Bulanık kümeler ve meteoroloji uygulamaları
Fuzzy sets and applications of meteorology
- Tez No: 66676
- Danışmanlar: PROF. DR. ZEKİ ŞEN
- Tez Türü: Yüksek Lisans
- Konular: Meteoroloji, Meteorology
- Anahtar Kelimeler: Bulanık kümeler, Fuzzy sets
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Meteoroloji Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmada, oldukça karmaşık yapılı ve belirsizlik içeren atmosferik çalışmalarında bulanık kümeler temelli bulanık sistem yaklaşımının meteoroloji alanında uygulanabilirliği araştırılmıştır. Dolayısıyla, bulanık kümelerin bilimsel serüveni ve tarihsel gelişimi açık literatüre bağlı olarak araştırılarak, adım adım bulanık sistem inşaası amacıyla,. Belirsizliğin tarihsel gelişimi ve modern tanımlaması ile felsefi boyutunun bilim dünyasındaki,. Belirgin kümeler ve belirgin kümelerin dayandığı mantık, belirgin kümelerin ardındaki matematik çerçeve ve gerçek-dünya sistemlerindeki başarısızlığı,. Belirgin kümelerin başarızlığından dolayı, belirgin kümeleri de kapsayan en genel anlamda bulanık kümelerin tanıtılması ve matematik çerçevesi,. En genel anlamda sistem tanıtımı ve doğrusal-olmayan sistemlerin zorunlu gerekliliği, dolayısıyla doğrusal-olmayan bulanık sistemlerin matematik çerçevesi ve uygulanan yöntemleri,. Bulanık sistemlerin adım adım inşaası için gerekli ve yeterli koşullar ve gözlemlerin (veri) bundaki rolü,. Meteoroloji gibi oldukça karmaşık yapılı ve fiziğinin izahı zor olduğu yer bilimlerinde bulanık kümeler temelli bulanık sistem yaklaşımlı modellemenin matematik çerçevesi ve sistem düzeneklerinin ortaya çıkartılması, olmak üzere izlenecek yollar tespit edilerek zaman serilerinin medellenmesi- öngörüsü ve Van Gölü'nün su seviyesinde meydana gelen düzensiz çalkantılarının hidro-meteorolojik büyüklükler ile olan bağlantıları, bulanık sistem yöntemleri ile incelenmiştir. Yapılan bu çalışma sonucu, bu yaklaşımın geleneksel yaklaşımlardan daha etkili ve doğal yapıya yatkm-yakm olduğu sonucuna varılmıştır. xıu
Özet (Çeviri)
SUMMARY FUZZY SETS AND APPLICATIONS OF METEOROLOGY Fuzzy logic was first subjected to technical scrutiny in 1965 when Dr. Zadeh published his seminal work,“ Fuzzy Sets,”in Information and Control journal. Since that time the subject has been the focus of many independent research investigations by mathematicians, scientists, and engineers from around the world. In this and subsequent chapters, fuzzy logic will focus on current summary of the thesis, and application part of it, case studies, and theoritical back ground of fuzzy sets and fuzzy systems for engineering related problems and mayor applications in meteorological field. The subsequent parts of the thesis are given as in the flow- diagram. Figure. S.l Flow-diagram of interconnections between individual chapters XIVAs can be seen from above flow-diagram, the thesis can be examined via independent chapters (and the chapter-5 alone) or as seen from flow-ordering related through chapters. Let us make a quick view of contents in the thesis based on the flow-diagram. Fuzzy Sets: In this part, fuzzy sets are introduced as generalizations of conventional set theory that were introduced in 1965 by Lütfi (or Lotfi) Asker Zadeh. Fuzzy interpretations of data structures are natural and intuitively plausible way to formulate and solve various problems in real-world situations. The basic idea of fuzzy sets is simple. Suppose one approaches a red light and must advise a learner driver when to apply the brakes. Would he/she say“begin braking 25 meters from the cross walk?”or“begin braking pretty soon?”or would her/his advice is too precise to be implemented?. This illustrates that vagueness does not necessarily weaken utility; the latter phrase is more useful than the former. Natural language is used, and is propagated in every-day life. Imprecision in data and information gathered from and about our environment is either statistical (e.g. coin toss)-the outcome is a matter of chance; or non statistical (e.g.“apply the brakes pretty soon”) this latter type of uncertainty is called fuzziness (Bezdek, 1993). Conventional sets contain objects that satisfy precise properties required for membership. The set of number A from 5 to 15 is crisp; we write A={x ?X 1 5< x <15}. Equivalently A is described by its membership (or characteristic value) function, p.^:x -» {0,1}, defined as l,5<x<15 ^H, otherwise/ <T The crisp set A and the graph of p.^ are shown in the left half Figure S.2. In logic, values of jj.^ are called“truth values”with reference to the question“is x in X?”. The answer is 'yes' if and only if ua(x)=1 an^ 'no' otherwise. Next, consider the set of F real numbers in X that are close to ten. Because the property“ close to 10”is fuzzy, XVthere is not a unique membership function for F. Rather, the modeler decides, based upon the potential application and properties that seem plausible for this particular fuzzy set might include: i) normality [jj.F(10) = lj; ii) monotonicity [the closer x is 10, the closer uF(x) is to 1, and conversely]; and iii) symmetry [numbers equally for left and right of 10 should have equal membership] (Figure S.2). Fuzzy Sytems: Since the introduction of fuzzy set theory by Zadeh (1965), there have been a number of applications based on this theory, such as“applications of fuzzy set theory (Maier and Sherif, 1985)”and“Fuzzy logic in control systems (Lee, 1990a,b)”. Figure S.2 Membership functions for crisp and fuzzy subsets of X In the most of these applications, the main design objective is to construct a fuzzy system to approximate a desired control or decision up to a given level of accuracy. Since the desired control or decision and the fuzzy system can be denoted as functions on some appropriate input and output spaces, such as a design objective is in function to find a mapping from the input space to output space which can approximate the desired control or decision function within a given accuracy. In other words, the problems of designing fuzzy systems can be considered as approximation problems (for detail, see Chapter 4). In pratical applications, when we start to design a fuzzy system, it is helpful if we know the basic mechanism how a fuzzy system approximates a desired control or decision function. In this thesis, it is tried to reduce basic problems of XVIdeciding on control or decision functions to find the“reasoning surface”which was first introduced to fuzzy systems, symbolically shown as g(x) by Kosko (1992). From beginning to end of this thesis, a pedagogical-order way is followed, step-by-step in order to build a fuzzy system. Thus the chapters can be examined alone for understanding fuzzy systems building if one has known fuzzy sets to a certain extant. First of all, a fuzzy system can be formulated via mathematical functions as fuzzy basis functions. A multi-input multi-output (MIMO) fuzzy systems can always be separated into a group of multi-input single-output (MISO) fuzzy systems (Lee, 1990a,b). It is assumed in this thesis that fuzzy systems are MISO systems f:UcR'->VcR where U=UixU2X...xUnczRn is the input space and Vc R is output space. Consider a fuzzy system which is comprised of four components (see, Chapter, 4), namely, fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzifier is the most commonly used singleton fuzzifier and that the rule base consist of N rules as in following form: Ri = IF xi is Aji and x2 is Aj2 and...and x;n is A;n THEN y is Q, i=l,2,...,N (S.2) where x; ( j=l,2,...,n ) are the input variables to the system, y is the output variable of the fuzzy system, and fuzzy sets Ay in U; and Cj in V are linguistic terms characterized by fuzzy membership functions Ajj(xj) andCj(y), respectively. Each Rj can be reviewed as a fuzzy implication (relation) (Zadeh, 1973). Aj =AjixAj2x...xAjn - »Q, which is a fuzzy set in U x V = Uj x U2x...xUn x V with membership function Ri(X,y) = Ail(x1)*Ai2(x2)*...*Ain(xn)*Ci(y), where '*' the T-norm (Gupta,1991), X= (xi,x2,...,xn)eU and ye V. The fuzzy inference engine is a decision making logic which employs, fuzzy rules from the fuzzy sets in input space U to the fuzzy sets in output space V. Let A be an arbitrary xviifuzzy set in U, then each Rj of (S.2) determines a fuzzy set V^0r in V based on the sup-star composition ( Lee, 1990a,b., Wang and Mendel, 1992): VAoR1(y)=Sup[A(X)*Ri(X,y)] = Sup[A(X)*Ail(x1)*Ai2(x2)*...*Ain(xn)*Ci(y)] (S3) XeU In this thesis, it is assumed that '*' is the (algebraic) product, one of the most commonly used T-norm in applications, then sup-star composition in the fuzzy inference engine becomes a sup-product composition which is the simplified form of (S.3) as; VAor. (Y) = Sup[A(X)Ail(x1)Ai2(x2)...Ata(xn)Ci(y)] (S.4) XeU Finally, the defuzzifier is a mapping from nth fuzzy sets in V to crisp points in V. It is choosen the most commonly used defuzzifier in applications, i.e. the center-average defuzzifier ( Lee, 1990a,b., Wang, 1993) which maps the fuzzy set Va0r in V to a crisp point y e V in the following way: N ŞyiVAoR,(y,) y = ^(s.5).JVAoRl(yl) where yj is the point in V at which Q (y) reaches its maximum value (if Q is a normal fuzzy set, then Q (yj )=1. It is always assumed that Q is a normal fuzzy set). Furthermore, from (S.4) fuzzy system can be expressed as following xvmy = f(X) = f(x1,x2,...,xn) N i=l n AjjCxj) Lj=i N z i=Hj=i IlAijCxj) (S.6) N = E i=l n Aij(xj) j=i N n z nAjj(xj) Li=lj=l J J. where X= (xi,x2,...,xn )?Ui XU2 x..xUn =UcRn, y; eVc R is the point for which Q (y) reaches its maximum value and Ay is a fuzzy set in Uj(j = 1,2,...,n; i = 1,2,...,N. Essentially, under assumptions that fuzzifier is singleton fuzzifier, the T-norm in fuzzy implication and inference is a product inference, the defuzzifier is center-average defuzzifier and Q (i=l,2,...,N) in the rule base are normal fuzzy sets, and the fuzzy system with rule base is expressed as, Ri = IF xi is An and...and xn is Aj“ THEN y is Q, i=l,2,..,N (S.7) Hence, (S.5) becomes y = f(X) = N i = l n Aij(xj) J=l N n s n a^xj) i=lj=l (S.8) From eq. (S.7), the function in the square-bracket can be defined as the basic functions of the fuzzy system (Wang and Mendel, 1992). An alternative form of (S.8) was introduced by Kosko (1992) as Fuzzy Associative Memory (FAM) in order to describe a type of fuzzy system modeling introduced by XIXZadeh(1973) and Mamdani (1974). In the FAM approaches, instead of (S.8), FAM is a combination of the outputs of individual primary stimulus-response pairs (A;, Bj ) (where Aj and Bj are normal fuzzy sets) that make up memory. In particular, if Xi = poss(Aj /D), one gets F(y) = max [B; (y)A t; ] or if xt is the degree of association between input D and the i-th stimulus then the output of the FAM is F, where F =Fj and where F; (y)=Xj ABj(y). One can see that F is a kind of ”weighted“ combination of the individual responses (Yager, 1993). Classification of Non-Linear Systems: In this section, non-linear systems are classified briefly, and they are necessary to set differences of linear systems through interacting with fuzzy systems as fuzzy inference as a non-linear mapping. The aim, therefore is to develop new non-linear models that are able to describe mathematically or verbally (i.e. fuzzy systems) the underlying physical action of a system better than conventional traditional approaches (such as classic or modern control-modeling techniques). There are number of reasons for the use of non-linear analysis techniques instead of purely linear or classic methods. Some are given here.. Nature is in general not linear: Nature and the environment about us by no means linear or purely defining mathematically in fact show strong tendencies towards human like recognition of non-linear behavior and are best than piece-wise linear and purely mathematical.. Clipping or Saturation: Physical systems that operate over a wide dynamic range often exhibit highly non-linear or like human face thinking behavior in the form of clipping or saturation.. Discrete Data: All data processed by digital systems are discrete in both temporal- spatial variations and amplitude. This quantisation is of course a non-linear transformation. XX. Better Representation: Computational recourses to support more complex schemes are available (e.g. see, Chapter-4 for details).. Failure of Linear and Traditional Models: There are definite cases where a ”linear and traditional systems"approaches will fail. The fact that many real-life systems connot be adequately represented by linear models or mathematical-models and like human systems can not be represented by traditional (classic or modern mathematical control like modeling) techniques, so why one needs fuzzy systems like modeling was tried to answer to this application of fuzzy set theory approaches in meteorology like geophysical fields. It was noticed as in the above section that nature is non-linear. So one of the most complex system is the atmosphere like geophysical systems. In this work, we concentrate on the time series analysis and zero-memory systems modeling via fuzzy techniques. Equation (S.8) is remodified fuzzy system so as to estimate the hydro- meteorological components which might be thought as influencing the irregular water level fluctuations of Van Lake. Final remodification yields N y = f(X) = I i=l n - r n Aij(xj) N n - r S n Ay(Xj) i=l j=l J J y i (s.9) where N is the number of rules for Van Lake system based on fuzzy algorithm, and n is the all components (i.e. number of the input) which might be thought as influencing Van Lake system, and r is the unknown components of lake model-system. If 'r' is changed then we can determine the influenced component by using a proper criterion which measures the quality of Van Lake model-system, named as simplified unbiasdness criterion (SUC). It is defined as, a proper criteria for the verification of fuzzy model structure as shown by Sugeno and Kang(1988) but in this thesis a simplified version of the unbiasedness criterion is employed as suggested by Kazuo, et. al (1995). The simplified UC is expressed as XXISUC=- £|y,Bf.Hi 00 (S.10) nB !=i yIB where y; b denotes real output system, ne is the number of data in the data set Na, yiBA is the estimated output for the data set Na from model identified using the data set NA. XXU
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