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1 Vol. 33, No. 1 (2004), 3 26 Power Transformation Both sides: PTB Nonparametric Transformation Both sides: NTB 3 2 NTB ρ PTB NTB PTB PTB NTB NTB PTB NTB NTB PTB NTB 2 ACE AVAS ACE AVAS

2 Power-Transformation Both sides: PTB: Carroll and Ruppert, 1984 PTB Nonparametric Transformation Both sides: NTB 3 Nychka and Ruppert, PTB NTB 4 NTB PTB NTB Alternating Conditional Expectation: ACE: Breiman and Friedman, 1985 Additivity and VAriance Stabilization: AVAS: Tibshirani, 1988 ACE AVAS Y = f (X; β) + ε (1) X X p (p = 1, 2,...,p 0 ) p 0 1 f (X; β) β i (i = 1, 2,..., I) ε 0 Y f (X; β) (1) t λ (t λ 1)/λ λ 0, H P (t; λ) = log t λ = 0 Box and Cox,

3 Vol. 33, No. 1 (2004) (Goto et al., 1983) (Gnanadesikan, 1977; 1986; 1991; 1996) Y f (X β) Y f (X; β) PTB (1) H P (Y; λ) = H P { f (X; β); λ} + ε P (2) (Carroll and Ruppert, 1984) λ (Goto et al., 1987; Goto, 1992; Goto 1995; Goto et al., 2000; 1997) Goto (1992); Goto (1995); Goto et al. (2000) 3 2 PTB σ 2 n N(0,σ 2 n), n = 1, 2,..., N σ 2 ε P N(0,σ 2 ) Bartlett (Bartlett, 1937) {(x n,y n ), n = 1, 2,...,N} N ( L(β,σ 2 1,λ)= 2 [H P(y n ; λ) H P { f (x n ; β),λ}] 2 /σ 2 +log d dt H P(y n ; λ) 1 2 log σ2) +C 0 (3) n=1 β, σ 2, λ (Carroll and Ruppert, 1984, 1988) C PTB NTB (3) H S (u) 5

4 β σ 2 H S (u) 2.1 (2) H P (u) L P (β,σ 2, H S (u)) = L(β,σ 2, H S (u)) ρj(h S (u)), ρ > 0 (4) L(β, σ 2, H S (u)) (3) H P (u) H S (u) J(H S (u)) H S (u) J(H S ) = uu u L (d 2 h S (u) ) 2du u L u U {y n } [u L, u U ] ρ h S (u) H S (u) h S (u) = log(dh S (u)/du) H S (u) = exp[h S (u)]du H S (u) L P (β, σ 2, H S (u)) H S (u) β 1 L P (β,σ 2, H S (u)) H S (u) 2 Ĥ S (u) L P (β,σ 2, H S (u)) ρ (Goto, 1979; Goto and Matsubara, 1979) Bayes ( 1999) H S (u) 3 (4) h S (u) PTB L P (β,σ 2, h S (u)) = 1 2 ρ N { ( f n n=1 uu u L du 2 y n exp h S (u)du ) 2/ σ 2 + 2h S (y n ) log σ 2} ( d 2 h S (u) du 2 ) 2du + C 0 (5) y n (n = 1, 2,..., N) f n = f (x n ; β) C0 (3) 6

5 Vol. 33, No. 1 (2004) h S (u) 3 h S (u) J 3 j ( j = 1, 2,...,J) (u j, Z j ) Z j = h S (u j ) + ν j u 1,..., u J [u L, u U ] u L < u 1 < < u J < u U {y n } [u 1, u J ] ν j 1/ω j N(0, 1/ω j ) S 1 [u L, u U ] [u L, u U ] 1 S 2 [u L, u U ] [u L, u U ] 2 ω j S (h S (u)) = J {Z j h S (u j )} 2 ω j + ρ j=1 uu u L ( d 2 h S (u)) 2du, ρ > 0 du 2 [u L, u U ] S 2 [u L, u U ] S (h S (u)) h S (u) ĥ S (u) ĥ S (u) u j 3 (O sullivan et al., 1986) Nychka and Ruppert (1995) h S (u) {y n } { f n } u 1,..., u J (5) 1 fn y n exp h S (u)du {Z j } (5) L PA (σ 2, h S (u)) = N [ { J W nj exp h S (u j ) } 2/ 2σ 2 + n=1 ρ j=1 uu u L J ζ nj h S (u j ) ] j=1 ( d 2 h S (u)) 2du (6) du 2 W nj n j fn y n exp h S (u)du J W nj exp(h S (u j )) j=1 ζ nj h S (y n ) J ζ nj h S (u j ) j=1 n j u 1,..., u J 3 h S (u) 7

6 h S = (h S (u 1 ),..., h S (u J )) T, d 2 h s (u) = (d 2 h du 2 S (u 1 )/du 2,..., d 2 h S (u J )/du 2 ) T h S (6) L PA (h S ) = 1 2 h T S Ωh S + ζt h T S ρht S Rh S (7) h S = (exp[h S(u 1 )],..., exp[h S (u J )]) T, Ω = W T diag(v)w W W nj V N N n = 1, 2,..., N V nn = σ 2 0 ζ = (ζ 1,..., ζ J ) T ζ j = N n=1 ζ nj R h S d2 h s (u) du 2 J J (7) {h S (u j )} L PA (h S (u)) h S (u) = h S (u j )[Ωh u=u S ] j + ζ j 2ρ[Rh S ] j = 0, j = 1, 2,..., J (8) j (8) {h S (u j )} ĥ S (u) β,σ 2 h S (u) 1 {u j } ( j = 1, 2,..., J) 2 Ω ζ 3 h S0 = 0 4 h S0, Ω, ζ Z = (Z 1, Z 2,...,Z J ) T w = (w 1,w 2,..., w J ) T 5 h S0 {w j }, {u j, Z j } 3 h S1 6 h S0 = ĥ S h S0 = ĥ S1 5 h S0 {w j } {u j, Z j } Ω jj D j (i) Ω jj = 0 Ω jj = 0 Z j = D j + h S0 (u j ), w j = 1 (ii) Ω jj > 0, D j = 0 (8) Ω jj j = 1, 2,..., J exp(2h S (u j ))Ω jj + exp h S (u j ) J j j 0 Ω jj0 exp h S0 (u j0 ) + 2ρ[Rh S ] j = 0 2 h S0 (u j0 ) S 0 S h S (u j0 ) 8

7 Vol. 33, No. 1 (2004) h S0 (u j ) 1 Taylor 1 exp(2h S (u j )) exp(2h S0 (u j )){1 + 2(h S (u j ) h S0 (u j ))} 2 exp h S0 (u j ) exp h S (u j ) exp(2h S0 (u j )){1 + 2(h S (u j ) h S0 (u j ))}Ω jj + exp h S0 (u j ) h S (u j) exp h S (u j ) J j j 0 Ω jj0 exp h S0 (u j ) + 2ρ[Rh S ] j = 0 2h S0 (u j) 2 Ω jj ( 1 2h S0 (u j)ω jj [Ωh S0 (u j )] j + h S0 (u j) h S (u j) ) + 2ρ[Rh S ] j = 0 2w j (Z j h S (u j )) + 2ρ[Rh S ] j = 0 w j Z j 2 (iii) Ω jj > 0, D j > 0 (8) Ω jj D j exp(2h S (u j )) { N } Ω jj + exp( h S (u j )) Ω jj0 exp h S (u j ) exp( 2h S (u j ))D j + 2ρ[RhS ] j = 0 j j 0 (ii) ( h S0 (u j )[Ωh S0 ] j 2D j 2D j + 1/2 + h S0 (u j ) h S(u j ) ) 2ρ[Rh S ] j = 0 w j = D j 1 Z j H S (u) exp[h S (u)] Ĥ S (u) {β,σ 2, H S (u)} H S (u) J(H S (u)) = 0 h S (u) γ 1, γ 2 u = log ω h S (u) = γ 1 + γ 2 u H S (u) H S (u 0 ) = {exp(γ 1 /γ 2 )}{exp(γ 2 u) exp(γ 2 u 0 )} 9

8 H S (u) H S (u 0 ) = H S (log ω) H S (log ω 0 ) = exp(γ 1 /γ 2 ){exp(γ 2 log ω) exp(γ 2 log ω 0 )} = exp(γ 1 /γ 2 )ω γ 2 exp(γ 1 /γ 2 )ω γ 2 0 = C 1 ω γ 2 C 2 C 1 C 2 γ 1, γ 2 γ 1, γ 2, u ACE Breiman and Friedman (1985) ACE Alternating Conditional Expectations Y X 1,..., X p0 H AC (Y), S AC1 (X 1 ),...,S ACp0 (X p0 ) E{Y X 1,..., X p0 } E{H AC (Y) X 1,..., X p0 } H AC (Y) p 0 p=1 S AC p (X p ) (e 2 ) e 2 (H AC, S AC1,..., S ACp0 ) = E{[H AC(Y) p 0 p=1 S AC p (X p )] 2 } E[H 2 AC (Y)] (9) (9) HAC, S AC 1,...,SAC p0 (H AC, S AC 1,..., S AC p0 ) = argmin HAC,S AC1,...,S ACp0 e 2 (H AC, S AC1,..., S ACp0 ) (9) H AC (Y) 0 p 0 p=1 S AC p (X p ) 0 (9) (HAC, S AC 1,..., S AC p0 ) Y, X 1,..., X p0 E[HAC 2 (Y)] = 1 H AC (Y) {S ACp (X p )} AVAS ACE ACE ACE AVAS 10

9 Vol. 33, No. 1 (2004) ACE p 0 p=1 Ŝ ACp (X p ) = E[H AC (Y) X 1,..., X p0 ], p 0 p 0 Ĥ AC (Y) = E[ S ACp (X p ) Y]/ E[ S ACp (X p ) Y] (10) p=1 Tibshirani (1988) (10) p 0 p=1 p=1 Ŝ AV p (X p ) = E[H AV (Y) X 1,..., X p0 ], (11) p 0 var[ĥ AV (Y) S AV (X p )] = const. (12) p=1 H AV S AV1,..., S AV p0 (11) (12) AVAS Additivity and VAriance Stabilization: AVAS H AV (t) p 0 H 0 AV (Y) = S 0 AV p (X p ) + ε AV p=1 H 0 AV (Y) ε AV 0 X H AV X 1,..., X p0 H AV (Y) p 0 p=1 Ŝ AV p (X p ) = E[H AV (Y) X 1,..., X p0 ] (12) Y s v(s) Y h AV (t) h AV (t) = t 0 1/ v(s)ds (Bartlett, 1947) Taylor h AV (t) H AV (Y) p 0 p=1 S AV p (X p ) H AV (Y) Ĥ AV (Y) var[ĥ AV (Y) p 0 p=1 S AV p (X p )] h AV (t) Ĥ AV (Y) Y h AV (H AV (Y)) u H AV (Y) AVAS 11

10 Ricker Beverton & Holt 28 x y Ricker (1954) f (x; β) = β 1 x exp( β 2 x) Beverton and Holt (1957) f (x; β) = 1 β 1 + β 2 /x, β 1 0,β 2 0 Beverton & Holt Ricker 1 1. Ricker 1 Ricker 2 Beverton & Holt PTB NTB L P L PA NTB 5 ρ NTB u 1,...,u J u 1 u J J = 100 NTB PTB 12

11 Vol. 33, No. 1 (2004) ˆ β 1 1. Ricker ˆβ 1 ˆβ 2 ˆ β 1 ˆ β 2 ˆ β 2 L P L PA PTB NTB:ρ = 10, NTB:ρ = NTB:ρ = NTB:ρ = NTB:ρ = ˆ β 1 2. Beverton & Holt ˆβ 1 ˆβ 2 ˆ β 1 ˆ β 2 ˆ β 2 L P L PA PTB NTB:ρ = 10, NTB:ρ = NTB:ρ = NTB:ρ = NTB:ρ = NTB ρ 1 ρ ρ = 10,000 PTB Ricker Beveton & Holt PTB ˆλ = , ˆλ = Ricker ρ ρ H S (u) ρ ρ = L PA = Ricker Beverton & Holt ρ ρ = 10,000 PTB Fortran ρ H S (u) ρ Spearman Shapiro-Wilk NTB ρ 5 ρ Shapiro-Wilk ACE AVAS ACE AVAS ACE AVAS ACE AVAS ACE AVAS Shapiro-Wilk ACE AVAS ACE AVAS 13

12 2. ρ NTB Ricker 3. ρ NTB Beverton & Holt 4. ρ Spearman Ricker 5. ρ Shapiro-Wilk Ricker 6. Ricker Beverton & Holt ACE AVAS NTB ρ = 10,000 Ricker Beverton & Holt AVAS PTB ACE

13 Vol. 33, No. 1 (2004) x 1 x 2 y ( 3 ) (Bruce and Schumacher, 1935) y = β 1 x 2 1 x 2 β 1 Atkinson and Rinai (2000) PTB λ H 0 β 1 = 0 β 1 ˆβ 1 λ = 0 Atkinson and Rinai (2000) NTB β 1 PTB NTB NTB ρ 100 ρ 70 (x 1, x 2,y) = (23.4, 104, 163.5) 1 53 (x 1, x 2,y) = (14.3, 77, 58.9) PTB NTB ˆβ 1 ( 70) ˆβ 1 3 ˆβ 1 ( 70) ˆβ 1 ρ = 100 ρ PTB NTB 3. ˆβ 1 ( 70) βˆ 1 ( 70) ˆβ 1 ( 70) ˆβ 1 βˆ 1 ( 70) l( 70) l PA ( 70) PTB NTB:ρ = NTB:ρ = NTB:ρ = NTB:ρ = NTB:ρ = NTB:ρ = NTB:ρ =

14 NTB 1 NTB PTB 2 1 NTB 3 NTB PTB 1 NTB PTB 2 3 NTB NTB Mahalanobis 2 β = (β 1,β 2 ) ˆβ 1, ˆβ 2 Σ [β E(ˆβ)] T Σ 1 [β E(ˆβ)]. (13) 1 2 NTB NTB f (x; β) = xβ 1 exp( β 2 x) (14) H S (y) = H S { f (x; β)} + ε S (15) ε S N(0,σ 2 ) (15) H 1 (u) = log(u) (16) (15) NTB ρ (15) (16) Y = [β 1 x exp( β 2 x)] exp(ε S ) 16

15 Vol. 33, No. 1 (2004) β 1, β 2 β 1 = 3, β 2 = x x [0, 1,000] 3 3 NTB 3 2 (14) H 2 (u) = log(u/ X) (17) NTB ρ 0 H S (u; ρ 0 ) = H 2 (u) (14) (17) Y = [β 1 x exp( β 2 x)] exp( xε S ) 1 β 1 = 3, β 2 = x x [0, 1,000] 3 3 NTB 5 3 (14) 2 V(X) = f (X) 2 N 1 (0, V(X)) N(0, av(x)) a a,α αn 1 (0, V(X)) + (1 α)n(0, av(x)) α a 4.2 x x [0, 1,000] 3 NTB 3 α a =27 [0, 1,000] 0 σ 2 N 1,000 N 0 < X < 500 A 500 < X < 1,000 B 2 Bartlett A B σ 2 A σ2 B H 0 σ 2 A = σ2 B H 1 σ 2 A σ2 B 0.05 σ 2 = 0.05 N = N = N = N = 29, 41, 63 3 σ 2 = 0.05, 0.075, ρ 0 ρ = 1,000,000 ρ = ρ = 0.01, ρ = NTB (13) (ˆβ 1, ˆβ 2 ) (β 1,β 2 ) Maharanobis (ˆβ 1, ˆβ 2 ) MSE Mean Squared Error NTB 17

16 E[sup H N Ĥ N ] N σ 2 ρ MSE 3 ρ i (i = 1,...,6) j j = 1, 2, 3 k k = 1, 2, 3 MSE E[sup H N Ĥ N ] M ijk, N ijk M ijk = µ + µ 1i + µ 2 j + µ 3k + (µ 1 µ 2 ) ij + (µ 2 µ 3 ) jk + (µ 3 µ 1 ) ki + ε ijk, N ijk = ν + ν 1i + ν 2 j + ν 3k + (ν 1 ν 2 ) ij + (ν 2 ν 3 ) jk + (ν 3 ν 1 ) ki + ε ijk i = 1,..., 6, j = 1, 2, 3, k = 1, 2, 3 µ 1i, µ 2 j, µ 3k iρ j k (µ 1 µ 2 ) ij, (µ 2 µ 3 ) jk, (µ 3 µ 1 ) ki ρ ρ ν 1i,ν 2 j,ν 3k, (ν 1 ν 2 ) ij, (ν 2 ν 3 ) jk, (ν 3 ν 1 ) ki = Bartlett 0.05 σ N = N = N = N = 26, 49, 64 3 σ 2 = 0.05, 0.075, ρ = ρ ρ = (1,000,000), 10, 1, 0.1, ρ 5 (ˆβ 1, ˆβ 2 ) MSE E[sup H S Ĥ S ] = Bartlett 0.05 α = 0.95, a = 1 N = N = N = N = 50, 60, 80 3 a a = 1, 3, 5, 7 4 a = 1 a = N 3 0.3% ρ ρ = 1,000,000 ρ = ρ = 0.1, ρ = a = 1, 3, 5, 7 NTB (14) (ˆβ 1, ˆβ 2 ) (ˆβ 1, ˆβ 2 ) MSE ,000 4 (ˆβ 1, ˆβ 2 ) MSE 5 2 (ˆβ 1, ˆβ 2 ) MSE 6 18

17 Vol. 33, No. 1 (2004) 4. 1 MSE F p % ρ N near σ near ρ N ρ σ N σ near F p % ρ near N near σ near ρ N near ρ σ near N σ MSE F p % ρ N near σ near ρ N ρ σ N σ (ˆβ 1, ˆβ 2 ) MSE ρ 0.05 MSE 79.54% 17.58% 5 ρ ρ ρ 0.05 ρ MSE ρ NTB PTB 5 ρ ρ MSE ρ NTB 7 σ 2 MSE 95% σ 2 = 0.075, σ 2 = 0.1 (N = 29) ρ MSE N = 63 MSE 95% 8 σ 2 ρ = σ 2 = 0.1 N = 29 19

18 σ 2 = 0.05 σ 2 = σ 2 = σ 2 = 0.05, 0.075, 0.1 MSE σ 2 = 0.05 σ 2 = σ 2 = σ 2 = 0.05, 0.075,

19 Vol. 33, No. 1 (2004) ρ = % N = 41, N = 63 NTB ρ PTB 2 6 (ˆβ 1, ˆβ 2 ) MSE ρ ρ 0.05 MSE ρ 38.97% 56.55% 7 ρ ρ % 1 MSE ρ NTB ρ 2 9 N = 26,σ 2 = ρ = 0.1, ρ = 1 MSE NTB 7. 2 F p % ρ near N σ ρ N ρ σ N σ MSE F p % ρ near N near a near ρ N ρ a N a near N = 26, σ 2 = ρ E{ˆβ 1 } E{ˆβ 2 } MSˆ E(ˆβ 1 ) MSˆ E(ˆβ 2 ) ˆβ 1 MS E ˆβ 2 MS E (ˆβ 1, ˆβ 2 ) MS E E[sup H N Ĥ N ]

20 3 6 (ˆβ 1, ˆβ 2 ) MSE ρ 0.05 MSE a N 53.6% 34.2% MSE MSE ρ a N MSE ρ MSE ρ N MSE a = 1 a = 3, a = 5, a = 7 MSE MSE 95% 9 50 ρ = 0.1 MSE ρ = MSE ρ MSE NTB MSE ρ = MSE 95% ρ MSE 95% 13 ρ a N MSE N 9. 3 MSE a = MSE a = MSE a = MSE a = 7 22

21 Vol. 33, No. 1 (2004) MSE N,ρ,a MSE N = 50, a = 7 ρ, ρ = 0.1 ρ = MSE 5. PTB NTB NTB ACE AVAS Beverton and Holt (1957) NTB PTB NTB NTB NTB PTB PTB PTB NTB NTB PTB 23

22 NTB 1 NTB NTB NTB NTB NTB 1 NTB 2 3 NTB NTB 1 2 ACE AVAS ACE AVAS Atkinson, A. and Riani, M. (2000): Robust Diagnostic Regression Analysis. Springer. Bartlett, M.S. (1937): Properties of sufficiency and statistical tests. Proc. Roy. Soc. A160, Bartlett, M.S. (1947): The use of transformations. Biometrics 3, Beverton, R.J. and Holt, S.J. (1957): On the Dynamics of Exploited Fish Populations. Her Majesty s Stationery Office, London. Box, G.E.P. and Cox, D.R. (1964): An analysis of transformations. J.R. Stat. Soc. B26, Breiman, L. and Friedman, J.H. (1985): Estimating optimal transformations for multiple regression and correlation (with discussion). J. Amer. Statist. Assoc. 80, Bruce, D. and Schumacher, F.X. (1935): Forest Mensuration. New York: McGraw-Hill. Carroll, R.J. and Ruppert, D. (1984): Power transformation when fitting theoretical models to data. J. Amer. Statist. Assoc. 79, Carroll, R.J. and Ruppert, D. (1988): Transformation and Weighting in Regression. Chapman and Hall. Gnanadesikan, R. (1977): Methods for Statistical Data Analysis of Multivariate Observations. John Wiley & Sons (1979). Goto, M. (1979): Choice of shrinkage factors in the generalized ridge regression. Math. Japonica. 24, Goto, M. and Matsubara, Y. (1979): Evaluation of ordinary ridge regression. Bull. Math. Statist. 19 (1-2), Goto, M., Matsubara, Y. and Tsuchiya, Y. (1983): Power-normal distribution and its applications. Rep. Stat. Appl. Res. JUSE, 30 (3), Goto, M., Inoue, T., and Tsuchiya, Y. (1987): Double power-transformation and its performances: An extensive version of Box-Cox transformation. J. Japan. Statist. Soc. 17 (2), Goto, M. (1992): Extensive views of power transformation: Some recent developments. Invited paper at Honolulu Conference on Computational Statistics as a memorial of the fifth anniversary of JSCS, JAIMS, December 1 5, Goto, M. (1995): Double power transformations and their applications. Invited paper of International Conference on Statis- 24

23 Vol. 33, No. 1 (2004) tical Computing for Quality and Productivity Improvement, Seoul, August, Goto, M., Isomura, T., and Hamasaki, T. (2000): Guinea pigs in statistical science. Proceedings of the Tenth Japan and Korea Joint Conference of Statistics 2000 (Invited paper), , December 4 5. B-Con Plaza, Beppu, Japan. (1984): 29 (8), (1986): 13 (2), (1991): 4 (1), (1996): 9 (1), Nychka, D. and Ruppert, D. (1995): Nonparametric transformations for both sides of a regression model. J.R. Statist. Soc. B 57 (3), O Sullivan, F., Yandell, B. and Raynor, W.J. (1986): Automatic smoothing of regression functions in generalized linear models. J. Am. Statist. Ass. 81, Ricker, W.E. (1954): Stock and recruitment. J. Fish. Res. Bd Can. 32, (1999): Tibshirani, R. (1988): Estimating transformations for regression via additivity and variance stabilization. J. Am. Statist. Ass. 83, (1997): ito.masanori@yamanouchi.co.jp gotoo@sigmath.es.osaka-u.ac.jp 25

24 Japanese J. Appl. Statist. 33 (1) (2004), 3 26 Various Types of Nonparametric Transformation and Its Diagnosis Masanori Ito 1 and Masashi Goto 2 1 Biometrics Depertment, Drug Development Devision, Yamanouchi Pharmaceutical Co., Ltd., 17 1, Hasune 3-chome, Itabashi-ku, Tokyo , Japan 2 Division of Statistical Science, Graduate School of Engineering Sciences, Osaka University, 1 3, Machikaneyama-cho, Toyonaka, Osaka , Japan Abstract In this paper, we introduce a Nonparametric Transform-Both-sides (NTB) approach as an alternative to the Power Transform-Both-sides (PTB) approach to inference for theoretical models and propose a method of parameter estimation by expressing the function transformation as a cubic spline curve. From the investigation of two examples, we suggest that the NTB could be an index for the validation of the PTB and is more robust than PTB to outliers. Furthermore, we verify these results by three simulation experiments. In the methodology for fitting the empirical model, we introduce Alternating Conditional Expectation (ACE) and Additivity VAriance Stabilization (AVAS) as two nonparametric transformation approaches that optimize the relationship between response and explanatory variables. We examine the validity of the theoretical models by fitting empirical models via ACE and AVAS to the example data. Both method, ACE and AVAS, improve the normality and homoscedasticity of the error. Key words: PTB, NTB, ACE, AVAS, transformation, normality, homoscedasticity address: ito.masanori@yamanouchi.co.jp Received December 26, 2002; Received in final form September 22, 2003; Accepted December 17,

Power Transformation and Its Modifications Toshimitsu HAMASAKI, Tatsuya ISOMURA, Megu OHTAKI and Masashi GOTO Key words : identity transformation, pow

Power Transformation and Its Modifications Toshimitsu HAMASAKI, Tatsuya ISOMURA, Megu OHTAKI and Masashi GOTO Key words : identity transformation, power-normal distribution, structured data, unstructured