H. 16, 551-560. (1980), Learning and reasoning by analogy, Communications of the ACM, 23(12), 689-703. [WS71] Woodley, W.L. and Sancho, B. (1971), A first step toward rainfall estimation from satellite cloud photographs, Weather 279-289. [Wd63] Wooldridge, D. (1963), An algebraic Simplify program in LISP, Memo AIM-11, Artificial Intelligence Laboratory, Stanford Uni v . , Stanford, CA. [Wd79] Wylie, D.P. (1979), An application of a geostationary satellite rain estimation technique to an extratropical area, J. Appl. Meteorol. [Yr83] Yager, R. 18, 1640-1648. (1983), Robot Planning with Fuzzy Sets, Robotics 1, 41-50. [Yr85] Yager, R. (1985), Inference in a multivalued logic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 system, International Journal of Man-Machine Studies, 23, 27-44. [Z165] Zadeh, L. A. (1965), Fuzzy sets, Information Control, 8, 338-353. [Z175a] Zadeh, L. A. (1975), The concept of a linguistic variable and its application to approximate reasoning, Information science 8, 199-249;301-357; 9, 43-80. [Z175b] Zadeh, L. A. (1975), Fuzzy logic and approximate reasoning, Synthese 30(1975), 407-428. [Z177a] Zadeh, L. A. (1977), Fuzzy sets and their application to classification and clustering, In classification and clustering (J. Van Ryzin, Ed.) Acad. Press, NY, 251-299. [Z177b] Zadeh, L. A. (1977), Information analysis applications in a possibilistic framework and its meaning in natural languages, Colloq. C N R S , Los Dev. Recents de la Theoric de I ’Inf et leurs Appl., Cachan, France. [Z177c] Zadeh, L. A. (1977), A theory of Approximate Reasoning, Electronics Research Laboratory Memorandum No. UCB/ERL M77/58, University of California, Berkeley, CA, also in Hayes, J.; Michie, D. and Mikulich, L . , Eds. Machine Intelligence, vol. 9, pp 149-194. [Z178] Zadeh, L. A. (1978), Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 vol. [Z179] 1, No. Zadeh, L. 1, pp. 3-28. A. (1979), On the Validity of Dempster's Rule of Combination of Evidence, Electronics Research Laboratory Memorandum No. UCB/ERL M79/24, Univ. of Califonia, Berkeley, CA. [Z181] Zadeh, L. A. (1981), Possibility theory and Soft data Analysis in: Cobb, L . , Thrall, R.M.(eds.) Mathematical Frontiers of the Social and Policy Sciences Boulder, CO 1981, 69-129. [Z183a] Zadeh, L. A. (1983), The role of fuzzy logic in the management of uncertainty in expert systems, Fuzzy Sets and Systems, [Z183b] Zadeh, L. A. 11, 199-227. (1983), A theory of commonsense knowledge, ERL Memorandum No. UCB/ERL M83/26, Univ. of California, Berkeley, California. [Z184] Zadeh, L. A. (-1984), Review of Books : A Mathematical Theory of Evidence, The AI Magazine, Vol.5, N o.3, 81-83. [Z184] Zadeh, L. A. (1984), A simple view of the Dempster- Shafer theory of Evidence, Berkely Cognitive Science Report No. 27, Institute of Cognitive Science, Univ. of California, Berkley, Oct. [Z184] Zadeh, L. A. knowledge, In (1984), A theory of commonsense H.J. Skala, S. Termini, and E. Trillas Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 (eds.), Aspects of Vagueness, R. Reidel, [Z186] Zadeh, L. A. 257-295. (1986), Outline of a theory of usuality based on fuzzy logic, A. Jones et al(eds.), sets theory and applications, Fuzzy 79-97, D. Reideh Publishing Co. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 Appendix A System configuration for image processing to retrieve precipitation from satellite microwave measurements Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 1. MIDGET OCNTCGJRAHCN 8 MgHz PC/AT WITH 32-BIT NUMBER NINE VIDEO BOARD RGB MONITOR HJ ItM U ' STORAGE _________ - m m 1. 2. 3. 4. PUCK I TABLET 0.64 BASE M E M O R Y I Mbyt* R A M M E M O R Y 20 Mbyt* HARO DISK 0.35 Mbyt* FLOPPY DISK SOFTWARE CONTENTS METEOROLOGY DEPARTMENT CY 930 GRAPHICS WORKSTATION SUN 3 /2 6 0 /xVAX GPX 2 £tx 1. 2. jtVAX 3. GPX I 4. HIGHER OROER 3. MS-DOS OP SYSTEM 8086 ASSEMBLER FORTRAN, C COMPILERS BSE EDITOR NUMBER NINE GRAPHICS INTERFACE LIBRARY 6. DECISION IMAGES GRAPHICS APPLICATIONS •SYSTEM 7. MATHTEXT W O R O PROCESSOR 8. OECNET, TELNET COMMUNICATION PROTOCOLS KEY mm ETHERNET EZZ3 LOOSELY COUPLEO NETWORK (CDC) C = 3 BROAD BAND CABLE <^> BUFFERED REPEATER EXTERNAL! WORLD CAMPUS I T TERMINAL SERVER SPAN J SUPERCOMPUTER COMPUTATIONS RESEARCH INSTITUTE ■ ■ SUN IRIS 3030 3 /2 6 0 COMPUTING CENTER S U P E R T COMPUTERS I DEC 11/780 DEC 8700 CYBER 205 FRONTI ENO I ETA 10 I CYBER CYBER 850 835 IBM 4381 BACK END FILE STORAGE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 2. System Flow to retrieve precipitation SSM/I ANTENNA TEMPERATURE TAPES a t 4 frequencies CYBER 205 EEA 10 'ssm/i bttemp : IMAGES a t 4 ^frequencies „ VAX 8700 INSAT IR IMAGES VAX 11/780 SSM/I INSAT-H OCMPOS. IMAGES VBAOOBHL-J see a menu in the next page fo r cloud targ e ts selection RXX: TABLET RUN GEFES TO RETRIEVE FKEEEFTIA11CN VAX 8700 FREapnmcN CATEGORY IMAGE MIDGET DISPLAY Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 3. MENU FCR OX3UD TARGETS SELECTION in MIDGET *** DISPLAY OPTIONS *** 1 2 3 4 R O P E 19GHz V/H/U SSM/I ULH SECTOR : INSAT SSM/I URH SECTOR : 37GHz V/H/U SSM/I LLH SECTOR : 85GHz V/H/U SSM/I ULH SECTOR : Resets 4 Frame Display Draw Continental Boundary Purge Overlay Exit CURSOR OPTIONS [MOVE CURSOR WITH PUCK] STATS : M VALUE : V MARK CLOUD OPTIONS STRATUS : S CONVECTIVE : C STDV VAR STATS AVG 11.57 133.86 136.42 LAT VALUE SECTOR LIN ELE -1.7 3 134 99 LDIM MARK SECTOR LIN ELE 30 3 134 99 TVPOS RAS= 495 PIX= 286 MIN MAX 164 118 COUNT TEMP LON 154 254 65.7 EDIM CLOUDTYPE 30 Con Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 Appendix B Interactive sample sessions In this appendix I show some examples of how GPFES runs on the Supercomputer Computations Research Institute Florida State University. The $ VAX 8700, the is the VAX system prompt. The symbols are explained in Section 5.6 and dummy-x clauses in reasoning processes are used for the connection in the infer ence network. $ run gpfes Welcome to GPFES which is a first application to retrieve precipitation from remote sensing microwave measurements of Air Force satellite SSM/I If you have to answer by a truth interval, then use the free format in integer type of [0,100]: e.g. 80 100,i.e. lower and upper bounds; 100 is yes, 0 is no and 50 is a medium certainty, y: yes, n: no and q: question about symbols continue ? (y/n) y there is a cloud over the point ? (y/n) n final conclusion; no rain truth; 100 100 match; 100 symm. dif= 0 Do you want to see the reasoning process ?(y,n) n If you have to answer by a truth interval, then use the free format in integer type of [0,100]: e.g. 80 100,i.e. lower and upper bounds; 100 is yes, 0 is no and 50 is a medium certainty, y: yes, n: no and q: question about symbols Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 continue ? (y/n) y there is a cloud over the point ? (y/n) y area is water ? (y/n) y 37V-37H exceeds about 18 ? (y/n/q or interval) n 18H-37H exceeds about 20 ? (y/n/q or interval) 0 15 18U exceeds 195 ? (y/n/q or interval) 86 92 18U exceeds about 240 ? (y/n/q or interval) q U=(V+H)/2 18U exceeds about 240 ? (y/n/q or interval) 0 12 89U is high in the interval [267,281];1 90 93 19U is low in the interval [204,251];1 86 89 22U is medium in the interval [246,268];1 78 87 37U is medium in the interval [240,273] ;1 90 96 final conclusion; rain is very light; H ( 1,6,1) truth; 81 90 match; 100 symm. dif= 4 Do you want to see the reasoning process ?(y,n) y a set of clauses in the premise continue truth of premise^ 100 match: 100 sym. dif: 100 0 intermediate conclusion; dummy-0 truth : 100 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a set of clauses in the premise there is a cloud over the point dummy-0 truth of premises 100 matchs 100 sym. difs 100 0 intermediate conclusion; dummy-1 truth = 100 100 a set of clauses in the premise dummy-1 area is water truth of premises 100 matchs 100 sym. difs 100 0 intermediate conclusion; dummy-8 truth s 100 100 a set of clauses in the premise dummy-8 18H-37H is less than about 20 truth of premises 85 matchs 100 sym. difs 100 0 intermediate conclusion; decide a rain class truth s 85 100 a set of clauses in the premise decide a rain class 18U exceeds 195 truth of premises 85 matchs 100 sym. difs 92 0 intermediate conclusion; rain truth s 85 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 188 a set of clauses in the premise rain 18U is less than about 240 truth of premise= 85 match: 100 sym. dif: 92 0 intermediate conclusion; light rain over water; tier2 truth : 85 92 a set of clauses in the premise light rain over water; tier2 truth of premise: 85 match: 100 sym. dif: 92 0 intermediate conclusion; tier2 truth : 85 92 a set of clauses in the premise tier2 89U is high in the interval [267,281];1 19U is low in the interval [204,251];1 22U is medium in the interval [246,268] ;1 37U is medium in the interval [240,273];1 truth of premise: 78 87 match: 100 sym. dif: 4 intermediate conclusion; rain is very light; H ( 1,6,1) truth : 81 90 If you have to answer by a truth interval, then use the free format in integer type of [0,100]: e.g. 80 100,i.e. lower and upper bounds; 100 is yes, 0 is no and 50 is a medium certainty, y: yes, n: no and q: question about symbols continue ? (y/n) y there is a cloud over the point ? (y/n) y Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 area is water ? (y/n) n do you know the cloud type (cumuli or strati) n ? (y/n) a tutoring session to help you about the cloud type ! homogeneous over large area ?(y/n) n stable ?(y/n) n v. vs. h. development is small ?(y/n) n long evolution ?(y/n) n cotton-like appearence ?(y/n) y cloud type is cumuliform Now you are ready to answer 18H-37H exceeds about 20 ? (y/n/q or interval) 10 23 85PCT is less than 255 ? (y/n/q or interval) 8 19 ctype is cumuliform ? (y/n) y 85PCT is less than about 275 ? (y/n/q or interval) 90 100 37U is less than .9(F)*ST-10 ? (y/n/q or interval) 87 93 final conclusion; light rain over land truth; 77 90 match; 100 symm. dif= 0 Do you want to see the reasoning process ?(y,n) y a set of clauses in the premise continue truth of premise= 100 match= 100 sym. dif= 100 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 intermediate conclusion; dummy-0 truth = 100 100 a set of clauses in the premise there is a cloud over the point dummy-0 truth of premises 100 match= 100 sym. dif= 100 0 intermediate conclusion; dummy-1 truth = 100 100 a set of clauses in the premise area is land dummy-1 truth of premises 100 matchs 100 sym. difs 100 0 intermediate conclusion; dummy-2 truth s 100 100 a set of clauses in the premise "you d o n ’t know the cloud type" is true dummy-2 truth of premises 100 100 matchs 100 sym. difs 0 intermediate conclusion; find a cloud type; go to tutoring section truth s 100 100 a set of clauses in the premise find a cloud type; go to tutoring section truth of premises 100 100 matchs 100 sym. difs 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 intermediate conclusion; dummy-3 truth = 100 100 a set of clauses in the premise dummy-3 ctype is cumuliform truth of premises match= 100 sym. 100 difs 100 0 intermediate conclusion; dummy-5 truth s 100 100 a set of clauses in the premise 18H-37H is less than about 20 85PCT exceeds 255 truth of premises 77 matchs 100 sym. difs * ; not A * ; not B 90 0 intermediate conclusion; dummy-7 truth s 77 90 a set of clauses in the premise dummy-7 truth of matchs premises 100 sym. 77 difs go 0 intermediate conclusion; (not A) and (not B) truth s 77 90 a set of clauses in the premise (not A) and (not B) truth of premises matchs 100 sym. 77 difs go 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 intermediate conclusion; dummy-6 truth = a set of 77 90 clauses in the premise 18H-37H is less than about 20 85PCT is less than about 275 37U is less than .9(F)*ST-10 truth of premises 77 matchs 100 sym. difs * * * not A J K 90 0 intermediate conclusion; (not A) and J and K truth s 77 a set of clauses in 90 the premise dummy - 6 (not A) and J and K truth of premises 77 matchs 100 sym. dif= 90 0 intermediate conclusion; light rain over land truth s 77 90 If you have to answer by a truth interval, then use the free format in integer'type of [0 ,1 0 0 ]: e.g. 80 1 0 0 ,i.e. lower and upper bounds; 100 is yes, 0 is no and 50 is a medium certainty, y: yes, n: no and q: question about symbols continue ? (y/n) n Thank you ! FORTRAN STOP Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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