Dempster-Shafer teorisinin değerlendirme problemine uygulanması
Application of Dempster-Shafer theory to an evaluation problem
- Tez No: 46471
- Danışmanlar: DOÇ.DR. GAZANFER ÜNAL
- Tez Türü: Yüksek Lisans
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 35
Özet
ÖZET Birden fazla kriter ve birden fazla uzmanın olduğu karar verme problemlerinde nihai bir sonuca ulaşmak oldukça güç olabilir. Çünkü bir uzmanın birden fazla kritere dayalı olarak ürün, kişi,...vs. değerlendirmesi belirsizlik içerebilir. Karar verici tarafından verilecek nihai karar, uzmanın değerlendirmesinin belirsizliklerini de yansıtacaktır. Bu nedenle problem, nihai karara özgü belirsizliğin nasıl indirgenebileceğidir. Bu problemi çözmek için bu tezde Dempster-Shafer teorisine başvurul muştur. Pratikte, bu metotda kullanılan hesaplamalar oldukça uzun ve yorucu olabilir. Bu yüzden, burada, karar vericiye yardımcı olmak için Mathe- matica'da öneri üreten bir uzman sistem programı geliştirilmiştir. Burada geliştirilen program sadece değerlendirme tabanlı karar verme problemleriyle değil, algılayıcıların verilerinden elde edilen delillerle de uğraşır. Tezin birinci bölümünde uzman sistemlerle ilgili genel bilgiler verilmektedir. Karar verme problemlerindeki belirsizlik çeşitleri ve gösterilim yaolları ikinci bölümde anlatılmıştır. Üçüncü bölümde Dempster-Shafer teorisi hakkında bilgi verilip teoriyle ilgili tanımlar ve kullanılan formüller anlatıl maktadır. Son bölüm olan dördüncü bölümde ise programın uygulandığı problemlerin tanıtımı, programda kullanılan temel fonksiyonlar' ve programın nasıl işlediği örneklerle anlatılmaktadır.
Özet (Çeviri)
SUMMARY APPLICATION OF DEMPSTER-SHAFER THEORY TO AN EVALUATION PROBLEM To make a final conclusion in decision making problems with multi crite ria and multi experts can be quite difficult. Because the expert's evaluation of product, person,... etc., based on multi criteria may involve uncertainti es. Final decision which will be given by the decision maker will reflect the uncertainties of the expert's evaluation. Therefore, the problem be comes how to reduce down the uncertainty inherent to final decision. To solve this problem we have resorted to Dempster-Shafer theory in this the sis. In practice, the calculations involved in this method can be quite tedio us. Therefore, here, we have developed an advisory expert system program in Mathematica to help the decision maker. The expert system developed here not only handles the evaluation based decision making problems but also the evidences obtained from sensory information. Evidential reasoning, which is the process of inferring the likelihood of some hypotheses by collecting and combining related evidence for or against these hypotheses, is essential to many computer systems that help users in decision making, diagnosis, pattern recognition, and speech understanding. The problem of evidential reasoning is complicated by information being conveyed by a piece of evidence is often not only uncertain, but also impreci se, incomplete, and vague [3]. For example, a sensor's output may indicate that a flying object is about 50 miles from Ankara and that it belongs to a general class of missiles. But the sensor gives no further information about the specific type of the missile. Therefore, an evidential reasoning mecha nism that can cope with all these different kinds of uncertainties in a firm manner is highly desirable. The traditional numerical approaches to uncertain decision making were based on Bayesian reasoning, and require a very large database of conditional probabilities, even if ad hoc simplifying assumptions such as independence of individual observations are made although they may be hard to justify physically. Shafer demonstrated by simple examples, that the concept of ignorance is hard to represent in the Bayesian framework. The method of assigning equal prior probabilities often produces counter intuitive results, especially when the number of hypotheses being considered is more than two. Most real life decision making involves complex problem solving in situations where facts and data available are insufficient, and knowledge of the domain is incomplete, therefore, a rigorous probabilistic analysis is not possible. The major drawbacks of the Bayesian and other schemes have appea led attention to the Dempster Shafer theory of evidence combination. The main advantages of this theory are its capabilities of incorporating explicitly the concept of ignorance in the decision making process, assigning belief tosubsets of hypotheses in addition to singleton hypothesis, and modeling the narrowing of the hypotheses set with the merging of evidence. This offers a better framework for modeling the human expert's inference mechanism. Belief functions and their combining rules defined by the scheme are well suited to represent the incremental merging of evidence and the results of its aggregation. Dempster-Shafer reasoning is a generalization of Bayes reasoning that allows confidences to be assigned to a set of hypotheses rather than to just N mutually exclusive hypotheses [10]. D-S theory was developed by Arthur dempster m the 1960's and extended by Glen Shafer in the 1970's. Two difficulties the researchers had with probability theory : the representation of ignorance, and the idea that the subjective beliefs assigned to an event and its negation must sum to one, motivated the development of D-S theory Ever since the probability theory originated in the 17th century, se veral aspects of the field have been extensively argued, including the rep resentation of ignorance. The traditional method represents ignorance by indifference or by uniform probabilities. Some authors argue on this appro- achjbecause uniform probabilities seem to represent more information than is given - one can attribute equal prior beliefs to either complete ignorance or equal belief in all hypotheses. Furthermore, the new evidence obscures the original ignorance expressed in the prior belief. Another heavily argued point involves fixing the probability of a hypot hesis 's negation once the probability of its occurrence is known, because P(a) + -P(a) = 1- Shafer claimed that, in many situations, evidence that only partially favors a hypothesis should not be construed as also partially supporting its negation. The D-S formulation is based on the evidence and hypothesis spaces. The evidence space S consists of a set of mutually exc lusive and exhaustive evidential elements. This space is called the universe of discourse [11]. The hypothesis space © consists of a set of exclusive and exhaustive hypotheses. Assume that one of the hypothesis in the subsets of this finite set is true. 0 is called the frame of discemmet. Any subset of 0 will be represented by hi. The expert system developed in this thesis is a shell in a manner. Beca use, its interpreter is separated from the database. The database is stored in a file. The first problem that the system applied is a student evaluation problem. A student is avaluated according to two criteria: Written and oral exam. The jury members can make their avaluations for the criteria from the set {very good, good, medium, bad} by choosing one element or two elements which are neighbor. When the jury's avaluations are considered, the required result is to find if the student exceeded the exam or not. Namely, 0 is determined as {exceeded, failed}. Secondly, the expert system developed here applied to a submarine problem. There are four criteria related with the submarine- :Area, position, speed and course. Each criterion can take its possible value from the following lists: Criterion Value List area {usual, unusual} position {submerged, surfaced} speed {quiet, normal} course {topedo range, course change} VIAccording to the evidences, the submarine is either in innocent passa ge or in a hostile intent. Therefore 0 becomes {innocent, hostile}. The evidences in S space will be represented by ey. Each evidence has a cer tainty degree, which we'll represent by Wj. Wj can take its value from 0 to 1. Wj is called the data quality measure and defined by the equation Wj=P(ej true) [14]. Wy=l means that the information from the jth. so urce ( evidence ey) is absolutely correct. Wy=0 means the information is absolutely false. Suppose we have two questions related with each other: X=What is the evidence? Y= What is the result? These questions take their possible values from the evidence and hypothesis space [13]. If ey is an answer to question X and hi is an answer to question Y, in other words if ey and hi are true simultaneously according to their questions, it is said that ey and hi are compatible. This is represented by ejRhi. A compatibility relation R over S and 0 is a subset of S x 0 cartesian product and provides a connection between S and 0. Any R over S x ® can be represented as a multivalued mapping ^e G:S such that G(ey) = {hi | (ey,/i,) ? R} Any compatibility relation R can get in the following form [3]: If X = ey then Y = h{ Formally the theory is concerned with belief structures, which are defi ned in the following. Suppose that 0 = {xi,x2,...,xn} is a finite set of elements, the frame of discernment. Let m be a measure on the subset of 0 such that (1) 0 < m(A) < for each A C 0 (2) m(0) = 0 (3) £ m(A) = 1 AC© m is called a basic probability assignment function. Any subset A of 0 such that m(A) > 0 is called a focal element [5]. The quantity m(A), the basic probability number of A, can be viewed as the portion of total belief assigned exactly to A. Let hi be a subset of 0. When S and 0 are given the m(hi) quantity is calculated by the following equation [14]: m(hi) = Wj x P(hi\ej) m and its associated values are called a belief structure. These belief structures are used to represent a piece of evidence or information about viithe potential special element. The measure of total belief committed to a subset A is defined as: Bel{A) = J2 m{B) BÇA A belief function satisfies the following: (1) 5e/(0) = 0 (2) J5e/(e) = 1 _ (3) Bel(A) + Bel(A) < 1 Thus, Bel equals m for singletons, but Bel is greater than or equal to m for sets that contain more than one element. The quantity 1 - Bel(A) is called the plausibility of A, providing the maximum amount of belief that can possible be assigned to A. Plausibility is defined as: Pl(A)= £ m(B) Br\A±% The functions Bel and PI can be interpreted as the lower and upper pro babilities induced by a multivalued mapping. Since Bel(A) + Bel(A) < 1, Pl(A) - Bel(A) > 0. Plausibility function satisfies the following: (1) Pl{%) = 0 (2) Pl{@) = 1 (3) Pl{A) + Pl{Ac) > 1 (4) Pl(A)>Bel(A),VA In the classical probability model the probability mass function, a map ping from Q, into [0,1] totaling one, indicates how the probability mass is assigned to the elements. In the more general Dempster- Shafer framework, rather than knowing exactly how the probabilities are distributed to each element x C 6, we only know that a certain quantity of probability mass is somehow divided among the elements of focal sets. When current evidence leads to multiple beliefs regarding the same hypothesis, the beliefs should be combined to provide an overall belief in the hypothesis. To propagate belief, D-S theory usually combines diffe rent belief functions by computing their orthogonal sum. Assume we have two independent sources of evidence. Assume these sources have associated belief structures m\ and m-i, respectively, over 6. The problem of concern is to find a combined belief structure m over 0 reflecting the“ANDing”of the two pieces of evidence(£i, E2). The approach suggested by Shafer is to use Dempster's Combination rule [5]. Assume mi and mi are two belief structures on 0 with focal elements Ai,...,Ak and Bı,...,Bı respectively. Then their combination, called the viiiorthogonal sum and denoted m = mi0 ra2, is another belief structure over 0 such that 2 m1(Ai)rri2(Bj) Aj represents hypotheses subsets that are supported by E\, Bj repre sents hypotheses subsets supported by Ü?2, and C represents the hypotheses subsets that are supported by the observation of both E\ and ü?2. The denominator is a normalizing factor to ensure that no belief is committed to the null hypothesis. In a way, this function provides a measure of confiiict between the two pieces of evidence. The operation of orthogonal sum of be lief structures satisfies commutativity and associativity properties. These two properties allow us to combine multiple belief structures by repeated application of Dempster's rule. IX
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