Research Article Analyzing Big Data with the Hybrid Interval Regression Methods

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1 Hidawi Publishig Corporatio e Scietific World Joural Volue 204, Article ID 24392, 8 pages Research Article Aalyzig Big Data with the Hybrid Iterval Regressio Methods Chia-Hui Huag, Keg-Chieh Yag, 2 ad Ha-Yig Kao 3 Departet of Busiess Adiistratio, Natioal Taipei Uiversity of Busiess, No. 32, Sectio, Jia Road, Zhogzheg District, Taipei City 00, Taiwa 2 Departet of Iforatio Maageet, Hwa Hsia Istitute of Techology, No., Gogzhua Road, Zhoghe District, New Taipei City 235, Taiwa 3 Departet of Coputer Sciece ad Iforatio Egieerig, Natioal Dog Hwa Uiversity, No.23,Hua-ShiRoad,Hualie97063,Taiwa Correspodece should be addressed to Chia-Hui Huag; leohkkio@gail.co Received 9 May 204; Accepted 7 July 204; Published 20 July 204 Acadeic Editor: Jug-Fa Tsai Copyright 204 Chia-Hui Huag et al. This is a ope access article distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. Big data is a ew tred at preset, forcig the sigificat ipacts o iforatio techologies. I big data applicatios, oe of the ost cocered issues is dealig with large-scale data sets that ofte require coputatio resources provided by public cloud services. How to aalyze big data efficietly becoes a big challege. I this paper, we collaborate iterval regressio with the sooth support vector achie (SSVM) to aalyze big data. Recetly, the sooth support vector achie (SSVM) was proposed as a alterative of the stadard SVM that has bee proved ore efficiet tha the traditioal SVM i processig large-scale data. I additio the soft argi ethod is proposed to odify the excursio of separatio argi ad to be effective i the gray zoe that the distributio of data becoes hard to be described ad the separatio argi betwee classes.. Itroductio Big data has becoe oe of ew research frotiers. Geerally speakig, big data is a collectio of large-scale ad coplex data sets that it becoes ore difficult to process usig curret database aageet systes ad traditioal data processig applicatios. I 202, Garter Ic. gave a defiitio of big data as Big data is high volue, high velocity, ad/or high variety iforatio assets that require ew fors of processig to eable ehaced decisio akig, isight discovery ad process optiizatio []. The tred of big data sets is due to the additioal iforatio derivable fro aalysis of a sigle large set of related data, as copared to separate saller sets with the sae total aout of data. Oeoftheajorapplicatiosofthefutureparallel, distributed, ad cloud systes is i big data aalytic [2 5]. Most cocered issues are dealig with large-scale sets which ofte require coputatio resources provided by public cloud services. How to aalyze big data efficietly becoes a big challege. The support vector achie (SVM) has show to be a efficiet approach for a variety of data iig, classificatio, aalysis, patter recogitio, ad distributio estiatio [6 4]. Recetly, usig SVM to solve the iterval regressio odel [5] has becoe a alterative approach.hog ad Hwag [6] evaluated iterval regressio odels with quadratic loss SVM. Bisserier et al. [7] proposed a revisited fuzzy regressio ethod where a liear odel is idetified fro Crisp-Iputs Fuzzy-Outputs (CISO) data. D Urso et al. [8] preseted fuzzy clusterwise regressio aalysis with LR fuzzy respose variable ad ueric explaatory variables. The suggested odel is to allow for liear ad oliear relatioship betwee the output ad iput variables. Jeg et al. [9] developed a support vector iterval regressio etworks (SVIRNs) based o both SVM ad eural etworks. Huag ad Kao [20] proposed a soft-argi SVM for iterval

2 2 The Scietific World Joural regressio aalysis. Huag [2] solved iterval regressio odel with reduced support vector achie. However, there are several ai probles while usig SVM odel. () Big data: whe dealig with big data sets, the solutio byusigsvmwithaoliearkerelaybedifficult to be foud. (2) Noises ad iteractio: the distributio of data becoes hard to be described ad the separatio argi betwee classes becoes a gray zoe. (3) Ubalace: the uber of saples fro oe class is uch larger tha the uber of saples fro other classes. It causes the excursio of separatio argi. Uder this circustace, developig a efficiet ethod to aalyze big data becoes iportat. The sooth support vector achie (SSVM) has bee proved ore efficiet tha the traditioal SVM i processig large-scale data [22 24]. The ai idea of SSVM is solved by a fast Newto-Arijo algorith [25] ad has bee exteded to oliear separatio surfacesbyusigaoliearkereltechology[24]. I this study, we collaborate iterval regressio [5]with SSVM to aalyze big data. The ai idea of SSVM is solved by a fast Newto-Arijo algorith ad has bee exteded to oliear separatio surfaces by usig a oliear kerel techology. Additioally, to odify the excursio of separatio argi ad to be effective i the gray zoe, the soft argi ethod is proposed. The experiet results show thattheproposedethodsareoreefficietthaexistig ethods. This study is orgaized as follows.sectio 2 reviews the curret ethods for iterval regressio aalysis. Sectio 3 proposes the soft argi ethod ad the forulatio of iterval regressio with SSVM to aalyze big data. Sectio 4 gives a uerical exaple by the proposed ethods dealig with big data which is extracted fro Taiwa Stock Exchage Capitalizatio Weighted Stock Idex (TAIEX) [26]. Fially, Sectio5 gives the cocludig rearks. 2. Literature Review Sice Taaka et al. [27] itroduced the fuzzy regressio odel with syetric fuzzy paraeters, the properties of fuzzy regressio have bee studied extesively by ay researchers. Fuzzy regressio odel ca be siplified to iterval regressio aalysis which is cosidered as the siplest versio of possibilistic regressio aalysis with iterval coefficiets. A iterval liear regressio odel is described as Y(x j )=A 0 A x j A x j, () where Y(x j ), j =,2,...,q, is the estiated iterval correspodig to the real iput vector x j =(x j,x 2j,...,x j ) t. A iterval coefficiet A i is defied as (a i,c i ),wherea i is the ceter ad c i is the radius. Hece, A i ca also be represeted as A i = [a i c i,a i c i ] = a i c i a a i c i }. (2) The iterval liear regressio odel () ca also be expressed as Y(x j )=A 0 A x j A x j =(a 0,c 0 )(a,c )x j (a,c )x j =(a 0 a i x ij,c 0 c i x ij ). For a data set with crisp iputs ad iterval outputs, two iterval regressio odels, the possibility ad ecessity odels, are cosidered. By assuptio, the ceter coefficiets of the possibility regressio odel ad the ecessity regressio odel are the sae [5]. For this data set, the possibility ad ecessity estiatio odels are defied as Y (x j )=A 0 A x j A x j Y (x j )=A 0 A x j A x j, where the iterval coefficiets A i ad A i are defied as A i = (a i,c i ) ad A i =(a i,c i ),respectively.theitervaly (x j ) estiated by the possibility odel ust iclude the observed iterval Y j ad the iterval Y (x j ) estiated by the ecessity odel ust be icluded i the observed iterval Y j. I this sectio, we review the curret ethods which are ordiarily used for iterval regressio aalysis. 2.. Taaka ad Lee s Approach. Taaka ad Lee [5] proposedaitervalregressioaalysiswithaquadraticprograig (QP) approach which gives ore diverse spread coefficiets tha a liear prograig (LP) oe. The iterval regressio aalysis by QP approach uifyig thepossibilityadecessityodelssubjecttotheiclusio relatios, Y (x j ) Y j Y (x j ), ca be represeted as i q j= (d 0 d i x ij ) 2 φ i=0 (a 2 i c 2 i ) s.t. Y (x j ) Y j Y (x j ), j=,2,...,q c i,d i 0, i=0,,...,, where φ is a extreely sall positive uber ad akes the ifluece of the ter φ i=0 (a2 i c 2 i ) o the objective (3) (4) (5)

3 The Scietific World Joural 3 fuctio egligible. The costraits of the iclusio relatios are equivalet to (λ 3i λ 3i )(λ 3j λ 3j )K( x i, x j ) Y (x j ) Y j y j e j (a 0 (a 0 Y j Y (x j ) a i x ij )(c 0 a i x ij )(c 0 c i x ij ) c i x ij ) y j e j, (6) 2 (λ 4i λ 4i )(λ 4j λ 4j )K( x i, x j ) (λ 3i λ 3i )(λ 4j λ 4j )K( x i, x j ) λ i λ j K( x i, x j ) (a 0 (d 0 a i x ij )(c 0 c i x ij ) d i x ij ) y j e j y j e j (a 0 a i x ij )(c 0 (d 0 d i x ij ), c i x ij ) where x j is the jthiputvectorady j is the correspodig iterval output that cosists of a ceter y j ad a radius e j deoted by Y j =(y j,e j ) Hog ad Hwag s Approach. Hog ad Hwag [6] evaluated iterval regressio odel cobiig the possibility ad ecessity estiatio forulatio with the priciple of quadratic loss support vector achie (QLSVM). This versio of SVM utilizes the quadratic loss fuctio. The QLSVM perfors iterval oliear regressio aalysis by costructig a iterval liear regressio fuctio i highdiesioal feature space. With the priciple of QLSVM, the iterval oliear regressio odel is give as follows: ax 2 ( (λ 2i λ 2i )(λ 2j λ 2j )K(x i, x j ) (λ 3i λ 3i )(λ 3j λ 3j )K(x i, x j ) (λ 4i λ 4i )(λ 4j λ 4j )K(x i, x j ) (λ 2i λ 2i )(λ 3j λ 3j )K(x i, x j ) (λ 2i λ 2i )(λ 4j λ 4j )K(x i, x j ) (λ 3i λ 3i )(λ 4j λ 4j )K(x i, x j ) (7) 2 λ i (λ 3j λ 3j )K( x i, x j )) λ 2 i 2C (λ 2 2i 2C λ 2 2i ) (λ 2i λ 2i )y i (λ 4i λ 4i )y i (λ 3i λ 3i )ε i (λ 3i λ 3i )y i (λ 4i λ 4i )ε i s.t. λ i,λ 2i,λ 2i,λ 3i,λ 3i,λ 4i,λ 4i 0, (8) where λ i, λ 2i, λ 2i, λ 3i, λ 3i, λ 4i, ad λ 4i are Lagrage ultipliers. K( ) is a oliear kerel. The followigs are well-kow oliear kerels, where σ, γ, r, h, adθ are kerel paraeters: () Gaussia (radial basis) kerel: e x ix j 2 /2σ 2, σ > 0 [0], (2) hyperbolic taget kerel: tah(γx i x t j θ), γ>0[2], (3) polyoial kerel: (γx i x t j r)h, h N, γ>0,ad r 0[4]. The advatage of Hog ad Hwag s approach is a odelfree ethod i the sese that there is o eed to assue the uderlyig odel fuctio for iterval oliear regressio odel with crisp iputs ad iterval output Huag s Approach. There are two probles while usig thetraditioalsvmodel.()largescale:whedealigwith large-scale data sets, the solutio ay be difficult to be foud whe usig SVM with oliear kerels; (2) Ubalace: theuberofsaplesfrooeclassisuchlargertha the uber of saples fro the other classes. It causes the excursio of separatio argi. To resolve these probles, Huag [2] proposed a reduced support vector achie (RSVM) approach i evaluatig iterval regressio odels. RSVM has bee prove

4 4 The Scietific World Joural ore efficiet tha the traditioal SVM i processig largescale data. With the priciple of RSVM, the iterval oliear regressio odel is listed as follows: ax 2 2 λ i λ j Q t,k Q,K (λ 2i λ 2i )(λ 2j λ 2j )K(x i, x j ) 2 2 (λ 3i λ 3i )(λ 3j λ 3j )K(x i, x j ) λ i Q,K (λ 2j λ 2j ) x j λ i Q,K (λ 3j λ 3j ) x j (λ 2i λ 2i )(λ 3j λ 3j )K(x i, x j ) (λ 2i λ 2i )(λ 2j λ 2j )K( x i, x j ) (λ 2i λ 2i )(λ 3j λ 3j )K( x i, x j ) (λ 3i λ 3i )(λ 3j λ 3j )K( x i, x j ) λ 2 i 4C (λ 2i λ 2i )y i (λ 3i λ 3i )y i (λ 2i λ 2i )ε i (λ 3i λ 3i )ε i s.t. λ i,λ 2i,λ 2i,λ 3i,λ 3i 0, (9) where λ i, λ 2i, λ 2i, λ 3i,adλ 3i are Lagrage ultipliers. Q is a positive seidefiite atrix i RSVM. K( ) is a oliear kerel. The advatage of Huag s approach is to reduce the uber of support vectors by radoly selectig a subset of saples. While processig with large-scale data sets, the solutiocabefoudeasilybytheproposedethodwith oliear kerels. 3. Proposed Methods Figure : Soft argi. I this sectio we first propose the soft argi ethod to odify the excursio of separatio argi ad to be effective i the gray zoe. The the forulatio of iterval regressio withssvmtoaalyzebigdataisitroduced. 3.. Soft Margi. I a covetioal SVM, the sig fuctio is used as the decisio-akig fuctio. The separatio threshold of the sig fuctio is 0, which results i a excursio of separatio argi for ubalaced data sets. The ai of the hard-argi separatio argi is to fid a hyperplae with the largest distace to the earest traiig data. However, the liitatios of the hard-argi forulatio are as follows: () there is o separatig hyperplae for certai traiig data; (2) coplete separatio with zero traiig error will lead to suboptial predictio error; (3) it is difficult to deal with the gray zoe betwee classes. Thus, the soft argi ethod is proposed to odify the excursio of separatio argi ad to be effective i the gray zoe. The soft argi is defied as f (δ) = f (δ) = arcta (δ sθ s) π arcta (δ sθ s) π 0.5, 0.5, (0) where δ is the decisio value. θ ad s are offset paraeter ad scale paraeter which eed to be estiated usig statistical ethod. With the soft argi as show i Figure, the predicatio of the class labels ca be deteried as follows: y (x) =, if (] r <f (δ),δ<θ) or (] r >f (δ),δ>θ), if (] r >f (δ),δ<θ) or (] r <f (δ),δ>θ), () where ] r is a rado uber betwee 0 ad.

5 The Scietific World Joural Iterval Regressio with SSVM. The ai idea of sooth supportvectorachie(ssvm)issolvedbyafastnewto- Arijo algorith [25] ad has bee exteded to oliear separatio surfaces by usig a oliear kerel techology [24]. Suppose that traiig data x i,y i }, i =,2,...,,are give, where x i R are the iput patters ad y i,} are the related target values of two-class patter classificatio case. The the stadard support vector achie with a liear kerel [4]is i w,b,ξ s.t. ξ 2 i 2 w 2 C y i (w t x i b) ξ i ξ i 0,,2,...,, (2) where b is the locatio of hyperplae relative to the origi. The regularizatio costat C is a positive paraeter to cotrol the tradeoff betwee the traiig error ad the part of axiizig the argi that is achieved by iiizig w 2. ξ i is the slack variable with weight C/2. w is the Euclidea or of w which is the oral to the followig hyperplaes: w t x i b=, for y i =, (3) w t x i b=, for y i =. (4) The first hyperplae (3) bouds the class} ad the secod hyperplae (4) bouds the class}. The liear separatig hyperplae is w t x i b=0. (5) I Lee ad Magasaria s approach [24], b 2 /2 is added to the objective fuctio of (2). This is equivalet to addig a costat feature to the traiig data ad fidig a separatig hyperplae through the origi. Cosider i w,b,ξ s.t. ξ 2 i 2 ( w 2 b 2 ) C 2 y i (w t x i b) ξ i ξ i 0,,2,...,, (6) where ξ i =y i (w t x i b)} for all i ad the fuctio is defied as x = ax0, x}. The(2) cabereforulated as the followig iiizatio proble by replacig ξ i with y i (w t x i b)} : i w,b 2 ( w 2 b 2 ) C y 2 i (w t x i b)} 2. (7) The objective fuctio i (7) is ot twice differetiable ad ca be solved by usig a fast Newto-Arijo ethod [25]. Thus the fuctio i SSVM is approxiated by a sooth fuctio, p(x, α), as follows: p (x, α) =x α log ( eαx ), α>0, (8) where α > 0 is the sooth paraeter. /( e αx ) is the itegral of the sigoid fuctio of eural etworks [28]. The p(x, α) with a soothig paraeter α is to replace the fuctio of (7) to obtai the followig sooth support vector achie (SSVM) with a liear kerel: i w,b 2 ( w 2 b 2 ) C p( y 2 i (w t x i b)},α) 2. (9) For specific data sets, a appropriate oliear appig x φ(x) cabeusedtoebedtheorigialr features ito ahilbertfeaturespacef, φ:r F, with a oliear kerel K(x i,x j ) φ(x i ) t φ(x j ).Thus,(9) ca be exteded to the SSVM with a oliear kerel: i w,b 2 ( w 2 b 2 ) C 2 p( y i ( j= V j K(x i,x j )b) } },α), } (20) where j= V jk(x i,x j )bis the oliear SSVM classifier. The coefficiet V j is deteried by solvig a optiizatio proble (20) ad the data poits with correspodig ozero coefficiets. With the priciple of SSVM, we ca forulate the iterval liear regressio odel as follows: i a,c,d 2 (at ac t cd t db 2 ) C 2 s.t. ax i c x i y i e i p( y i (w t x i b)},α) 2 ax i c x i y i e i ax i c x i d x i y i e i ax i c x i d x i y i e i,2,...,, 2 (2) where a, c, add are the collectios of all a i, c i,add i, i=,2,...,,respectively.

6 6 The Scietific World Joural Give (2), the correspodig Lagragia objective fuctio is (λ 3i λ 4i )(λ 3j λ 4j ) x i t x j b2 ) L:= 2 (at ac t cd t db 2 ) C 2 p( y i (w t x i b)},α) 2 λ i (y i e i ax i c x i ) λ 2i (ax i c x i y i e i ) λ 3i (ax i c x i d x i y i e i ) λ 4i (y i e i ax i c x i d x i ), (22) C 2 p( y i (w t x i b)},α) 2 s.t. λ i,λ 2i,λ 3i,λ 4i 0. (24) Siilarly, we ca obtai the iterval oliear regressio odel by appig x φ(x) to ebed the origial R features ito a Hilbert feature space F, φ : R F, with a oliear kerel K(x i,x j ) φ(x i ) t φ(x j ) as discussed i Sectio 2.2. The we obtai the optiizatio proble as (25) byreplacigx t i x j ad x i t x j i (24) withk(x i, x j ) ad K( x i, x j ),respectively: ax 2 ( (λ i λ 2i λ 3i λ 4i ) (λ j λ 2j λ 3j λ 4j )K(x i, x j ) where L is Lagragia ad λ i, λ 2i, λ 3i,adλ 4i are Lagrage ultipliers. The idea to costruct a Lagrage fuctio fro the objective fuctio ad the correspodig costraits is toitroduceadualsetofvariables.itcabeshowthatthe Lagragia fuctio has a saddle poit with respect to the prial ad dual variables i the solutio [29]. The Karush-Kuh-Tucker (KKT) coditios that the partial derivatives of L withrespecttotheprialvariables (a, c, d) for optiality L a =0 a= (λ i λ 2i λ 3i λ 4i ) x i, L c =0 c= (λ i λ 2i λ 3i λ 4i ) x i, L d =0 d= (λ 3i λ 4i ) x i. (23) Substitutig (23) i(22) yields the followig optiizatio proble: ax 2 ( (λ i λ 2i λ 3i λ 4i ) (λ j λ 2j λ 3j λ 4j ) x t i x j (λ i λ 2i λ 3i λ 4i ) (λ j λ 2j λ 3j λ 4j ) x i t x j C 2 (λ i λ 2i λ 3i λ 4i ) (λ j λ 2j λ 3j λ 4j )K( x i, x j ) (λ 3i λ 4i ) p( y i ( s.t. λ i,λ 2i,λ 3i,λ 4i Nuerical Exaple (λ 3j λ 4j )K( x i, x j )b2 ) j= V j K(x i, x j )b) } },α) } 2 (25) To illustrate the ethods developed i Sectio 3,thefollowig exaple is preseted. Exaple. To illustrate the proposed ethods dealig with big data sets, we use the data sets fro Taiwa Stock Exchage Capitalizatio Weighted Stock Idex (TAIEX) [26] which icluded the highest, lowest, ad closed data ad the rages are fro 0/02/202 to 2/28/202,fro0/02/20 to 2/28/202, fro 0/02/200 to 2/28/202, ad fro 0/02/2009 to 2/28/202, respectively. For these data sets, the Gaussia kerel [0] is used where σ = 2.5 ad the regularizatio costat C = 300. The results are illustrated fro Figure 2 to Figure 5. The copariso is show by usig the easure of fitess [5] as(26), which defies how closely the possibility output

7 The Scietific World Joural 7 Table : Copariso results of the easure of fitess. Taaka ad Lee [5] HogadHwag[6] Huag[2] Proposedethods φ Y (Figure 2) φ Y (Figure 3) φ Y (Figure 4) φ Y (Figure 5) TWSE TAIEX (0/02/202~2/28/202) TWSE TAIEX (0/02/200~2/28/202) Highest Lowest TAIEX Figure 2: TAIEX [26] fro 0/02/202 to 2/28/202. Highest Lowest TAIEX Figure4: TAIEX [26]fro 0/02/200 to 2/28/ TWSE TAIEX (0/02/20~2/28/202) TWSE TAIEX (0/02/2009~2/28/202) Highest Lowest TAIEX Figure 3: TAIEX [26]fro 0/02/20 to 2/28/ Highest Lowest TAIEX Figure 5: TAIEX [26]fro0/02/2009 to 2/28/202. for the jth iput approxiates the ecessity output for the jth iput. Cosider q φ Y (x i )= q j= c 0 c i x ij c 0 c i x ij d 0 d i x ij, (26) where q is a saple size ad 0 φ Y. Table presets the proposed ethods with a Gaussia kerel alog with the results coputed by Taaka ad Lee [5], Hog ad Hwag [6], ad Huag [2]. We ca fid that the proposed ethods are ore efficiet tha other ethods. 5. Coclusios I this paper, we collaborate iterval regressio with SSVM to aalyze big data. I additio, the soft argi ethod is proposed to odify the excursio of separatio argi ad to be effective i the gray zoe. The ai idea of SSVM is solved by a fast Newto-Arijo algorith ad has

8 8 The Scietific World Joural bee exteded to oliear separatio surfaces by usig a oliear kerel techology. The experiet results show thattheproposedethodsareoreefficietthaexistig ethods. I this study, we estiate the iterval regressio odel with crisp iputs ad iterval output. I future works, both iterval iputs-iterval output ad fuzzy iputs-fuzzy output will be cosidered. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgets The authors appreciate the aoyous referees for their useful coets ad suggestios which helped to iprove thequalityadpresetatioofthispaper.theorigialversio was accepted by Iteratioal Coferece o Busiess, Iforatio, ad Cultural Creative Idustry, 204. Also, special thaks are due to the Natioal Sciece Coucil, Taiwa, for fiacially supportig this research uder Grats os. NSC H-4-02-MY2 (C. H. Huag) ad NSC H (H. Y. Kao). Refereces [] D. Laey, The Iportace of Big Data: A Defiitio, Garter, 202. 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