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- みそら いさやま
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1 K-Means
2 1 5 2 K-Means K-Means K-Means K-Means K-Means K-Means K-Means LVQ K-Means K-Means
3 K =3 K-Means K=3,m =2 K-Means K =3 LVQ K=3,m =2 LVQ K-Means LVQ K-Means K-Means K-Means Km, ( , ),( , ) Km, ( , ),( , ) Km, ( , ),( , ) Km, ( , ),( , ) Km, ( , ),( , ) Km, ( , ),( , ) K-Means m =2 K-Means m =3 K-Means {R }, {R (m) } {D (m) } {ρ (m) } {d( c (m =2),p )} {d(r (m =2),p )} LVQ m =2 LVQ {R }, {R (m) } {D (m) } {ρ (m) }
4 13 {d(r (m =2),p )} Km,Km
5 1 2 [1][2][3] K-Means [5][8][9] 1 N ( ) 2 1 N K-Means c- [4][7] K-Means (Learning Vector Quantization:LVQ)[6][11] 2 K-Means LVQ K-Means LVQ K-Means [16][17] 5
6 K-Means K-Means K-Means K K-Means [13][14] K-Means 2 K-Means LVQ 3 K-Means 4 K-Means LVQ 5 K-Means, K-Means,LVQ 2, K-Means 6,7 6
7 2 K-Means K-Means K K-Means n x i =(x i1,...,x id ),i=1,...,n X K X, =1,...,K J = min { c,=1,...,k} i=1 n x i X x i c 2. (1) x i c 2 = D d=1 (x id c d ) 2 c =(c 1,..., c D ) (1) K-Means m (m1) { c (t) } x i x i X (t) α α = arg min x i c (t) 2. (2) (m2) {X (t) } c (t+1) = 1 n (t) x i X (t) x i, =1,...,K. (3) n (t) X(t) c (t+1) X (t) ( +1) ɛ c (t+1) c (t) <ɛ (m1) (m2) { c (t) } { c } (1) R R = { x x c 2 < x c i 2 for all i } (4) 7
8 if x R, x Class (5) { c } 1: 2.2 K-Means K-Means n X p x X q x X p X q X p X p { x} X q X q { x} c p, c q c p, c q X p N p 1, Xq N q +1 J [1] m (m 1) c (t), =1,...,K (m 2) J x i,i=1,...,n (m 2-1) x i X p N p 1 8
9 J p = J l = N p N p 1 x i c p 2 N l N l +1 x i c l 2, l p J q J l, for all l x i X p X q c p = c p 1 N p 1 ( x i c p ) c q = c q 1 N q +1 ( x i c q ) N p = N p 1 N q = N q +1 J J = J + J q J p X p Xp = X p { x i } X q Xq = X q { x i } x X p x X q x c p 2 = x c q 2 = x X p x c p 2 x X q x c q 2 N p N p 1 x i c p 2 N q N q +1 x i c q (LVQ) K-Means t t =1, 2,... x(t) R p (t = 1, 2,...) (VQ) m R p, =1,...,K x(t) x(t) 9
10 m l m l (t) = arg min x(t) m (t). (6) x(t) m l (t) m l (t +1)=m l (t)+α(t)[x(t) m l (t)]. (7) α(t) α(t) =, α 2 (t) <, t =1, 2,... (8) t=1 t=1 α(t) α(t) = /t x(t) x(t) lvq (lvq1) m, =1,...,K (lvq2) t =1, 2,... (lvq2-1) m l = arg min 1 K x(t) m (t) (9) (lvq2-2) m 1 (t),...,m (t) m l (t +1) = m l (t)+α(t)[x(t) m l (t)] m (t +1) = m (t), l x(t) X l 10
11 3 K-Means K-Means X {R } { c } R R R R K-Means R R K-Means R 3.1 K-Means K = m (2 m M) R R m {R (m),p,p= 1,...,m}, { c (m),p,p=1,...,m} M 2 3 R D (m=1) R (m),p D (m),p = x i R (m),p = m R (m) x i R x i c 2. (10) x i c (m),p 2,m=2,...,M (11) D (m) = m p=1 D (m),p. (12) R ρ (m) =D (m) /D (m 1),m=2,...,M. (13) ρ (m) R m R m R m 1 R m 11
12 D (m) D (m 1) ρ (m ) = min {ρ (m), m=2,...,m} (14) m η ρ (m ) <η (15) R m R η η 3.2 R {R (m ),p,p =1,...,m } {R (m ),p } { c (m ),p,p =1,...,m } K-Means R m 1 c (m ),p R d( c (m ),p ) ˆd( c (m ),p ) = max p {d( c(m ),p ),p=1,...,m } (16) R (m ),p R p p R (m ),p d( c (m ),p ) d(r (m ),p ) = min x i R (m ),p, x j R l,l d( x i, x j ) (17) ˆd(R (m ),p ) = max p {d(r(m ),p ),p=1,...,m } (18) R (m ),p R R (m ),p ) d(r (m ),p ρ (m ) <η 12
13 4 SVM(Support Vector Machine)[15] H Φ :R p H H K( x, y) = Φ( x), Φ( y) H (19) Φ( x) RBF K( x, y) = exp( C x y 2 ) (20) K( x, y) = (1 + x, y ) d (21) 4.1 K-Means K-Means (1) K-Means K J = Φ( x i ) m 2 (22) =1 x i X m X m = x X Φ( x) n. (23) Φ Φ Φ( x i ) m 2 K( x i, x j ) D i = Φ( x i ) m 2 H. (24) (22) K J = D i (25) =1 x i X 13
14 D i = Φ( x i ) m 2 = Φ( x i ) x j X Φ( x j ), Φ( x i ) n x l X Φ( x l ) n = Φ( x i ), Φ( x i ) 2 Φ( x i ), Φ( x j ) + 1 n n 2 Φ( x j ), Φ( x l ) x j X x j, x l X = K( x i, x i ) 2 K( x i, x j )+ 1 n n 2 K( x j, x l ) x j X x j, x l X (26) Km (Km1) X K y j (j = 1,..., K) x i x i X (t) α α = arg min Φ( x i ) Φ( y j ) 2 = arg min K( x i, x i ) 2K( x i, y j )+K( y j, y j ). (27) (Km2) x i (Km2-1) x i α = arg min Φ( x i ) m 2 (28) (Km2-2) Φ( x i ) m 2 (26) 14
15 4.2 K-Means 4.1 n (25) Km (Km 1) X K y j (j = 1,..., K) x i (27) (Km 2) x i (Km 2-1) x i α = arg min Φ( x i ) m 2 (Km 2-2) x i (Km 2-2) J = D i (29) x i X J = K J. (30) =1 x i X p Xq X p,x q, m p, m q N p,n q Xp = X p { x i }, Xq = X q { x i }, m p = m q = N p N p 1 Φ( x i) m p 2, N q N q +1 Φ( x i) m q 2. J,J p,j q J,J p,j q J = J 15
16 = J p + J q + J p,q J p = x X p Φ( x) m p 2 Φ( x i ) m p 2 = Φ( x) m p + Φ( x i) m p 2 N p N x X p 1 N p 1 Φ( x i) m p 2 p = J p N p N p 1 Φ( x i) m p 2 = J p N p N p 1 D ip (31) J q = J q + N q N q +1 D iq. (32) J = J p + J q + p,q J = J p N p N p 1 D ip + J q + N q N q +1 D iq + p,q = J N p N p 1 D ip + N q N q +1 D iq (33) J 4.3 LVQ LVQ m l (t) = arg min Φ( x h ) m (t) (34) m l (t +1) = m l (t)+α(t)[φ( x h ) m l (t)] (35) m m t D i (t) = Φ( x i ) m (t) 2 (36) 16
17 (34) D il (t) = arg min D i (t) (37) (35) D i (t +1) = Φ( x i ) m (t +1) 2 = Φ( x i ), Φ( x i ) 2 Φ( x i ), m l (t +1) + m l (t +1), m l (t +1) (38) (35) α = α(t) D i (t +1) = Φ( x i ), Φ( x i ) 2{(1 α) Φ( x i ), m l (t) + α Φ( x i ), Φ( x h ) } + {(1 α) 2 m l (t), m l (t) +2α(1 α) Φ( x h ), m l (t) } + α 2 Φ( x h ), Φ( x h ) (39) D il (t +1) = (1 α)d il (t) α(1 α)d hl (t) + α{k( x i, x i ) 2K( x i, x h )+K( x h, x h )} (40) LVQ Klvq (Klvq1) D i,i=1,...,n, =1,...,K (Klvq2) t =1, 2,... (Klvq2-1) x i X l D il (t) = min D i (t) (Klvq2-2) D il (40) 17
18 5 5.1 K-Means (x 1,x 2 )= (0, 0), (x 1,x 2 )=(0.1, 0.1) [10] 2 (x 1,x 2 )=(5, 0), (x 1,x 2 )=(2, 2) 3 (x 1,x 2 )=(1, 4), (x 1,x 2 )=(0.2, 0.2) (0.0861, 0.113), (4.98, 0.163), (1.10, 4.04) x 1 x class1 class2 class3 centroid x x1 2: 3 18
19 3 K-Means line1,line2,line3 3 R 1 line1 line3 c 1 =(1.67, 0.383), c 2 =(5.36, 0.146), c 3 =(1.55, 3.86) K-Means c1 c2 c3 cluster-center line1 line2 line3 3 2 x x1 3: K =3 K-Means 1: K-Means R 1 23 c 1 =(1.67, 0.383) R 2 83 c 2 =(5.36, 0.146) R 3 24 c 3 =(1.55, 3.86) 19
20 2 K-Means D D c D p K-Means 3 2 K-Means M=3 K=m K-Means m m =2, 3 {R } K-Means 3,4 2: D D D p R R R : m =2 K-Means R (m=2) 1,1 10 c (m=2) 1,1 =(0.0861, 0.113) R (m=2) 1,2 13 c (m=2) 1,2 =(2.89, 0.590) R (m=2) 2,1 45 c (m=2) 2,1 =(4.72, 0.579) R (m=2) 2,2 38 c (m=2) 2,2 =(6.16, 1.05) R (m=2) 3,1 20 c (m=2) 3,1 =(1.10, 4.04) R (m=2) 3,2 4 c (m=2) 3,2 =(3.80, 2.98) 20
21 4: m =3 K-Means R (m=3) 1,1 10 c (m=3) 1,1 =(0.0861, 0.113) R (m=3) 1,2 2 c (m=3) 1,2 =(2.48, 1.37) R (m=3) 1,3 11 c (m=3) 1,3 =(2.96, 0.947) R (m=3) 2,1 29 c (m=3) 2,1 =(4.70, 1.10) R (m=3) 2,2 34 c (m=3) 2,2 =(5.15, 1.46) R (m=3) 2,3 20 c (m=3) 2,3 =(6.78, 0.218) R (m=3) 3,1 19 c (m=3) 3,1 =(1.05, 3.97) R (m=3) 3,2 1 c (m=3) 3,2 =(2.06, 5.21) R (m=3) 3,2 4 c (m=3) 3,2 =(3.80, 2.98) (10) (12) {R } {R (m) } {D (m) } 5 6 (13) 6 {ρ (m) } ρ (m=2) =1 ρ (m=2) =3 (15) η 0.4 m =2 R 1 R 3 2 5: {R }, {R (m) } {D (m) } D (m=1) D (m=2) D (m=3) = = = : {ρ (m) } ρ (m=2) ρ (m=3) = = =
22 7: {d( c (m =2),p )} R 1 R 2 R 3 d( c (m =2),p ) R (m =2) 1, d( c (m =2) 1,1 )=1.81 R (m =2) 1, d( c (m =2) 1,2 )=0.688 R (m =2) 3, d( c (m =2) 3,1 )=2.31 R (m =2) 3, d( c (m =2) 3,2 )= : {d(r (m =2),p )} R 1 R 2 R 3 d(r (m ),p ) R (m =2) 1, d(r (m =2) 1,1 )=3.04 R (m =2) 1, d(r (m =2) 1,2 )=0.366 R (m =2) 3, d(r (m =2) 3,1 )=1.97 R (m =2) 3, d(r (m =2) 3,2 )=0.628 (16) {R (m =2),p, =1,p =1, 2} {R (m =2),p, =3,p =1, 2} {d( c (m =2),p )} 7 R (m =2) 1,2 R 2 R (m =2) 3,2 R 2 8 (17) R (m =2) 1,2 R 2 R (m =2) 3,2 R 2 4 K-Means R 1 line11 2 R 2 R 3 line33 2 R
23 c11 c12 c21 c22 c31 c32 cluster-center line1 line2 line3 line11 line33 x x1 4: K=3,m =2 K-Means c1 c2 c3 line1 line2 line3 3 2 x x1 5: 23
24 5.2 LVQ K-Means K-Means LVQ LVQ K-Means 2 LVQ LVQ LVQ LVQ [12] 6 5 "c1" "c2" "c3" "first_centroid" x x1 6: K =3 LVQ 24
25 9: LVQ R 1 17 c 1 =(0.695, 0.073) R 2 89 c 2 =(5.23, 0.131) R 3 24 c 3 =(1.27, 3.97) 10: m =2 LVQ R (m=2) 1,1 10 c (m=2) 1,1 =(0.0861, 0.113) R (m=2) 1,2 7 c (m=2) 1,2 =(2.52, 0.391) R (m=2) 2,1 50 c (m=2) 2,1 =(4.54, 0.662) R (m=2) 2,2 39 c (m=2) 2,2 =(6.09, 1.03) R (m=2) 3,1 20 c (m=2) 3,1 =(1.10, 4.04) R (m=2) 3,2 4 c (m=2) 3,2 =(3.80, 2.98) 11: {R }, {R (m) } {D (m) } D (m=1) D (m=2) D (m=3) = = = : {ρ (m) } ρ (m=2) ρ (m=3) = = = M=3 K=m LVQ K- Means m m =2 {R } K-Means 10 {R } {R (m) } {D (m) } (13) η 0.4 {ρ (m) } m =2 R 1 R
26 LVQ (17) 13 R (m =2) 1,2 R 2 R (m =2) 3,2 R 2 LVQ K-Means 8 13: {d(r (m =2),p )} R 1 R 2 R 3 d(r (m ),p ) R (m =2) 1, d(r (m =2) 1,1 )=2.66 R (m =2) 1, d(r (m =2) 1,2 )=0.333 R (m =2) 3, d(r (m =2) 3,1 )=1.97 R (m =2) 3, d(r (m =2) 3,2 )=
27 6 5 4 "c11" "c12" "c21" "c22" "c31" "c32" "centroid" 3 2 x x1 7: K=3,m =2 LVQ K-Means 6 5 "cluster1" "cluster2" "cluster3" x x1 8: LVQ K-Means 27
28 5.3 K-Means 9 2 (ball) (ring) 200 K-Means 10 K-Means 11 C =0.1 RBF 5 4 "b" "r" x x1 9: 2 28
29 5 4 "c1" "c2" x x1 10: K-Means 5 4 "b15" "r15" x x1 11: K-Means 29
30 ,14 15,16 17, : K-Means (Km) K-Means (Km ) : Km,Km method Km Km
31 5 4 "output/b15" "output/r15" "output/init_pnt15" x x1 13: Km, ( , ),( , ) 5 4 "output/b71" "output/r71" "output/init_pnt71" x x1 14: Km, ( , ),( , ) 31
32 5 4 "output/b27" "output/r27" "output/init_pnt27" x x1 15: Km, ( , ),( , ) 5 4 "output/b41" "output/r41" "output/init_pnt41" x x1 16: Km, ( , ),( , ) 32
33 5 4 "output/b18" "output/r18" "output/init_pnt18" x x1 17: Km, ( , ),( , ) 5 4 "output/b72" "output/r72" "output/init_pnt72" x x1 18: Km, ( , ),( , ) 33
34 6 K-Means K-Means K-Means LVQ K-Mean LVQ LVQ 7 K-Means K-Means K-Means K-Means 34
35 [1] Duda R.O., Hart P.E., Stor D.G., Pattern Classification (2nd Edition), John Wiley & Sons, INC., [2] Jain A.K., Dubes R.C., Algorithms for Clustering Data, Prentice-Hall, Englewood Cliffs, NJ, [3] Gordon A.D., Classification (2nd Edition), Chapman & Hall/CRC, [4] Bezde J.C., Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, NY, [5] MacQueen J., Some Methods for Classification and Analysis of Multivariate Observations, Proc. 5th Bereley Symp. on Math. Stat. and Prob. 1, Univ. of California Press, Bereley and Los Angeles, pp , [6] Linde Y., Buzo A., Gray R.M., An Algorithm for Vector Quantizer Design, IEEE Trans. Commun., Vol.28, pp , [7], :,,, [8] Tarsitano A., A Computational Study of Several Relocation Methods for K-Means Algorithm, Pattern Recognition, Vol.36, pp , [9] Yu J., General C-Means Clustering Model, IEEE Trans. PAMI., Vol.27, No.8, pp , [10] Press W.H., Flannery B.P., Teuolsy S.A., Vetterling W.T., Numerical Recipes in C, Cambridge University Press, [11] T.Kohonen, Self-Organizing Maps (2nd Edition), Springer, Berlin, [12],,, Vol.46, pp , [13] M.Girolami, Mercer ernel based clustering in feature space, IEEE Trans. on Neural Networs, Vol.13, No3, pp ,
36 [14] S.Miyamoto, Y.Naayama, Algorithms of hard c-means clustering using ernel functions in support vector machines, J. of Advanced computational Intelligence and Intelligent Informatics, Vol.1.7, No.1, pp.19-24, [15] V.Vapni, Statistical Learning Theory, Wiley, New Yor, [16] F.Morii, K.Kurahashi, Clustering by the K-Means Algorithm Using a Split and Merge Procedure, Proceedings of SCIS&ISIS, SA-F2-6, pp , [17],, K-Means,, PRMU, vol.106, No.470, pp.67-71,
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