A Study of Fuzzy Linear Regression - InterStat

A Study of Fuzzy Linear Regression Dr. Jann-Huei Jinn Department of Statistics Grand Valley State University Allendale, Michigan, 49401 and Dr. Chwan-Chin Song and Mr. J. C. Chao Department of Applied Mathematics National Cheng-Chi University Taipei, Taiwan, R.O.C. 1. Introduction We often use regression analysis to model the relationship between dependent (response) and independent (explanatory) variables. In traditional regression analysis, residuals are assumed to be due to random errors. Thus, statistical techniques are applied to perform estimation and inference in regression analysis. However, the residuals are sometimes due to the indefiniteness of the model structure or imprecise observations. The uncertainty in this type of regression model becomes fuzziness, not random. Since Zadeh (1965) proposed fuzzy sets, fuzziness has received more attention and fuzzy data analysis has become increasingly important. In order to consider the fuzziness in regression analysis, Tanaka et al. (1982) first proposed a study of fuzzy linear regression (FLR) model. They considered the parameter estimations of FLR models under two factors, namely the degree of the fitting and the vagueness of the model. The estimation problems were then transformed into linear programming (LP) based on these two factors. Since the measure of best fit by residuals under fuzzy consideration is not presented in Tanaka’s approach, Diamond (1988) proposed the fuzzy least-squares approach, which is a fuzzy extension of the ordinary least squares based on a new defined distance on the space of fuzzy numbers. In general, the fuzzy regression methods can be roughly divided into two categories. The first is based on Tanaka’s LP approach. The second category is based on the fuzzy least-squares approach. In section 2, we introduced the fuzzy number and its operation, a simple distance formula, two fuzzy linear regression models and their least squares estimates. In section 3, we introduced LR type fuzzy number and nonsymmetrical doubly linear adaptive fuzzy regression model, Yang and Ko’s distance formula, and least squares estimates which relate to membership functions. In section 4, we applied traditional methods of detecting possible outliers and influence points to derive the leverage values, residuals and Cook distance formula for the fuzzy linear regression models..

In section 5, we used the theoretical results in the previous chapters to analyze the Tanaka’s (1987) data. The derivation of some important formulas is given in appendices. 2. Introduction to Fuzzy Linear Regression 2.1 Fuzzy umber and Its Operation Fuzzy data is a natural type of data, like non-precise data or data with a source of uncertainty not caused by randomness. This kind of data is easy to find in natural language, social science, psychometrics, environments and econometrics, etc. Fuzzy numbers have been used to represent fuzzy data. These are also used to model fuzziness of data. Let ℜ be a one-dimensional Euclidean space with its norm denoted by . . A fuzzy number is an upper semicontinuous convex function F: ℜ → [0,1] with {x ∈ ℜ F ( x) = 1} non-empty. In other words, a fuzzy number A is defined as a convex normalized fuzzy set of the real line ℜ so that there exists exactly one xo ∈ ℜ with F( xo )=1, and its membership F(x) is piecewise continuous. Definition 2.1 (Zimmermann [pp.62-63]) Let L (and R) be decreasing, shape functions from ℜ + to [0,1] with L(0)=1; L(x)0; L(x)>0 for all x0, β > 0 in ℜ ,  m− x  L  α  x ≤ m,   A(x)=    R x − m  x ≥ m,   β 

where m is called the center (mean or mode)value of A

and α and β are called left and right spreads, respectively. Symbolically, A is denoted by (m, α , β )LR . If α = β , A=(m, α , α ) LR is called symmetrical fuzzy number, denoted by A=(m, α ) LR . For instance, the algebraic and geometric characteristics of the membership function of the more utilized LR fuzzy number, the triangular fuzzy number, are shown in the following:  m−x 1− m − α ≤ x ≤ m,  α A(x)=  x−m 1 − m ≤ x ≤ m + α,  β Another example, the exponential fuzzy number, its membership function

   m − x n  exp −       s   A(x)=  n exp −  x − m      s      Definition 2.2 (Dubois(1980))

x ≤ m, where s is the spread. x ≥ m,

Let A= (ma , α a , β a )LR and B= (mb , α b , β b )LR be two LR-type fuzzy numbers. Then by the extension principle, the following operations are defined: 1. A+B= (ma + mb , α a + α b , β a + β b )LR 2. λ A= λ (ma , α a , β a )LR = (λm a , λα a , λβ a )LR when λ >0

3. λ A= λ (ma , α a , β a )LR = (λm a ,−λβ a ,−λα a )RL when λ 0 , wα > 0 , and wβ > 0 are arbitrary weights.

2.2 General Fuzzy Linear Regression Model Consider the following general fuzzy linear regression model and call it Model I: yi = Ao + A1 xi1 + A2 xi 2 + ... + A p xip i=1,2,…,n (2.3) where xij are real numbers, y i = [ci − s i , ci + si ] are fuzzy numbers, ci is the center (mean or mode) value, s i is the spread, and Am = [a m − rm , a m + rm ] are the fuzzy regression parameters, which has the same membership function as y i . How should we estimate Am if the distance between two fuzzy numbers are undefined? We may treat y Li = ci − s i and y Ri = ci + s i as the left and right end points of the sample data,

{(y x , x ,..., x )i = 1,2,..., n}(and similarly to ,..., x )i = 1,2,..., n} we may use the linear regression

respectively. For the left end data points the right end data points

{(y

Ri

, xi1 , xi 2

Li , i1

i2

ip

ip

model y = β o + β 1 x1 + ... + β p x p to obtain the following estimates, respectively, ) ) ) ) ) y Li = Lo + L1 xi1 + L2 xi 2 + ... + L p xip , i = 1,2,..., n

) ) ) ) ) y Ri = Ro + R1 xi1 + R2 xi 2 + ... + R p xip , i = 1,2,..., n ) ) ) ) R m − Lm ) Lm + R m ) ) ) ) ) ) , rm = . Then, Am = [a m − rm , a m + rm ] where a m = 2 2 Using this way to estimate the regression parameters, Am , didn’t consider the advantage of using the membership function to describe the data. The fuzzy concept were not used in the estimation of parameters. In order to obtain more appropriate estimates of fuzzy regression parameters, Am , the least squares method and the distance between two fuzzy numbers should be considered. Based on the definition 2.3, we can use ordinary least-squares method to estimate the fuzzy parameters in the general fuzzy linear regression model ((2.3), Model I). Assuming that y i = (ci , si ) and Am = (a m , rm ) have same membership function, after appropriate translation, we can make all of xij > 0 . Then (2.3) can be expressed as (ci , s i ) = (a o , ro ) + (a1 , r1 ) xi1 + (a 2 , r2 ) xi 2 + ... + (a p , rp ) xip According to the Euclidean distance formula of (2.1), the least-squares estimates of ai and ri are the values of ai , ri which minimize the value of D 2 where n

[

D 2 = ∑ (ci − (a o + a1 xi1 + ... + a p xip )) 2 + ( s i − (ro + r1 xi1 + ... + rp xip )) 2

]

i =1

r r Let v denote the length of vector v , then by using vector and matrix expressions 2

D 2 can be rewritten as D 2 = Χa − C + Χr − S

2

where Χ is a n × ( p + 1) design

matrix, a = (a o , a1 ,..., a p )' , r = (ro , r1 ,..., r p )' , C = (c1 , c 2 ,..., c n )' , and S = ( s1 , s 2 ,..., s n )' . ∂D 2 ∂D 2 = 0 and = 0 then the solutions of a and r which minimize D 2 are as Let ∂a ∂r follows: aˆ = ( Χ ' Χ ) −1 Χ ' C rˆ = ( Χ ' Χ ) −1 Χ ' S (2.4)

The above method used regression with respect to center and spread. The estimation results are not related to the membership functions. But, in the later real data analyses, this method provided better results in the estimation of fuzzy parameter values.

2.3 Symmetrical Doubly Linear Adaptive Fuzzy Regression Model Under the structure of model I, if we use the Euclidean distance formula and the leastsquares method to do linear regression with respect to center and spread respectively, then the estimates show that the centers and spreads are not related. But, D’Urso and Gastaldi (2000) think that the dynamic of the spreads is somehow dependent on the magnitude of the (estimated) centers. Therefore, they proposed the doubly linear adaptive fuzzy regression model (call it Model II) to obtain the parameter estimates.

They considered symmetrical fuzzy numbers with triangular membership function. Where a fuzzy number, y i = (ci , si ) , is completely identified by the two parameters c (center) and s (left and right spread). Model II is defined as follows: C = C* + εc C * = Χa (2.5) S = S* + εs S * = C *b + 1d (2.6) where Χ is a n × ( p + 1) matrix containing the input variables (data matrix), a = (a o , a1 ,..., a p )' is a column vector containing the regression parameters of the

first model (referred to as core regression model), C = (c1 , c 2 ,..., c n )' and C * = Χa are the vector of the observed centers and the vector of the interpolated centers, respectively, both having dimensions n × 1 , and S = ( s1 , s 2 ,..., s n )' and S * are the vector of the assigned spreads and the vector of the interpolated spreads, respectively, both having dimension n × 1 , 1 is a n × 1 -vector of all 1's, b and d are the regression parameters for the second regression model (referred to as spread regression model). Apparently, the above model is based on two linear models. The first one interpolates the centers of the fuzzy observations, the second one yields the spreads, by building another linear model over the first one. Observe that predictive variables X are taken into consideration in Eq. (2.6) through the observed centers. The model is hence capable to take into account possible linear relations between the size of the spreads and the magnitude of the estimated centers. This is often the case in real world applications, where dependence among centers and spreads is likely (for instance, the uncertainty or fuzziness with a measurement could depend on its magnitude). D’Urso and Gastaldi used the Euclidean distance formula of (2.1) and the leastsquares method to obtain the estimates of a , b and d such that the value of D 2 is minimized, where

D2 = C − C*

2

+ S − S*

2

= C ' C − 2C ' Χa + a ' Χ ' Χa (1 + b 2 ) + S ' S − 2 S ' Χab − 2 S '1d + 2a ' Χ '1bd + nd 2

∂D 2 ∂D 2 ∂D 2 = 0, = 0 and = 0 , they obtained the following equations: Let ∂a ∂b ∂d ∂D 2 = 0 = − Χ ' C + Χ ' Χa (1 + b 2 ) − Χ ' Sb + Χ '1bd ∂a ∂D 2 = 0 = a' Χ' Χab − S ' Χa + a' Χ'1d ∂b ∂D 2 = 0 = − S '1 + a ' Χ'1b + nd (2.7) ∂d Based on the equations in (2.7), they obtained the following least-squares iterative solutions of a , b and d :

a=

1 (( Χ ' Χ ) −1 Χ ' (C + Sb − 1bd )) 2 (1 + b )

b = (a ' Χ ' Χa ) −1 ( S ' Χa − a ' Χ '1d ) 1 d = ( S '1 − a ' Χ '1b) (2.8) n The derivation of the recursive solutions of a , b and d : from the first equation of (2.7), 1 we can easily obtain a = (( Χ ' Χ ) −1 Χ ' (C + Sb − 1bd )) , substituting it into the 2 (1 + b ) second and third equations of (2.7), we obtained: 2

2

Cˆ b + C ' Sˆb 2 − nC b 2 d − S ' Cˆ − Sˆ b + 2nS bd + nC d − nbd 2 = 0 and bC − S + d = 0

(2.9) (2.10)

1 1 where Χ ( Χ ' Χ ) −1 Χ ' C = Cˆ , Χ ( Χ ' Χ ) −1 Χ ' S = Sˆ , 1'C = C , 1'S = S . n n From (2.10), we obtained d = S − bC , substituting it back into (2.9), we obtained a simplified quadratic equation of b: M 1b 2 + M 2 b + M 3 = 0 Where M 1 = C ' Sˆ − nC S , M 2 = Cˆ

2

− Sˆ

2

+ nS 2 − nC 2 , M 3 = nC S − S ' Cˆ .

By solving the quadratic equation of b, we obtained

− M 2 ± M 22 − 4 M 1 M 3 bˆ = , and the corresponding solutions of 2M 1 dˆ = S − bˆC , 1 aˆ = (( Χ' Χ) −1 Χ' (C + Sbˆ − 1bˆdˆ )) 2 ˆ (1 + b ) The least-squares estimates were obtained by substituting these two sets of aˆ , bˆ , and dˆ into D 2 such that the value of D 2 is minimized. Based on the equations of aˆ , bˆ , and dˆ , we can conclude that no matter what kind of membership function of the response fuzzy number, y i = (ci , si ) , the estimates of parameters are the same. Therefore, these least squares estimates do not consider other possible shapes of fuzzy numbers.

3. LR type of Fuzzy Linear Regression 3.1 onsymmetrical Doubly Linear Adaptive Fuzzy Regression Model When we have numerical (crisp) explanatory variables X j ( j = 1,2,..., k ) and a LR fuzzy dependent variable Y ≡ (c, p, q ) (where c is the center, p and q , respectively, the left and right spreads), a model capable to incorporate the possible influence of the magnitude of the centers on the spreads, can be taken into account (D’Urso and Gastaldi, 2000, 2001, 2002). If the fuzzy response numbers yi = [ci − q i , ci + p i ] are

nonsymmetrical with triangular membership function. D’Urso (2003) proposed a fuzzy regression model (call it Model III) which is expressed in the matrix form: C = C* + ε P = P* + λ

C * = Χa P * = C *b + 1d

(3.1) (3.2)

q = q* + ρ q * = C * g + 1h (3.3) where Χ is a n × (k + 1) matrix containing the vector 1 concatenated to k crisp input

variables; C , C * are n × 1 vectors of the observed centers and interpolated centers, respectively; P , P * are n × 1 vectors of observed left spreads and interpolated left spreads, respectively; q , q * are n × 1 vectors of observed right spreads and interpolated right spreads, respectively; a is a (k + 1) × 1 vector of regression parameters for the regression model for C ; b, d , g , h are regression parameters for the regression models for P and q ; 1 is a n × 1 vector of all 1’s; ε , λ , ρ are n × 1 vectors of residuals. This model is based on three sub-models. The first one interpolates the centers of the fuzzy data, the other two sub-models are built over the first one and yield the spreads. This formulation allows the model to consider possible relations between the size of the spreads and the magnitude of the estimated centers, as it is often necessary in real case studies. Model III can be called a nonsymmetrical doubly linear adaptive fuzzy regression model. D’Urso used the Euclidean distance formula of (2.2) and the least-squares method to obtain the estimates of a , b, d , g , h such that the value of D 2 is minimized, where 2

2

2

D 2 = C − C * π c + P − P * π p + q − q * π q = (C ' C − 2C ' Χa + a ' Χ ' Χa )π c

+ ( P' P − 2 P' ( Χab + 1d ) + a ' Χ' Χab 2 + 2a ' Χ'1bd + nd 2 )π p + (q ' q − 2q ' ( Χag + 1h) + a ' Χ' Χag 2 + 2a ' Χ'1gh + nh 2 )π q

(3.4)

and π c , π p , π q are arbitrary positive weights. Recursive solutions to the above system are found by equating to zeros the partial derivates with respect to the parameters a , b, d , g , h : 1 aˆ = ( Χ ' Χ ) −1 Χ ' (Cπ c + ( P − 1d )bπ p ) + (q − 1h) gπ q 2 2 πc + b π p + g πq bˆ = (a ' Χ ' Χa ) −1 (a ' Χ ' P − a ' Χ '1d )

[

]

1 dˆ = ( P '1 − a ' Χ '1b) n gˆ = (a ' Χ ' Χa ) −1 (a ' Χ ' q − a ' Χ '1h) 1 1 hˆ = (q '1 − a ' Χ '1g ) = (q '1 − a ' Χ ' q − a ' Χ '1h) (3.5) n n Where aˆ , bˆ, dˆ , gˆ , hˆ are the iterative least-squares estimates (obtained at the end of the iterative process). The optimization procedure does not guarantee the attainment of the global minimum, only a local one. For this reason, it is suggested to initialize the iterative

algorithm by considering several possible starting points in order to check the stability of the solution. Based on the equations of aˆ , bˆ, dˆ , gˆ , hˆ , we can conclude that the estimates of parameters are not related to the membership function of the response fuzzy number.

3.2 Yang and Ko’s Distance Formula Under the structure of Model I, II, III and the use of Euclidean distance, all the leastsquares estimates are not able to consider the possible effect of the membership function of fuzzy response numbers. In this section, we will adapt the Yang and Ko’s (1996) distance formula to try to find the least-squares estimates which are related to the membership function of fuzzy response numbers. Definition 3.1 (Yang and Ko’s distance formula(1996)) Let FLR (ℜ) denote the set of all LR-type fuzzy numbers. Define a new type of distance for any A = (m a , α a , β a ) LR , B= (mb , α b , β b )LR in FLR (ℜ) as follows: 2 d LR ( A, B ) = (ma − mb ) 2 + ((m a − lα a ) − (mb − lα b )) 2 + ((m a + rβ a ) − (mb + rβ b )) 2 (3.6)

1

1

0

0

where l = ∫ L−1 (ω )dω and r = ∫ R −1 (ω )dω Yang and Ko (1996) also proved that ( FLR (ℜ) , d LR ) is a complete metric space. If A and B are symmetrical LR type fuzzy numbers then l = r and 2 d LR ( A, B ) = 3(m a − mb ) 2 + 2l 2 (α a − α b ) 2 . If A and B are symmetrical triangular type of 1 1 1 fuzzy numbers then l = ∫ L−1 ( x)dx = ∫ (1 − x)dx = . If A and B are exponential type of 0 0 2 1 1 1 fuzzy numbers then l = ∫ L−1 ( x)dx = ∫ (− ln x)1 / m dx = Γ(1 + ) . Compare with the 0 0 m distance formulas of (2.1) and (2.2), the distance formula of (3.6) can avoid the subjective choice of the weights ( wm > 0 , wα > 0 , and wβ > 0 are arbitrary weights).

3.3 The Least Squares Estimates (Based on Yang and Ko’s Distance) In this section, we will consider LR type of response fuzzy numbers and use the distance formula of (3.6) to find least squares estimates of regression parameters. Under the structure of Model I, if we have symmetrical LR type fuzzy response numbers y i = (ci , s i ) LR , then l = r in (3.6). The sum of squared error D 2 can be expressed in vector form: D 2 = Χa − C

2

2

+ ( Χa − lΧr ) − (C − lS ) + ( Χa + lΧr ) − (C + lS )

= 3a ' Χ ' Χa − 6a ' Χ ' C + 2l 2 r ' Χ ' Χr − 4l 2 r ' Χ ' S + 2l 2 S ' S ∂D 2 ∂D 2 Let = 0 and = 0 then ∂a ∂r

2

∂D 2 = 0 = 6 Χ ' Χa − 6 Χ ' C ∂a ∂D 2 = 0 = 4l 2 Χ ' Χr − 4l 2 Χ ' S ∂r and the solutions of a and r which minimize D 2 are as follows: aˆ = ( Χ ' Χ ) −1 Χ ' C rˆ = ( Χ ' Χ ) −1 Χ ' S (3.7) Therefore, under the structure of Model I, no matter whether we used the distance formula of (2.1) or (3.6), we obtained the same least squares estimates and they are are not related with their class membership functions. Next, let us consider Model II (D’Urso and Gastaldi (2000), doubly linear adaptive fuzzy regression model), the sum of squared error D 2 can be expressed in vector form: D 2 = Χa − C

2

2

+ [ Χa − l ( Χab + 1d )] − (C − lS ) + [ Χa + l ( Χab + 1d )] − (C + lS )

2

= 3a ' Χ ' Χa − 6C ' Χa + 3C ' C + 2l 2 b 2 a ' Χ ' Χa + 4l 2 bda ' Χ '1 − 4l 2 ba ' Χ ' S + 2l 2 nd 2 − 4l 2 dnS + 2l 2 S ' S

∂D 2 ∂D 2 ∂D 2 = 0, = 0 and = 0 , after lengthy tedious and complicated calculations ∂a ∂b ∂d (see Appendix I) we obtained the following least squares estimates: aˆ , bˆ , and dˆ Let

− K 2 ± K 22 − 4 K 1 K 3 ˆ b= 2K1 dˆ = S − bˆC , 1 aˆ = ( Χ ' Χ ) −1 (3Χ ' C + 2l 2 bˆΧ ' S − 2l 2 bˆS Χ '1 + 2l 2 bˆ 2 C X '1) 2 ˆ2 3 + 2l b where K 1 = 2l 2 (Cˆ ' S − nC S ) , K 2 = 3( Cˆ

2

− nC 2 ) − 2( Sˆ

2

(3.8)

− nS 2 ) , K 3 = 3nS C − 3C ' Sˆ ,

1 1 and Χ ( Χ ' Χ ) −1 Χ ' C = Cˆ , Χ ( Χ ' Χ ) −1 Χ ' S = Sˆ , 1'C = C , 1'S = S . n n The least-squares estimates were obtained by substituting these two sets of aˆ , bˆ , and dˆ into D 2 such that the value of D 2 is minimized. Based on the equations of aˆ , bˆ , and dˆ , we can conclude that these least squares estimates do relate to the membership function of the response fuzzy number, y i = (ci , s i ) LR . Under the structure of Model III (D’Urso (2001)) and consider nonsymmetrical LR type of response fuzzy numbers, the sum of squared error D 2 can be expressed in vector form: D 2 = Χa − C

2

2

+ [ Χa − l ( Χab + 1d )] − (C − lP ) + [ Χa + l ( Χag + 1h)] − (C + rq )

2

= 3C ' C − 6C ' Χa + (3 − 2lb + l 2 b 2 + 2rg + r 2 g 2 )a ' Χ ' Χa + (2lb − 2rg )C ' Χa + (2 − 2lb) P ' Χa − (2 + 2rg )q ' Χa + (2l 2 db − 2ld + 2rh + 2r 2 gh)a ' Χ '1 − 2C ' P + 2C ' q + (2ld − 2rh)nC − 2ldnP − 2rhnq + l 2 d 2 n + r 2 h 2 n + P ' P + q ' q

∂D 2 ∂D 2 ∂D 2 ∂D 2 ∂D 2 = 0, = 0, = 0, = 0 and = 0 , after lengthy tedious and ∂a ∂b ∂d ∂g ∂h complicated calculations we obtained the following equations: Let

− 6 Χ ' C + (6 − 4lb + 2l 2 b 2 + 4rg + 2r 2 g 2 ) Χ ' Χa + 2lbΧ ' C + 2 Χ ' P − 2lbΧ ' P − 2ldΧ '1 + 2l 2 dbΧ '1 − 2rgΧ ' C − 2 Χ ' q − 2rgΧ ' q + 2rhΧ '1 + 2r 2 ghΧ '1 =0 2lC ' Χa − 2lP ' Χa − 2la ' Χ ' Χa + 2l 2 ba ' Χ ' Χa + 2l 2 da ' Χ '1 =0 2lC '1 − 2lP '1 − 2la ' Χ '1 + 2l 2 ba ' Χ '1 + 2l 2 dn =0

− 2rC ' Χa − 2rq ' Χa + 2ra ' Χ ' Χa + 2r 2 ga ' Χ ' Χa + 2r 2 ha ' Χ '1 =0 − 2rC '1 − 2rq '1 + 2ra ' Χ '1 + 2r 2 ga ' Χ '1 + 2r 2 hn =0 Since the equations are too complicated to find general solutions of a, b, d , g , h we just list the following recursive equations and try to use mathematics software to find possible solutions. 1 a= ( Χ ' Χ ) −1 [3Χ ' C − 2lbΧ ' C − 2lΧ ' P + 2l 2 bΧ ' P + 2ldΧ '1 2 2 2 2 3 − 2lb + 2rg − l b + r g

− 2l 2 bdΧ '1 + 2rgΧ ' C + 2rΧ ' q + 2r 2 gΧ ' q − 2r 2 ghΧ1 + 2rhΧ '1 ] 1 b = (a ' Χ ' Χa ) −1 (C ' Χa − lP ' Χa − a ' Χ ' Χa + lda ' Χ '1) l 1 g = (a ' Χ ' Χa ) −1 (C ' Χa + rq ' Χa − a ' Χ ' Χa + rha ' Χ '1) r 1 d = (−C + lP + a ' Χ '1 − lba ' Χ '1) l 1 h = (C + rq − rga ' Χ '1 + a ' Χ '1) r From the above equations, it is obvious that the least squares estimates are related to the membership function of the response fuzzy number, y i = (ci , s i ) LR .

4. Diagnostic of Outliers and Influences 4.1 Diagnostic of Outliers and Influences in Linear Regression Model Although a residual analysis is useful in assessing model fit, departures from the regression model are often hidden by the fitting process. For example, there may be “outliers” in either the response or explanatory variables that can have a considerable effect on the analysis. Observations that significantly affect inferences drawn from the data are said to be influential. Methods for assessing influence are typically based on the change in the vector of parameter estimates when observations are deleted.

The leverage h jj = x 'j ( Χ' Χ) −1 x j is associated with the j th data point and measures, in the space of the explanatory variables, how far the j th observation is from the other n-1 observations. For a data point with high leverage, h jj approaches 1 (0 ≤ h jj ≤ 1) , indicates it is a possible outlier. The residuals ei = yi − yˆ i are used to detect possible outliers for the response variable y, where yˆ i is the i th predicted y value. A large value of

ei indicates the i th data point could be an outlier. One may also use e(i ) = yi − yˆ (i ) =

ei to detect possible outliers, where yˆ (i ) is the predicted y value when 1 − hii

the i th observation is dropped from the analysis. A large value of e(i ) also indicates the i th data point could be an outlier.

In traditional linear regression analysis, one may use the Cook distance, 2 Υˆ − Υˆ ( i ) ei2 hii CDi = = 2 to detect possible influential data points where Υˆ (i ) is 2 ks ks (1 − hii ) 2 the predicted Y vector value when the i th observation is dropped from the analysis, k is n

∑e

2 i

the number of parameters, and s 2 =

i =1

is the mean square error. A large value of CDi n−k indicates that i th data point could be an influential observation. One of the advantages of using Cook distance is that no matter what measurement units are used in the explanatory and response variables, the value of CDi will not be affected.

4.2 Diagnostic of Outliers and Influences in Fuzzy Linear Regression Model In this section, we will consider the Model I (see (2.3)) and derive the corresponding formulas of ei , e(i ) , and CDi to detect possible outliers and influential data points. For Model II (see (2.5) and (2.6)) and Model III (see (3.1), (3.2), and (3.3)), we were not able to derive any formulas of ei , e(i ) , and CDi to detect possible outliers and influential data points. Based on the Euclidean distance, we obtained (see the derivations in Appendix A.2) ei2 = (ci − xi aˆ ) 2 + ( s i − xi rˆ) 2 = (eic ) 2 + (eis ) 2 (4.1) 2

 e  e = (ci − xi aˆ ( i ) ) + ( si − xi rˆ(i ) ) =  i  (4.2) 1 h − ii   c ˆ where ei = ci − xi a is the residual from the center of a fuzzy number and eis = si − xi rˆ is the residual from the spreads of a fuzzy number. aˆ and rˆ are defined in (2.4). 2 (i )

2

2

Similarly, based on the Yang and Ko’s distance we obtained (see the derivations in Appendix A.2) 2 ei2 = d LR ( y i , yˆ i ) = 3(eic ) 2 + 2l 2 (eis ) 2 (4.3)

 e ec es = d ( y i , yˆ ( i ) ) =3 ( i ) 2 + 2l 2 ( i ) 2 =  i 1 − hii 1 − hii  1 − hii

2

  (4.4) e  From (4.2) and (4.4), the relation between ei and e( i ) are the same as in general linear 2 (i )

2 LR

regression model. That is, a large value of e( i ) , indicates the i th data point could be an outlier. In order to derive a formula similar to the Cook’s distance under the fuzzy environment, we need to define a new type of distance between fuzzy vectors. Let FLR (ℜ) denote the set of all LR-type fuzzy numbers, and ~ FLR (ℜ) = {( X 1 , X 2 ,..., X k )' X i ∈ FLR (ℜ)} is the set of all fuzzy k dimensional vectors. ~ Based on the distance definition in FLR (ℜ) , we can define a new distance in FLR (ℜ) . Lemma 4.1 Let d : FLR (ℜ) × FLR (ℜ) → ℜ be a metric, for any two fuzzy vectors ~ Χ = ( X 1 , X 2 ,..., X k )' , Υ = (Y1 , Y2 ,..., Yk )'∈ FLR (ℜ) , define

~ d LR ( Χ, Υ ) =

k

∑d

2

( X i , Yi )

(4.5)

i =1

~ ~ ~ then d LR is a metric in FLR (ℜ) . If d is a complete metric then so does d LR (see the proof in Appendix 3). When d is a simple metric, define Cook’s distance CDi as follows: 2 2 ~2 ˆ ˆ d LR (Υ , Υ( i ) ) Χaˆ − Χaˆ ( i ) + Χrˆ − Χrˆ( i ) CDi = = ks 2 ks 2 then we obtained (see the derivation in Appendix 4) ei2 hii 1 CDi = 2 ks (1 − hii ) 2

(4.6)

n

∑e

2 i

where s 2 =

i =1

and ei2 = (eic ) 2 + (eis ) 2 .

n−k When d is Yang and Ko’s metric, define Cook’s distance CDi as follows: ~2 ˆ ˆ d LR (Υ , Υ( i ) ) CDi = ks 2

{

2 2 1 Χaˆ − Χaˆ ( i ) + ( Χaˆ − lΧrˆ) − ( Χaˆ ( i ) − lΧrˆ( i ) ) + ( Χaˆ + lΧrˆ) − ( Χaˆ ( i ) + lΧrˆ(i ) 2 ks then we obtained (see Appendix A.4)

=

2

}

ei2 hii 1 CDi = 2 ks (1 − hii ) 2

(4.7)

n

∑e

2 i

where s 2 =

i =1

n−k

and ei2 = 3(eic ) 2 + 2l 2 (eis ) 2 .

Although the formulas (4.6) and (4.7) looks the same, the values of ei2 and s 2 are different. In general, s 2 in (4.7) is larger than the value of s 2 in (4.6), therefore the Cook distance calculated in (4.6) is larger than the Cook distance calculated in (4.7). From (4.6) and (4.7) we knew that CDi is affected by the leverage value h jj and residual ei . This is the same as in the traditional regression analysis. Since we were not able to derive similar formulas as (4.1) – (4.4) for Model II and III, the best we can do is to delete a data point (per time) and recalculate the values of e( i ) , CDi , etc.

5. Data Analysis In this section, we will use the Tanaka’s data (1987, see Table 1) to illustrate the theoretical results which we obtained in the previous sections. The data set contains three independent variables, one fuzzy response variable and ten data points. We only consider exponential fuzzy response values. The advantage of using exponential membership function is that we only need to choose appropriate m value ( Note: m is the mean value of LR type fuzzy numbers) to reflect the distribution of response variable. If the values of response variable tend to fall outside the interval of existing data then we choose smaller m value. Otherwise, we will choose larger m value to describe the membership function. Since we were not able to derive the least squares estimates for model III and we only consider exponential membership function, we will use model I and II to do data analysis. Tables 2 – 11 show the results of using the Euclidean distance, Yang and Ko’s distance and different m values. In each table, it contains the least squares estimates, the sum of squared residuals, the leverage value h jj , the values of ei2 and e(2i ) , and the COOK distance, CDi . Since under the Euclidean simple distance formula, the m value will not affect the results of using model I and II, therefore we only give the results of m=2 (see Table 2 and 3).

Table 1: Tanaka’s Data (1987) Case # Predictors Fuzzy Response Variable xi1 xi 2 xi 3 Yi = (ci , ri ) 1 3 5 9 (96,42) 2 14 8 3 (120,47) 3 7 1 4 (52,33) 4 11 7 3 (106,45) 5 7 12 15 (189,79) 6 8 15 10 (194,65) 7 3 9 6 (107,42) 8 12 15 11 (216,78) 9 10 5 8 (108,52) 10 9 7 4 (103,44) Table 2: Model I, m=2, Least-Squares Estimates Under Euclidean Distance Case # (ci , si ) (cˆi , sˆi ) hii ei2 CDi e(2i ) 1 (96,42) (93.20, 44.62) 0.40 14.69 41.25 0.25 2 (120,47) (122.48, 49.13) 0.43 10.67 32.44 0.21 3 (52,33) (49.36, 32.11) 0.41 7.75 21.90 0.13 4 (106,45) (104.82, 43.01) 0.26 5.35 9.75 0.04 5 (189,79) (191.79, 76.71) 0.55* 13.06 63.57* 0.52* 6 (194,65) (193.64, 67.67) 0.39 7.25 19.38 0.11 7 (107,42) (109.77, 40.85) 0.60* 9.08 55.55* 0.50* 8 (216,78) (211.65, 77.08) 0.42 19.73 58.34* 0.37 9 (108,52) (110.89, 53.24) 0.37 9.91 25.12 0.14 10 (103,44) (103.36, 42.58) 0.18 2.14 3.22 0.01 2 aˆ = (−1.39,3.25,7.92,5.03)' rˆ = (8.01,1.64,1.20,2.85)' ∑ ei = 99.63 Table 3: Model II, m=2, Least-Squares Estimates Under Euclidean Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (93.86, 42.38) (122.04, 50.63) (50.11, 29.56) (104.13, 45.38) (193.31, 71.51) (192.58, 71.30) (108.12, 46.55) (211.71, 76.90) (112.48, 47.86) (102.67, 44.96)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

4.71 17.34 15.41 3.65 74.67* 41.67 21.97 19.64 37.44 1.02

aˆ = (−3.14,3.43,7.62,5.40)' , bˆ =0.29, dˆ =14.88,

∑e

11.08 31.44 92.87* 6.55 90.38* 118.34* 38.00 78.55* 82.31* 5.34 2 i

=236.98

CDi 0.36 0.38 0.66* 0.35 0.38 0.63* 0.39 0.61* 0.51 0.34

Table 4: Model I, m=1.2, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (93.20, 44.62) (122.48, 49.13) ( 49.36, 32.11) (104.82, 43.01) (191.79, 76.71) (193.64, 67.67) (109.77, 40.85) (211.65, 77.08) (110.89, 53.24) (103.36, 42.58)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

CDi

35.61 100.02 26.43 80.41 22.26 62.94 11.17 31.59 32.71 159.18* 12.99 34.72 25.61 156.77* 58.16 171.95* 27.84 70.53 3.95 5.94

aˆ = ( −1.39,3.25,7.92,5.03)' , rˆ = (8.01,1.64,1.20,2.85)' ,

∑e

2 i

0.24 0.20 0.15 0.03 0.51* 0.08 0.55* 0.42 0.16 0.01

=256.73

Table 5: Model II, m=1.2, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (94.91, 37.94) (122.32, 49.77) (52.54, 19.65) (105.01, 42.30) (191.09, 79.46) (190.68, 79.29) (108.98, 44.01) (209.07, 87.22) (112.82, 45.67) (103.52, 41.69)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

32.79 29.66 316.42* 15.89 13.48 394.20* 18.96 294.63* 140.58 10.51

60.26 80.87 360.47* 24.84 63.15 536.55* 104.09 498.27* 207.55 13.72

∑e

=1267.13

aˆ = (1.28,3.30,7.41,5.19)' , bˆ =0.43, dˆ = - 3.03,

2 i

CDi 0.02 0.04 0.08 0.01 0.04 0.07 0.07 0.13* 0.05 0.002

Table 6: Model I, m=2, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (93.20, 44.62) (122.48, 49.13) (49.36, 32.11) (104.82, 43.01) (191.79, 76.71) (193.64, 67.67) (109.77, 40.85) (211.65, 77.08) (110.89, 53.24) (103.36, 42.58)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

34.24 25.53 22.10 10.38 31.66 11.57 25.35 57.99 27.53 3.55

CDi

96.18 77.68 62.49 18.91 154.08* 30.93 155.17* 171.46* 69.75 5.34

aˆ = ( −1.39,3.25,7.92,5.03)' , rˆ = (8.01,1.64,1.20,2.85)' ,

∑e

2 i

0.23 0.20 0.15 0.03 0.51* 0.07 0.55* 0.43 0.16 0.01

=249.91

Table 7: Model II, m=2, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (94.76, 37.78) (122.33, 49.76) (52.28, 19.29) (105.00, 42.22) (191.12, 79.67) (190.93, 79.59) (109.07, 43.99) (209.27, 87.57) (112.65, 45.54) (103.59, 41.60)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

32.73 28.30 295.52* 15.12 14.15 362.74* 19.09 279.81* 130.24 10.06

61.86 77.95 336.34* 23.74 67.62 493.27* 107.17 476.79* 194.90 13.27

∑e

=1187.75

aˆ = (1.11,3.29,7.45,5.17)' , bˆ =0.43, dˆ = - 3.45,

2 i

CDi 0.02 0.04 0.09 0.01 0.06 0.07 0.08 0.13* 0.05 0.002

Table 8: Model I, m=3, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (93.20, 44.62) (122.48, 49.13) (49.36, 32.11) (104.82, 43.01) (191.79, 76.71) (193.64, 67.67) (109.77, 40.85) (211.65, 77.08) (110.89, 53.24) (103.36, 42.58)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

34.41 25.64 22.12 10.48 31.79 11.74 25.38 58.01 27.57 3.60

CDi

96.65 78.01 62.54 19.08 154.70* 31.39 155.36* 171.52* 69.85 5.41

aˆ = ( −1.39,3.25,7.92,5.03)' , rˆ = (8.01,1.64,1.20,2.85)' ,

∑e

2 i

0.23 0.20 0.15 0.03 0.51* 0.07 0.55* 0.43 0.16 0.01

=250.74

Table 9: Model II, m=3, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (94.78, 37.79) (122.33, 49.76) (52.31, 19.34) (105.00, 42.23) (191.11, 79.64) (190.90, 79.55) (109.06, 43.99) (209.24, 87.52) (112.67, 45.56) (103.59, 41.61)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

32.73 28.46 298.04* 15.21 14.06 366.55* 19.08 281.59* 131.49 10.11

61.65 78.30 339.26* 23.87 67.06 498.51* 102.80 479.11* 196.43 13.32

∑e

=1197.32

aˆ = (1.13,3.29,7.44,5.17)' , bˆ =0.43, dˆ = - 3.39,

2 i

CDi 0.02 0.04 0.08 0.01 0.05 0.07 0.08 0.13* 0.05 0.002

Table 10: Model I, m=10, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (93.20, 44.62) (122.48, 49.13) (49.36, 32.11) (104.82, 43.01) (191.79, 76.71) (193.64, 67.67) (109.77, 40.85) (211.65, 77.08) (110.89, 53.24) (103.36, 42.58)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

35.88 26.62 22.29 11.33 32.92 13.28 25.67 58.19 27.90 4.02

CDi

100.80 80.97 63.03 20.64 160.21* 35.49 157.10* 172.06* 70.69 6.06

aˆ = ( −1.39,3.25,7.92,5.03)' , rˆ = (8.01,1.64,1.20,2.85)' ,

∑e

2 i

0.24 0.20 0.15 0.03 0.51* 0.08 0.54* 0.42 0.15 0.01

=258.10

Table 11: Model II, m=10, Least-Squares Estimates Under Yang and Ko’s Distance Case # 1 2 3 4 5 6 7 8 9 10

(c i , s i ) (96,42) (120,47) (52,33) (106,45) (189,79) (194,65) (107,42) (216,78) (108,52) (103,44)

(cˆi , sˆi ) (94.93, 37.97) (122.31, 49.77) (52.59, 19.71) (105.01, 42.31) (191.09, 79.43) (190.63, 79.23) (108.96, 44.02) (209.03, 87.16) (112.86, 45.70) (103.59, 41.70)

hii 0.40 0.43 0.41 0.26 0.55* 0.39 0.60* 0.42 0.37 0.18

ei2

e(2i )

32.80 29.93 320.68* 16.05 13.36 400.57* 18.93 297.67* 142.67 10.61

59.96 81.46 365.36* 25.07 62.29 545.32* 103.48 502.71* 210.11 13.82

∑e

=1283.29

aˆ = (1.31,3.30,7.40,5.19)' , bˆ =0.43, dˆ = - 2.96,

2 i

CDi 0.01 0.04 0.08 0.01 0.04 0.07 0.07 0.13* 0.05 0.002

5.1 Discussion From Table 2 and 3, the estimates of center and spread under model I are better than those estimates in model II. In theory, if we useYang and Ko’s distance, the estimates of center and spread under model II should be affected by the value of m. But, based on Tables 5,7,9,11, we found that different m values do not affect very much on the estimates. In theory, the distance formula and m values do not affect the estimates of model I parameters. But, they do affect the parameter estimates in model II. Based on Tables 3 and 5, the usage of different formula has more effect on the parameter estimates in model II. Case #5 and #7 have larger leverage values hii , they are possible outliers from the predictors. In model I, based on the value of ei it seems no possible outliers from the response variable. However, based on the values of e( i ) in Tables 2,4,6,8, case #5,7,8 are possible outliers from the response variable. In model II under the Euclidean distance, Tale 3 shows that case #3,5,6,8,9 are the five possible outliers from the response variable. But, under the Yang and Ko’s distance, Tables 5,7,9,11 show that only case #3,6,8 are the three possible outliers from the response variable. Under model I, based on Tables 2,4,6,8,10, case #5,7 have larger CDi values and they are influential observations. Under model II and Euclidean distance, table 3 shows that case #3,6,8 have larger CDi values. But, in model II and use Yang and Ko’s distance, only case #8 has large CDi value and is an influential point (see Tables 5,7,9,11). If we use exponential membership function for our fuzzy numbers and useYang and Ko’s distance, how to best choose the m value to do fuzzy liner regression under model II? The simplest rule is to choose the m value such that the residual sum of squares, ei2 , is smallest. Based on tables 5,7,9,11, we can see the best choice is m=2.

APPENDIX A.1: The derivation of aˆ , bˆ , and dˆ in (3.8) D 2 = Χa − C

2

2

+ [ Χa − l ( Χab + 1d )] − (C − lS ) + [ Χa + l ( Χab + 1d )] − (C + lS )

2

= a ' Χ ' Χa − C ' Χa − a ' Χ ' C + C ' C + 2a ' Χ ' C + l 2 b 2 a ' Χ ' Χa + 2l 2 bda ' Χ '1 − 2l 2 ba ' Χ ' S + 2l 2 bd1'Χa + 2l 2 d 2 1'1 − 2l 2 d1'S − 2C ' Χa + 2C ' C − 2l 2 bS ' Χa − 2l 2 S ' d1 + 2l 2 S ' S = 3a ' Χ ' Χa − 6C ' Χa + 3C ' C + 2l 2 b 2 a ' Χ ' Χa + 4l 2 bda ' Χ '1 − 4l 2 ba ' Χ ' S + 2l 2 nd 2 − 4l 2 dnS + 2l 2 S ' S

∂D 2 ∂D 2 ∂D 2 = 0, = 0 and = 0 , we obtained ∂a ∂b ∂d ∂D 2 = 0 = 3Χ ' Χa − 3Χ ' C + 2l 2 b 2 Χ ' Χa + 2l 2 bdΧ '1 − 2l 2 bΧ ' S (A.1.1) ∂a ∂D 2 = 0 = ba' Χ' Χa + da' Χ'1 − a' Χ' S (A.1.2) ∂b ∂D 2 = 0 = ba' Χ'1 + nd − 1'S (A.1.3) ∂d From (A.1.1), we obtained 1 aˆ = ( Χ ' Χ ) −1 (3Χ ' C + 2l 2 bˆΧ ' S − 2l 2 bˆS Χ '1 + 2l 2 bˆ 2 C X '1) , substituting aˆ into 2 ˆ2 3 + 2l b (A.1.2) and (A.1.3), we obtained 2 ∂D 2 = 0 = 9b Cˆ + 6l 2 b 2 Cˆ ' S − 6l 2 b 2 dnC + 9nC d ∂b 2 + 12l 2 nS bd − 6l 2 nbd 2 − 9C ' Sˆ − 6l 2 S b (A.1.4) Let

∂D 2 = 0 = bC + d − S ∂d

(A.1.5)

1 1 where Χ ( Χ ' Χ ) −1 Χ ' C = Cˆ , Χ ( Χ ' Χ ) −1 Χ ' S = Sˆ , 1'C = C , 1'S = S . n n ˆ ˆ From (A.1.5), we obtained d = S − bC and substituting it into (A.1.4) we obtained a quadratic equation of b, K 1b 2 + K 2 b + K 3 = 0 . The solution is

− K 2 ± K 22 − 4 K 1 K 3 bˆ = 2K1 A.2: The derivation of (4.2), (4.3), and (4.4) I. Based on Euclidean distance formula, we have e(2i ) = (ci − xi aˆ ( i ) ) 2 + ( si − xi rˆ(i ) ) 2

since aˆ (i )

eic eis −1 ' = aˆ − ( Χ ' Χ ) xi , and rˆ( i ) = rˆ − ( Χ ' Χ ) −1 xi' therefore 1 − hii 1 − hii

eis eic −1 ' 2 e = (ci − xi aˆ + xi ( Χ ' Χ ) −1 xi' ) 2 xi ( Χ ' Χ ) xi ) + ( si − xi rˆ + 1 − hii 1 − hii 2 (i )

eic 2 eis 2 ) +( ) =( 1 − hii 1 − hii 2

 e  =  i   1 − hii  II. Based on Yang and Ko’s distance formula, we have ei2 = (ci − xi aˆ ) 2 + [(ci − ls i ) − ( xi aˆ − lxi rˆ)] 2 + [(ci + ls i ) − ( xi aˆ + lx i rˆ)] 2 = 3(ci − xi aˆ ) 2 + 2[l ( si − xi rˆ)] 2 = 3(eic ) 2 + 2l 2 (eis ) 2

e(2i ) = (ci − xi aˆ ( i ) ) 2 + [(ci − lsi ) − ( xi aˆ ( i ) − lxi' rˆ(i ) )]2 + [(ci + lsi ) − ( xi' aˆ ( i ) + lxi' rˆ( i ) )]2 = 3(ci − xi aˆ (i ) ) 2 + 2(lsi − l ( xi rˆ( i ) )) 2 ec es =3 ( i ) 2 + 2l 2 ( i ) 2 1 − hii 1 − hii

 e =  i  1 − hii

  

2

A.3: Proof of Lemma 4.1 ~ In order to prove d LR is a metric, we need to prove the following three properties: ~ ~ ~ 1. ∀ Χ, Υ ∈ FLR (ℜ) , d LR ( Χ, Υ ) ≥ 0 . If d LR ( Χ, Υ ) =0 then Χ = Υ . ~ ~ ~ 2. ∀ Χ, Υ ∈ FLR (ℜ) , d LR ( Χ, Υ ) = d LR (Υ , Χ ) . ~ ~ ~ ~ 3. ∀ Χ , Υ , Ζ ∈ FLR (ℜ) , d LR ( Χ, Υ ) ≤ d LR ( Χ, Ζ) + d LR ( Ζ, Υ ) . Since d is a metric, it’s easy to show that properties 1 and 2 are satisfied. We need to show that property 3 is satisfied: k k k ~ d LR ( Χ, Υ ) = ∑ d 2 ( X i , Yi ) ≤ ∑ d 2 ( X i , Z i ) + ∑ d 2 ( Z i , Yi ) i =1

i =1

i =1

~ ~ ≤ d LR ( Χ, Ζ) + d LR ( Ζ, Υ ) +2

(

)

k

∑ d 2 (X i , Zi ) i =1

2 ~ ~ = d LR ( Χ, Ζ) + d LR ( Ζ, Υ ) ~ ~ ~ Therefore, d LR ( Χ, Υ ) ≤ d LR ( Χ, Ζ) + d LR ( Ζ, Υ ) .

k

∑d i =1

2

( Z i , Yi )

{ }

∞

Assume that ( FLR (ℜ) , d LR ) is a complete metric. Let Χ m m =1 be a Cauchy ~ ~ sequence in FLR (ℜ) , i.e., ∀ ε > 0 , ∃ l ∈ Ν ∋ m, m' > l ⇒ d LR ( Χ m , Χ m ' ) < ε . Then, for ∀ m, m' > l , d ( X mj , X mj ' )