Journal of East China Normal University (Natural Science) : ramp Amijo-Newton
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1 ( ) Journal of East Cina Normal University (Natural Science) No. 2 Mar : (2016) ramp 1, 2 (1., ; ) : ramp Amijo-Newton... : ; ; ; : TP181; TP391 : A DOI: /j.issn Support vector macine in te primal space based on te ramp loss function Abstract: YUAN Yu-ping 1, AN Zeng-long 2 (1. College of Sciences, Heilongjiang Bayi Agricultural University, Daqing Heilongjiang , Cina; 2. College of Economics & Management, Heilongjiang Bayi Agricultural University, Daqing Heilongjiang163319, Cina) Aiming at te problem of standard support vector macine being sensitive to te noise, a new metod of support vector regression (SVR) macine based on dissymmetry quadratic and controlled-insensitive loss function is proposed. Using te concave and convex process optimization and te smoot tecnology algoritm, te problem of non-convex optimization is transformed into te problem of te continuous and twice differentiable convex optimization. Using te Amijo-Newton optimized algoritm of finite : : ( ) (XDB ); (HNK11A-14-07) :,,,. byndyyps@sina.com. :,,,. anzl@2001@163.com.
2 2, : ramp 21 iteration termination, te establised optimization model is solved, and te convergence of te algoritm is analyzed. Te algoritm can not only keep te sparse nature of support vector, but also control te abnormal values of te training sample. Te results of te experiment sowed tat te support vector regression macine model proposed kept good generalization ability, and te model could fit better bot te simulated data and te standard data. Compared wit te standard support vector macine (SVM) model, te proposed model not only can reduce te effects of noise and outliers, but also as stronger robustness. Key words: support vector regression; outliers; loss function; concave-convex procedure 0,,,,,.,.,, [1-4]... Lin 2002 (Fuzzy Support Vector Macine, FSVM) [5-6],,,,,,,. [7],, [8-9] Xu Crammer Hinge [10], Hinge,, ramp Wang, ramp, concave-convex procedure (CCCP), [11]., Zao,, [12]. Xu inge, [13]., [12] ramp, [13].,,,,, ramp,,.
3 22 ( ) Ramp 1.1 : T = {(x i, y i )} N, x i R n, y i. ε- : min w,b 1 2 w 2 + C (ξ i + ξi ), y i (w x i + b) ε + ξ i, w x i + b y i ε + ξ i, (1) ξ i, ξ i 0, i = 1, 2,, N,, C, (1), ξ i, ξ i, i = 1, 2,, N, : min w,b L ε(w, b) = 1 2 w 2 + C H θ (z i ). (2) ramp H θ (z i ) = min(θ 2, H A (z i )),, 0, ε 1 z i ε 2, H A (z i ) = (z i ε 2 ) 2, z i > ε 2, ( z i ε 1 ) 2, z i < ε 1, z i = w x i + b y i, ε 1 < ε 2. ramp : H θ (z i ) = min{θ 2, max(0, z i ε) 2 }, (0, θ 2 ), θ,., b, H, : min L(f) = 1 N f 2 f 2 H + C H θ (z i ). (3) [14], (3) f : (4) (3), : min β L(β) = 1 2 j=1 f(x) = β i k(x, x i ). (4) ( N β i β j k(x i, x j ) + C H β j k(x i, x j ) y i ). (5) j=1
4 2, : ramp 23 β = [β 1, β 2,, β N ] T, K, K ij = k(x i, x j ), i, j = 1, 2,, N, (5) : H uber 2 (z) = min L(β) = 1 β 2 βt Kβ + C H(z i ), (6) z i = Ki Tβ y i, i = 1, 2,, N. 1.2 H θ (z),,,, uber : θ[ 2z (2ε 1 + θ)], z ε 1 θ, ( z ε 1 ) 2, ε 1 θ < z < ε 1, H1 uber 0, ε 1 z ε 2, (z) = (z ε 2 ) 2, ε 2 < z < ε 2 + θ, ( z i ε 1 ) 2, z < ε 1, θ[2z (2ε 2 + θ)], z ε 2 + θ, θ[ 2z (2ε 1 + 2θ + )], z ε 1 θ, θ( z ε1 θ)2, ε 1 θ < z ε 1 θ, 0, ε 1 θ z ε 2 + θ, (z ε2 θ)2, ε 2 + θ < z < ε 2 + θ +, ( z i ε 1 ) 2, z < ε 1, θ[2z (2ε 2 + 2θ + )], z ε 2 + θ +, uber, ( < θ 2 ). Huber 1 H2 uber Hθ, uber (z): H uber θ, (z) = H1 uber + H2 uber (z) θ 2 + θ, z ε 1 θ, θ[ 2z (2ε 1 + θ)] θ( z ε1 θ)2, ε 1 θ < z ε 1 θ, ( z ε 1 ) 2, ε 1 θ < z < ε 1, = 0, ε 1 z ε 2, (z ε 2 ) 2, ε 2 < z ε 2 + θ, θ[2z (2ε 2 + θ)] (z ε2 θ)2, ε 2 + θ < z ε 2 + θ +, θ 2 + θ, z > ε 2 + θ +. 0, Hθ, uber (z) H θ (z), : min L(β) = 1 β 2 βt Kβ + C (H uber 1 (z i ) + H uber 2 (z i )), (7)
5 24 ( ) 2016 z i = K T i β y i, i = 1, 2,, N. (7),. 1. Fig. 1 1 Smoot non-convex loss function 1.3 (Concave-Convex Procedure, CCCP) [15]. E(θ) E cav (θ) E vex (θ), E(θ) = E cav (θ) + E vex (θ), E(θ). : (i) θ; (ii) θ i+1 = arg min θ (E vex (θ) + E cav(θ i ) θ), θ i ; (iii) θ = θ i. CCCP, (7) : min L(β) = 1 β 2 βt Kβ + C (H uber 1 (z i ) + H uber 2 (z i )), L vex = 1 2 βt Kβ + C N Huber 1 (z i ), L cav = C N Huber 2 (z i ), (7) β, (8),. β n+1 = arg min{l vex (β) + L cav (β n ) β}, (8) β β n CCCP, L cav (β n ) L cav (β) β β n. L cav (β n H2 uber (β n ) ) = C z i z i β = C N ηi n Ki T, (9) 2θ, zi n ε 1 θ, ηi n = Huber 2 (β n ) 2θ( zn i ε1 θ), ε 1 θ < zi n < ε 1 θ, = 0, ε z 1 θ zi n ε 2 + θ, i 2θ(zn i ε2 θ), ε 2 + θ < zi n < ε 2 + θ +, 2θ, zi n ε 2 + θ +. (8) (9), min L(β) = 1 β 2 βt Kβ + C ( H uber 1 (z i ) + η n i K T i β ). (10)
6 2, : ramp 25 CCCP, (7) (10), (10). 2 Amijo-Newton 2.1 Amijo-Newton [16] z i = Ki Tβ y i, ( 1). (1) ε 1 < z i < ε 2, NSV, NSV ; (2) ε 2 < z i < ε 2 + θ + ε 1 θ < z i < ε 1., 4 : ε 2 < z i < ε 2 + θ, ε 1 θ < z i < ε 1, SV 1 SV 2, SV 1 SV 2 ; ε 2 + θ < z i < ε 2 + θ +, ε 1 θ < z i < ε 1 θ, SV 3 SV 4, SV 3 SV 4 ; (3) z i > ε 2 + θ + z i < ε 1 θ ESV 1 ESV 2, ESV 1 ESV 2., SV 1, SV 2, SV 3, SV 4, ESV 1, ESV 2, NSV, I SV1, I SV2, I SV3, I SV4 N N, I SV1 = diag{i SV1, 0 SV2, 0 SV3, 0 SV4, 0 ESV1, 0 ESV2, 0 NSV }, I SV1 SV 1, 0 SV2 SV 2 0., I SV2 = diag{0 SV1, I SV2, 0 SV3, 0 SV4, 0 ESV1, 0 ESV2, 0 NSV }, I SV3 = diag{0 SV1, 0 SV2, I SV3, 0 SV4, 0 ESV1, 0 ESV2, 0 NSV }, I SV4 = diag{0 SV1, 0 SV2, 0 SV3, I SV4, 0 ESV1, 0 ESV2, 0 NSV }, (10) β L(β) Hesse H: L(β) =Kβ + CK( ) I SV1 + I SV2 Kβ + 2CK [ ] (z ε 2 )I SV1 ( z ε 1 )I SV2 2 [( + CK 2θ θ ) ( I SV3 + 2θ θ ] )I SV4, (11) H = 2 L(β) = K + CK( ) I SV1 + I SV2 K. (12) 2 : d = H 1 L(β) ( = β C I N + CK(I ) 1{ SV 1 + I SV2 )K 2[(z ε 2 )I SV1 ( z ε 1 )I SV2] 2 [( + 2θ θ ) ( I SV3 + 2θ θ ]} )I SV4. (13)
7 26 ( ) 2016 : β n+1 =β n + λd n = β n λ(h n ) 1 n L(β) ( = Cλ I N + C( ) ) I SV1 + I SV2 K 1 { 2 [ ] (z ε 2 )I SV1 ( z ε 1 )I SV2 2 [( + 2θ θ ) ( I SV3 + 2θ θ ]} )I SV4, (14) λ, (10)., N, I N + C(ISV 1 +ISV 2 )Kβ 2,,,, : ( I N + C I SV1 +I SV2 )K 2 0 I SV1 + C 2 KSV 1,SV C 1 2 KSV 1,SV C 2 2 KSV 1,SV C 3 2 KSV 1,SV C 4 2 KSV 1,ESV C 1 2 KSV 1,ESV C KSV 1,NSV C 2 KSV 2,SV C 1 2 KSV 2,SV C 2 2 KSV 2,SV C 3 2 KSV 2,SV C 4 2 KSV 2,ESV C 1 2 KSV 2,ESV C 2 2 KSV 2,NSV 0 I SV3 = 0 I SV4. B 0 I ESV1 0 I ESV2 A 0 I NSV. (14) β n+1, β n+1 ESV 1 = 0, β n+1 ESV 2 = 0, β n+1 NSV, : f n+1 (x) = = 0, β n+1 i k(x i, x). (15),,,. Amijo-Newton (10), : Step 1. χ = {(x i, y i ) n }, ρ, ε 1, ε 2, θ,, C, k = 0, χ f 0 (x); Step 2. z 0 = f 0 (x) y : SV 1, SV 2, SV 3, SV 4, ESV 1, ESV 2, NSV ; Step 3. n L(β), n L(β) ρ,, ; Step 4. λ k = max{1, 1/2, 1/4, } Amijo, L(β k ) L(β k + λ k d k ) b 1 λ k d T k L(βk ), b 1 (0, 1 2 ), ; Step 5. (14) β n+1, (15) f n+1 (x), step Amijo-Newton (10), β n. (7), min β L(β) = 1 2 βt Kβ + C N (Huber 1 (z i ) + H2 uber (z i )), L vex = 1 2 βt Kβ + C N Huber 1 (z i ), L cax = C N Huber 2 (z i ). β n+1 (10) n, : L vex (β n+1 ) + CΣ N ηn i K iβ n+1 L vex (β n ) + CΣ N ηn i K iβ n. (16)
8 2, : ramp 27, L cav (β n+1 ) L cav (β n ) L cav (β n )(β n+1 β n ) = CΣ N ηn i K iβ n+1 CΣ N ηn i K iβ n, (17) L cav (β n ) = C N ηn i K i. (16) (17), : L vex (β n+1 ) + L cav (β n+1 ) L vex (β n ) + L cav (β n ), (18). L(β) 0,,. 3, UCI.,. (RBF): K(x, y) = exp( x y 2 /σ 2 ). (19) 6 ε 1, ε 2, θ, C, σ,, (C, σ), {2 5, 2 4,, 2 4, 2 5 } {2 5, 2 4,, 2 4, 2 5 }. ramp, ; θ, θ,,,. θ,,,. (RMSE). Matlab 7.0, Windows XP, 2 GB, 2.99 GHz. 3.1 T = {(x 1, y 1 ), (x 2, y 2 ),, (x 300, y 300 )},, x i, i = 1, 2,, 300, [ 4, 4], y i = sin(3x i )/(3x i ) + γ i, i = 1, 2,, 300, γ i N(0, ), 200,., (LS-SVR), (LS-SVR) Fig. 2 Te algoritm of least square support vector regression (LS-SVR) 3 Fig. 3 Algoritm in tis paper, y i = sin(3x i )/(3x i ).,
9 28 ( ) 2016,. 1.,,,,. 1 Tab. 1 Te experimental results on artificial data sets RMSE NSV /s UCI,, UCI (ttp://arcive.ics.uci.edu/ml/) StatLib (ttp://lib.stat.cmu. edu/datasets/) Tab. 2 Statistical information on data set AutoMPG Boston ousing Bodyfat , NP-RSVR-NCLF(N-R-N) [10] LS-SVR, 3. 3 NP-RSVR-NCLF LS-SVR UCI Tab. 3 Comparison of te experimental results of standard UCI data set wit NP-RSVR-NCLF algoritm, LS-SVR algoritm and te algoritm in tis paper C σ ε 1 ε 2 RMSE NSV /s LS-SVR / / AutoMPG N-R-N / Boston- LS-SVR / / N-R-N LS-SVR / Bodyfat LS-SVR / / N-R-N E / E , LS-SVR, NP-RSVR-NCLF.,, LS-SVR,,,,.
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