Fuzzy mathematical models for the analysis of fuzzy systems with application to liver disorders

Transcription

1 IOSR Journal of Computer Engineering (IOSR-JCE) e-issn: ,p-ISSN: , Volume 6, Issue, Ver. VII (Sep Oct. 04), PP Fuzzy mathematical models for the analysis of fuzzy systems with application to liver disorders Rana Waleed Hndoosh, Saneev Kumar, M. S. Saroa 3 Department of Mathematics, Ph.D Student, India. 3 Dept. of Mathematics, Dr. B. R. Ambedar University, India. Dept. of Mathematics, Maharishi Marandeshawar University, India. Abstract: he main obective of this model is to focus on how to use the model of fuzzy system to solve fuzzy mathematics problems. Some mathematical models based on fuzzy set theory, fuzzy systems and neural networ techniques seem very well suited for typical technical problems. We have proposed an extension model of a fuzzy system to N-dimension, using Mamdani's minimum implication, the minimum inference system, and the singleton fuzzifier with the center average defuzzifier. Here construct two different models namely a fuzzy inference system and an adaptive fuzzy system using neural networ. We have extended the theorem for accuracy of the fuzzy system to N- dimensions, and provided a medical application of the fuzzy mathematics models. Since, liver is the largest internal member in the human body, so diagnosing liver disorder disease is a high interest to researchers of the fuzzy modeling and the fuzzy system. herefore, the fuzzy mathematical models are applied on a real data to the Liver Disorder disease. Consequently, a comparison between three models: the FS with Mamdani model, S model, and the ANFIS is made. herefore, we have obtained the best result with the ANFIS. Finally, the programs of these models by using MALAB created and performed. Keywords: Fuzzy system; FIS; ANFIS; Neural networs; Accuracy of the fuzzy system; Liver disorders. I. Introduction Fuzzy mathematics provides the starting point and basic language for fuzzy systems (FSs) and fuzzy modeling (FM) [Ruan and Wang (997)], while the fuzzy mathematical principles are developed by replacing the sets in classical mathematical theory with fuzzy sets (FSs) [Samandar (0)]. he concepts and principles in fuzzy mathematics are useful in FSs and adaptive neuro-fuzzy systems (ANFSs) [Singh et al. (009)], [Shing and Jang (993)]. Fuzzy variables are processed using a system called a fuzzy inference system (FIS) which involves fuzzification, fuzzy inference, and defuzzification [Wang (997)], [Ai and Jayaumar (008)]. he FIS collects the rules in the fuzzy rule-base into a mapping from fuzzy set A X to fuzzy set B Y [Sivanandam and Deepa (008)]. We must construct interfaces that are the fuzzifier and defuzzifier, between the FIS and the environment because in most applications the input and output of the FS are real valued numbers such our application in this model to Liver Disorders [Sug (0) and Gulia et al. (04)]. he reason to represent a fuzzy system in terms of a neural networ is to utilize the learning ability of neural networs to improve performance, such as adaptation of FS [Rameshumar and Arumugam (0)]. When the expert is demonstrating, we measure the inputs and the outputs; that is, we can collect a set of input-output data pairs [Naya (004)], [Hndoosh et al. (0) and (03)]. herefore, the nowledge is transformed into a set of inputoutput pairs. he tas in this wor is to model a FS that describes the input-output behavior represented by the input-output pairs and apply the model to Liver Disorders. We will model the FS by first assigning its structure and then tuning its parameters [Jose et al. (999)]. o simulate the modeling system, need a mathematical model of the Liver Disorders that is described by linguistic variables and its membership functions (MFs) [Chai et al. (009)]. We note that the fuzzy modeler can successfully control and handle the real data of any problem. As well as, the prediction accuracy is improved by defining more FSs for each input variable [Marza and Seyyedi (008)]. he advantage of using the FS is that the parameters of MFs have clear physical meanings and we have models to choose good initial values for them [Doğan et al. (007). We can recover the fuzzy if-then rules that model the FS [Belohlave and Klir (0)]. hese recovered fuzzy if-then rules may help to demonstrate the modeled FS in a user-friendly manner. he wor is divided into four Sections. Section introduces the fundamental concepts and principles in the general field of fuzzy theory that are particularly useful in FSs and ANFIS [Sivanandam and Deepa (008)]. In Section, we have provided the detailed mathematical formulas of the FIS, and we construct interfaces between the FIS and the environment using fuzzifier and defuzzifier models [Jandaghi et al. (00)]. In the first part of this Section, we propose and extend the model of the FS, the wor of Hndoosh et al. (03) and Wang (998), from -dimention to N-dimension using Mamdani's minimum implication with the minimum inference system, the singleton fuzzifier, and center average defuzzifier [Roas (996)]. In the second 7 Page

2 part, we have provided the theorem for accuracy of a proposed model [Hndoosh et al. (0), (03), and (04)]. his approach requires N-pieces of information in order to model FS to satisfy any pre-specified degree of accuracy [Kamel and Hassan (009)]. As well as, we have adapted the structure of the FS and modeled of an adaptive FS using a neural networ through the third part of this Section [Singh et al. (009)], [Shing, and Jang (993)]. In Section 3, we have applied all the previous concepts and models on a real application to Liver Disorders [Sug (0) and Gulia et al. (04)], and we have structured of the applied model at the first part. Second and third subsection, provided discussion and results for the model of the FS with Mamdani and S models and the adaptive FS using neural networ, respectively. Consequently, obtained good results of the models, and created programs for the different models using MALAB. Concluding remars are present in Section 4. Finally, Appendix is provided the representation of results of the FIS with the Mamdani and S models, and the results of the ANFIS with their errors through able 3. II. Proposal Of A New Model Of A Fuzzy System On N-Dimensions In this section, proposed a model of a fuzzy system, that is extension of the wor of Hndoosh et al. (03) and Wang (997), from -dimention to N-dimension [Ruan and Wang (997)]. Consider the general membership function of fuzzy set, A, is a continuous function in R given by: 0 ifx < a a(x) ifa x < b μ x; a, b, c, d = ifb x c () d(x) ifc < x d 0 ifd < x where a, d R and a b c d, 0 a x a non-decreasing function is a, b and 0 d x is a non-increasing function (c, d]. If fuzzy sets A, A,, A N W R then, they are called:. Complete on W, if there exists A such that μ A (x) > 0, for any x W.. Consistent on W if μ A x = for some x W implies that μ A x = 0, for all. 3. Normal, consistent and complete with general MFs, μ A x; a, b, c, d. If A < A < < A N, then c a + < d b +, for =,,, N. In the next section, we will mode a particular type for medical application of Liver Disorders that have some properties [Belohlave and Klir (0)], and consider N-inputs fuzzy systems. Now the proposed model is as the following:. he Proposed Model Let G x be defined G x : X R n R, that is, a function on the compact set X = α, β [α n, β n ] and the analytic formula of G x be unnown. Suppose that for any x U, we can obtain G x. Now, to model a fuzzy system that approximates G x is main tas and model of a fuzzy system as follows: Step : Define N =,,.., n fuzzy sets A, A,, A N α, β, which are normal, consistent, and complete with triangular MFs μ A x ; a, b, c,, μ A N x ; a N, b N, c N, and A < A < < A N with a = b = α and b N = c N = β, which, e N = α, e = β, and e = b for =, 3,, N, e N = α, e = β, and e = b for =, 3,, N, () : e N n = α n, e n n = β n, and e = b for =, 3,, N n. Step : Construct I = N N N n fuzzy if-then rules in the following form: R n X IF x is A and x is A and and xn is A n n hen y is B n, (3) where =,,, N, =,,, N,, n =,,, N n, and the center of the fuzzy set B n, denoted by y n, is chosen as: y n = G e,, e n n (4) his is the case when (3) depends on the Mamdani fuzzy rule [Chai et al. (009)], and the antecedent of our model is connected by and [Sivanandam and Deepa (008)], [Marza and Seyyedi (008)], then the truth-value evaluation is given by: 7 Page

3 θ i = τ μ A n,i x, μ A n,i x,, μ An n,i x n () herefore, from μ B i y = t θ i, μ B i y, y R, the fuzzy inference produces the fuzzy set of output by: μ B n,i y = t θ i, μ B n,i y y R (6) he consequents of all the rules are aggregated in the consequents by the max function as: μ B n y = s(μ B n, y, μ B n, y,, μ B n,i y ) (7) However, when the consequent of rule is a linear function, then the output of the Sugeno rule depends on function as follows: y n = f i x, x,, x n, (8) where f i is linear function base on x, that is defined as: f i x, x,, x n = a i x + a i x + + a i i n x n + a n+, (9) i where a are the parameters, and can be computed by the least square model. Step3: Constructing the fuzzy system f x from the N N N n rules of (4) using Mamdani's minimum implication (MMI) (0a) with the minimum inference system (MIS) (0b), the Singleton fuzzifier (SF) (), and the Center Average Defuzzifier (CAD) (), are as follows: μ QIM x, y = min[μ A x, μ A y ], Q IM X Y (0a) μ B y = max sup min μ A x, μ i A i x,, μ An i x n, μ B i y x X if x = x μ A x = 0 otherwice I y i w y i= i = I i= w i herefore, we obtain: (0b) () () f x = N Nn y n min μ A x,,μ A n (x n ) = n = n N Nn min μ A x,,μ A n (x n ) = n = n (3) Since the fuzzy sets A,, A N are complete at every x X, then there exist,,, n such that: min μ A x, μ A x,, μ An n (x n ) 0. (4) Consequently, the fuzzy system (3) is well defined. From step, we note that the antecedent of the rules (4) constitute all the possible sets of the fuzzy sets defined for each input variable [Hndoosh (03)]. he total number of rules is N n, that increases exponentially with the dimension of the input space [Jandaghi (00)]. In the second part, we explain the accuracy of the f x modeled above on the unnown function G x, is explained [Naya (004)], [Kamel and Hassan (009)]. Here extended the accuracy of the fuzzy system from - dimention to N-dimention, as well as changed the type of fuzzy inference system to a minimum inference system to suit any application [Wang (997)]... heorem (he Fuzzy System Accuracy for the Proposed Model) Let f x be the fuzzy system in (3) and G x be the unnown function in (4). If G x is continuously differentiable on X = α, β α, β [α n, β n ], then: G f G x h + G x h + + G x n h n, () where the infinite norm. is defined as: d(x) = sup x X d(x) and h = max N e + e, =,,,n. Proof: Let X n = [e, e + ] [e, e + ] [e n n, e n n + ], where =,,, N, =,,, N,, n =,,, N n. Since α, β = [e, e ] [e, e 3 ] [e N, e N ], =,,, n., then: N N N n X = α, β α, β α n, β n = X n, (6) = = n = 73 Page

4 which implies that for any x X, there exists X n such that x X n. Now suppose x X n, that is x [e, e + ], x [e, e + ],, x n [e n n, e n n + ] (since x fixed, n are also fixed in the sequel). Since the fuzzy sets A, A,, A N are normal, consistent, and complete, at least one and at most two, μ A x are non-zero for =,, N. From the definition of e =,,, N, these two possible non-zero μ A x s are μ A x and μ A + x. Similarly upto, the two possible non-zero μ An n x n s for n =,, N n are μ An n x n and μ An n + x n. Hence, the fuzzy system f x simplified as the following: f x = + = From (4), we obtain: n + n = n y n min μ A x, μ A x,, μ An n x n of (6) is 7 + n + min μ A x, μ A x,, μ An n = x n n = n + n + min μ A x,, μ An n x n f x = G e,, e n n 8 + n + = n = n min μ A x,..., μ An n x n + n + = n = n min μ A x,.., μ An n (x n ) + n + = n = n min μ A x,.., μ An n = n = n (x n ) = (9) we have: + n + min μ A x,, μ An n (x n ) G x f x G x G e + n +,, e n n = min μ A x,, μ An n (x n ) n = n = n = n max = : + G x G e,, e n n (0) : n = n : n + From the Mean Value model, may be written (0) as: G G(x) f(x) x x e G + x max =, + : n = n, n + x e + + G x n x n e n n () Since x X n, means that x [e, e + ], x [e, e + ] x n [e n n, e n + n ], we have, x e + e e, x e + e e, and xn e n n n + en e n n for =, +, =, +,, and n = n, n + hen () becomes: G x f x G e + e G + e + e G + + e n + n e n n () x Since d(x) = sup x X G f G x x d(x) then G f = sup x X + e G + + max e N, G x G G f, we get: x n max n N n x n e n n + e n n G f G h x + G h x + + G h x n (3) n From (), we can conclude that fuzzy systems in the form of (7). G Specifically, since,, G are finite numbers for any given ε > 0, we can choose h x x, h,.., h n n small enough such that h x + G h x + + G h x n < ε. Hence from () we have: n sup x X G f = G f < ε (4) From (3), we can show that, in order to model a fuzzy system with a pre-specified accuracy, we must now the G bounds of the derivatives of G(x) with respect to x, x,, x n, that is, G,, G. In the x x x n model process, we need to now the value of G x at x, where x = e, e,, e n n for =,,, N, =,,, N,, n =,,, N n. () 74 Page

5 herefore, this approach requires these N pieces of information in order for the model fuzzy system, to satisfy any pre-specified degree of accuracy..3. Model of an Adaptive Fuzzy System Using a Neural Networ In this section, we adapt the structure of the fuzzy system that is specified with the structure of some parameters [Singh et al. (009)], [Shing and Jang (993)]. We specify the structure of the fuzzy system to be modeled. Here, we choose the fuzzy system with a MIS, a SF, a CAD, and a riangular MF [Samandar (0)], then, we obtain: f x = I i= I i= y i min μ i A x min μ A i x = I i= I i= y i min min max min max min i x a b i i a, c i x c i i b, 0 i x a b i i a, c i x c i i b, 0 where I is fixed, and y i, a i, b i i, c are free parameters. he fuzzy system (6) has not been modeled because the parameters y i, a i, b i i, c are not specified [Wang (997)]. In order to determine these parameters in some optimal manner, it is helpful to represent the fuzzy system f x of (6) as a feed-forward networ [Jose et al. (999)]. Specifically, the mapping from the input x U R n to the output f x V R can be performed according to operations [Doğan et al. (007)]. Note that, the input x is passed through a minimum triangular operator to become: i x a b i a i, c i x z i = max min, 0, where c i b i zi are passed through a summation operator. Let K = I i= z i and L = I i= y i z i ; therefore, the output of the fuzzy system is computed as f x = L K. Consequently, we summarize the procedures to model a fuzzy system that depends on layers of networ as the following: Step: Structure specification and initial parameters Select the fuzzy system (6) and determine the number of rule [Rameshumar and Arumugam (0)]. he larger number of rule, results more parameters and more computation, but gives better accuracy. Specify the initial parameters y i (0), a i (0), b i (0), c i (0), then the initial fuzzy system becomes as in (7). hese initial para-meters may be determined according to the linguistic rules from human experts as in our application. (6) f x = N = N n n N = y n (0) min max min N n n min max min x p 0 a n (0) b 3 (0) a 3 (0), x p 0 a n (0) b n (0) a n (0), c n 0 p x0 c n (0) b n (0), 0 c n 0 p x0 c n (0) b n (0), 0 (7) x μ A μ An min z y L f x = L K f y I K x n μ A min z I μ An Figure : Networ representation of the fuzzy system 7 Page

6 Step: Calculating the outputs of the fuzzy system For a given inputs-output pair x p 0 ; y p 0, p =,,.., =,,.., n and at the q th training stage, q = 0,,, present x p 0 to the input layer of the fuzzy system in Figure and compute the outputs of layers, and therefore, we compute: First output: Every node produces MF of an input parameter. he node output o i is explained by: o = μ A x ; o = μ A x,, and o n = μ An n x n, (8) where x, x,, x n are the inputs, μ A, μ A,, μ An n are linguistic fuzzy sets related with nodes, and o i is the degree of MFs of a fuzzy set. Second output: Every node is a fixed node, whose output is the minimum of all MFs: o n = z n, z n = min μ A n x, (9) where μ A n is declared by triangular MF, and then we obtain: z n = min max min x p 0 a n (q) b n (q) a n (q), c n p q x0 c n (q) b n (q), 0 (30) hird output: Depending on (30), the n th node calculates all rules as: o 3 n = z n x = N = z n N n z n n = (3) Fourth output: Every node n is an adaptive node with a node MF of output. o n 4 = z n y n, (3) where y n = f n x, x,, x n, and from (9), we get: y n = α n x + α n x + + α n n x n + α n n+, (33) where α n, =,, n +., is the parameter set of the node. Fifth output: he single node is a fixed node labeled all result o n 4 o n = Suppose, N N = N n n = K = z n = N N n n = N n N = z n N n n z n L = z n α n x + α n x + α n n x n + α n n+ = n = Consequently, the final output is obtained as:, which computes the final output as the summation of α n x + α n x + α n n x n + α n n+ (34) f x = L K. (36) Here, noted that y n are free parameters to be modeled. When select the initial parameters θ(), and there are linguistic rules from experts, then choose y n to be the centers of the then part fuzzy sets in these linguistic rules; otherwise, choose θ() arbitrarily in the output space Y R. In this way, we can say that the initial fuzzy system is constructed from experts. Step3: Update the parameters Use the training algorithm to compute the updated parameters y n q +, a n q +, b n q +, c n (q + ), where y = y 0 p, and z n, K, L and f equal to those that computed in step, i.e., compute the new parameters θ using the least squares model as: (3) 76 Page

7 θ p + = θ p + t p + y p p 0 z x 0 in which, t p + = P p + = P p p P p + z x 0 p P p + z x 0 p P p z x 0 p z x 0 p P p z x 0 p z x 0 θ p 37 +, (38) p P p z x 0 + when p =, note that θ is chosen using step, and P is a large constant. he modeled fuzzy system in (6) with the parameters y n is equal to the corresponding elements in θ p. (39) < ε, or until the q equals a pre- Step4: Repeat by going to step with q = q +, until the error f y p 0 specified number. Step: Repeat by going to step with p = p +, p =,, ; that is, update the parameters using the next input-output pair x p+ 0 ; y p+ 0. Keep in mind that the parameters y n are the centers of the fuzzy sets in the consequent parts of the rules, and the parameters a n and n c are the left and right base points, n b, the centers of the triangular fuzzy sets in the antecedent parts of the rules [Roas (996)]. We can improve the fuzzy if-then rules that modeled the fuzzy system, and improved fuzzy if-then rules may help to explain the model fuzzy system in a user-friendly manner [Ai and Jayaumar (008)]. III. Application Liver is the largest internal member in the human body, and it is nown that the member is responsible for more than one hundred functions of human body. he complexity of this member maes it easily affected by disease of disorder. herefore, diagnosing liver disorder disease (LDD) is a high interest to researchers and doctors [], and fuzzy system has been a good intelligent model to diagnose such disease [Sug (0) and Gulia et al. (04)]. he fuzzy system has very good property that the model is easy to understand. his property of fuzzy system is important in case that human should understand the nowledge structures fully. his is one of the main reasons why fuzzy system is accepted in medical domain. here are six continuous attributes as dependent attributes, (able for detail of the attributes). he first five variables are all blood tests that are thought to be sensitive to liver disorders that might result from excessive alcohol consumption. Each line in the LDD_data constitutes the record of a single male individual. Variable Variable name able : he meaning of variables Meaning Range x mcv mean corpuscular volume [79,03] x alphos alaline phosphotase [3,09] x 3 sgpt alamine aminotransferase [,] x 4 sgot aspartate aminotransferase [,68] x gammagt gamma-glutamyl transpeptidase [,97] x 6 drins number of half-pint equivalents of alcoholic beverages drun per day [0,0] 3.. Structure of the Model Consider the multi-inputs x i, i =,,,6, with output y (disorder types of liver that contains simple liver disorder or acute liver disorder). A fuzzy inference system (FIS) can be defined as: FIS: X Y, where X R n and Y R. (40) he fuzzy system is composed of a fuzzifier, fuzzy rule-base, fuzzy inference, and defuzzifier. In order to apply a steps of FIS systematically, the inputs x, x,, x 6 with output y must be described as the following: 77 Page

8 . Inputs Inputs-data have treated and measured, and it becomes restricted between zero and one. he first five variables have five different degrees of linguistic variables: Low (L), Low Medium (LM), Medium (M), High Medium (HM) and High (H), while sixth input is represented by five linguistic variables, Less (Le), Less Average (LA), Average (A), More Average (MA) and More (Mo).. Output Output-data are represented the disorder types of liver that contains two types: simple liver disorder (SLD) or acute liver disorder (ALD), see Figure. 3.. Discussion and Results of the Fuzzy System with Mamdani and S Models In this part, we build the proposed fuzzy system step by step on our application: Step : Define N, =,,,6, where N =,,3. Let A i,, A i 6 is a fuzzy sets for x i, with triangular MFs, μ A x ; a, b, c,, μ A6 x 6 ; a 6, b 6, c 6, and A < A < < A N with a = b = 0 and b N = c N =. herefore, we can define: e = 0, e =, and e = b, e 3 = b 3, e 4 = b 4 e = 0, e =, and e = b, e 3 = b 3, e 4 = b 4, e 3 = 0, e 3 =, and e 3 = b 3, e 3 3 = b 3 3, e 3 4 = b 3 4, (4) e 4 = 0, e 4 =, and e 4 = b 4, e 4 3 = b 4 3, e 4 4 = b 4 4, e = 0, e =, and e = b, e 3 = b 3, e 4 = b 4, e 6 = 0, e 6 3 =, and e 6 = b 6. Figure a: he FIS of LD using Mamdani model Figure b: he FIS of LD using S model Here the Singleton fuzzifier is defined as: SF: x A, where x = x,, x 6, and the softwaer MALAB is used to create the programs of our application. Six vectors of inputs with one vector of output are loaded for 00 observations. For example, the first observation is represented as [mcv=0.844, alphos= 0.088, sgpt=0.903, sgot= 0., gammagt=0.0303, and drins =0.0]. Consequently, we have built the first rule such as [If (mcv is M) and (alphos is M) and (sgpt is L) and (sgot is LM) and (gammagt is L) and (drins is L) hen y is SLD]. Step: Note that, the fuzzy rule-base consists of I = 6 fuzzy if-then rules and the centers of (f 6,i (x,, x 6 )) are evaluated at the 79 points, and therefore, we obtain: If x is A, and x is A, and and x 6 is A 6, 6 hen y is B 6, R i : If x is A, and x is A, and and x 6 is A 6, 6 hen y is B 6, : If x is A,I and x is A,I and and x 6 is A 6,I 6 hen y is B 6,I antecedent consequent (4a) If x is A, and x is A, and and x 6 is A 6 6, hen y is f 6, x,.., x 6 R i : If x is A, and x is A, and and x 6 is A 6, 6 hen y is f 6, x,.., x 6 : If x is A,I and x is A,I and and x 6 is A 6,I 6 hen y is f 6,I x,.., x 6 consequent antecedent (4b) 78 Page

9 he model (4a) is a system of Mamdani fuzzy rules, while the model (4b) is a system of Sugeno fuzzy rules. he sets A and B are a fuzzy sets, x h (h =,,6) are an input variables, y is the output variable, and i, i =,, I, is the number of rules. he fuzzy set A consists of, A,, A 6 6, fuzzy subsets. It is called linguistic terms that represented by triangular MFs such as () with b = c. he fuzzy operator and (t-norm) is used to connecting between linguistic terms in each rule of the model. he function f... 6,i (x,, x 6 ) is a linear function depends on inputs x that defined using (9). he first five linguistic terms are represented by A h i i =,,, that depends on linguistic variable x h (h =,,), see Figure 3, for e.g. the linguistic term for first input variable defined as the following: (A) he MFs of an I (B) he MFs of an I (C) he MFs of an I3 (D) he MFs of an I4 (E) he MFs of an I (F) he MFs of an I6 (G) he MFs of an output Figure 3: he representation of membership functions of inputs and output A Low μ A (x ; 0,0,0.8), A Low Mediam μ A (x ; 0.,0.,0.4), A 3 Mediam, μ A 3 (x ; 0.3,0.,0.6), A 4 High Medium μ A 4(x ; 0.6,0.73,0.8) and A High μ A (x ; 0.8,,). While, we have represented the sixth linguistic term A 6 i i =,, that depends on linguistic variable x 6 as the following: A 6 Less μ A6 (x 6 ; 0,0,0.), A 6 Less Average μ A6 (x 6 ; 0.08,0.,0.3), A 6 3 Average μ A6 3 (x 6 ; 0.,0.,0.7), A 6 4 More Average μ A6 4 (x 6 ; 0.6,0.7,0.83) and A 6 More μ A6 (x 6 ; 0.8,,). Moreover, the output is described by triangular MFs, (μ B 6 (y)), and its linguistic term that represented as the following: B SLD μ B (y; 0,0,0.7) and B ALD μ B (y; 0.,,). he fuzzy inference process defines as the following: FI: A B 6,i, (43) where A is an input fuzzy set in the input space X, and B 6,i the fuzzy sets in the output space Y. Each one of the rules specifies a fuzzy set B 3,i Y that is given by the compositional rule of inference: B 6,i = A A 6,i B 6,i, where A 6,i = A 6,i A 6,i A 6,i 6 (44) herefore, from μ A i An i x,, x n = μ A i x μ An i x n, we obtain μ A 6,i x = μ A 6,i A 6,i A6 6,i x,, x 6, =,,,6 (4) From μ B y = sup μ B 6,i y = x X x X sup μ A x μ A (x) μ QI x, y t, the fuzzy sets B 6,i are described by MF: t μ A 6,i B x, y (46) 6,i Consequently, can be re-express (46) as the following μ B 6,i y = μ A 6,i B 6,i x, y = Im μ A 6,i x, μ B 6,i y, 47a where Im(. ) is an implementation. Since, we used Mamdani's minimum implication (0a), therefore, we obtained: 79 Page

10 Im μ A 6,i x, μ B 6,i y = min μ A 6,i x, μ B 6,i y 47b or (47) may be written as: μ B 6,i y = min μ A 6,i x, μ B 6,i y (48) he aggregation operator has been applied in order to get the fuzzy set B that uses the functions max (s-norm) or min (t-norm) depending on the type of fuzzy implication. he aggregation operator is denoted by: B I = B 6, i i=, (49) herefore, the membership function of B is computed using the max function as the following: μ B y = max μ i B 3, i y (0) Stepe3: he defuzzifier performs a mapping as the following: def = B f x, () where B is a fuzzy set, f(x) is a crisp point f(x) (Y R). he center of the area is a final system output that is defined by the following formula: f x = = y 6 min max min = 6= 6= min max min =:6 =:6 x a 6 b 6 a 6, x a 6 b 6 a 6, c 6 x c 6 b 6, 0 c 6 x c 6 b 6, 0 () Figure 4a: Representation of the rules editor Figure 4b: he rule view of the FIS he model of fuzzy system f x has been built from the I rules of (4a) using Mamdani's minimum implication (0a) with the MIS (0b), the SF (), and the CAD () (see Figure 4). We have created two programs; the first program depends on the Mamdani model, while the second program depends on the S model. he programs of FIS have applied, that is given by (), and obtained a good result of target. herefore, we have applied the measure for accuracy of all data. he deference between the actual and target outputs is given by the formula as the following: Error = Rv FISv, (3) (a) between I & I4 (b) between I & I6 (c) between I6 & I (d) between I & I3 (e) between I6 & I4 Figure : he output surface for the different inputs 80 Page

11 where Rv is actual output values and FISv is the output target values. he value of accuracy is a very small, where the average error of Simple LD=0.047 and the average error of Acute LD=0.8, when FIS is used the Mamdani fuzzy rule model. While, when the FIS is used the S fuzzy rule model, then the average error of Simple LD=0.463 and the average error of Acute LD= able represents all the results. he surf view tool is the surface viewer that helps view the input-output surface of the FIS. his conception is very helpful to understand how the system is going to behave for the entire range of values in the inputs space. Figure shows the output surface for the different inputs Discussion and Results of the Adaptive Fuzzy System Using Neural Networ In this Section, specify the structure of the fuzzy system to be modeled. Here, we choose the fuzzy system with a MIS, a SF, the CAD, and a riangular MF that given by the model (). Model () has not been modeled, because the free parameters y 6, a 6, b 6, and c 6 are not specified. hese parameters should be determined in order to represent the f x. he Model of the adaptive fuzzy system using neural networ may be express as the following: Step: Determine the initial parameters y 6 0, a 6 0, b 6 0, and c 6 0 according to the linguistic rules from experts, such as when = h =, h =,,6, then a 0 = 0, b 0 = 0, c 0 = 0.8, similarly for h, and and for each input. Step: From a given inputs-output pair x p,0,, x p 6,0 ; y p 0, p (observe), compute the outputs of layers as the following: (i) he node of the first output o i is represented by: o = μ A x, =,, ; o = μ A x, =,,0 ; o 3 = μ A x, 3 = 0,, ; o 4 = μ A x, 4 =,,0 ; o = μ A x, = 0,, and o 6 = μ A x, 6 =,,30, where o i is the degree of MFs of a fuzzy set. (ii) o compute the second output, we should use the minimum function of all MFs as: o 6 = min 6 μ A 6 x, μ A 6 x,, μ A6 6 x 6. (4) Since μ A 6 is a riangular MF, therefore, we obtain: o 6 = min 6 max min x p 0 a 6 (q) b 6 (q) a 6 (q), c 6 p q x0 c 6 (q) b 6 (q), 0. () (iii) For all rules, we have calculated the 6 th node as: o 3 6 = = min 6 max min min max min x p 0 a 6 q b 6 q 6, a q x p 0 a 6 q b 6 q 6, a q c 6 p q x0 c 6 q 6, 0 b q c 6 q p x0 c 6 q 6, 0 b q. (6) (iv) he fourth output depends on a node MF of output z 6 with an adaptive node y 6, therefore, we obtain o 4 6 = o 3 6 f 6 x, x,, x 6 (7) Since f 6 x, x,, x 6 = α 6 x + α 6 x + + α 6 6 x 6 + α 6 7, then we should determine the initial parameters of α 6, =,,7.. (v) In order to compute the final output, we must tae the summation of all results o 4 6 as the following: o 6 = = 6 = = o 6 o 6 6 α 6 x + α 6 x + + α 6 6 x 6 + α 7 6 (8) 8 Page

12 Step3: In step, it is specified the initial parameters of y 6 q,a 6 q, b 6 (q), c 6 (q) and q = 0. Consequently, we update these parameters at q +, using the least squares model and repeat all the procedure in step for compute K, L and f. From (37) to (39), we compute the new parameters as follows: θ = θ + t 0.46 o 6 θ t = P = P o 6 P o 6 o 6 + (9) P o 6 P o 6 o 6 + o 6 Similarly, we can update all parameters of θ for all training data. Repeat all procedures in step with q = q + ttil the last specific number of q is reached. From the program of the AFS using a Neural Networ that is given by model (), we have obtained better results with the AFS using a Neural Networ than the result of the FIS. We have applied the measure of accuracy for all data, and obtained the average error of Simple LD= , while the average error of Acute LD=0.00, (see able ). Additionally, obtained the best average testing error of training data ( ) with epoch 00, and the average testing error of checing data (0.0380), see Figure 6. As well as, able 3 presents the representation of results of the FIS with Mamdani model, the FIS with S models, and the ANFIS with their errors through Appendix. Figure 6a: Representation of the errors of training data with 00 epoch Figure 6b: Representation of the errors of checing data with 00 epoch able : Representation errors of Simple, and Acute Liver Disorder and average error for different models. ype of model Average error of Simple LD Average error of Acute LD Average error of training data FIS (Mamdani model) FIS (S model) ANFIS (S model) IV. Conclusion his wor focused on how to use the fuzzy models to solving fuzzy mathematics problems. Here constructed two different models, namely fuzzy inference system, and adaptive neuro-fuzzy inference system. Further, suggested an extension FS and ANFS at N-dimension those depended on the MMI with the MIS, the SF, and the CAD. It is provided the theorem for accuracy of proposed models, as well as, adapted the model of an adaptive FS using a neural networ. In addition, we have provided a medical application of the fuzzy mathematics models. We have diagnosed the liver disorder disease that is a high interest to researchers of fuzzy modeling and fuzzy system because the liver is the largest internal member in the human body. herefore, we have applied the fuzzy mathematical models on the real data to the liver disorder disease. We have presented discussion and results for the model of the FS with Mamdani and S models, and the ANFIS, respectively. Additionally, we have used the software MALAB in order to perform the results of different models. Consequently, we have gotten good results for accuracy of these models. he comparison between the three models the FS with Mamdani model, the S model, and the ANFIS had presented. We have obtained the best 8 Page

13 result with the ANFIS. Finally, we have presented the representation of results of the different models with their errors through Appendix. In the future wor, we can develop these models of fuzzy system to generate many outputs or extending the number of input variables. As well as, we can change the fuzzy inference system with another types, or with different type of fuzzifier or defuzzifier. Acnowledgements he author wishes to than the reviewers for their excellent suggestions that have been incorporated into this paper. I would lie to acnowledge my profound gratitude to my research supervisors DR. Saneev Kumar, a well-nown mathematician and researcher and DR. M. S. Saroa for their vigilant supervision as well as for pouring many invaluable pearls from the deep ocean of nowledge. No volume of words is enough to express my gratitude towards my guide. APENDIX In this Section, present the representation of results of the FIS with Mamdani model, the FIS with S models, and the ANFIS with their errors. N0. Real able 3: Representation of results of FIS and ANFIS with their errors FIS with (Mamdani model) E(FIS M ) FIS with ( S model) E(FIS S ) ANFIS with (S model) E(ANFIS S ) E E E E E E E E E E E E E E E E E-07.8E E-0 -.8E E E E-0.00E E-06 -.E E E E E E E E E E E E E E-08.00E E E E E E E E E E E E E E-08.0E E-08.0E E E E E E-07.07E E E E E E E E E E E E E E E E-07.37E E E Page

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