Two-stage Framework for Visualization of Clustered High Dimensional Data
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- Horatio Alexander
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1 T- F Vulzn Clu H Dnnl D Jul C Cll Cun G Inu Tnl 266 F Dv, Aln, GA 3332, USA Sn Bn Nnl Vulzn n Anl Cn P N Nnl L 92 Bll Blv, Rln, WA 99354, USA Hun P Cll Cun G Inu Tnl 266 F Dv, Aln, GA 3332, USA ABSTRACT In, u nn un 2D vulzn nnl lu. W vulzn u n nn un. In, n u nnl ln uv nn un u ln nn nl v nl lu uu n. T uln l u nn n n zn n n l n 2. In n, nn u u 2 vulzn u n nn un u nl nn nl. T l n- nz l nn u un nn ll 2. Un, vl -, n n l ll xnl n n l n l-l x. K: nn un, ln nn nl, nl nn nl, nl n, 2D n, lu, ulzn, nlz nul vlu n Inx T: H.5.2 [INFORMATION INTERFACES AND PRE- SENTATION]: U In T n 1 INTRODUCTION Wn vul nl un, vu nn nn n un nnl nu. T nu uul n n n n l nn (.., 2D 3D) v vul nn u l. Mn n un v u n nn un nqu,.., l-nzn (SOM) [12], nl nn nl (PCA) [11], ulnnl ln (MDS) [2],. Hv, n l l n nqu n n v n, ll n u. Tll, l n v n : n n u. En n u un, u u n l qun. Mn ul l qun u u ln vn n nnl u nnl. N nqu l nl n nl. T n nn n u. Sll, n l ll n vn n n un nl. N nl un nqu ll v l v vu u l. W v nn ll v ln vn. Hv, l : ul n uu vulzn. Oul null (.., vn n l), u ul qun, -l: {ull, }@..u -l:n.n@nl.v n nl n ul n nu u. Cn nqu (.., PCA) n ul l, v n u xn n ul. O nqu (.., SOM) xz u ll, u xn n vn n ul. L, uu nnl vl un un. M uu l un (.., lu) x n nl nnl. W nz n n nn un n n vul nl un. Hv, u n nn ll vulzn lu [2, 13, 3]. W l u n n l ll n nl u ull u/nl unn ln n n nl. 2 MOTIVATION T u unnl nn un nqu vulzn nnl n 2D l n lu uu. T l nn un v 2-nnl nn l vn lu uu u l. T n, uv nn un n lu nn u ln nn nl (LDA) [4] nl n (OCM) [1] n null n. Hv, n u n nn un n v lu uu n, ll l u nn, ll nn l zn nn un, uull l n 2. F xl, n LDA, nu u nn v lu uu qul u n xzn l n l n nu lu n n nl [8, 7]. In, n l nn nu u u. Hv, nl nn, u u nnl ll n nl qun nn un. T ul n l nn n vulzn n l ln u lu ln. A l un u n un PCA vulzn n vn lu uu. Evn u PCA n nl x xll u vn, n uln 2-nnl nn nn nl ll n l vn, ln n 2 nn ll l nnn n nl nn. Su l nn nvl n nn u 2. Ou n vn l u l ull n l-nun nl nn un. B n, - nn un vulzn.
2 In, uv nn un l n nl nn u nu nn vl l vn qul lu u n n nn un. T u nn v n n l n 2. Tu n n, n n nn un nn nz l nu n u un nn ll 2. T - v u n lxl l n nn un nqu n n ll nlz, v unnn vll nn un vulzn lu. T u n n nn un, ln zn, n nn lun. In, n l n l n u. Sll, vl - ulzn ln nn un u LDA, nl n (OCM), n nl nn nl (PCA), n n l un ln zn v l lun. Al, llu n vn n l vulzn n n n l-l.alu nnln nn un u MDS nl lnn u u n [16] n lll ln n [14] l ulz v 2D vulzn nnl, u u n n ln. T ln unll n n nl, n unl nl lnn, l v nn un nn n l n vulz unn n n xn vulz. Ou uvl l nn un ul n vu [18, 19, 21] n uull vn unl n, ll, ll n nn un,.., LDA. T nz ll. In Sn 3, LDA, OCM, n PCA n un n zn l. In Sn 4, ul - nn un, n n Sn 5, vl - vulzn n ln. Exnl n vn un l n l-l n Sn 6, n nlun n uu n Sn 7. 3 DIMENSION REDUCTION AS TRACE OPTIMIZATION PROBLEM In n, nu nn u n nn lu qul n zn nn un. Su nn un ln nn G T R l n -nnl v x v z n n l- nnl ( > l): G T : x R 1 z = G T x R l 1. (1) Su l x A = [ 1 2 n ] R n vn lun, = 1,..., n, A n n n n -nnl, n n n lu. Wu l nl, l nn, u u A n A = [A 1 A 2 A ], A R n n n = n. =1 L N n lun n ln lu, n n z N. T - lu n () n ll n n, vl, () = 1 n n = 1 n n. N =1 T x n - lu S (), n-lu x S, n-lu x S, n l ( xu) x S n [9, 15], vl, S () = N ( () )( () ) T, S = S = S = = 1 n S () = ( () )( () ) T, (2) =1 =1 N ( () )( () ) T = n ( () )( () ) T =1 N =1 1 n n ( () ( ) )( () ( ) ) T, n (3) =1 =+1 n ( )( ) T. (4) =1 N l x S l S n S [9] S = S + S. (5) Wn G T n Eq. (1) l x A, S, S, n S n nl nnl u l l G T S G, G T S G, n G T S G, vl. B un (S ) = = (S ) = = (S ) = = 1 n ( () ) T ( () ) =1 N () 2 2, (6) =1 N ( () ) T ( () ) =1 N n () 2 2 (7) =1 1 =1 =+1 n n () ( ) 2 2, n (8) n ( ) T n ( ) = 2 2, (9) =1 =1 n vlu n u u lu qul. N Eq. (7) n (8), (S ) n v qu u n n lu n ll n n n n ll n. T lu uu qul n n nlzn ll lu n n. H qul lu uull v ll (S ) n l (S ), ln ll vn n lu n l n n lu. Suqunl, nn un nn xz (G T S G) n nz
![alu nnln nn un u MDS nl lnn u u n [16] n lll ln n [14] l ulz v 2D vulzn nnl, u u n n ln. T ln unll n n nl, n unl nl lnn, l v nn un nn n l n vulz unn n n xn vulz.](/docs-images/52/13108430/images/page_2.jpg "Ou uvl l nn un ul n vu [18, 19, 21] n uull vn unl n, ll, ll n nn un,.., LDA. T nz ll. In Sn 3, LDA, OCM, n PCA n un n zn l. In Sn 4, ul - nn un, n n Sn 5, vl - vulzn n ln.")
3 (G T S G) n u nnl. T ulnu zn n x nl n J / (G) = x((g T S G) 1 (G T S G)), (1) n LDA. In n, n u n xzn n n lu, n n n OCM,.., J (G) = x G T G=I (GT S G). (11) On n, l lu nn, S n S, l x S n xz J (G) = x G T G=I (GT S G), (12) un u n PCA. In Eq. (11) n (12), u nn, G T G = I, J (G) n J (G) n l l. In ll, LDA, OCM, n PCA u n u xzn, n lvn vulzn n. 3.1 Ln Dnn Anl (LDA) Cnull, n LDA, ln nn un nn n-lu ln l xzn (G T S G) l n n lu ln l nzn (G T S G). A n n Eq. (1), n LDA n n J / (G) = x((g T S G) 1 (G T S G)). (13) I n n n G R l > l, ((G T S G) 1 (G T S G)) (S 1 S ), (14) nn lu uu qul u (S 1 S ) nn n nn un [4]. B n vv Eq. (13) G z, v l nn, n n lun LDA, n G LDA, lun ln nlz nv u nlz nvlu l, S u = λs u. (15) Sn n S 1, LDA v u un ((G T S G) 1 (G T S G)) n Eq. (14) n l u l 1,.., ((G T LDA S G LDA ) 1 (G T LDA S G LDA )) = (S 1 S ) l 1, (16) n (S 1 S ) v n nl n u nnl n G LDA. 3.2 Onl Cn M (OCM) Onl n (OCM) [1] u nl n xzn (G T S G) un nn G T G = I. T n OCM n J (G) = x G T G=I (GT S G). (17) I nn n G R l > l u G T G = I, (G T S G) (S ), (18) n lu uu qul u (S ) nn n nn un. T lun Eq. (17) n n n lun G ln nv S. Sn S 1 nnz nvlu, u un (G T S G) n Eq. (18) n v n l u l 1,.., (G T S G) = (S ) l 1. (19) Eq. (19) n (S ) v n nl n u nnl. An vn OCM v n u un (G T S G) nl un QR n, vn nn. T l OCM ll. F n x C lun C ] lu n v,.., C = [ 1 2. Tn u QR n [5] C u C = Q R Q R Q T Q = I n R R u nul. T lun OCM, G OCM, un G OCM = Q. N lun G OCM nl u nn n, n l = n. Fnll, OCM v (G T OCM S G OCM ) = (S ), l =. B un quvln n Eq. (3) n (3), n n v n n lu n l v n u nnl n OCM. An n OCM n n u nn n, l n n ul n n n v n Euln n n n l u [1, 7]. In, n v q R 1 n lu n () n ( ), v q () 2 < q ( ) 2 G T OCM q GT OCM () 2 < G T OCM q GT OCM ( ) 2, n q T () q 2 () < qt ( ) 2 q 2 ( ) 2 ( G T OCM q ) T G T OCM () G T OCM q 2 G T OCM () 2 < 3.3 Pnl Cnn Anl (PCA) ( G T OCM q ) T G T OCM ( ) G T OCM q 2 G T OCM ( ) 2. PCA ll-nn nn un u xl vn n. T n PCA n n J (G) = x G T G=I (GT S G). (2) F n G R l > l u G T G = I, v (G T S G) (S ), (21) n (S ) nn n nn un. T lun Eq. (2), n G PCA, n n n lun G ln nv S. Sn n S n(, n), PCA v u un (G T S G) n Eq. (21) n l u l n(, n),.., (G T PCA S G PCA ) = (S ) l n(, n).
4 Tl 1: Cn nn un. I u S n S ull n. LDA OCM PCA Ozn Cn (x R 1 GT R l 1 ) J / (G) = x((g T S G) 1 (G T S G)) J (G) = x G T G=I (GT S G) J (G) = x G T G=I (GT S G) Al nlz nn QR n nn Sll nn vn n u un 1 n(, n) In n ln PCA, v, l uull n x vlu l n n S u nn un n un. T n u n ll nv S, n v PCA nn. Alu S l S n S n Eq. (5), S n nn n nn n lu ll. T, unl LDA n OCM, PCA n lu uu n S n/ S, PCA n n unuv nn un. Uull, PCA u ll n z un l n v. T n n n A T, n- nnl v nn ll 1. PCA unqu, vn x l, u u nnl nn nz n n n x A T n n u nnl GG T (A T ) G nl lun,.., G PCA = n GG T (A T ) (A T ), G,G T G=I l x n Fnu n Euln n. T u uz n Tl 1. 4 FORMULATION OF TWO-STAGE FRAMEWORK FOR VISU- ALIZATION Su n n nn un ln nn V T R 2 n -nnl v x v z n 2-nnl ( 2): V T : x R 1 z = V T x R 2 1. (22) Fu u nn un ll. In, nn un ln nn G T R l n -nnl v x v n l-nnl (l ): G T : x R 1 = G T x R l 1, (23) l x nu l nn - n. Wn l 2, v n u nn un. Hv, n l l n n n n l n 2, n u l > 2. In n, n nn un ln nn H T R 2 l n l-nnl v v z n 2-nnl (l > 2): H T : R l 1 z = H T R 2 1. (24) Su nuv nn un G T ll H T n n, uln n nl nn un nn V T V T = H T G T. (25) In nx n, un ll u n vu n nn un nn G n n nn nn H u nu n nn un nn V T = H T G T 2-nnl vulzn n vu zn. 5 TWO-STAGE METHODS FOR 2D VISUALIZATION All - n uv nn un u LDA OCM n lu. In ( G T R l n Eq. (23)), nn u LDA OCM ll nn Eq. (16) (19), vl. T n, lu uu qul u (S 1 S ) (S ) v. Tn n- nn un ( H T R 2 l n Eq. (24)) nz l nn ln n u n un J n Eq. (2),.., PCA. A n n Sn 3.3, Eq. (2) v xn - ul nz n n Fnu/Euln n. In ll, - n l, n v quvln nl- ( V T R 2 n Eq. (22)) n x. 5.1 Rn-2 LDA In, LDA l n, n (S 1 S ) v n l-nnl l = 1. In n, n J / (H) u u l- nnl - ul 2-nnl. T n n- nn un x H n ul H / = x H R l 2 ((H T (G T LDA S G LDA )H) 1 (H T (G T LDA S G LDA )H)). (26) Aun lun G LDA, nlz nv Eq. (15), n n nn nlz nvlu,.., G LDA = [ u 1 u 2 u 1 ] λ 1 λ 2 λ 1, lun Eq. (26) H / = [ 1 2 ], 1 n 2 n n n un v,.., 1 = [ 1 ] T R l 1 n 2 = [ 1 ] T R l 1. T lun quvln n nn ln nlz nvlu ul, n uln - n n nl- nn un V R 2 l xz J /,.., V / = x V R 2 J /(V) = x V R 2 ((V T S V) 1 (V T S V)). (27)
5 S 1: Pvn (x R 1 GT R l 1 ) S 2: Mxzn ( Rl 1 HT z R 2 1 ) Tl 2: Su zn - nn un. Rn-2 LDA LDA + PCA OCM+PCA Rn-2 PCA n S ((G T S G) 1 (G T S G)) = (S 1 S ) (G T S G) = (S ) ((H T (G T S G)H) 1 (H T (G T S G)H)) H T H=I (HT (G T S G)H) H T H=I (HT (G T S G)H) H T H=I (HT (G T S G)H) T lun Eq. (27) V / = G LDA H / = [ u 1 u 2 ], u 1 n u 2 ln nlz nv Eq. (15). T lun l nn u-n ln nn nl [6]. 5.2 LDA ll PCA In, LDA l n, n (S 1 S ) v n l-nnl l = 1. In n, PCA l n n xn l-nnl - ul n Fnu/Euln n. T n- nn un x H n lvn H = x (H T (G T LDA S G LDA )H), (28) H R l 2,H T H=I lun ln nv l x - ul, G T LDA S G LDA. F Eq. (5), v G T LDA S G LDA = G T LDA (S + S )G LDA. (29) Sn LDA null xz (G T S G) n nz (G T S G), ul x (G T LDA S G LDA ) (G T LDA S G LDA )), n G T LDA S G LDA n G T LDA S G LDA,.., G T LDA (S + S )G LDA G T LDA S G LDA. In, nl x PCA v n n l n-lu x - ul, G T LDA S G LDA, n n un n lu ll n 2-nnl. In n, LDA ll PCA v l lu uu ll xn - ul. 5.3 OCM ll PCA In, OCM l n, n (S ) v n l-nnl l =. In n, PCA l n n xn l-nnl - ul n Fnu/Euln n. A n Sn 5.2, n- nn un x H n lvn H = x (H T (G T OCM S G OCM )H), (3) H R l 2,H T H=I lun ln nv l x - ul, G T OCM S G OCM. F Eq. (5), v G T OCM S G OCM = G T OCM (S + S )G OCM. (31) Unl LDA, OCM n nz (G T S G) n n Eq. (17). T lln n : (G T OCM S G OCM ) (G T OCM S G OCM ), n G T OCM S G OCM n nl n G T OCM S G OCM. Tn nl x G T OCM S G OCM n PCA n n n l l G T OCM S G OCM, n n lu, vnull vnn vulzn ul n l lu uu. 5.4 Rn-2 PCA n S In, OCM l n, n (S ) v n l-nnl l =. In n, n J (H) u u l-nnl - ul 2-nnl. T n- nn un x H n lvn H = x (H T (G T OCM S G OCM )H), (32) H R l 2,H T H=I lun ln nv n x - ul, G T OCM S G OCM. T lun G OCM u nn n, n u nlu n S. Anl, n n l nv u Y R l 1 G T OCM S G OCM l nv u R 1 S u Y = G T OCM u nn nvlu ll,.., λ Y = λ. Hn, lun Eq. (32) n n H = [ u Y 1 u Y ] [ ] 2 = G T OCM u1 u 2. (33) Un Eq. (33) n ln n n Eq. (25), nl nn un nn V n ul [ ] V T = H T u GT OCM = T 1 u T G OCM G T OCM 2 [ ] u T = 1 u T (34) 2 = x J (V) V R 2 = x V R 2 (V T S V). (35) Eq. (34) l n nv S, u 1 n u 2, n n G OCM. T n Eq. (35) n u n n uul vul nl, IN-SPIRE, 2D nn un [17]. T u - uz n Tl 2. 6 EXPERIMENTS In n, n vulzn ul un vl, ll un n unl x vulzn n n -nnl n nu n l n ( > n).
![G)H) T lun Eq. (27) V / = G LDA H / = [ u 1 u 2 ], u 1 n u 2 ln nlz nv Eq. (15). T lun l nn u-n ln nn nl [6]. 5.2 LDA ll PCA In, LDA l n, n (S 1 S ) v n l-nnl l = 1.](/docs-images/52/13108430/images/page_5.jpg "In n, PCA l n n xn l-nnl - ul n Fnu/Euln n. T n- nn un x H n lvn H = x (H T (G T LDA S G LDA )H), (28) H R l 2,H T H=I lun ln nv l x - ul, G T LDA S G LDA. F Eq.")
6 Tl 3: Dn. GAUSSIAN MEDLINE NEWSGROUPS REUTERS Onl nn, Nu, n Nu lu, Rulzn n LDA unl In unl, LDA n n n Eq. (13) nn l l u S nul. In v nul l, Hln l. unvl l LDA un nlz nul vlu n (LDA/GSVD) [8, 7]. Sll, n n, uul unl l, LDA/GSVD v lun G u G T S G = l nnn xu vlu (G T S G). T lun n n LDA n ul nz (G T S G). Hv, n ll n lnn lu n nl n n u nnl, ln nlzn l ln ll vulzn nvul ln n lu. On n, LDA G T S G = n v n vn vulzn u n LDA l ull nz (G T S G). B n ulzn n S n n nl (G T S G), n n n lu. In ulz LDA nll v nul S n ln nx, S l nnnul x S + γi I n n x, n γ nl. In nl, γ n, n-lu n, (G T S G), l l n n un nn n. A γ, n-lu n ll, n n l un n. Su nuln γ n xl n vulzn nx u n n n vlu γ n- u PCA, xz (G T S G) = (G T S G+G T S G), n u u n (G T S G). T ul ll n u γ. 6.2 D S T n ll-n Gun-xu (GAUSSIAN) n l-l x (MEDLINE, NEWSGROUPS, n REUTERS) lu n. All x un n -un nn n ul, n vlu n nn n qun nn n n un. E v n qul nu lu, n v n z n un ll n. (S Sn 6.3.) T n, l uz n Tl 3, ll. T GAUSSIAN nl n Gun xu 1 lu. E lu u 1 v, u 1 n l, n nn 11, ll n nu. In vulzn n n F. (1), lu ll un l,,..., n. T MEDLINE un u l l n Nnl Inu Hl 1. T nl nn 2295, n nu lu 5, lu- 1 ://...u/ /.l 1 un. T lu ll n un n ( ), ln n ( ), ( ), l n ( ), n ( ), l n n u n vulzn n n F. (2). T NEWSGROUPS [1] lln nu un, n nll 2. Hv, v n 11 vulzn, n lu v 7 un. T nl nn 1672, n lu ll n... ( ),... ( ),.l ( ),..ll ( ), ( ),.ln ( ),. ( ),.ln.n ( ), l.l.un ( ), l.l. ( ), n l.ln. ( ), l n n u n vulzn n n F. (3). T REUTERS [1] un lln n Ru n n 1987, n nll un. An, 1 l n u n vulzn, n lu 8 un. T nl nn 397, n lu ll n n ( ), qun ( ), n-x ( ), n ( ), u ( ), ( ), n ( ), ( ), ( ), n n ( ), l n n u n vulzn n n F. (4). 6.3 E D Cnn F. 5 xl ln OCM+PCA MEDLINE n u nn. On MEDLINE n -un x, v nn nn-nv vlu, ul n ll n nnl n. Tn n PCA u nn v nl x n ln ll n n nn lu. I n n n znl n vl x n F. 5,, n nl x, n l n n lu uu ll, n nl vl x, n n nl x PCA, n lu. W v un u unl v n n n u nn, u n uu. Anl, ll ul n n F 1-4 n nn. 6.4 Cn Vulzn Rul T ul u - n n F In ll, LDA- lu uu ll n OCM-. T v vn LDA n n- n n-lu u l OCM nl n un l. Du n, OCM nll u l- nn n lu. A ul, n NEWSGROUPS, u n-lu vn nnl 2 T u n l n u ln lun n ln vn.
7 lu uu vulzn vn OCM ll xz n-lu n. In MEDLINE n REUTERS, ll u u lvl l ul. Hv, v nll n-lu vn n LDA- un ulzn γi. In n, n-2 LDA n LDA+PCA u l nl ul n G T LDA S G LDA n G T LDA S G LDA LDA l n x. Rn-2 LDA n lu l nzn n-lu n n n. Hv, u n-lu n ll u J / xz nul u. A n n n LDA- l NEW- GROUPS, l n-2 LDA nz n-lu, l l n l n LDA+PCA. Du, n l n n-2 LDA n LDA+PCA n n vulz. Ovll, OCM+PCA n Rn-2 PCA n S l ul. I n G T S G G T S G n n n l n PCA l G T S G G T S G n n. Sn n PCA n G T S G unll n n PCA n G T S G, Rn-2 PCA n S n lnv OCM+PCA n n un n. Fnll, vulzn ul vl nn lu ln unln n. In F. (2), lu ln n ( ) n l n ( ) n l. In F. (3), lu.ln.n ( ) n l.ln. ( ),... ( ) n... ( ), n. ( ) n. ( ) ll l vl n LDA-. In n, lu,.l ( ) n..ll ( ), n nv, n u qu lvn. In F. (4), lu n ( ), ( ), n n ( ) ll n-x ( ) n n ( ) vulz v l. 7 CONCLUSION AND FUTURE WORK An u ul, LDA- n u OCM- n n- n nlu ln n n un n LDA. Ell, n PCA n n, LDA+PCA v l nn n lu ll xn ul LDA n n n u n Fnu/Euln n. Hv, n l x ll unl n vl ll n ln l nu n n lu. T nnl u nu n- nn un n nl x n l nu n n ll-n. Su v n x n n ul lu, n ln n n vulzn qu. In MEDLINE n REUTERS, vulzn ul v l- ln n. W n un nnn u n n. I ll unl u n vulzn,.. nn l n n. Fnll, n n u l nn nu, u nl n vu qunv u u n-lu n n n-lu ul nu. ACKNOWLEDGEMENTS T u u n Nnl Sn Funn n CCF n CCF An nn, nn n nlun nn x n l u n n nl l v Nnl Sn Funn. REFERENCES [1] A. Aunn n D. Nn. UCI n lnn. Unv Cln, Ivn, Sl Inn n Cu Sn, 27. [2] T. F. Cx n M. A. A. Cx. Mulnnl Sln. Cn & Hll, Lnn, [3] I. S. Dlln, D. S. M, n W. S. Snl. Cl vulzn -nnl ln. Cunl S & D Anl, 41(1):59 9, 22. [4] K. Fuun. Inun Sl Pn Rnn, n n. A P, Bn, 199. [5] G. H. Glu n C. F. vn Ln. Mx Cun, n. Jn Hn Unv P, Bl, [6] T. H, R. Tn, n J. Fn. T Eln Sl Lnn: D Mnn, Inn, n Pn. Sn, 21. [7] P. Hln, M. Jn, n H. P. Suu vn nn un lu x n nlz nul vlu n. SIAM Junl n Mx Anl n Aln, 25(1): , 23. [8] P. Hln n H. P. Gnlzn nn nl un nlz nul vlu n. Pn Anl n Mn Inlln, IEEE Tnn n, 26(8):995 16, Au. 24. [9] A. Jn n R. Du. Al lun. Pn-Hll, In. U Sl Rv, NJ, USA, [1] M. Jn, H. P, n J. B. Rn. Dnnl un n n n l qu n n x. In Pn F SIAM Innnl W n Tx Mnn. C, IL, 21. [11] I. Jll. Pnl nn nl. Sn, 22. [12] T. Knn. Sl-nzn. Sn, 21. [13] Y. Kn n L. Cl. Vulzn ll un ln nn. In Inn Vulzn, 23. INFOVIS 23. IEEE Su n, 23 3, O. 23. [14] S. T. R n L. K. Sul. Nnln Dnnl Run Lll Ln En. Sn, 29(55): , 2. [15] D. L. S n J. J. Wn. Un nn nu vl. IEEE Tnn n Pn Anl n Mn Inlln, 18(8): , [16] J. B. Tnnu, V.. Slv, n J. C. Ln. A Gll G F Nnln Dnnl Run. Sn, 29(55): , 2. [17] J. A. W. T ll x vulzn. Junl An S Inn Sn, 5(13): , [18] J. Y n Q. L. A - ln nn nl v qn. Pn Anl n Mn Inlln, IEEE Tnn n, 27(6): , Jun 25. [19] H. Yu n J. Yn. A LDA l -nnl ln nn. Pn Rnn, 34:267 27, 21. [2] X. Zn, C. M, n S. Kun. C- nn nl vulzn n. In Au, S, n Snl Pn, 24. Pn. (ICASSP 4). IEEE Innnl Cnn n, vlu 5, V vl.5, M 24. [21] W. Z, R. Cll, n A. Kn. Dnn nl nl nn nn. Au F n Gu Rnn, IEEE Innnl Cnn n, :336, 1998.
8 Fu 1: Cn - n GAUSS. () Rn-2 LDA () LDA+PCA () OCM+PCA () Rn-2 PCA n S Fu 2: Cn - n MEDLINE. () Rn-2 LDA () LDA+PCA () OCM+PCA () Rn-2 PCA n S Fu 3: Cn - n NEWSGROUPS. () Rn-2 LDA () LDA+PCA () OCM+PCA () Rn-2 PCA n S Fu 4: Cn - n REUTERS. () Rn-2 LDA () LDA+PCA () OCM+PCA () Rn-2 PCA n S Fu 5: Exl nn n MEDLINE. ()OCM+PCA nn ()OCM+PCA u nn
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