On the Limited Sample Effect of the Optimum Classifier by Bayesian Approach he Case of Independent Sample Size for Each Class Xuexian HA, etsushi WAKA

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1 Journal Article / 学術雑誌論文 ベイズアプローチによる最適識別系の有限 標本効果に関する考察 : 学習標本の大きさ がクラス間で異なる場合 (< 論文小特集 > パ ターン認識のための学習 : 基礎と応用 On the limited sample effect of bayesian approach : the case of each class 韓, 雪仙 ; 若林, 哲史 ; 木村, 文隆 ; 三宅, 康二 Han, Xuexian; Wakabayashi, etsushi; Kimura, Fumitaka; Miyake 電子情報通信学会論文誌. D-II, 情報 システム, II-パターン he transactions of the Institute o Communication Engineers. D-II

2 On the Limited Sample Effect of the Optimum Classifier by Bayesian Approach he Case of Independent Sample Size for Each Class Xuexian HA, etsushi WAKABAYASHI, Fumitaka KIMURA, and Yasuji MIYAKE [1] Faculty of Engineering, Mie University, su-shi, Japan [] [5] [6] Aitchson Dunsmore [7] [8] [6].. 1 θ D II Vol. J8 D II o. 4 pp

3 99/4 Vol. J8 D II o. 4 p(x χ = p(x θp(θ χdθ (1 p(x χ χ X p(x θ θ X p(θ χ θ p(θ χ θ p(θ [9] (1 ( p(θ. X p(x p(x θ θ ˆθ(χ ( θ ˆθ(χ χ θ (1 ( ˆθ(χ X (3 [10] t 1 1 (1 p(x θ p(θ χ p(x χ t p(x θ p(θ χ p(x χ [6] p(x χ =( π n Σ 1 = (X M Σ 1 Σ =(1 ασ + ασ 0 0 α = + 0 Γ( +1 Γ( n+1 (X M } +1 (3 X n M Σ Σ 0 X 0 Σ 0 Γ p(x χ t (3 g(x = lnp(xp (ω =( +1ln 1+ (X } M Σ 1 (X M D =( π n +ln Σ lnd lnp (ω (4 Γ( +1 Γ( n+1 P (ω ω 0 =0 = Σ =Σ Σ 0 0 = 0 Σ 0 X Σ 0 = σ I I σ = = 0 σ M 3. (4 6

4 [11] [ 1.] g(x [ =( ln 1+ 1 X M 0σ + k (1 αλ i (1 αλ i + ασ [ Φ i (X M ] k ln ( (1 αλ i + ασ lnp (ω ]} (5 λ i Φ i Σ i i k 3. 3 Σ P (ω 0 (5 3 1 ( g(x = X M k (1 αλ i (1 αλ i + ασ Φ i (X M } (6 i < = k λ i ( 0/σ (6 [1] g(x = n i=k+1 = X M Φ i (X M } k Φ i (X M } (7 X k K-L X 1 Fig. 1 Decision boundaries of projection distance and modified projection distance. 1 1 X g(x =(X M Σ 1 (X M +ln Σ lnp (ω (8 63

5 99/4 Vol. J8 D II o. 4 (% 1 + =6,σ1 = σ =1.0, 1 Fig. heoretical mean error rate (% v.s. sample size with fixed total sample size ( 1 + =6,σ1 = σ =1.0, univariate case. 3 (% 1 + =6,σ1 =4.0,σ =0.5, 1 Fig. 3 heoretical mean error rate (% v.s. sample size with fixed total sample size ( 1 + =6,σ1 =4.0,σ =0.5, univariate case σ 1 = σ =1.0 3 σ 1 =4.0,σ = =6 t [.] (% 1 + =40,8 Fig. 4 Mean error rate (% v.s. sample size with fixed total sample size ( 1 + = 40, 8-variate case

6 8 8 diagσ =(8.41, 1.06, 0.1, 0., 1.49, 1.77, 0.35,.73 (9 8 M 1 =(0, 0, 0,..., 0, M =(3.86, 3.10, 0.84, 0.84, 1.64, 1.08, 0.6, 0.01 ( = [11] 3, 6, 9, 1, 0, 3, 48, 64, 100, 144, 196, (1 7 7 ( r r=5 (3 0 1 (4 Roberts (5 π / able 1 Sample size of each class (Case of nearly common learning sample size otal able Sample size of each class (Case of independent learning sample size otal ( (5 3 (7 [14641] (8 y = x u u = n n = 3, 6, 9, 1, 0, 3, 48, 64, 100, 144, 196, 56,

7 99/4 Vol. J8 D II o. 4 5 Fig. 5 Recognition rate of handwritten numeral recognition (Case of nearly common learning sample size. [11], [13], [14] 3 14,946 44,838 9,877 14, = α 1 α (11 α 5 6 Fig. 6 Recognition rate of optimum discriminant function (Case of independent learning sample size % % [15]

8 able 3 3 Ratio of computation cost Fig. 7 Recognition rate of handwritten numeral recognition (Case of independent learning sample size. [], [16] (1 ( (3 (1 ( (3 (4 (5 0 Σ 0(= σ I t 67

9 99/4 Vol. J8 D II o. 4 [11] 6. (1 0 0 ( 4.3 (3 (4 (5 [1] [] J.M. Van Campenhout, On the peaking of Hughes mean recognition accuracy : he resolution of an apparant paradox, IEEE rans. Syst., Man & Cybern., vol.smc-8, no.5, pp , May [3] W.G. Waller and A.K. Jain, On the monotonicity of the performance of Bayesian classifiers, IEEE rans. Info. heory, vol.i-4, pp , [4] J.M. Van Campenhout, opics in Measurement Selection, Handbook of Statistics, vol., orth- Holland Publishing Company, pp , 198. [5] D. Lindley, he Bayesian approach, Scand. J. Statist., vol.5, pp.1 6, [6] D.G. Keehn, A note on learning for Gaussian properties, IEEE rans. Inform. heory, vol.i-11, no.1, pp.16 13, Jan [7] B.D. Ripley, Pattern Recognition and eural etworks, p.5, Cambridge University Press, [8] S.J. Raudys and A.K. Jain, Small sample size effects in statistical pattern recognition : Recommendations for practitioners, IEEE rans. Pattern Analysis & Machine Intelligence, vol.13, no.3, pp.5 64, March [9] R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis, p.5, John Wiley & Sons, Inc., ew York, [10] vol.77, no.8, pp , Aug [11] D-II vol.j77-d-ii, no.10, pp , Oct [1] vol.4, no.1, pp , Jan [13] PRU9-33, Sept [14] PRU-93-46, Sept [15] K. Fukunaga and R.R. Hayes, Effects of sample size in classifier design, IEEE rans. Pattern Analysis & Machine Intelligence, vol.pami-11, no.8, pp , Aug [16] G.F. Huges, On the mean accuracy of statistical pattern recognizers, IEEE rans. Info. heory, vol.i- 14, no.1, pp.55 63, Jan Σ 0 = σ I Σ =(1 ασ + ασ I (A 1 (1 ασ + ασ I}Φ i =(1 ασφ i + ασ Φ i =(1 αλ i + ασ }Φ i (i =1,,,n (A Σ (1 αλ i + ασ Φ i(i =1,,,n Y =(X M Σ 1 (X M n 1 = Φ (1 αλ i + ασ i (X M } (A 3 k i > k (1 αλ i ασ (A 3 68

10 k 1 Y Φ (1 αλ i + ασ i (X M } + n i=k+1 n Φ i (X M } i=k+1 = X M 1 ασ Φ i (X M } k Φ i (X M } (A 4 [ Y 1 X M ασ k (A 4 (A 5 (1 αλ i (1 αλ i + ασ Φ i (X M } n ln Σ = ln(1 αλ i + ασ } k ln(1 αλ i + ασ } + n i=k+1 ] (A 6 ln(ασ (A 7 (4 (A 6 (A 7 α = 0/( + 0 ( (4 g i(x =σ +1 i 1+ 1 ( } x mi σ i (i =1, (A 8 h(x 0 h(x =g 1(x g (x = σ +1 1 σ +1 ( 1+ 1 x m1 σ 1 } ( 1+ 1 x m σ =(a bx (am 1 bm x + am 1 bm + c a = 1 σ +1 1, b = 1 σ +1, c = σ h(x =0 α = β = +1 1 σ m1 + m +1 (σ 1 = σ } (A 9 α, β = am1 bm (m 1 m ab (a bc a b (σ 1 = σ (A 10 σ 1 > = σ ε = P (ω 1ε 1 + P (ω ε = P (ω 1P (error χ, ω 1+P (ω P (error χ, ω = B + A + C A = B = α p(x χ, ω P (ω dx ( α m = 1 Φ β α σ p(x χ, ω 1P (ω 1dx ( ( = 1 β m1 Φ 1 α m1 σ 1 Φ σ 1 C = β p(x χ, ω P (ω dx = 1 ( } β m 1 Φ (A 11 σ Φ (x 0 t t (x Φ (x 0= x0 t (xdx (A 1 69

11 99/4 Vol. J8 D II o. 4. σi ( } i +1 g i(x =( 1+ 1 x mi D i σ i D i =( i π 1 Γ i ( i +1 Γ ( i (i =1, (A 13 h(x =0 ewton x k+1 = x k h(x k h (x k σ1 h(x =( D ( } 1 +1 x m1 σ 1 ( σ ( 1+ 1 x m D σ ( h (x = ( 1 +1 ( x m1 σ1 1 σ 1 σ 1 D 1 ( } x m1 1 σ 1 } ME ME ( ( +1 ( x m σ σ σ D ( } 1+ 1 x m (A 14 σ (A

Accuracy Improvement by Compound Discriminant Functions for Resembling Character Recognition Takashi NAKAJIMA, Tetsushi WAKABAYASHI, Fumitaka KIMURA,

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