A new fuzzy regression model based on interval valued fuzzy neural network and its applications to management

A new fuzzy regression model based on interval valued fuzzy neural network and its applications to management Accepted Manuscript A new fuzzy regression model based on interval valued fuzzy neural net[.]

Accepted Manuscript A new fuzzy regression model based on interval-valued fuzzy neural network and its applications to management Somaye yeylaghi, Mahmood Otadi, Niloofar Imankhan PII: DOI: Reference: S2314-8535(16)30163-9 http://dx.doi.org/10.1016/j.bjbas.2017.01.004 BJBAS 175 To appear in: Beni-Suef University Journal of Basic and Applied Sciences Received Date: Revised Date: Accepted Date: 19 November 2016 January 2017 16 January 2017 Please cite this article as: S yeylaghi, M Otadi, N Imankhan, A new fuzzy regression model based on intervalvalued fuzzy neural network and its applications to management, Beni-Suef University Journal of Basic and Applied Sciences (2017), doi: http://dx.doi.org/10.1016/j.bjbas.2017.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ŶĞǁĨƵnjnjLJƌĞŐƌĞƐƐŝŽŶŵŽĚĞůďĂƐĞĚŽŶŝŶƚĞƌǀĂůͲǀĂůƵĞĚĨƵnjnjLJŶĞƵƌĂů ŶĞƚǁŽƌŬĂŶĚŝƚƐĂƉƉůŝĐĂƚŝŽŶƐƚŽŵĂŶĂŐĞŵĞŶƚ ^ŽŵĂLJĞLJĞLJůĂŐŚŝ

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 ĞƉĂƌƚŵĞŶƚŽĨDĂŶĂŐŵĞŶƚ͕&ŝƌŽŽnjŬŽŽŚƌĂŶĐŚ͕/ƐůĂŵŝĐnjĂĚ hŶŝǀĞƌƐŝƚLJ͕&ŝƌŽŽnjŬŽŽŚ͕/ƌĂŶ  ĞƉĂƌƚŵĞŶƚŽĨDĂƚŚĞŵĂƚŝĐƐ͕&ŝƌŽŽnjŬŽŽŚƌĂŶĐŚ͕/ƐůĂŵŝĐnjĂĚ hŶŝǀĞƌƐŝƚLJ͕&ŝƌŽŽnjŬŽŽŚ͕/ƌĂŶ ďƐƚƌĂĐƚ͗ In this paper, a novel hybrid method based on interval-valued fuzzy neural network for approximate of interval-valued fuzzy regression models, is presented The work of this paper is an expansion of the research of real fuzzy regression models In this paper interval-valued fuzzy neural network (IVFNN) can be trained with crisp and interval-valued fuzzy data Here a neural network is considered as a part of a large field called neural computing or soft computing Moreover, in order to find the approximate parameters, a simple algorithm from the cost function of the fuzzy neural network is proposed Finally, we illustrate our approach by some numerical examples and compare this method with existing methods  A new fuzzy regression model based on interval-valued fuzzy neural network Abstract In this paper, a novel hybrid method based on interval-valued fuzzy neural network for approximate of interval-valued fuzzy regression models, is presented Here a neural network is considered as a part of a large field called neural computing or soft computing Moreover, in order to find the approximate parameters, a simple algorithm from the cost function of the fuzzy neural network is proposed Finally, we illustrate our approach by some numerical examples and compare this method with existing methods Keywords: Interval-valued fuzzy neural networks; Interval-valued fuzzy regression model; Feedforward neural network; Learning algorithm Introduction The concept of fuzzy numbers and fuzzy arithmetic operations were first introduced by Zadeh,1975, Dubois and Prade,1978 Also, fuzzy systems are used to study a variety of problems ranging from (Zhang et al.,2014a,Zhang et al., 2014b) We refer the reader to (Kaufmann and Gupta, 1985) for more information on fuzzy numbers and fuzzy arithmetic Regression analysis is of the most popular methods of estimation It is applied to evaluate the functional relationship between the dependent and independent variables Fuzzy regression analysis is an extension of the classical regression analysis in which some elements of the model are represented by fuzzy numbers Fuzzy regression methods have been successfully applied to various problems such as forecasting (Chang,1997,Chen and Wang,1999, Kao,2003, Tanaka,1989,Tseng,2002) and engineering (Lai and Chang,1994) Thus, it is very important to develop numerical procedures that can appropriately treat fuzzy regression models Sakawa and Yano (1992) proposed a mathematical programming model to estimate the parameters of a fuzzy linear regression Yi = A1 xi1 + A2 xi2 + An xin , where xij ∈ R and A1 , A2 , , An , Yi are symmetric fuzzy numbers for i = 1, 2, , m, j = 1, 2, , n Ishibuchi et al (1995) proposed a learning algorithm of fuzzy neural networks with triangular fuzzy weights and Hayashi et al (1993) fuzzified the delta rule Buckley and Eslami (1997) consider neural net solutions to fuzzy problems The topic of numerical solution of fuzzy polynomials by fuzzy neural network investigated by Abbasbandy et al (2006,2008), consists of finding solution to polynomials like a1 x+a2 x2 + .+an xn = a0 where x ∈ R and a0 , a1 , , an are fuzzy numbers, and finding solution to systems of s fuzzy polynomial equations where x1 , x2 , , xn ∈ R and all coefficients are fuzzy numbers Then Mosleh and Otadi (2010,2011,2012,2014) proposed a fuzzy neural network model to estimate the parameters of a fuzzy regression models In this paper, we first propose an architecture of interval-valued fuzzy neural network with interval-valued fuzzy weights for real input vectors and interval-valued fuzzy targets to find approximate coefficients to intervalvalued fuzzy linear regression model ˜ i = A˜ ˜0 + A˜ ˜1 xi1 + + A˜ ˜n xin , Y˜ where i indexes the different observations, xi1 , xi2 , , xin ∈ R, all coeffi˜ i are interval-valued fuzzy numbers cients and Y˜ 2.1 Basic concepts of fuzzy numbers Generalized fuzzy numbers In this section, we briefly review basic concepts of generalized fuzzy numbers Chen and Wei (1999,2009) represented a generalized triangular fuzzy number represented a generalized triangular fuzzy number A˜ as A˜ = (a1 , a2 , a3 ; w) where a1 , a2 and a3 are real values and < w ≤ 1, as shown in Fig The membership function µA˜ of a generalized fuzzy number A˜ satisfies the following conditions: (1) µA˜ is a continuous mapping from the universe of discourse R to the closed interval in [0, 1]; (2) µA˜ = 0, where −∞ < x ≤ a1 ; (3) µA˜ is monotonical increasing in [a1 , a2 ]; (4) µA˜ = w, where x = a2 ; (5) µA˜ is monotonical decresing in [a2 , a3 ]; (6) µA˜ = 0, where a3 ≤ x < +∞ If w = 1, then the generalized fuzzy number µA˜ is a normal fuzzy number, denoted as µA˜ = (a1 , a2 , a3 ) If a1 = a2 = a3 and w = 1, then µA˜ is a crisp value 2.2 Interval-valued fuzzy numbers and their arithmetic operations Gorzalczany (1987) proposed the concept of IVFS Then, Yao and Lin ˜ = [A˜ ˜L , A˜ ˜U ] shown in Fig where A˜ ˜L denotes (2002) represented the IVFS A˜ ˜ ˜ ¯U denotes the upper IVFS, where ≤ A˜L ≤ A˜U ≤ 1, and the lower IVFS, A˜ ˜L ⊂ A˜ ˜U Thereby, the minimum and maximum membership value of A˜ ˜ are A˜ ˜ ˜ L U ˜ ˜ A and A , respectively We denote the set of all IVFS with EI ˜ conFrom Fig 3, we can see interval-valued triangular fuzzy number A˜ ˜ sists of the lower triangular fuzzy number A˜L and the upper triangular fuzzy ˜U be two generalized triangular fuzzy numbers, ˜U Let A˜ ˜L and A˜ number A˜ ˜L and A˜ ˜U , respectively A IVFN and let w L˜˜ and W U˜˜ denote the heights of A˜ A A A defined in the universe of discourse X is represented by the following: L L L U U U ˜ = [(aL A˜ ), (aU )] , a2 , a3 ; w ˜ , a2 , a3 ; w ˜ ˜ ˜ A A ˜L and A˜ ˜U can be expressed as follows: The membership functions of A˜  L w (x−aL )   A˜˜L L1 , f or aL ≤ x ≤ aL ,   a2 −a1        L  f or x = aL  wA˜˜ , 2, ˜ L ˜ A =   wU˜ (aL −x)  ˜  L A  f or aL  L −aL , ≤ x ≤ a3 , a        0, otherwise; ˜U = A˜ (1)  U w ˜ (x−aU )  ˜ U A  , f or aU  ≤ x ≤ a2 ,  aU −aU        U  f or x = aU  wA˜˜ , 2, (2)   wU˜ (aU −x)  ˜  U A  , f or aU  U ≤ x ≤ a3 , a3 −aU        0, otherwise ˜ ˜ h = [[A˜ ˜L ]h , [A˜ ˜U ]h ] as follows: Denote the h-level sets of [A]  ˜L ]r ], [[A˜ ˜U ]l , [A˜ ˜U ]r ]], f or < h ≤ w L ,  [[[A˜˜L ]l , [A˜ ˜ h h h h ˜L ] , [A˜ ˜U ] ] = ˜ A [[A˜ h h  [[A˜˜U ]l , [A˜ ˜U ]r ], f or wL ≤ h ≤ w U , h ˜ ˜ A h ˜ ˜ A (3) ˜ ˜ ˜ U l L r L l ˜ ˜ ˜ where [A ]h and [A ]h are left hand side of the h-cut, and [A ]h and L L ˜U ]r are right hand side of the h-cut Also, [A˜ ˜L ]l = aL + (a2 −aL1 )h , [A˜ h h w ˜L ]r = aL − [A˜ h L (aL −a2 )h , wL˜ ˜ A ˜U ]l = aU + [A˜ h U (aU −a1 )h wU˜ ˜ A ˜U ]r = aL − and [A˜ h ˜ ˜ A U (aU −a2 )h wU˜ ˜ A ˜ ˜ and B ˜ are two interval-valued fuzzy numbers We Definition Let A˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜L = B ˜ L and A˜ ˜U = B ˜U say that A is equal to B if and only if A˜ ˜ Assume that there are two interval-valued triangular fuzzy numbers A˜ ˜ ˜ and B, where ˜ = [(aL , aL , aL ; wL˜ ), (aU , aU , aU ; wU˜ )], A˜ 3 ˜ ˜ A A ˜ L L L U U U ˜ = [(bL B ), (bU )], , b2 , b3 ; w ˜ , b2 , b3 ; w ˜ ˜ ˜ B L L L U U U L L L U U a1 , a2 , a3 , a1 , a2 , a3 , b1 , b2 , b3 , b1 , b2 wU˜˜ ≤ and ≤ w L˜˜ ≤ wU˜˜ ≤ B B A B and bU are real values, ≤ w L˜˜ ≤ A The arithmetic operations between ˜ ˜ and B ˜ are reviewed from the interval-valued triangular fuzzy numbers A˜ (1985,1985) as follows: (1) Interval-valued fuzzy numbers addition: ˜ ˜+B ˜ = [(aL + bL , aL + bL , aL + bL ; min{w L , wL }) A˜ 1 2 3 ˜ ˜ ˜ ˜ A (4) B U U U U U U , w U })] , (aU + b1 , a2 + b2 , a3 + b3 ; min{w ˜ ˜ ˜ ˜ A B (2) Interval-valued fuzzy numbers subtraction: ˜ ˜−B ˜ = [(aL − bL , aL − bL , aL − bL ; min{w L , wL }) A˜ 2 ˜ ˜ ˜ ˜ A (5) B U , w U })] U U U U U , (aU − b3 , a2 − b2 , a3 − b1 ; min{w ˜ ˜ ˜ ˜ A B (3) Interval-valued fuzzy numbers multiplication by a crisp number (q is a nonzero number):  L L L L U U U U   [(q.a1 , q.a2 , q.a3 ; wA˜˜ ), (q.a1 , q.a2 , q.a3 ; wA˜˜ )], if q ≥ 0, ˜= q.A˜ (6)   (q.aL , q.aL , q.aL ; hL ), (q.aU , q.aU , q.aU ; hU )], if q < ¯ ¯ 3 A A 4) Interval-valued fuzzy numbers by an increasing function: ˜et) = f ([N˜ ˜etL , N˜ ˜etU ]), f (N˜ (7) with the h−level sets:  ˜ ˜ ˜ ˜ ˜ Ll ˜ Lr ˜ U l ˜ U r    [[f ([N et ]h ), f ([N et ]h )], [f ([N et ]h ), f ([N et ]h )]], ˜ f or ≤ h ≤ w L˜˜ , [f (N˜et)]h = N et    [f ([N˜ ˜etU ]l ), f ([N˜ ˜etU ]r )], f or wL ≤ h ≤ w U h h ˜ ˜ A ˜ ˜ A (8) Definition Let I be a real interval A mapping g : I → EI is called a interval-valued fuzzy process We denote ( [[[g˜ ˜L (x)]lh , [g˜ ˜L (x)]rh ], [[g˜ ˜U (x)]lh , [g˜ ˜U (x)]rh ]], f or < h ≤ wg˜˜L(x) , ˜(x)]h = [g˜ [[g˜ ˜U (x)]lh , [g˜ ˜U (x)]rh ], f or wg˜˜L(x) ≤ h ≤ wg˜˜U(x) , for x ∈ I Regression model ˜ ˜ We have postulated that the dependent interval-valued fuzzy variable Y, is a function of the independent real variables x1 , x2 , , xn More formally ˜ f˜ : Rn −→ EI, ˜(xi1 , xi2 , , xin ), ˜i = f˜ Y˜ where i indexes the observations The objective is to estimate a interval-valued fuzzy linear regression (IVFLR) model, express as follows: ˜ + A˜ ˜ x + + A˜ ˜ x ˜ i = A˜ Y˜ i1 n in (9) ˜ : The extension principle leads to the following definition of Y˜ i ˜ (τ ), A˜ ˜ (τ ), , A˜ ˜ (τ )}|y = τ + τ x + + τ x } ˜ i (y) = sup{min{A˜ Y˜ 0 1 n n i1 n in ˜ When f˜ : R −→ EI, we might it by eye-fitting the line that looks best to us Unfortunately, different people will draw different lines and it would be nice to have a formal method for finding the line that would consistently provide us with the best line possible What would a “best possible line” look like? Intuitively, it would seem to have to be a line that fit the data well That is, the distance of the line from the observations should be as ˜1 , , A˜ ˜n denote the list of regression coeffismall as possible Let A˜˜0 , A˜ ˜ ˜1 , , A˜ ˜n cients (parameters) A˜0 is an optional intercept parameter and A˜ are weights or regression coefficients corresponding to xi1 , , xin Then interval-valued fuzzy linear regression is given by Eq (9) where i indexes ˜0 , A˜ ˜1 , , A˜ ˜n are interval-valued fuzzy numthe different observations and A˜ ˜ ˜ ˜n of interval-valued fuzzy bers We are interested in finding A˜0 , A˜1 , , A˜ ˜ linear regression such that Y˜ i approximates Yi for all i = 1, 2, , m, closely enough according to some norm k.k, i.e., L L ˜ i ]l k, ˜ i ]l − [Y˜ k[Y˜ h h < h ≤ w L˜˜ , Yi L L ˜ i ]r k, ˜ i ]r − [Y˜ k[Y˜ h h < h ≤ w L˜˜ , Yi U U ˜ ]l k, ˜ ]l − [Y˜ k[Y˜ i h i h < h ≤ w L˜˜ , Yi U U ˜ i ]r k, ˜ i ]r − [Y˜ k[Y˜ h h < h ≤ w L˜˜ , Yi U U ˜ i ]l k, k[Y˜˜ i ]lh − [Y˜ h U U ˜ i ]r k, k[Y˜˜ i ]rh − [Y˜ h wL˜˜ Yi ≤ h ≤ w U˜˜ , wL˜˜ Yi ≤ h ≤ w U˜˜ Yi Yi (10) Then, it becomes a problem of optimization A IV F N N4 (interval-valued fuzzy neural network with interval-valued fuzzy weights, output signals and real inputs) solution to Eq (9) is given in Fig The input neurons make no change in their inputs and the input signals interact with the weights, so the input to the output neuron is ˜ x + + A˜ ˜ x , ˜0 + A˜ A˜ i1 n in and the output, in the output neuron, equals its input, so ˜ i = A˜ ˜0 + A˜ ˜1 xi1 + + A˜ ˜n xin Y˜ How does the IV F N N4 solve the interval-valued fuzzy linear regression? The training data are {(1, x11 , , x1n ), , (1, xm1 , , xmn )} for inputs and target (desired) outputs are {Y1 , , , Ym } We proposed a learning algorithm from the cost function for adjusting fuzzy number weights Following Section, we proposed a learning algorithm such that the network can approximate the fuzzy A0 , A1 , , An of Eq (9) to any degree of accuracy Learning algorithm Consider the learning algorithm of the two-layer fuzzy feedforward neural network with inputs and one output as shown in Fig Let the h-level sets of the target output Yi , i = 1, , m be denoted ˜i ]h = [[Y˜ ˜iL ]h , [Y˜ ˜iU ]h ], [Y˜ i = 1, , n, (11) where YiL (h) shows the left-hand side and YiU (h) the right-hand side of the h-level sets of the desired output A cost function to be minimized is defined for each h-level sets as follows: ˜ ˜ ˜ ˜ ,W ˜ , ,W ˜ )]l + [E(W ˜ ,W ˜ , ,W ˜ )]r , ˜ 0, W ˜ 1, , W ˜ n )]h = [E(W ˜ ˜ ˜ ˜ ˜ ˜ [E(W n h n h (12) where L U L U m X ˜ i ]l − [Y˜ ˜ i ]l − [Y˜ ˜ L ]l )2 ([Y˜ ˜ U ]l )2 ([Y˜ ˜ ˜ ˜ i h i h h h ˜ 0, W ˜ 1, , W ˜ n )]l = [E(W + , h 2 i=1 for < h ≤ w L˜˜ , Yi m ˜ L ]r )2 ([Y˜ X ˜ i ]r − [Y˜ ˜ i ]r − [Y˜ ˜ U ]r )2 ([Y˜ ˜ ˜ ˜ i h i h r h h ˜ ˜ ˜ + , [E(W0 , W1 , , Wn )]h = 2 i=1 for < h ≤ w L˜˜ , Yi U m X ˜ U ]l )2 ˜ i ]l − [Y˜ ([Y˜ ˜ ˜ ˜ i h l h ˜ ˜ ˜ , [E(W0 , W1 , , Wn )]h = i=1 for wL˜˜ ≤ h ≤ w U˜˜ , Yi Yi U m X ˜ U ]r )2 ˜ i ]r − [Y˜ ([Y˜ ˜ ˜ ˜ i h h ˜ 0, W ˜ 1, , W ˜ n )]r = , [E(W h i=1 ˜i ) for wL˜˜ ≤ h ≤ w U˜˜ The total cost function for the input-output pair (xi , Y˜ Yi is obtained as Yi e= X [E(W0 , W1 , , Wn )]h (13) h ˜ ˜ ˜ ˜ 0, W ˜ 1, , W ˜ n )]L denotes the error between the left-hand sides of Hence [E(W h ˜ ˜ ˜ ˜ 0, W ˜ 1, , W ˜ n )]U the h-level sets of the desired and the computed output, and [E(W h denotes the error between the right-hand sides of the h-level sets of the desired and the computed output Clearly, this is a problem of optimization of quadratic functions without constrains that can usually be solved by gradient descent algorithm In fact, denoting ˜ ˜ ˜ ˜ ˜ ˜ )]h = ([ ∂E(W ) ]h , , [ ∂E(W ) ]h )T , [∇E(W ˜ ˜ ˜0 ˜n ∂W ∂W in order to solve Eq (10), assume k iterations to have been done and get ˜ ˜ k the k th iteration point W REMARK Since the Eq (12) are quadratic functions, supposing ≤ oij = xij for i = 1, , m, j = 0, , n, we rewrite these as follows: ˜ ˜ )]l [E(W h Pm ( Pn ˜ ˜ ˜L l ˜Ll j=0 [oij Wj ]h −[Yi ]h ) i=1 + ( Pn ˜ ˜ ˜U l ˜U l j=0 [oij Wj ]h −[Yi ]h ) ˜ ˜ ˜ ˜ ˜ L ]l )T Q[W ˜ L ]l + ([B ˜ L ]l )T [W ˜ L ]l + [C˜ ˜ L ]l + = 12 ([W h h h h h ˜ ˜ ˜U l T ˜U l ([W ]h ) Q[W ]h ˜ ˜ ˜ U ]l )T [W ˜ U ]l + [C˜ ˜ U ]l , + ([B h h h where Pm Pm P m P m i=1 oi1 i=1 oi2 i=1 oin P P P m m m m  o o o o i1 i2 i1 i1 i=1 i=1 oi1 oin i=1 i=1  Q=  Pm Pm Pm Pm o o o o o i=1 in i=1 in i1 i=1 in i2 i=1 oin       ˜ ˜b0 ]h , [˜ ˜b1 ]h , , [˜ ˜bn ]h )T , ˜ h = ([˜ [B] ˜ ˜h= [C] ˜ ˜ i=1 ([Yi ]h ) , Pm P ˜ ˜˜ with [˜bj ]h = − m i=1 oij [Yi ]h We have ˜ ˜ ˜ ˜ ˜ ˜ )]l = Q[W ˜ L ]l + [ B ˜ L ]l + Q[W ˜ U ]l + [ B ˜ U ]l , [∇E(W h h h h h (14) and we can obtain ˜ ˜ ˜ ˜ ˜ ˜ )]r = Q[W ˜ L ]r + [ B ˜ L ]r + Q[W ˜ U ]r + [ B ˜ U ]r , [∇E(W h h h h h (15) for < h ≤ w L˜˜ Also, we have Yi ˜ ˜ ˜ ˜ )]l = Q[W ˜ U ]l + [ B ˜ U ]l , [∇E(W h h h (16) ˜ ˜ ˜ ˜ )]r = Q[W ˜ U ]r + [ B ˜ U ]r , [∇E(W h h h (17) for wL˜˜ ≤ h ≤ w U˜˜ Now we consider its explicit scheme Hence we have Yi [11, ?, 19] Yi ˜ ˜ ˜ ˜ k+1 = W ˜ k + ∆W ˜ k, W ˜ ˜ ˜ k = −µ∇E(W ˜ k ), ∆W (18) where k indexes the number of adjustments and µ is a learning rate (a positive real number) Comparison with other methods This study would not be completed without comparing it with the recent papers (Kao,2003,Mosleh,2010,2011,2012,Otadi,2014,Tanaka,1989,1982,Sakawa,1992) Some comparisons are as follows: • Fuzzy linear regression was first introduced by Tanaka et al (1989) The objective was to minimize the total spread of the fuzzy parameters subject to the support of the estimated values that cover the support of the observed values for a certain h-level Although this approach was later improved by Tanaka et al (1989), their model is very sensitive to outliers Moreover,it can produce infinite solutions and the spread of the estimated values become wider as more data are included in the model Then Mosleh and et al (2010) used fuzzy neural network and compared the performance of these two methods in estimation Example three in (2010) shown that the fuzzy neural network method is better than of the previous studies • Sakawa and Yano (1992) formulated a fuzzy linear regression model with fuzzy output and fuzzy parameters as a mathematical programming problem Recently Otadi in (2014) shown fuzzy neural network is better than of the mathematical programming problem • Kao et al (2003) formulated a nonlinear programming problem to determine the linear regression coefficients Mosleh and et al (2010) used fuzzy neural network and compared the performance of these two methods in estimation • Mosleh and et al (2010,2011,2012) used fuzzy neural network for estimation of fuzzy regression In this paper, we consider dependent ˜i , i = 1, 2, , m and we used intervalinterval-valued fuzzy variable Y˜ valued fuzzy neural network for estimation Numerical examples To illustrate the technique proposed in this paper, consider the following examples Example 6.1 Consider the fuzzy data for a dependent fuzzy variable ˜ and two independent real variables x1 and x2 in table Y˜ ¯ ˜ = Using these data, develop an estimated fuzzy regression equation Y˜ ˜ ˜ ˜ A˜0 + A˜1 x1 + A˜2 x2 In the computer simulation of this example, we use the following specifications of the learning algorithm (1) Number of input units: units (2) Number of output units: unit (3) Stopping condition: K = 100 iterations of the learning algorithm ˜ ˜ ˜ ˜ (1) = W ˜ (1) = W ˜ (1) = [(1, 1.5, 2; 0.5), (0.5, 1.5, 2.5; 1)] The training starts with W Applying the proposed method to the approximate solution of problem (9), therefor we have interval-valued fuzzy regression: ¯ ˜ = [(2.6987, 3.4393, 4.4393; 0.5), (1.6987, 3.4393, 5.4393; 1)]+ Y˜ [(0.6067, 0.6569, 0.6569; 0.5), (0.6067, 0.6569, 0.6569; 1)]x + [(−0.5188, −0.4100, −0.4100; 0.5), (−0.5188, −0.4100, −0.4100; 1)]x Example 6.2 Consumer Reports uses a survey to collect data on the annual cost of repairs for makes and models of automobiles The intervalvalued fuzzy data are the interval-valued fuzzy annual repair cost (dollar) and the age of the automobile (year), see table ¯ ˜ = A˜ ˜0 + Develop the estimated interval-valued fuzzy regression equation Y˜ ˜ A˜1 x1 In the computer simulation of this example, we use the following specifications of the learning algorithm (1) Number of input units: units (2) Number of output units: unit (3) Stopping condition: K = 100 iterations of the learning algorithm ˜ ˜ ˜ (1) = [(50, 55, 60; 0.75), (45, 55, 65; 1)] and W ˜ (1) = The training starts with W [(10, 15, 20; 0.75), (5, 15, 25; 1)] Applying the proposed method to the approximate solution of problem (9), therefore we have interval-valued fuzzy regression: ¯ ˜ = [(62.5, 67.5, 72.5; 0.75), (57.5, 67.5, 77.5; 1)] + 11.5x1 Y˜ Summary and conclusions In this paper we consider solving interval-valued fuzzy linear regression (IVFLR) by using universal approximators (UA), that is, IVFNN To obtain the “Best-approximated” solution of IVFLRs, the adjustable parameters of IVFNN are systematically adjusted by using the learning algorithm The effectiveness of the derived learning algorithm was demonstrated by computer simulation of numerical examples Computer simulation in this paper was performed for two-layer feedforward neural networks using the back-propagation-type learning algorithm The proposed method possesses several properties which makes it better, or at least more suitable, than the existing methods Firstly, the proposed method is able to handel interval-valued fuzzy observations Secondly example three in [15] shown that the fuzzy neural network method is better than of the previous studies, in 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number U A˜˜ (x) L A˜˜ (x) 0.5 X Fig An interval-valued fuzzy set 14 U A˜˜ (x) L A˜˜ (x) wU˜˜ A wL ˜˜ A aU aL1 aL2 = aU2 aL3 X aU3 Fig An interval-valued triangular fuzzy number Input units W˜˜0 xi1 xin Output unit W˜˜1 Y˜˜ i ˜ W˜ n Fig Interval-valued fuzzy neural network for approximating interval-valued fuzzy regression 15 Table Captions Table 1: Inputs and output data for example 5.1 Table 2: Inputs and output data for example 5.2 16 ˜ Y˜ x1 x2 [(2,3,4;0.5),(1,3,5;1)] [(3,4,5;0.5),(2,4,6;1)] 4 [(2,4,5;0.5),(1,4,6;1)] 5 [(4,5,6;0.5),(3,5,7;1)] [(2,4,5;0.5),(1,4,6;1)] Table Inputs and output data for example 6.1 Age Repair [(80,85,90;0.75),(75,85,95;1)] [(85,90,95,0.75),(80,90,100;1)] [(90,95,100;0.75),(85,95,105;1)] [(100,105,110;0.75),(95,105,115;1)] [(130,135,140;0.75),(125, 135,145;1)] Table Age and repair for example 6.2 17 ... research of real fuzzy regression models In this paper interval- valued fuzzy neural network (IVFNN) can be trained with crisp and interval- valued fuzzy data Here a neural network is considered as... that can appropriately treat fuzzy regression models Sakawa and Yano (1992) proposed a mathematical programming model to estimate the parameters of a fuzzy linear regression Yi = A1 xi1 + A2 ... paper, a novel hybrid method based on interval- valued fuzzy neural network for approximate of interval- valued fuzzy regression models, is presented Here a neural network is considered as a part