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Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 1958 – 1962 www.elsevier.com/locate/procedia
The representation and processing of uncertain problems Qinghua Zhanga,b, Yu Xiaob,Yuke Xingb a* a
b
College of Mathematics & Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Institute of Computer Science & Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
Abstract Uncertainty is everywhere, and there are many researches on uncertain problems. Soft Computing combined intelligent paradigms as Probabilistic Reasoning, Fuzzy Logic to deal with pervasive imprecision and uncertainty of the real-world problems. In this paper, four main Soft Computing methods, namely, Probability Theory, Fuzzy Set Theory, Rough Set Theory and Cloud Model are introduced briefly, and their representation and measure of uncertainty are discussed respectively. © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
Selection and/or peer review under responsibility of [CEIS 2011]
Keywords: uncertain problems; probability theory; fuzzy sets; rough sets; cloud model
1. Introduction Probability theory has been the only well-founded theory of uncertainty for a long time. It was viewed either as a powerful tool for modeling random phenomena, or as a rational approach to the notion of degree of belief[1-3]. During the last thirty years, in areas centered around decision theory, artificial intelligence and information processing, numerous approaches extending or orthogonal to the existing theory of probability and mathematical statistics have come to the front. The common feature of those attempts is to allow for softer or wider frameworks for representing uncertain information. Zadeh in 1965 [5] solved the problem of modeling vagueness in some degree by allowing partial memberships. He proposed the concept of fuzzy sets to resemble human reasoning in the use of approximate information and uncertainty to generate decision. The main contribution of the fuzzy sets is to propose a concept of membership function. Rough set theory was proposed by Pawlak in 1982 [6] primarily to represent and process “vague” information. The theory of rough set is motivated by practical needs to interpret, characterize, represent, and process indiscernibility of individuals. For bridging the gap of uncertain transforming between a linguistic term of a qualitative concept and its quantitative representation, using the cloud model to represent commonsense concepts and using logical sentences (rules) for reasoning which is different traditional probability reasoning. Cloud model theory was proposed by Li Deyi in 1995 * Corresponding author. Tel.: 023-62460540. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.365
Qinghua Zhang et al. /Procedia Procedia Engineering Engineering 00 15 (2011) (2011) 000–000 1958 – 1962 QinghuaZhang/
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[7]. In the cloud model, a piece of cloud is made up of lots of cloud drops, visible shape in a whole, but fuzzy in detail, which is similar to the natural cloud in the sky. Any one of the cloud drops is a mapping in the discourse universe from qualitative concept. The rest of this paper is organized as follows. Section 2 deals with relationship between Probability Theory and Uncertain problems. In Section 3, the fuzzy sets model is discussed to handle uncertainty. Section 4 focuses on Rough Set Theory. And in Section 5, Cloud model is introduced. Finally, the conclusion is given in Section 6. 2. Probability Theory The typical mathematical method for solving the stochastic problem is probability Theory. Based on measure theory, Kolmogorov proposed and established the axiomatic method of Probability Theory in his paper in 1933, which makes it possible to study Randomness with mathematical methods which quantify “Randomness” with the concept of “Probability”. With the distribution function of random variables, one can study all the statistical characteristics of random phenomena. Random experiment is an experiment whose result would not be predicted due to the insufficient conditions and interference of chance factors but the list of possible outcomes are known. The result of random experiments may not be predicted exactly but the result must be in the list of predicted outputs. We study random phenomena through the study of random experiment [8]. For random experiment, the outcome cannot be predicted with certainty before the experiment is run, which presents uncertainty, but the set of all possible outcomes of a random experiment are determined. The random experiment manifests the uncertainty in the form of “true” or “false” or in the form of “consistency” or “inconsistency” with objective criteria. Let “1” denotes “true” or “consistency” and “0” denotes “false” or “inconsistency”, then {0,1} could denote the measurement range of information. The first measurement is self-information. The uncertainty of random event is equal to self-information quantitatively. Shannon [2] derived a measure of information content called the self-information of an event m as follows, 1 I ( m) = log( ) = − log( p ( m)) . p (m) p ( m) pr = ( M m) is a probability that event m happens in the sample space M. The second Where = measurement is entropy. It is defined as the average self-information of an event m from that sample space as follows, H ( M ) = E{I ( M )} = p ( m) I ( m) = − p ( m) log p (m) .
∑
m∈M
∑
m∈M
Where E{} denotes the expect value operation. 3. Fuzzy Set Theory 3.1. The Representation of Fuzzy Set
In the fuzzy set theory[10], the member ship function represent the affiliation between element x and set A, and the value of the membership function represent the “grade of membership” of x in A. In the fuzzy sets theory, using the fuzzy degree (or fuzzy entropy) to deal with uncertainty, and using “Ambiguity” to characterize the fuzzy degree of the fuzzy concept. Let X be any set, as the universe of discourse, a fuzzy set (FS) A in X is characterized by a membership function μ : X → [0,1] . The value μ ( x ) represents the grade of membership of x in A . A
A
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3.2. The uncertainty measure of Fuzzy Set
Measures of fuzziness estimate the average ambiguity in fuzzy sets in some well-defined sense. We begin by considering properties that seem plausible for such a measure. The fuzziness of a crisp set using any measure should be zero, as there is no ambiguity about whether an element belongs to the set or not. If a set is maximally ambiguous (for any x , μ A ( x) =0.5), then its fuzziness should be maximum. A fuzzy * set A is called a sharpened version of A if μ A* ( x ) ≤ μ A ( x ) when μ A ( x ) ≤ 0.5 or μ A* ( x ) ≥ μ A ( x ) * when μ A ( x ) ≥ 0.5 . For a sharpened version A of A , the measure of fuzziness should decrease because sharpening reduces ambiguity. Membership functions of fuzzy sets are never unique. Different individuals might define various μ A ( x ) for the same fuzzy set. For example, the membership function of a fuzzy set TALL defined by an American for Americans would probably be different from that defined by an Oriental for Orientals. So many new fuzzy theories are proposed combining with some other theories, such as, In 1975 Kaufmann introduced an index of fuzziness, its name is Mingkowski ambiguity and the Equation as follows,
= d p ( A)
2 n ( | A(ui ) − A1/2 (ui ) p |)1/ p 1/ p ∑ n i =1
(1)
2 n ( | A(ui ) − A1/2 (ui ) |2 )1/2 1/2 ∑ n i =1
(2)
If p = 2 , the equation (1) becomes equation (2), this equation (2) is Euclid ambiguity.
= d 2 ( A)
Kaufmann also mirrored Shannon's probabilistic entropy by using its form to define the entropy of a discrete fuzzy set. Shannon ambiguity：
1 n ∑ s( A(ui )) n ln 2 i =1 In the equation (3), s ( x) = − x ln x − (1 − x) ln(1 − x) is the Shannon function [11]. H ( A) =
(3)
4. Rough Set Theory 4.1. The Representation of Rough Set
A decision information system[6] is defined as S =< U , R , V , f > , where U is a non-empty finite set of objects, called universe, R is a non-empty finite set of attributes, R = C ∪ D , where C is the set of condition attributes and D is the set of decision attributes, D ≠ φ . With every attribute a ∈ R , Va denotes the domain of attribute a. Each attribute has a determine function f : U × R → V . Given a decision information system S =< U , R , V , f > , each subset of attribute B ⊆ R determines an indiscernibility = IND ( B ) {( x , y ) | ( x , y ) ∈ U × U , ∀b∈B ( b ( x ) = b ( y ))} . Obviously, the indiscernibility relation relation IND ( B ) is an equivalence relation on U . The quotient set of equivalence classes induced by IND ( B ) , denoted by U / IND ( B ) , forms a partition of U , and each equivalence class of the quotient set is called an elementary set. Let S =< U , R , V , f > be a decision information system, for any subset X ⊆ U and indiscernibility relation IND (B), the B lower and upper approximation of X is defined as follows, ∪ Yi . B− ( X ) = ∪ Yi , B − ( X ) = Yi ∈U / IND ( B ) ∧Yi ⊆ X
Yi ∈U / IND ( B ) ∧Yi ∩ X ≠φ
If B− ( X ) = B ( X ) , X is definable with respect to IND (B). Otherwise, X is rough with respect to IND (B). The lower approximation B− ( X ) is the union of elementary sets which are subsets of X , and the upper approximation B ( X ) is the union of elementary sets which have a non-empty intersection −
−
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with X . That is, B− ( X ) is the greatest definable set contained by X , while B ( X ) is the least definable set containing X .The lower approximation B− ( X ) is also called the positive region, the complement of the upper approximation B ( X ) is called the negative region, and the difference of the upper approximation B ( X ) with the lower approximation B− ( X ) is called boundary region. −
−
−
4.2. The uncertainty measure of Rough Set
Let (U , R) be an approximation space, X be an arbitrary concept, i.e., a subset of U . Then the accuracy of X in (U , R) is defined as follows,
α(X ) =
| R− ( X ) |
| R− ( X ) |
.
Where | . | denotes the cardinality of a set. For the empty set φ , we define α (φ ) = 1 . The accuracy has the following properties: • 0 ≤ α(X ) ≤ 1 • α ( X ) = 0 , if the lower approximation of X is empty; • α ( X ) = 1 , if X is exact in (U , R ) . The roughness of knowledge is defined based on indiscernibility relation and inclusion degree, but its intension is not clear. For this reason, information entropy was applied into rough set theory to measure the roughness of knowledge by Miao [9] as follows,
H ( R) = −
∑
Yi ∈U / R
| Yi |
|U |
log 2
| Yi |
|U |
.
5. Cloud Model Theory
In the cloud model[11], let U be the set, as the universe of discourse, and T is a linguistic term associated with U . The membership degree of x in U to the linguistic term T , CT ( x ) is a random variable with a probability distribution. A membership cloud is a mapping from the universe of discourse U to the unit interval [0, 1]. That is, CT ( x ) :U → [0,1], ∀x ∈ U , x → CT ( x ) .The normal cloud is based on normal distributions, which have been supported by results in every branch of both social sciences (such as psychology, economics, and linguistics) and natural sciences. A normal compatibility cloud can be characterized by three digital parameters: Ex (expected value), Ex (Entropy) and He (hyper Entropy). The following will introduce the uncertainty of three digital parameters [8].The Ex (Expected value) denotes the center of gravity of a cloud. In other words, the element Ex is the best sample point which can represents the qualitative concept. It is very easy to determine Ex in practical applications. The En (entropy) is a measure of the fuzziness of the concept over the universe of discourse, the uncertainty of the entropy shown in three parts as follows, It is a measure of the concept coverage. The measure of the fuzziness, which indicates how many elements could be accepted to the qualitative linguistic concept; It is a measure of the cloud droplets which can represent the qualitative concept appearance. The measure of the randomness, and it shows the probability density of the qualitative concept that represented by cloud droplets. Entropy shows the relationship between the fuzziness and the randomness. The He (Hyper Entropy) is a measure of the dispersion on the cloud drops, which can also be considered as the entropy of Ex. If Hyper Entropy （He） increase the cloud drops dispersing and the cohesive of the cloud drops will deteriorate, the cloud model will transform form normal cloud to general cloud. According to algorithm shows, 99.7% of the cloud drops will included between curve y1 and curve y2 , the expressions as following,
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Qinghua Zhang /etProcedia al. / Procedia Engineering 15 (2011) 1958 – 1962 QinghuaZhang Engineering 00 (2011) 000–000
( x − Ex)2 ( x − Ex)2 , y2 = exp(− ). ) 2( En − 3He) 2 2( En + 3He) 2 Overall, the precise function is not the core of the cloud model, but using three digital parameters express the uncertainty of the qualitative concept, through a specific method to achieve uncertain transition between a qualitative concept and its quantitative representation, meanwhile reveals the relationship between fuzziness and randomness. Cloud model sketched out the contours of the uncertainty, to make similar and flexibility conclusions.
y1 = exp(−
6. Conclusion
The earliest and most mature approach to deal with uncertainty is probability theory. It started from the inevitability and occasionality of an incident. In fuzzy sets, membership degree was introduced to describe the uncertainty. Rough set used classification of knowledge to discuss the uncertainty [12]. However, cloud model is a qualitative concept to quantitative description of the uncertain transformation model. This paper mainly talked about the four main soft computing methods to characterize uncertain problems. We hope these work can contribute to better study other methods, such as type-2 fuzzy sets, quotient space theory, and so on. Acknowledgements
This work is supported by the National Natural Science Foundation of China(No.61073146), Science & Technology Research Program of Chongqing Education Commission (No.KJ110512, No. KJ110522 ) and Doctor Foundation of CQUPT(No. A2010-06). References [1]Bayes, Thomas , Price. An Essay towards solving a Problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society of London 1763;53: 370-418. [2]C.E. Shannon. A Mathematical Theory of Communication (Part I). Bell Syst. Techn.J 1948; 27:379-423. [3]G. Frege. Correspondence with Russell, in N. Salmon and S. Soames (eds.), Propositions and Attitudes; 1904. [4]Shafer, Glenn. A Mathematical Theory of Evidence. Princeton University Press;1976. [5]Zadeh, L. A. Fuzzy sets. Information and Control1965;8:338-353. [6]Pawlak, Z. Rough sets. International Journal of Computer and Information Sciences1982;11:341-356 . [7]Li Deyi, Meng Haijun, Shi Xuemei.membership clouds and membership cloud generators. Computer R&D1995;32(6):15-20. [8]Sheng, Z., Xie, S.Q. & Pan, C.Y. Probability Theory and Mathematical Statistics. Beijing, Higher Education Press (in Chinese), 2001. [9]Miao, D. Q., & Wang, J. On the relationships between information entropy and roughness of knowledge in rough set theory. Pattern Recognition and Artificial Intelligence1998; 11(3):34-40. [10]Zadeh, L. A. Fuzzy sets. Information and Control1965;8: 338-353. [11] Li Deyi, Li Changyu, Du Yi, Han Xu.Indeterminate Artificial Intelligence. Chinese Journal of Software2004; 15 (9): 15831594. [12] G.Y.Wang, D.Q. Miao,W.Z. WU, J.Y. Liang. Uncertain knowledge representation and processing based on rough set. Journal of Chongqing University of Posts and Telecommunications (Natural Science Edition) 2010;22(5): 541-544.
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