Support Vector Machine (SVM) 4 SVM SVM 2 80% 100% SVM SVM SVM 4 SVM 2 2 SVM 4

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1 Analysis of Groove Feelings of Drums Plays

2 Support Vector Machine (SVM) 4 SVM SVM 2 80% 100% SVM SVM SVM 4 SVM 2 2 SVM 4

3 Support Vector Machine SVM SVM i

4 SVM SVM SVM SVM SVM SVM ii

5 iii

6 1.1 [2] /100BPM/ [10] /100BPM/ [10] /100BPM/ [10] /100BPM/ [10] [15] [18] iv

7 v

8 vi

9 SVM SVM SVM SVM SVM SVM SVM SVM vii

10

11 : [2] MIDI Musical Instrument Digital Interface MIDI [1] MIDI 2

12 1 1.1 DTM DeskTop Music DTP DeskTop Publishing 3

13 Led Zeppelin [3]

14 : Support Vector Machine (SVM) 3 4 Support Vector Machine 3 5 5

15 Support Vector Machine (SVM)

16 2 2.1 Bass Drum Snare Drum 1 Floor Tom Tom-Tom 2 Hihat Cymbal Ride Cymbal Crash Cymbal Snare 2 High Tom Low Tom 7

17 : 8

18 A B B 9

19 : 8 2.3: 16 10

20 Friberg [4] :1 3:1 8 Friberg 3.5:1 1:1 Waadeland [5] [6] 11

21 Draguna [7, 8, 9] [10, 11, 12, 13] BPM Beat Per Minute BPM 12

22 %

23 : 8 /100BPM/ [10] 2.5: 8 /100BPM/ [10] 14

24 : 8 /100BPM/ [10] 2.7: 8 /100BPM/ [10] 15

25 2 2.3 Support Vector Machine 2.3 Support Vector Machine Support Vector Machine (SVM) 1992 Vaptik [14] 2 SVM SVM SVM SVM 2 f : X R n R x = (x 1,, x n ) T f(x) 0 f(x) x X f(x) = w T x + b (2.1) f(x) = 0 X 2 w b SVM 2 SVM x 1,, x n y 1,, y n x i A y i = 1 B y i =

26 2 2.3 Support Vector Machine 2.8: 2 w, b min w T x i + b = 1 (2.2) i=1,,n w T x i + b min i=1,,n w (2.3) 1 w w b min w w 2 (2.4) ) y i (w T x + b 1 (2.5) 17

27 2 2.3 Support Vector Machine 1 w α i ( 0) L(w, b, α) = 1 2 w 2 n i=1 { )} α i y i (w T x i + b (2.6) L b = L w = 0 n α i y i = 0 (2.7) i=1 n w = α i y i x i i=1 (2.4)(2.5) α (2.8) max α n α i 1 n n α i α j y i y j x T i x j (2.9) 2 i=1 i=1 j=1 α i 0, Karush-Kuhn-Tucker n α i y i = 0 (2.10) i=1 ) α i (y i w T x i 1 = 0 (2.11) α 0 SV (2.9) α w w b w = αi y i x i (2.12) x i SV b = y k w T x k (x k SV ) (2.13) 18

28 2 2.3 Support Vector Machine f(x) f(x) = w T x + b = αi y i x T i x + b (2.14) x i SV αi y i x T i x + b = 0 (2.15) x i SV (2.5) w, b C-SVC [14] ξ i 0, i = 1, 2,, n (2.16) (2.4) (2.5) 1 n min w 2 w 2 + C ξ i i=1 (2.17) ) y i (w T x + b 1 ξ i (2.18) (2.17) C C C C 19

29 2 2.3 Support Vector Machine 2.9: α i (2.9) (2.10) max α n α i 1 n n α i α j y i y j x T i x j (2.19) 2 i=1 i=1 j=1 0 α i C, n α i y i = 0 (2.20) i=1 α i > 0 7 ξ i SV (2.19) (2.9) (2.15) 7 α i > 0 20

30 2 2.3 Support Vector Machine αi y i x T i x + b = 0 (2.21) x i SV C C 3.7 C φ(x) x = (x 1, x 2,, x n ) φ(x) = (φ 1 (x), φ 2 (x),, φ N (x)) (2.22) φ : R n R N N N n φ : R 2 R 3 (x 1, x 2 ) (z 1, z 2, z 3 ) = (x 1 2, 2x 1 x 2, x 2 2 ) (2.23) 21

31 2 2.3 Support Vector Machine 2.10: 2 [15] φ K(x, z) x, z K(x, z) = φ(x) T φ(z) (2.24) K K (2.15) x i SV α i y i K(x i, x) + b = 0 (2.25) φ φ(x) T φ(z) K(x, z) = ( γ x T z + r) d, γ > 0 (2.26) 22

32 2 2.3 Support Vector Machine Radial Basis Function (RBF) ( K(x, z) = exp γ x z 2), γ > 0 (2.27) ( d K(x, z) = tanh γ x T z + r) (2.28) γ, r, d C SVM SVM SVM 2 SVM SVM SVM 23

33 SVM SVM 3.8 SVM

34 3 3.2 Support Vector Machine (SVM) SVM

35 3 3.2 SVM 4 SVM SVM SVM SVM 2 2 SVM

36 : 27

37 [10, 11, 12, 13] 2 Digidesign Pro Tools 1 [17] Pro Tools KHz 16bit WAVE BPM / 132 BPM DAW 28

38 :

39 : 1 3.4: 2 3.5: 3 3.6: 4 30

40 : (BPM)

41 3 3.4 SVM 3.4 SVM SVM SVM 2.2 SVM WAVE 32

42 3 3.5 WAVE [18] 1. FFT 2. FFT DC IFFT x a (n) r(n) h(n) h(n) 3.7 A(n) A(n) = r(n) 2 + h(n) 2 (3.1) π/2 [19]

43 : [18] 34

44 : n 2n k n = 250 k =

45 : 3.10: 36

46 w 2. h w = 1000 h =

47 3 3.6 N t 1, t 2,, t N t d m(t, d) = N [t k {t + (k 1)d}] 2 (3.2) k=1 m t d m t d 2 m t = 0 (3.3) m d = 0 (3.4) t = N N 4(N + 1) t k 6 kt k k=1 k=1 N(N + 1) (3.5) d = N N 12 kt k 6(N + 1) t k k=1 k=1 N(N + 1)(N 1) (3.6) 2 N N (3.5) (3.6) t k kt k 2 k=1 k=1 n k = k=1 n(n + 1), 2 n k 2 = k=1 n(n + 1)(2n + 1) 6 38

48 3 3.7 SVM 3.11: (3.5) (3.6) t d 3.7 SVM SVM C SVM C overfitting C (cross validation) n (n-fold cross validation) 1. SVM 39

49 3 3.7 SVM 3.2: 1 A, B, C D 2 A, B, D C 3 A, C, D B 4 B, C, D A = 75% C 2. n 3. n 1 SVM 4. 1 SVM SVM A,B,C,D C 40

50 : SVM SVM SVM SVM p 1 p 2 p 1 2 SVM 41

51 p 1 +5p 2 5 = 0 p 1 1 p 2 5 p 2 42

52 4 SVM SVM SVM 4.2 SVM 4.4 SVM 4.1 CPU : Intel Pentium GHz 2 RAM : 2.00GB OS : Microsoft Windows XP Service Pack 2 C++ Python SVM SVM LIBSVM [20] 43

53 4 SVM 4.2 SVM SVM 4 17 SVM SVM 2. SVM 3. SVM 4. SVM

54 4 SVM 4.2 SVM SVM y i y i = 1 y i = SVM

55 4 SVM 4.2 SVM 4 4.1: SVM C % C 2 5, 2 4,, 2 15 C 17 C C 4.1 C C SVM SVM [ 1, 1]

56 4 SVM 4.2 SVM 4 4.2: SVM % % % 80% 90% f(x) f(x) = a 1 x 1 + a 2 x 2 + a 3 x 3 + a 4 x 4 + b = 0 (4.1) x 1, x 2, x 3, x 4 47

57 4 SVM 4.2 SVM 4 4.3: a 1 a 2 a 3 a 4 b x 1 x 2 x 3 x

58 4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.1: 1 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.2:

59 4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.3: 2 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.4:

60 4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.5: 3 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.6:

61 4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.7: 4 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.8:

62 4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.9: 1 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.10: 3 53

63 4 SVM 4.2 SVM 4 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.11: 2 Coefficient (Absolute Value) of the Separating Hyperplane Time(Mean) Volume(Mean) Time(SD) Dimensions of Input Vectors Volume(SD) 4.12: 4 54

64 4 SVM 4.2 SVM 4 4.4:

65 4 SVM 4.2 SVM

66 4 SVM 4.3 SVM SVM 4 17 SVM 4.2 SVM SVM 2. SVM 3. SVM 4. SVM 5. SVM SVM

67 4 SVM 4.3 SVM SVM y i 4.2 y i = 1 y i = SVM

68 4 SVM 4.3 SVM 4 Radial Basis Function (RBF) ( K(x, z) = exp γ x z 2), γ > 0 (4.2) RBF [21] RBF γ (2.28) RBF [22] RBF SVM C RBF (4.2) γ SVM C RBF (4.2) γ 2 C : 2 5, 2 4,, 2 15 γ : 2 15, 2 14,, 2 3 C γ 17 C γ 59

69 4 SVM 4.3 SVM 4 4.5: SVM 4 C γ % C γ 4.5 RBF C γ C γ SVM % % 80% 90% 60

70 4 SVM 4.3 SVM 4 4.6: SVM % SVM 4.6 SVM SVM SVM SVM 4.2 SVM 80% 90% SVM 61

71 4 SVM 4.3 SVM 4 4.7: SVM SVM SVM SVM % %

72 4 SVM 4.4 SVM SVM SVM SVM 2. SVM 3. SVM 4. SVM 5. SVM

73 4 SVM 4.4 SVM SVM y i y i = 1 y i = SVM

74 4 SVM 4.4 SVM 2 Radial Basis Function (RBF) ( K(x, z) = exp γ x z 2), γ > 0 (4.3) RBF SVM C RBF (4.3) γ SVM C RBF (4.3) γ 2 C : 2 5, 2 4,, 2 15 γ : 2 15, 2 14,, 2 3 C γ 17 C γ C γ 4.8 RBF C γ C γ RBF SVM SVM 65

75 4 SVM 4.4 SVM 2 4.8: SVM 2 C γ % % % SVM % 70% 80% 4.2 SVM SVM SVM 2 66

76 4 SVM 4.4 SVM 2 4.9: SVM % RBF f(x) 67

77 4 SVM 4.4 SVM : ( f(x) 0.1 ) (f(x) < 0.1 ) (f(x) > 0.1 ) f(x) = x i SV ( αi y i exp γ x i x 2) + b (4.4) f(x) 0.1 f(x) y i f(x) < 0 f(x) > 0 68

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