Ł\”ƒ.dvi
|
|
|
- せいごろう しまむね
- 7 years ago
- Views:
Transcription
1 ,
2 ,
3 PLM CCM PLCM BTM 54
4 MCMC MCMC BTM PLM PLM GRM BTM CCM 129
5 3 535 CCM Bias
6 4 11 N J 9 21 CCM (1,1),(1,0),(0,1),(0,0) ( ) PLCM (1,1),(1,0),(0,1),(0,0) ( ) PLM Bias,RMSE,cor(θ, ˆθ) CCM Bias,RMSE,cor(θ, ˆθ) CCM Bias, RMSE, cor(θ, ˆθ) PLCM Bias,RMSE,cor(θ, ˆθ) PLCM Bias, RMSE, cor(θ, ˆθ) BTM Bias,RMSE,cor(a,â) PLM Bias,RMSE,cor(a,â) Bias, RMSE, cor(a,â) Bias/σ a j RMSE/σ a j Bias,RMSE,cor(a,â) Bias, RMSE, cor(a,â) Bias/σ a j RMSE/σ a j BTM Bias,RMSE,cor(b,ˆb) 75
7 5 39 2PLM Bias,RMSE,cor(b,ˆb) Bias, RMSE, cor(b,ˆb) I BTM Bias,RMSE PLM Bias,RMSE PLM Bias, RMSE PLM Bias/I, RMSE/I PLM Bias,RMSE,cor(θ, ˆθ) GRM Bias,RMSE,cor(θ, ˆθ) GRM Bias, RMSE, cor(θ, ˆθ) BTM Bias,RMSE,cor(θ, ˆθ) BTM Bias, RMSE, cor(θ, ˆθ) CCM Bias,RMSE,cor(θ, ˆθ) CCM Bias, RMSE, cor(θ, ˆθ) σγ 2 d(j) = BTM (1,1),(1,0),(0,1),(0,0) (100 ) b jk = 1714 CCM (1,1),(1,0),(0,1),(0,0) (100 ) σγ 2 d(j) = 0155 BTM (1,1),(1,0),(0,1),(0,0) (100 ) b jk = 0612 CCM (1,1),(1,0),(0,1),(0,0) (100 ) 140
8 6 21 CCM Bias CCM RMSE CCM cor(θ, ˆθ) CCM Bias CCM RMSE CCM cor(θ, ˆθ) PLCM Bias PLCM RMSE PLCM cor(θ, ˆθ) PLCM Bias PLCM RMSE PLCM cor(θ, ˆθ) Bias/σ a j RMSE/σ a j cor(a,â) Bias/σ a j RMSE/σ a j cor(a,â) Bias/σ a j RMSE/σ a j cor(a,â) Bias RMSE cor(b,ˆb) Bias 90
9 7 314 RMSE cor(b,ˆb) Bias RMSE cor(b,ˆb) Bias/I RMSE/I Bias/I RMSE/I Bias/I RMSE/I GRM Bias GRM RMSE GRM cor(θ, ˆθ) GRM Bias GRM RMSE GRM cor(θ, ˆθ) GRM Bias GRM RMSE GRM cor(θ, ˆθ) BTM Bias BTM RMSE BTM cor(θ, ˆθ) BTM Bias BTM RMSE 154
10 8 515 BTM cor(θ, ˆθ) BTM Bias BTM RMSE BTM cor(θ, ˆθ) CCM Bias CCM RMSE CCM cor(θ, ˆθ) CCM Bias CCM RMSE CCM cor(θ, ˆθ) 4 164
11 J ( A), N, ( 11) 11 N J 1 2 J N 1 1 1, 11 1, 0 (item response theory, IRT),,,,, A,, A,,,,, ( )
12 1 10,,,,,,,,,,, (eg, Kan, van der Ven, Breteler, & Zitman, 2001; Simms, Goldberg, Roberts Watson, Welte, & Rotterman, 2011) 12 j θ θ j (item characteristic function, ICF) 11,,, 2 (two-parameter logistic model, 2PLM) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (11), (11) P j (θ i ) θ θ i j, a j,b j j a j θ i = b j (11), j, a j j, (11) θ i = b j P j (θ i ) = 05, j 05 θ, b j j, (11) a j 1 (one-parameter logistic model, 1PLM) (11) c j P j (θ i ) P j (θ i ) = c j + 1 c j 1+exp[ 17a j (θ i b j )] (12)
![i b j )] (11), (11) P j (θ i ) θ θ i j, a j,b j j a j θ i = b j (11), j, a j j, (11) θ i = b j P j (θ i ) = 05, j 05 θ, b j](/docs-images/47/23770548/images/page_12.jpg "j, (11) a j 1 (one-parameter logistic model, 1PLM) (11) c j P j (θ i ) P j (θ i ) = c j + 1 c j 1+exp[ 17a j (θ i b j )]")
13 (three-parameter logistic model, 3PLM),, (12) θ i θ i =, P j (θ i ) = c j, c j j, j, A B, A, A 100%, B 0%, i j ( ) π ij,, θ i π ij, Lord (1980, pp ), P j (θ i ), 1 j, θ i 2 θ i j 3 j, θ i 3, (1984), P j (θ i ) π ij,, θ i π ij, P j (θ i ),, P j (θ i ) = E i θi [π ij ] (13) 13 (local independence) (Lord & Novick, 1968, p360), 2 j k, θ j k u j,u k,, U j, j U j = 1, j U j = 0,
![j, θ i 3, (1984), P j (θ i ) π ij,, θ i π ij, P j (θ i ),, P j (θ i ) = E i θi [π ij ] (13) 13](/docs-images/47/23770548/images/page_13.jpg "(local independence) (Lord & Novick, 1968, p360), 2 j k, θ j k u j,u k,, U j, j U j = 1, j U j =")
14 1 12, j k Prob(U j = u j,u k = u k θ i ) = Prob(U j = u j θ i )Prob(U k = u k θ i ) = P j (θ i ) u j (1 P j (θ i )) 1 u j P k (θ i ) u k (1 P k (θ i )) 1 u k (14), (14) Prob(U j = u j,u k = u k θ i ) θ θ i U j,u k u j,u k, Prob(U j = u j θ i ) Prob(U k = u k θ i ) θ i U j,u k u j,u k j k, (14), θ i u k u j Prob(u j u k,θ i ) Prob(u j u k,θ i ) = Prob(u k,u j θ i ) Prob(u k θ i ) = Prob(u j θ i )Prob(u k θ i ) Prob(u k θ i ) = Prob(u j θ i ) (15), j k, θ u j u k, (experimental independence) (Lord & Novick, 1968, p 44), j,k i ( ) u ij,u ik, u ij u ik Prob(u ij,u ik ), Prob(u ij ),Prob(u ik ), j, k, Prob(u ij,u ik ) = Prob(u ij )Prob(u ik ) = π u ij ij (1 π ij ) 1 u ij π u ik ik (1 π ik) 1 u ik (16), i θ i, θ i *1, P j (θ i ) = π ij (17) P k (θ i ) = π ik (18) *1 Lord & Novick (1968, p 539)
15 1 13, θ i j,k u j,u k Prob(u j,u k θ i ) Prob(u j,u k θ i ) = P j (θ i ) u j (1 P j (θ i )) 1 u j P k (θ i ) u k (1 P k (θ i )) 1 u k (19), j k,,, j k, 12,, j k, (2000), j,k, 2,, π ij,π ik Π j,π k, θ i Π j,π k π j,π k Prob(Π j = π j,π k = π k θ i ) θ i Π j,π k π j,π k Prob(Π j = π j θ i ),Prob(Π k = π k θ i ),, Prob(Π j = π j,π k = π k θ i ) = Prob(Π j = π j θ i )Prob(Π k = π k θ i ) (110) j,k, (2000), (,, ), (14), θ i 1 π i1, i θ,, P 1 (θ), 2 1, 1 π i1, (110), 2 π i2,,, 1, 2, 2 1 P 2 (θ),, 1, 2, P 1 (θ)p 2 (θ) 3 θ i
16 ,, Ferrara, Huynh, & Michaels (1999), Hoskens & De Boeck (1997), Kreiner & Christensen (2004), Yen (1993), (local dependence),, 3 ( I) ( II) ( III) I,,, 1 X 2 X, X N x 1,,x N, 2 X X M X S 2 X S 2 X = 1 N N (x i M X ) 2 (111) i=1,, 1 2, I, 13, II, 1 ( ),, j k, ( ) j k ( ), θ i
17 1 15 j k j k, II, 13, III, θ θ,,,,, *2, j k θ θ, θ θ j,k, θ i π ij π ik, Π j Π k, III, 13, 15 14,,,, 3 1 ( a) ( b) ( c) a, J J d(j), a, (graded response model, GRM) *2,, θ i, (Differential Item Functioning, DIF)
18 1 16 (Samejima, 1969) P d(j) (r θ i ) = P d(j) (r θ i) P d(j) (r+1 θ i) (112), P d(j) (r θ i ) θ θ i d(j) r, Pd(j) (r θ i) θ θ i d(j) r, Pd(j) (r θ i) 2PLM ((11) ), (112) P d(j) (r θ i ) P d(j) (r θ i ) = 1 ] 1+exp [ 17a d(j) (θ i b r ) 1 ] (113) 1+exp [ 17a d(j) (θ i b r+1 ), (113) b r,b r+1, b r b r+1, d(j),,, a, a,, (Bock, 1972) (Muraki, 1992),, Sireci, Thissen & Wainer (1991), b, 2PLM, b, 2 (Baeysian testlet model, BTM) (Bradlow, Wainer, & Wang, 1999) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ id(j) ) ] (114), P j d(j) (θ i ) θ i d(j) j, a j d(j),b j d(j), 2PLM, j, a j d(j) γ id(j) 0 θ i = b j d(j) (114), b j d(j) γ id(j) 0 P j d(j) (θ i ) 05 θ, (114) exp 17a j d(j) (θ i b j d(j) γ id(j) ) = 17a j d(j) ( (θi γ id(j) ) b j d(j) ) (115), θ i γ id(j) j,
![2PLM, b, 2 (Baeysian testlet model, BTM) (Bradlow, Wainer, & Wang, 1999) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ id(j) ) ] (114), P j d(j) (θ i ) θ i d(j) j, a j d(j),b j d(j),](/docs-images/47/23770548/images/page_18.jpg "2PLM, j, a j d(j) γ id(j) 0 θ i = b j d(j) (114), b j d(j) γ id(j) 0 P j d(j) (θ i ) 05 θ, (114) exp 17a j d(j) (θ i b j d(j) γ id(j) ) = 17a j d(j) ( (θi γ id(j) ) b j d(j) ) (115), θ i γ")
19 1 17 (114) γ id(j), θ i i d(j) γ id(j) N(0,σγ 2 d(j) ) *3, σγ 2 d(j) 0, d(j), σγ 2 d(j) 0, d(j), σγ 2 d(j) d(j), d(j),, θ d(j), θ i,,, b, b, a, 2 BTM, 3 (three parameter Bayesian testlet model, 3PBTM) (Wang, Bradlow, & Wainer, 2002), (multidimensional item response model, MIRM),, Nandakumar (1990) Li, Bolt & Fu (2006), MIRM MIRM,, c, 2PLM, c,, (constant combination model, CCM) (Hoskens & De Boeck, 1997) exp[u j Z j +u k Z k u j u k b jk ] P(u j,u k θ i ) = 1+exp[Z j ]+exp[z k ]+exp[z j +Z k b jk ] (116) Z j = 17a j d(j) (θ i b j d(j) ) (117) Z k = 17a k d(j) (θ i b k d(j) ) (118), P(u j,u k θ i ) θ i j k u j,u k, (117) a j d(j),b j d(j) j, (118) a k d(j),b k d(j) k, a j d(j) a k d(j), b jk 0, θ i = b j d(j) θ i = b k d(j), b j d(j) *3, 223
![CCM) (Hoskens & De Boeck, 1997) exp[u j Z j +u k Z k u j u k b jk ] P(u j,u k θ i ) = 1+exp[Z j ]+exp[z k ]+exp[z j +Z k b jk ] (116) Z j = 17a j d(j) (θ i b j d(j) ) (117) Z k = 17a k](/docs-images/47/23770548/images/page_19.jpg "d(j) (θ i b k d(j) ) (118), P(u j,u k θ i ) θ i j k u j,u k, (117) a j d(j),b j d(j) j, (118) a k d(j),b k d(j) k, a j d(j) a k d(j), b jk 0, θ i = b j d(j) θ i = b k d(j), b j d(j) *3,")
20 1 18 b k d(j), b jk 0, 05 θ (116), θ i j k ω jk, ω jk = ln ( / ) P(Uj = 1,U k = 1 θ i ) P(Uj = 0,U k = 1 θ i ) P(U j = 1,U k = 0 θ i ) P(U j = 0,U k = 0 θ i ) = ln P(U j = 1,U k = 1 θ i )P(U j = 0,U k = 0 θ i ) P(U j = 1,U k = 0 θ i )P(U j = 0,U k = 1 θ i ) = b jk (119) j k ω jk 0, (116) b jk j k, b jk j,k ( ), j,k, c,, 2 (two-parameter logistic copula model, 2PLCM) 2PLCM Braeken, Tuerlinckx, & DeBoeck (2007) (Rasch copula model) 2PLM, 2PLCM, 2 j,k u j,u k P(u j,u k θ i ) =u j u k +( 1) 2 u j u k Q j (θ i )+( 1) 2 u k u j Q k (θ i ) +( 1) u j+u k C(Q j (θ i ),Q k (θ i )) (120), (120) Q j (θ i ), θ i j,, Q j (θ i ) = exp[ 17a j (θ i b j )] (121), C(Q j (θ i ),Q k (θ i )) Q j (θ i ),Q k (θ i ) ( ),, C(Q j (θ i ),Q k (θ i )) = 1 [ log 1 W (Q ] j(θ j ))W (Q j (θ k )) δ jk W(1) (122) W(x) = 1 exp[ δ jk x] (123) (Frank, 1979), (122) (123) δ jk, θ i j k
21 1 19, δ jk, θ i j k, δ jk 0 ω jk 0, δ jk j k, d(j),,, θ i, c, c,, 3 (three parameter constant combination model, 3PCCM, Chen & Wang, 2007) hybrid kernel (Ip, 2002), (locally dependent linear logistic test model, LDLLTM, Ip, Smits, & De Boeck, 2009), (conjunctive item response model, CIRM, Jannarone, 1986),, (2005), CIRM 16, (, 2005;, 1992;, 2001),,, (eg, ),,,,,, 14,,, 15,,,,,, (eg, Yang & Gao, 2008 ),,,,
22 1 20,,,,,,,,,,,,,,, 2,, 3,,, 4,, 5,,,,,, 6,
23 ,, *1,, Bradlow et al (1999), 2 BTM, 2PLM,,,,, 95% (mean 95% posterior interval width, M95%PIW),, M95%PIW,, *1, (2010),
24 2 22, Junker (1991),,, (2013),, 2PLM,,,,,,,,,,,,, Bradlow et al (1999), 1000, 60, (2013), 1000, 12,,,,,,,, ( ),,,,,,,,,,,
25 ,, 2,,, (Bias(ˆθ i )), (RMSE(ˆθ i )),, (cor(θ, ˆθ)), 2 221,, (11) 2PLM ( ) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (21),,,, 15 c, (116) CCM (120) 2PLCM (CCM, 2PLCM ) CCM ( ) P(U j,u k θ i ) = exp[u j Z j +U k Z k U j U k b jk ] 1+exp[Z j ]+exp[z k ]+exp[z j +Z k b jk ] Z j = 17a j d(j) (θ i b j d(j) ) Z k = 17a k d(j) (θ i b k d(j) ) (22)
26 2 24 2PLCM ( ) P(U j,u k θ i ) =U j U k +( 1) 2 U j U k Q j (θ i )+( 1) 2 U k U j Q k (θ i ) +( 1) U j+u k C(Q j (θ i ),Q k (θ i )) (23) 1 Q j (θ i ) =1 1+exp[ 17a j (θ i b j )] C(Q j (θ i ),Q k (θ i )) = 1 δ jk log W(x) = 1 exp[ δ jk x] [ 1 W (Q j(θ j ))W (Q j (θ k )) W(1) ] 222,,,, 100, 300, 500, , 10, 30, 50 3, 12,, 2j 1 2j (j = 1,2,,J/2, J ), j,k, CCM b jk = 2, 2PLCM δ jk = 30 b jk δ jk,,,,, 2 j,k (1,1),(0,0),, 100, 10, 2 1,2, , 21 22, CCM 1,2 (1,1),(0,0) (100 ) 84% (73%+11%), 2PLCM 88% (52%+36%)
27 CCM (1,1),(1,0),(0,1),(0,0) ( ) U 2 = 1 U 2 = 0 U 1 = 1 73% 10% U 1 = 0 6% 11% 22 2PLCM (1,1),(1,0),(0,1),(0,0) ( ) U 2 = 1 U 2 = 0 U 1 = 1 52% 12% U 1 = 0 0% 36%, Hoskens & De Boeck (1997), b jk 2-2,, b jk -2, δ jk 30,, 221 3, 2PLM, (Markov chain Monte Carlo, MCMC) *2, R 1 θ i (i = 1,2,,N) j (j = 1,2,,J) a j b j, N(0,1), U(03,15) U( 20,20) *3 2, 1, 2PLM *2 MCMC 223 *3, R
28 2 26 ((21) ), N J A 3 U(0,1) NJ, N J B 4 A,B, U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 j,k (0,0),(1,0),(0,1),(1,1), 1, CCM ((22) ), N (J/2) D,E,F,G 6 U(0,1) NJ/2, N (J/2) H 7 D,E,F,G H, N J U, d ij h ij (u i(2j 1),u i(2j) ) = (0,0), e ij+d ij h ij > d ij (u i(2j 1),u i(2j) ) = (1,0), 8 2PLCM 5 7, N J U 9 U,U,U 2PLM, MCMC , Bias(ˆθ i ) = RMSE(ˆθ i ) = 100 r= cor(θ, ˆθ) = ˆθ ir θ i (24) 100 ) 2 (ˆθir θ i (25) r=1 100 cor(θ, ˆθ r ) (26) r=1, θ i 1, ˆθ i θ i, ˆθ ir 10 θ i r, θ N θ i, ˆθ θ, ˆθ r 10 θ r, (26) cor(θ, ˆθ r ) θ ˆθ r, Bias(ˆθ i )
29 2 27 ˆθ i, RMSE(ˆθ i ) ˆθ i, cor(θ, ˆθ) ˆθ θ ,, 2PLM, MCMC, MCMC (2008),, t(t = 0,1,2, ), t X t, x t,, t Prob(X t+1 = x t+1 X 0 = x 0,X 1 = x 1,,X t = x t ) = Prob(X t+1 = x t+1 X t = x t ) (27), X t Ω, X t (t = 0,1,2, ) Ω, Prob(X t+1 = x t+1 X t = x t ), p x t+1 x t, t, p x t+1 x t xt x t+1 P, t X t x t, Ω x t+1 π t+1, X t+1 π t+1, π t+1 = π t P (28),, π t+1 = π 0 P t+1 (29), t+1 X t+1 π t+1 t = 0, 1 2 ω(ω Ω),
30 2 28, π 0, t π t+1 π, (28), π = πp (210),, π, λ, λ ((211) ) λ = (θ 1,θ 2,,θ N,a 1,a 2,,a J,b 1,b 2,,b J ) (211) λ U,, *4 λ Prob(λ U), (MAP ) (EAP ), λ ((212) ) Prob(λ U) = L(U λ)prob(λ) Prob(U) = L(U λ)prob(λ) (212) L(U λ)prob(λ)dλ, (212) L(U λ), λ, U (λ ),, L(U λ) = N i=1 j=1 J P j (θ i ) u ij Q j (θ i ) 1 u ij (213), U λ, (212) λ,,, MCMC, U λ, *4, A B Prob(B A) = Prob(A B)Prob(B) Prob(A), Prob(B A), A B, B, Prob(B), A B, B
31 2 29 λ, MAP EAP λ MCMC, (Gibbs sampler, Geman & Geman, 1984), (data augmentation and Gibbs sampling, Tanner & Wong, 1987), (Metropolis-Hastings algorithm, Hastings, 1970),, Patz & Junker (1999) - (Metropolis-Hastings within Gibbs algorithm), 222 9, N J U λ, λ, λ (EAP ) (U,U ), MCMC,, θ i,a j,b j, N(0,1), N(1,025), N(0,1) MCMC 1 U z i r j, p j, z i,r j, 1 p j θ i,a j,b j θi 0,a0 j,b0 j, λ λ0 λ 0 = (λ 0 1,λ0 2,,λ0 N+2J ) = (θ 0 1,θ 0 2,,θ 0 N,a 0 1,a 0 2,,a 0 J,b 0 1,b 0 2,,b 0 J) = (z 1,z 2,,z N,r 1,r 2,,r J,1 p 1,1 p 2,,1 p J ) (214) 2 λ 1 1 λ 1 ((215) ) h(λ 1 λ 0 1) = [ ] 1 exp (λ 1 λ0 1 )2 2πσ 2 2σ 2 (215) 3 λ 1 λ0 1, ((216) ) ( Prob(λ α(λ 1 λ 0 1) = min 1 λ 0 1,U)h(λ0 1 λ 1 ) ) Prob(λ 0 1 λ0 1,U)h(λ 1 λ0 1 ),1 (216), (216) λ 0 1 λ0 λ 0 1 λ 0 1 = (λ0 2,,λ0 N+2J ) (217)
32 U(0,1) (216) λ 1 1 = λ 1, λ 1 1 = λ0 1 5 λ 0 2,,λ 0 N+2J 2 4, λ 1 = (λ 1 1,λ 1 2,,λ 1 N+2J) (218) 6 λ 1 2 5, λ λ, (215) σ 2, 4 (l = 1,,N +2J) λ l,,, λ l 25% 50% σ 2,, , 1 (burn-in ), 20000,,, 3000 burn-in, λ 23,, 2PLM CCM, 2PLCM Bias,RMSE,cor(θ, ˆθ), Bias Bias(ˆθ i ) N, 222,, RMSE RMSE(ˆθ i ) N, 222,,,, 222, cor(θ, ˆθ), 2PLM CCM, 2PLCM Bias
33 2 31 RMSE, θ i 222,, θ i N(0,1), θ i 1, Bias RMSE,, θ i 231 2PLM, 2PLM Bias,RMSE,cor(θ, ˆθ),, CCM, CCM Bias,RMSE,cor(θ, ˆθ),, 24, CCM, 2PLM, Bias,RMSE,cor(θ, ˆθ),, 25, Bias, RMSE, cor(θ, ˆθ), Bias,RMSE,cor(θ, ˆθ) 2PLM, CCM Bias, RMSE, cor(θ, ˆθ), 21, 22, 23, CCM Bias, RMSE, cor(θ, ˆθ), 24, 25, 26, CCM,,, 25, N = 100,, 2PLM, CCM Bias 01 (θ i 10%),, N = 300 N = 500, ( 10 ), CCM Bias 01, 21 24,,, Bias
34 2 32,,, Bias, Bias, 25, N = 100 J = 50, 2PLM, RMSE 015 (θ i 15%),, 22, N = 100, 2PLM RMSE, 22, N 300, J = 10 J = 30 RMSE, J = 30 J = 50 RMSE, RMSE 2,, J = 10 J = 30, J = 30 J = 50 25,, CCM, 2PLM,,,,, 26,, J = 10 J = 30 cor(θ, ˆθ), J = 30 J = 50 cor(θ, ˆθ),, 3,,,, 11,,,,,
35 2 33,,,, N = 100,,,,,,,,,, 233 2PLCM, 2PLCM Bias,RMSE,cor(θ, ˆθ),, 26, 2PLCM Bias, RMSE, cor(θ, ˆθ),, 27, 2PLCM Bias, RMSE, cor(θ, ˆθ), 27, 28, 29, 2PLCM Bias, RMSE, cor(θ, ˆθ), 210, 211, 212, 2PLCM,,, 27,, Bias 2PLM 27,, 2PLM, RMSE, J = 10, RMSE 010 (θ i 10%),,,
36 ,, 2PLM, cor(θ, ˆθ),, J = 10,, 010 cor(θ, ˆθ), 212,, cor(θ, ˆθ) 2PLM, 29, cor(θ, ˆθ), J = 10, N = 1000, 2PLM cor(θ, ˆθ),,,,,,,,,,,,, ,,,,,,, Bradlow et al (1999) 2, 2 BTM
37 2 35 Bradlow et al (1999),, CCM,,, Bias,, M95%PIW,,,, M95%PIW, M95%PIW,, 3,, Junker (1991),, CCM, 2PLCM,,, 235, ( ),, Bradlow et al (1999),,, CCM, 2PLCM, (2013),,,, Bias,,, RMSE, cor(θ, ˆθ)
38 2 36, 3,, CCM, 2,, 3, 4,,, (eg,, ),,,,,, N = 100,,,, 23 2PLM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = N = 100,J = N = 100,J = N = 300,J = N = 300,J = N = 300,J = N = 500,J = N = 500,J = N = 500,J = N = 1000, J = N = 1000, J = N = 1000, J =
39 CCM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = N = 100,J = N = 100,J = N = 300,J = N = 300,J = N = 300,J = N = 500,J = N = 500,J = N = 500,J = N = 1000, J = N = 1000, J = N = 1000, J = CCM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100, J = N = 100,J = N = 100,J = N = 300, J = N = 300,J = N = 300, J = N = 500, J = N = 500,J = N = 500, J = N = 1000, J = N = 1000, J = N = 1000, J =
40 PLCM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100,J = N = 100,J = N = 100,J = N = 300,J = N = 300,J = N = 300,J = N = 500,J = N = 500,J = N = 500,J = N = 1000, J = N = 1000, J = N = 1000, J = PLCM Bias, RMSE, cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 100, J = N = 100, J = N = 100, J = N = 300, J = N = 300, J = N = 300, J = N = 500, J = N = 500, J = N = 500, J = N = 1000, J = N = 1000, J = N = 1000, J =
41 CCM Bias
42 CCM RMSE
43 CCM cor(θ, ˆθ)
44 CCM Bias
45 CCM RMSE
46 CCM cor(θ, ˆθ)
47 PLCM Bias
48 PLCM RMSE
49 PLCM cor(θ, ˆθ)
50 PLCM Bias
51 PLCM RMSE
52 PLCM cor(θ, ˆθ)
53 ,,, *1,, Bradlow et al (1999), 2 BTM, 2PLM BTM,,, 2PLM BTM 2PLM BTM, M95%PIW,, 2PLM, BTM, M95%PIW,, 2PLM, BTM, 2PLM,, Chen & Wang (2007), 3PCCM, 3PLM, *1, (2012b),
54 3 52,,,,, Jiao Kamatani, Wang, & Jin (2012),,, (Rasch, 1960),,, Looney & Spray (1992),,,, Tuerlinckx & De Boeck (2001), CCM, 2PLM,,,, Wainer & Wang (2000), TOEFL, 3PLM 3PBTM,,, (2012), 2PLM GRM, 2PLM GRM,,,,,,,, (Braeken, 2011; Braeken, et al, 2007; Ip, 2010; Ip, Smits, & De Boeck, 2009), Bradlow et al (1999), 2 BTM, 2PLM BTM, 15, BTM 2PLM,,, Chen & Wang (2007) Tuerlinckx & De
55 3 53 Boeck (2001), CCM, 2PLM 3PLM, 12 15,,,,,,,, Ip (2010), 2 BTM 2PLM,,,,,,,,,, 2 BTM, 2PLM, Ip (2010),,,,, 32,,,, 321, 31, (114) 2 BTM ( ) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ ad(j) ] (31)
56 , 31, (11) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (32) 2 BTM ( (31) ) BTM 15, (32) a j θ i = b j (32), (32) b j P j (θ i ) = 05 θ i, (31) a j d(j) γ ad(j) 0 θ i = b j d(j) (31), b j d(j) γ ad(j) 0 P j d(j) (θ i ) = 05 θ i, 2PLM 2 BTM, 31 Ip (2010), (31) γ ad(j) 2 BTM 2PLM, (31) 2 BTM γ ad(j) (marginalized testlet item response function, MIRF), MIRF
57 3 55, (31) a j d(j) a j = τa j d(j) (33) τ = 1 κ 2 (17a j d(j) ) 2 σγ 2 d(j) +1 (34) κ = π (35) b j = b j d(j) (36), 2 BTM 2PLM a j,b j *2, 31, (33) (36) a j d(j),b j d(j) a j,b j,,,, 324,, 4, 5, 31,,,,,, 4,, 300, ,, 3, 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2), 4, 4 5, 4 ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) *2 MIRF, Ip (2010)
58 ( d(j) 1) 4 ( d(j) 0),, 2 BTM Bradlow et al (1999) 2 BTM 3 (Bradlow et al, 1999; Li, Bolt, & Fu, 2005; Li, Bolt, & Fu, 2006), 3 17 σγ 2 d(j) ˆσ γ 2 d(j) 4 (14075) σγ 2 d(j), 4 (0155) σγ 2 d(j),, (2 ) (3 ) (5 ) 30, *3 1 θ i (i = 1,2,,N) γ ad(j) (a = 1,2,,N; d(j) = 1,2,3,4) a j d(j) b j d(j) (j = 1,2,,20) N(0,1), N(0,σγ 2 d(j) ), U(05,15), N(0,1) *4 2 1 (31), N 20 A 3 U(0,1) N 20 N 20 B 4 A,B U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 U R Ip (2010), 2 BTM 2PLM 8 7 R *3 R *4, R
59 3 57 Bias(ˆλ j ) = 1 R ˆλ jr λ j (37) R r=1 RMSE(ˆλ j ) = 1 R ) 2 (ˆλjr λ j (38) R cor(λ,ˆλ) = 1 R r=1 R cor(λ, ˆλ r ) (39) r= Bias(ˆλ j ) RMSE(ˆλ j ), Bias(ˆλ j ) RMSE(ˆλ j ) Bias,RMSE, λ j j, ˆλ j λ j, ˆλ jr ˆλ j r, λ 20 a j b j, ˆλ, ˆλ r r, 6 R, ˆλ ir,, 1000 R = R = MCMC MCMC (MCMC 223 ) MCMC WinBUGS 14 (Spiegelhalter, Thomas, & Best, 2003), MCMC (slice sampling) (Neal, 1997),, 2PLM θ i N(0,1) a j N(1,025) b j N(0,1) 2 BTM
60 3 58 θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) γ ad(j) N(0,σγ 2 d(j) ) σγ 2 d(j) Γ(3, 1), MCMC,, burn-in 2PLM 1000 burn-in 2 BTM 1000 burn-in, MCMC, WinBUGS User Manual (Spiegelhalter, Thomas, Best, & Lunn, 2003), 5% 2PLM BTM 5000, MCMC, θ i 0 a j,a j d(j) 1 b j,b j d(j) 0 γ ad(j) 0 σγ 2 d(j) 3
61 ,,,, Bias,RMSE,cor(a,â),, 2PLM 2 BTM Bias RMSE, a j, 324, a j d(j) U(05,15), a j a j d(j) 323 (33) (35), a j,, a j a j d(j) U(05,15), a j 0289, Bias RMSE, 0289, a j 2 BTM, 2 BTM Bias,RMSE,cor(a,â),, 31 2PLM, 2PLM Bias,RMSE,cor(a,â),, 32, 2PLM, 2 BTM Bias,RMSE,cor(a,â),, 33, Bias, RMSE, cor(a,â), 2PLM Bias,RMSE,cor(a,â) 2 BTM, 2PLM, Bias RMSE a j,, 34, Bias/σ a j RMSE/σ a j,, Bias RMSE a j, 2PLM Bias/σ a j, RMSE/σ a j, cor(a,â)
62 3 60, 31, 32, 33, 2PLM Bias/σ a j, RMSE/σ a j, cor(a,â), 34, 35, 36, 2PLM Bias/σ a j, RMSE/σ a j, cor(a,â) 4, 37, 38, 39, 2PLM,,, 34,, 2PLM Bias σ a j 01 (a [ j ] 10% ), 2 BTM Bias, 34,,, Bias/σ a j,, 2PLM Bias, 2 BTM, 34,, 2PLM RMSE BTM RMSE,, N = 300, σ a j 01 (a j 10% ), 32,, RMSE/σ a j, 2PLM, 2PLM RMSE, 2 BTM, 2PLM,, 35,,, RMSE/σ a j, 2PLM RMSE, BTM
63 ,, 2PLM cor(a,â) BTM,, 5,,, cor(a,â) -015, 36,,, cor(a,â),,,, 2PLM cor(a,â), BTM,, 39, 5,, cor(a,â) 0,, cor(a,â), 4,, 2PLM, BTM, 4,, 2PLM, 5,,,,,,,,,,,,,,,,, 2PLM, a j d(j) a j, Bias,RMSE,cor(a,â)
64 3 62,, 35,, Bias, RMSE, cor(a,â),, 36, Bias/σ a j RMSE/σ a j,, 37,,,, 37,,, 2PLM Bias BTM Bias,, 37,,, 2PLM RMSE BTM RMSE,, N = 300, σ a j 01 (a j 10% ),,, 36,,, cor(a,â),, 3 2, cor(a,â) 01,,,
65 3 63,,,,,, 332,,,, Bias,RMSE,cor(b,ˆb),, 2PLM 2 BTM Bias RMSE, b j, 324, b j d(j) N(0,1), 323 (36), b j N(0,1), Bias,RMSE,, b j 2 BTM, 2 BTM Bias,RMSE,cor(b,ˆb),, 38 2PLM, 2PLM Bias,RMSE,cor(b,ˆb),, 39, 2PLM, Bias, RMSE, cor(b,ˆb),, 310, 2PLM, Bias, RMSE, cor(b,ˆb), 310, 311, 312, 2PLM, Bias, RMSE, cor(b,ˆb), 313, 314, 315, 2PLM,
66 3 64 Bias, RMSE, cor(b,ˆb) 4, 316, 317, 318, 2PLM,,, 310,, Bias, 2PLM 2 BTM 310,, 2PLM 2 BTM, RMSE, J d(j) = 5, d(j) 4,3,2, 2PLM 2 BTM, RMSE 005 (b j 5% ), 314,, RMSE,, 2PLM RMSE, 2 BTM, 317,, RMSE, 2PLM RMSE, BTM 310,, cor(b,ˆb), 2PLM BTM, 318, 5, cor(b,ˆb) 0,, cor(b,ˆb), 4,,2PLM, BTM,,,, 5,
67 3 65,,,,,,, J d(j) = 5, d(j) 4,3,2,,,,, 2PLM, 323 (36), b j d(j) b j b j b j,, 333,, Lord & Novick (1968), 2PLM a j,, j u j s r bj a j = r bj 1 r 2 bj (310), 2 BTM θ i d(j) γ id(j), γ id(j) θ i, 2PLM, θ i,, r bj
68 3 66, (310), a j r bj, 2 BTM 2PLM, â j, 2PL M Bias, 2, 2PLM RMSE,, â j ,,,, 2 BTM, 2PLM, BTM, Ip (2010),, 2 BTM 2PLM Bradlow et al (1999),,,, Bradlow et al (1999),,,,,, M95%PIW,,, Chen & Wang (2007) Jiao et al (2012),,,,
69 3 67,,,
70 BTM Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
71 PLM Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
72 Bias, RMSE, cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
73 Bias/σ a j RMSE/σ a j Bias/σ a j RMSE/σ a j N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
74 Bias,RMSE,cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
75 Bias, RMSE, cor(a,â) Bias RMSE cor(a,â) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
76 Bias/σ a j RMSE/σ a j Bias/σ a j RMSE/σ a j N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
77 BTM Bias,RMSE,cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
78 PLM Bias,RMSE,cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
79 Bias, RMSE, cor(b,ˆb) Bias RMSE cor(b,ˆb) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
80 Bias/σ a j
81 RMSE/σ a j
82 cor(a,â)
83 Bias/σ a j
84 RMSE/σ a j
85 cor(a,â)
86 Bias/σ a j 4
87 RMSE/σ a j 4
88 cor(a,â) 4
89 Bias
90 RMSE
91 cor(b,ˆb)
92 Bias
93 RMSE
94 cor(b,ˆb)
95 Bias 4
96 RMSE 4
97 cor(b,ˆb) 4
98 ,,,,, Ip (2010), National Assessment of Educational Progress (NAEP),,, (2001), 2000 GRM 2PLM, 2PLM GRM, Keller, Swaminathan, & Sireci (2003),,, Wainer & Wang (2000), TOEFL,,,, Ip (2000),,, Lee (2000),,,, Wainer et al (2007, p 182),,
99 4 97,,, Anastasi (1961, p 121) Thorndike (1951, p 585), Guilford (1936, p417), Kelly (1924),,, Wainer (1995), Low School Admission Test (LSAT),, Wang & Wilson (2005), ,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 42,,,,,,,,,,
100 , (114) 2 BTM ( ) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ id(j) ] (41) 15, 2 BTM b a c,, a d(j),,, c, 3,,, 2 (Braeken et al, 2007; Hoskens & De Boeck, 1997; Ip, 2002; Ip et al, 2009),,, b 2 BTM 422,, 2,, (11) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (42),, 2 BTM ( (41) )
101 ,,,,,,, U i u i ( i ) i θ i,, θ i I(θ i ), I(θ i ) ((43) ) [ ( ) 2 ] I(θ i ) = E logl(u i θ i ) θ i θ i, ˆθ i,,, θ i ˆθ i σ 2ˆθi I(θ θ i ) (44) i (43) σ 2ˆθi θ i = 1 I(θ i ) (44),, I(θ i ) ˆθ i,,,, 2PLM 2PLM, (45) ( Lord & Novick (1968) (2002) ) J I(θ i ) = (17) 2 a 2 j P j(θ i )Q j (θ i ) (45) j=1, (45) P j (θ i ) 2PLM ((42) ), Q j (θ i ) 1 P j (θ i ), 2PLM (45),, 2 BTM 2 BTM,
102 4 100 (46) ( Wainer, Bradlow, & Du (2000) Ip (2010) ) I(θ i ) = J j=1 [ ] (17a j d(j) ) 2 exp(17a j d(j) (θ i b j d(j) γ id(j) )) (1+exp(17a j d(j) (θ i b j d(j) γ id(j) ))) 2 dγ id(j) (46), 2 BTM (46), 424,, 4, 5, 41,,,,,,, 4, 300, ,, 3 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2),, ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) 4 1 ( d(j) 1) 4 ( d(j) 0)
103 4 101,, 3, *1 1 θ i (i = 1,2,,N) γ id(j) (a = 1,2,,N; d(j) = 1,2,3,4) a j d(j) b j d(j) (j = 1,2,,20) N(0,1), N(0,σγ 2 d(j) ), U(05,15), N(0,1) * BTM ((41) ), N 20 A 3 U(0,1) N 20 N 20 B 4 A,B U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 U , R 8 7 R, Bias(Î(θ i)) = 1 R Î r (θ i ) I(θ i ) (47) R r=1 RMSE(Î(θ i)) = 1 R ) 2 (Îr (θ i ) I(θ i ) (48) R, (47), (48) Î(θ i ) I(θ i ), Îr(θ i ) I(θ i ) r r=1, 7 R,, 1000 R = 50, 300 R = 100 *1, R *2, R
104 MCMC, 5 MCMC MCMC (Neal, 1997), WinBUGS 14 (Spiegelhalter et al, 2003) MCMC,,, 2PLM θ i N(0,1) a j N(1,025) b j N(0,1) 2 BTM θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) γ id(j) N(0,σγ 2 d(j) ) σγ 2 d(j) Γ(3, 1),,, 2PLM θ i 0 a j 1 b j 0 2 BTM θ i 0 a j d(j) 1 b j d(j) 0 γ id(j) 0 σγ 2 d(j) 3,, λ,, burn-in
105 PLM 1000 burn-in 2 BTM 1000 burn-in,, WinBUGS User Manual (Spiegelhalter et al, 2003), 5%, 2PLM BTM ,,, 2PLM 2 BTM Bias,RMSE,, Bias, θ i = 300, 275,,275, Bias(Î(θ i)),, 425,, θ i N(0,1), N(0,1) θ i 0125 ( , ), Bias,, RMSE, Bias, θ i = 300, 275,,275,300 RMSE(Î(θ i)),,,,, 2PLM 2 BTM Bias RMSE, ( ),,, θ i = 300, 275,,275,300
106 4 104 Bias,RMSE, I, Bias,RMSE, I, 431,, I, BTM, 2 BTM Bias,RMSE,, PLM, 2PLM Bias,RMSE,, 43, 2PLM, 2 BTM Bias,RMSE,, 44, Bias, RMSE, 2PLM Bias,RMSE 2 BTM, 2PLM, Bias RMSE I,, 45, Bias/I RMSE/I, Bias RMSE I, 2PLM Bias/I, RMSE/I, 41, 42, 2PLM Bias/I, RMSE/I, 43, 44, 2PLM Bias/I, RMSE/I 4, 45, 46, 2PLM,,
107 ,, 2PLM, 2 BTM, Bias,, d(j) 2,1,0,,, Bias I 01, 43,, Bias/I, 45,,, Bias/I, Bias,, 41,, Bias/I,,, Bias 45,, 2PLM, 2 BTM, RMSE,, d(j) 2,1,0,,, RMSE I 01, 44,, RMSE/I, 46,,, RMSE/I, RMSE,, 42,, RMSE/I,, RMSE 423,,,,,,,
108 4 106,,, d(j) 2,1,0,,,,,,,,, ,,,,,,,,,, 4,, Ip (2010), (2001), Keller et al (2003), Wainer & Wang (2000),,,,,
109 I I N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
110 BTM Bias,RMSE Bias RMSE N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
111 PLM Bias,RMSE Bias RMSE N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
112 PLM Bias, RMSE Bias RMSE N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
113 PLM Bias/I, RMSE/I Bias/I RMSE/I N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
114 Bias/I
115 RMSE/I
116 Bias/I
117 RMSE/I
118 Bias/I 4
119 RMSE/I 4
120 ,,,,,,,, *1,, Keller et al (2003), Lee (2000), Lee, Kolen, Frisbie, & Ankenmann (2001), Reise, Horan, & blanchard (2011), Tuerlinckx & De Boeck (1999), Wainer & Wang (2000), Wang, Cheng, & Wilson (2005), Zhang (2010),,, Braedlow et al (1999), DeMars (2006), (2001), (2013), Wainer et al (2007, pp ), Bradlow et al (1999), 2 BTM, 2PLM BTM, BTM, 2PLM, BTM, 2PLM, M95%PIW *1, (2012a),
121 5 119 BTM, 2PLM, M95%PIW BTM 2PLM,, DeMars (2006), b, a, b,,,,, (2001), 2000 GRM 2PLM, 0987, (2013), 2PLM GRM, , Wainer et al (2007, pp ), b, b,,,,, 15, a, b, c 3,, 3,,,,, 3 ( d ),,, 4,,,,,,,,,
122 5 120 ( a, b, c) ( d),,,,, 52,, a d,, ˆθ ir ˆθ i Bias(ˆθ i ), RMSE(ˆθ i ) cor(θ, ˆθ), 521, 421, (114) 2 BTM ( ) P j d(j) (θ i ) = 1 1+exp [ 17a j d(j) (θ i b j d(j) γ id(j) ) ] (51) 522, a d, a a, (112) GRM ( ) P d(j) (r θ i ) = 1 ] 1+exp [ 17a d(j) (θ i b r ) 1 ] (52) 1+exp [ 17a d(j) (θ i b r+1 ) b b, 2 BTM ( (51) )
123 5 121 c c, (116) CCM ( ) P(U j,u k θ i ) = exp[u j Z j +U k Z k U j U k b jk ] 1+exp[Z j ]+exp[z k ]+exp[z j +Z k b jk ] (53) Z j = 17a j d(j) (θ i b j d(j) ) (54) Z k = 17a k d(j) (θ i b k d(j) ) (55) d d, (11) 2PLM ( ) P j (θ i ) = 1 1+exp[ 17a j (θ i b j )] (56) 523,, 4, 5, 51,,,,, 4, 300, ,, 3 5 (J d(j) = 5) 3 (J d(j) = 3) 2 (J d(j) = 2),, ( d(j) 4) 4 3 ( d(j) 3) 4 2 ( d(j) 2) 4 1 ( d(j) 1)
124 ( d(j) 0),, 3 *2 1 θ i (i = 1,2,,N) γ id(j) (i = 1,2,,N; d(j) = 1,2,3,4) a j d(j), b j d(j) (j = 1,2,,20) N(0,1), N(0,σγ 2 d(j) ), U(05,15), N(0,1) * BTM ((51) ), N 20 A 3 U(0,1) N 20, N 20 B 4 A,B U, a ij b ij u ij = 1, a ij < b ij u ij = 0 5 U R 7 6 R ˆθ ir,, Bias(ˆθ i ),RMSE(ˆθ i ),cor(θ, ˆθ) ( ) Bias(ˆθ i ) = 1 R ˆθ ir θ i (57) R r=1 RMSE(ˆθ i ) = 1 R ) 2 (ˆθir θ i (58) R cor(θ, ˆθ) = 1 R r=1 R cor(θ, ˆθ r ) (59) r=1, 6 R, ˆθ ir, 1000 R = 50, 300 R = 100, c, 421, 2, c,, J d(j) = 2, *2, R *3, R
125 , 5 MCMC MCMC (Neal, 1997), WinBUGS 14 (Spiegelhalter et al, 2003) MCMC,,, a θ i N(0,1) a d(j) N(1,025) b r N(0,1) b θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) γ id(j) N(0,σγ 2 d(j) ) σγ 2 d(j) Γ(3, 1) c θ i N(0,1) a j d(j) N(1,025) b j d(j) N(0,1) b jk N( 2,1) d θ i N(0,1) a j N(1,025) b j N(0,1),,, a θ i 0 a d(j) 1 b r r = 1,2,3,4,5, 0, 01, 02, 03, 04 b
126 5 124 θ i 0 a j d(j) 1 b j d(j) 0 γ id(j) 0 σγ 2 d(j) 3 c θ i 0 a j d(j) 1 b j d(j) 0 b jk -2 d θ i 0 a j 1 b j 0,, λ,, burn-in a 1000 burn-in b 1000 burn-in c 1000 burn-in d 1000 burn-in,, WinBUGS User Manual (Spiegelhalter et al, 2003), 5%, a J d(j) = , 4000 b 5000
127 5 125 c 4000 d ,,, 2PLM GRM, BTM, CCM Bias,RMSE,cor(θ, ˆθ), 2PLM GRM, BTM, CCM Bias RMSE, θ i 523,, θ i N(0,1), θ i 1, Bias RMSE,, θ i 531 2PLM, 2PLM Bias,RMSE,cor(θ, ˆθ),, GRM, GRM Bias,RMSE,cor(θ, ˆθ),, 52, GRM Bias, RMSE, cor(θ, ˆθ),, 53, Bias, RMSE, cor(θ, ˆθ), GRM Bias,RMSE,cor(θ, ˆθ) 2PLM Bias,RMSE,cor(θ, ˆθ), GRM Bias, RMSE, cor(θ, ˆθ), 51, 52, 53, GRM Bias, RMSE, cor(θ, ˆθ)
128 5 126, 54, 55, 56, GRM Bias, RMSE, cor(θ, ˆθ) 4, 57, 58, 59, GRM,,, 57,, 2PLM,, 2PLM GRM,, 51,, Bias, 2PLM,, GRM,, 57,, 2PLM,, 2PLM GRM,, 58, 2 3, RMSE, 5,,, RMSE 2 3, 2PLM RMSE, GRM, RMSE, 5, 57, 4 2PLM, GRM,, 2PLM GRM 57,, 2PLM,, 2PLM GRM,
129 5 127, 59, 2 3,, cor(θ, ˆθ), cor(θ, ˆθ), GRM 2PLM, 59, 5, 4 4, cor(θ, ˆθ), 4,, 2PLM, GRM, 4,,, 5,,,,, a,,,,, a BTM, BTM Bias,RMSE,cor(θ, ˆθ),, 54, BTM Bias, RMSE, cor(θ, ˆθ),, 55, BTM Bias, RMSE, cor(θ, ˆθ), 510, 511, 512,
130 5 128 BTM Bias, RMSE, cor(θ, ˆθ), 513, 514, 515, BTM Bias, RMSE, cor(θ, ˆθ) 4, 516, 517, 518, BTM,,, 59, N = 1000, 2PLM 2 BTM,, 2PLM 2 BTM,, 510,, Bias,, 2PLM,, 2 BTM 59,, 2PLM 2 BTM,, 2PLM 2 BTM,, 514,, RMSE, RMSE, 2 BTM 2PLM,, 517, 5, 4, 4 RMSE, 5, 59, 4 2PLM 2 BTM, 2PLM BTM
131 ,, 2PLM 2 BTM,, 2PLM 2 BTM,, 518, 5, 4 4, cor(θ, ˆθ), 4,, 2PLM, BTM, 4,,, 5,,,, b,,,,,,, b, 534 CCM, CCM Bias,RMSE,cor(θ, ˆθ),, 56, CCM Bias, RMSE, cor(θ, ˆθ),, 57, CCM Bias, RMSE, cor(θ, ˆθ), 519, 520, 521, CCM Bias, RMSE, cor(θ, ˆθ) 4, 522, 523,
132 , CCM,,, 511,, 2PLM CCM,, 2PLM CCM,, 519, Bias,,,, 2PLM Bias, CCM, 522,, Bias,, CCM Bias, 2PLM, 511,, 2PLM CCM,, 2PLM CCM,, 520,, RMSE, CCM 2PLM, RMSE 511,, 2PLM CCM,, 2PLM CCM,,,,, c
133 5 131,,,,,,, c,,,, CCM,, 535, 535 CCM Bias, CCM b jk, -1714, b jk -0612, 2 BTM CCM, 2 j,k 100, σγ 2 d(j) = BTM j,k (1,1),(1,0),(0,1),(0,0) (100 ) 58, b jk = 1714 CCM 59, 2 BTM CCM, 2 j,k (N = 100), σγ 2 d(j) = BTM 510, b jk = 0612 CCM ,, 59, 510, 511, 2 BTM,, j,k, j,k CCM,, j,k,, j,k,,, 2 BTM CCM,,, CCM Bias
134 ,,, 3, GRM, BTM, CCM, 2PLM,, BTM 2PLM, Bradlow et al (1999), BTM,, BTM, 2PLM, M95%PIW BTM 2PLM,, DeMars (2006), a, b, d,,,,,,, Wainer et al (2007, pp ), b d,,, BTM,,, 537,,, 3,,,
135 5 133,,,,,,,, 4,,,,,,,
136 PLM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
137 GRM Bias,RMSE,cor(θ, ˆθ) Bias RMSE cor(θ, ˆθ) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 5, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 3, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 300,J d(j) = 2, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 5, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 3, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j) N = 1000,J d(j) = 2, d(j)
1 1 1.1..................................... 1 1.2................................ 3 1.3................................ 4 I 6 2 7 2.1................................ 7 2.2 δ............................
t χ 2 F Q t χ 2 F 1 2 µ, σ 2 N(µ, σ 2 ) f(x µ, σ 2 ) = 1 ( exp (x ) µ)2 2πσ 2 2σ 2 0, N(0, 1) (100 α) z(α) t χ 2 *1 2.1 t (i)x N(µ, σ 2 ) x µ σ N(0, 1
t χ F Q t χ F µ, σ N(µ, σ ) f(x µ, σ ) = ( exp (x ) µ) πσ σ 0, N(0, ) (00 α) z(α) t χ *. t (i)x N(µ, σ ) x µ σ N(0, ) (ii)x,, x N(µ, σ ) x = x+ +x N(µ, σ ) (iii) (i),(ii) z = x µ N(0, ) σ N(0, ) ( 9 97.
12/1 ( ) GLM, R MCMC, WinBUGS 12/2 ( ) WinBUGS WinBUGS 12/2 ( ) : 12/3 ( ) :? ( :51 ) 2/ 71
2010-12-02 (2010 12 02 10 :51 ) 1/ 71 GCOE 2010-12-02 WinBUGS kubo@ees.hokudai.ac.jp http://goo.gl/bukrb 12/1 ( ) GLM, R MCMC, WinBUGS 12/2 ( ) WinBUGS WinBUGS 12/2 ( ) : 12/3 ( ) :? 2010-12-02 (2010 12
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
dvi
2017 65 2 235 249 2017 1 2 2 2016 12 26 2017 3 1 4 25 1 MCMC 1. SLG OBP OPS Albert and Benett, 2003 1 2 3 4 OPS Albert and Benett 2003 Albert 2008 1 112 8551 1 13 27 2 112 8551 1 13 27 236 65 2 2017 Albert
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x
SFGÇÃÉXÉyÉNÉgÉãå`.pdf
SFG 1 SFG SFG I SFG (ω) χ SFG (ω). SFG χ χ SFG (ω) = χ NR e iϕ +. ω ω + iγ SFG φ = ±π/, χ φ = ±π 3 χ SFG χ SFG = χ NR + χ (ω ω ) + Γ + χ NR χ (ω ω ) (ω ω ) + Γ cosϕ χ NR χ Γ (ω ω ) + Γ sinϕ. 3 (θ) 180
chap10.dvi
. q {y j } I( ( L y j =Δy j = u j = C l ε j l = C(L ε j, {ε j } i.i.d.(,i q ( l= y O p ( {u j } q {C l } A l C l
QMII_10.dvi
65 1 1.1 1.1.1 1.1 H H () = E (), (1.1) H ν () = E ν () ν (). (1.) () () = δ, (1.3) μ () ν () = δ(μ ν). (1.4) E E ν () E () H 1.1: H α(t) = c (t) () + dνc ν (t) ν (), (1.5) H () () + dν ν () ν () = 1 (1.6)
untitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
gr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
/22 R MCMC R R MCMC? 3. Gibbs sampler : kubo/
2006-12-09 1/22 R MCMC R 1. 2. R MCMC? 3. Gibbs sampler : kubo@ees.hokudai.ac.jp http://hosho.ees.hokudai.ac.jp/ kubo/ 2006-12-09 2/22 : ( ) : : ( ) : (?) community ( ) 2006-12-09 3/22 :? 1. ( ) 2. ( )
LLG-R8.Nisus.pdf
d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =
x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v
12 -- 1 4 2009 9 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 c 2011 1/(13) 4--1 2009 9 3 x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2
untitled
18 1 2,000,000 2,000,000 2007 2 2 2008 3 31 (1) 6 JCOSSAR 2007pp.57-642007.6. LCC (1) (2) 2 10mm 1020 14 12 10 8 6 4 40,50,60 2 0 1998 27.5 1995 1960 40 1) 2) 3) LCC LCC LCC 1 1) Vol.42No.5pp.29-322004.5.
main.dvi
SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1
.2 ( ) ( ) (?? ).3 *2 ( ) *2 2
. (22) * * (22) .2 ( ) ( ) (?? ).3 *2 ( ) *2 2 ( ) *3 n N n < N N ( n N ) N ( ) ICC(Item Characteristic Curve, ) 2 2. 7 3 ( ) 2 *4 x ( ) f (x) = 2π e.5x2 Φ( f (x)) = f (x) f (x)dx *3 ( ) *4 Excel 3 3 A
80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
ohpmain.dvi
fujisawa@ism.ac.jp 1 Contents 1. 2. 3. 4. γ- 2 1. 3 10 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 5.2. 5.5 5.6 +5.7 +5.4 +5.5 +5.8 +5.5 +5.3 +5.6 +5.4 +5.2 =5.5. 10 outlier 5 5.6, 5.7, 5.4, 5.5, 5.8,
untitled
17 5 13 1 2 1.1... 2 1.2... 2 1.3... 3 2 3 2.1... 3 2.2... 5 3 6 3.1... 6 3.2... 7 3.3 t... 7 3.4 BC a... 9 3.5... 10 4 11 1 1 θ n ˆθ. ˆθ, ˆθ, ˆθ.,, ˆθ.,.,,,. 1.1 ˆθ σ 2 = E(ˆθ E ˆθ) 2 b = E(ˆθ θ). Y 1,,Y
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
![24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x](/thumbs/94/118744578.jpg "24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x")
24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
X X X Y R Y R Y R MCAR MAR MNAR Figure 1: MCAR, MAR, MNAR Y R X 1.2 Missing At Random (MAR) MAR MCAR MCAR Y X X Y MCAR 2 1 R X Y Table 1 3 IQ MCAR Y I
(missing data analysis) - - 1/16/2011 (missing data, missing value) (list-wise deletion) (pair-wise deletion) (full information maximum likelihood method, FIML) (multiple imputation method) 1 missing completely
7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
![7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±](/thumbs/91/106462075.jpg "7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±")
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n
. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n
.. ( )T p T = p p = T () T x T N P (X < x T ) N = ( T ) N (2) ) N ( P (X x T ) N = T (3) T N P T N P 0
20 5 8..................................................2.....................................3 L.....................................4................................. 2 2. 3 2. (N ).........................................
2011 8 26 3 I 5 1 7 1.1 Markov................................ 7 2 Gau 13 2.1.................................. 13 2.2............................... 18 2.3............................ 23 3 Gau (Le vy
60 (W30)? 1. ( ) 2. ( ) web site URL ( :41 ) 1/ 77
60 (W30)? 1. ( ) kubo@ees.hokudai.ac.jp 2. ( ) web site URL http://goo.gl/e1cja!! 2013 03 07 (2013 03 07 17 :41 ) 1/ 77 ! : :? 2013 03 07 (2013 03 07 17 :41 ) 2/ 77 2013 03 07 (2013 03 07 17 :41 ) 3/ 77!!
201711grade1ouyou.pdf
2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2
untitled
1 (1) (2) (3) (4) (1) (2) (3) (1) (2) (3) (1) (2) (3) (4) (5) (1) (2) (3) (1) (2) 10 11 12 2 2520159 3 (1) (2) (3) (4) (5) (6) 103 59529 600 12 42 4 42 68 53 53 C 30 30 5 56 6 (3) (1) 7 () () (()) () ()
B
B YES NO 5 7 6 1 4 3 2 BB BB BB AA AA BB 510J B B A 510J B A A A A A A 510J B A 510J B A A A A A 510J M = σ Z Z = M σ AAA π T T = a ZP ZP = a AAA π B M + M 2 +T 2 M T Me = = 1 + 1 + 2 2 M σ Te = M 2 +T
kubo2015ngt6 p.2 ( ( (MLE 8 y i L(q q log L(q q 0 ˆq log L(q / q = 0 q ˆq = = = * ˆq = 0.46 ( 8 y 0.46 y y y i kubo (ht
kubo2015ngt6 p.1 2015 (6 MCMC kubo@ees.hokudai.ac.jp, @KuboBook http://goo.gl/m8hsbm 1 ( 2 3 4 5 JAGS : 2015 05 18 16:48 kubo (http://goo.gl/m8hsbm 2015 (6 1 / 70 kubo (http://goo.gl/m8hsbm 2015 (6 2 /
0406_total.pdf
59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
chap9.dvi
9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
³ÎΨÏÀ
2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p
1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π
![1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π](/thumbs/101/149329485.jpg "1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π")
. 4cm 6 cm 4cm cm 8 cm λ()=a [kg/m] A 4cm A 4cm cm h h Y a G.38h a b () y = h.38h G b h X () S() = π() a,b, h,π V = ρ M = ρv G = M h S() 3 d a,b, h 4 G = 5 h a b a b = 6 ω() s v m θ() m v () θ() ω() dθ()
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
linearal1.dvi
19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352
waseda2010a-jukaiki1-main.dvi
November, 2 Contents 6 2 8 3 3 3 32 32 33 5 34 34 6 35 35 7 4 R 2 7 4 4 9 42 42 2 43 44 2 5 : 2 5 5 23 52 52 23 53 53 23 54 24 6 24 6 6 26 62 62 26 63 t 27 7 27 7 7 28 72 72 28 73 36) 29 8 29 8 29 82 3
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1)
1 9 v..1 c (216/1/7) Minoru Suzuki 1 1 9.1 9.1.1 T µ 1 (7.18) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) E E µ = E f(e ) E µ (9.1) µ (9.2) µ 1 e β(e µ) 1 f(e )
構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
: (GLMM) (pseudo replication) ( ) ( ) & Markov Chain Monte Carlo (MCMC)? /30
PlotNet 6 ( ) 2006-01-19 TOEF(1998 2004), AM, growth6 DBH growth (mm) 1998 1999 2000 2001 2002 2003 2004 10 20 30 40 50 70 DBH (cm) 1. 2. - - : kubo@ees.hokudai.ac.jp http://hosho.ees.hokudai.ac.jp/ kubo/show/2006/plotnet/
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
p = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
~nabe/lecture/index.html 2
2001 12 13 1 http://www.sml.k.u-tokyo.ac.jp/ ~nabe/lecture/index.html nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 4 2. 10/11 3. 10/18 1 4. 10/25 2 5. 11/ 1 6. 11/ 8 7. 11/15 8. 11/22 9. 11/29 10. 12/ 6 1 11. 12/13
ＪＭＰ V4 による生存時間分析
V4 1 SAS 2000.11.18 4 ( ) (Survival Time) 1 (Event) Start of Study Start of Observation Died Died Died Lost End Time Censor Died Died Censor Died Time Start of Study End Start of Observation Censor
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
![II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re](/thumbs/94/118770263.jpg "II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re")
II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
( ) ( )
20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
.. F x) = x ft)dt ), fx) : PDF : probbility density function) F x) : CDF : cumultive distribution function F x) x.2 ) T = µ p), T : ) p : x p p = F x
203 7......................................2................................................3.....................................4 L.................................... 2.5.................................
ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
: , 2.0, 3.0, 2.0, (%) ( 2.
2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
OHP.dvi
t 0, X X t x t 0 t u u = x X (1) t t 0 u X x O 1 1 t 0 =0 X X +dx t x(x,t) x(x +dx,t). dx dx = x(x +dx,t) x(x,t) (2) dx, dx = F dx (3). F (deformation gradient tensor) t F t 0 dx dx X x O 2 2 F. (det F
TOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................
IV (2)
COMPUTATIONAL FLUID DYNAMICS (CFD) IV (2) The Analysis of Numerical Schemes (2) 11. Iterative methods for algebraic systems Reima Iwatsu, e-mail : iwatsu@cck.dendai.ac.jp Winter Semester 2007, Graduate
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)
19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
19 σ = P/A o σ B Maximum tensile strength σ 0. 0.% 0.% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional limit ε p = 0.% ε e = σ 0. /E plastic strain ε = ε e
,, Mellor 1973),, Mellor and Yamada 1974) Mellor 1973), Mellor and Yamada 1974) 4 2 3, 2 4,
Mellor and Yamada1974) The Turbulence Closure Model of Mellor and Yamada 1974) Kitamori Taichi 2004/01/30 ,, Mellor 1973),, Mellor and Yamada 1974) Mellor 1973), 4 1 4 Mellor and Yamada 1974) 4 2 3, 2
http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P
1 1.1 (population) (sample) (event) (trial) Ω () 1 1 Ω 1.2 P 1. A A P (A) 0 1 0 P (A) 1 (1) 2. P 1 P 0 1 6 1 1 6 0 3. A B P (A B) = P (A) + P (B) (2) A B A B A 1 B 2 A B 1 2 1 2 1 1 2 2 3 1.3 A B P (A
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
? (EM),, EM? (, 2004/ 2002) von Mises-Fisher ( 2004) HMM (MacKay 1997) LDA (Blei et al. 2001) PCFG ( 2004)... Variational Bayesian methods for Natural
SLC Internal tutorial Daichi Mochihashi daichi.mochihashi@atr.jp ATR SLC 2005.6.21 (Tue) 13:15 15:00@Meeting Room 1 Variational Bayesian methods for Natural Language Processing p.1/30 ? (EM),, EM? (, 2004/
鉄鋼協会プレゼン
NN :~:, 8 Nov., Adaptive H Control for Linear Slider with Friction Compensation positioning mechanism moving table stand manipulator Point to Point Control [G] Continuous Path Control ground Fig. Positoining
土地税制の理論的・計量的分析
126 312 1 126 312... 2... 4 I...12...12...12...14...14...16...16...17...20...22...22...24...25 II...31...33...33...33...36...36...38 2...41...41...42...50...50...51 III...54...54...54...54...55...55...57...57...58...60...60...60...63...65...67...67
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
18 2 20 W/C W/C W/C 4-4-1 0.05 1.0 1000 1. 1 1.1 1 1.2 3 2. 4 2.1 4 (1) 4 (2) 4 2.2 5 (1) 5 (2) 5 2.3 7 3. 8 3.1 8 3.2 ( ) 11 3.3 11 (1) 12 (2) 12 4. 14 4.1 14 4.2 14 (1) 15 (2) 16 (3) 17 4.3 17 5. 19
keisoku01.dvi
2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
N cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
第1章 微分方程式と近似解法
April 12, 2018 1 / 52 1.1 ( ) 2 / 52 1.2 1.1 1.1: 3 / 52 1.3 Poisson Poisson Poisson 1 d {2, 3} 4 / 52 1 1.3.1 1 u,b b(t,x) u(t,x) x=0 1.1: 1 a x=l 1.1 1 (0, t T ) (0, l) 1 a b : (0, t T ) (0, l) R, u
H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [
![H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [](/thumbs/99/141438442.jpg "H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [")
3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e
2002 11 21 1 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2002 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture nabe@sml.k.u-tokyo.ac.jp 2 1. 10/10 2. 10/17 3. 10/24 4. 10/31 5. 11/ 7 6. 11/14
9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P
9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)
磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論
email: takahash@sci.u-hyogo.ac.jp May 14, 2009 Outline 1. 2. 3. 4. 5. 6. 2 / 262 Today s Lecture: Mode-mode Coupling Theory 100 / 262 Part I Effects of Non-linear Mode-Mode Coupling Effects of Non-linear
N/m f x x L dl U 1 du = T ds pdv + fdl (2.1)
23 2 2.1 10 5 6 N/m 2 2.1.1 f x x L dl U 1 du = T ds pdv + fdl (2.1) 24 2 dv = 0 dl ( ) U f = T L p,t ( ) S L p,t (2.2) 2 ( ) ( ) S f = L T p,t p,l (2.3) ( ) U f = L p,t + T ( ) f T p,l (2.4) 1 f e ( U/
成長機構
j im πmkt jin jim π mkt j q out j q im π mkt jin j j q out out π mkt π mkt dn dt πmkt dn v( ) Rmax bf dt πmkt R v ( J J ), J J, J J + + T T, J J m + Q+ / kt Q / kt + ( Q Q+ )/ ktm l / ktm J / J, l Q Q
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t,
1 Gourgoulhon BSSN BSSN ϕ = 1 6 ( D i β i αk) (1) γ ij = 2αĀij 2 3 D k β k γ ij (2) K = e 4ϕ ( Di Di α + 2 D i ϕ D i α ) + α ] [4π(E + S) + ĀijĀij + K2 3 (3) Ā ij = 2 3Āij D k β k 2αĀikĀk j + αāijk +e
Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
2001 年度 『数学基礎 IV』 講義録
4 A 95 96 4 1 n {1, 2,,n} n n σ ( ) 1 2 n σ(1) σ(2) σ(n) σ σ 2 1 n 1 2 {1, 2,,n} n n! n S n σ, τ S n {1, 2,,n} τ σ {1, 2,,n} n τ σ σ, τ τσ σ n σ 1 n σ 1 ( σ σ ) 1 σ = σσ 1 = ι 1 2 n ι 1 2 n 4.1. 4 σ =
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
January 27, 2015
e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6
ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(
I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
, 1), 2) (Markov-Switching Vector Autoregression, MSVAR), 3) 3, ,, , TOPIX, , explosive. 2,.,,,.,, 1
2016 1 12 4 1 2016 1 12, 1), 2) (Markov-Switching Vector Autoregression, MSVAR), 3) 3, 1980 1990.,, 225 1986 4 1990 6, TOPIX,1986 5 1990 2, explosive. 2,.,,,.,, 1986 Q2 1990 Q2,,. :, explosive, recursiveadf,
q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices
Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple
1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
Γ Ec Γ V BIAS THBV3_0401JA THBV3_0402JAa THBV3_0402JAb 1000 800 600 400 50 % 25 % 200 100 80 60 40 20 10 8 6 4 10 % 2.5 % 0.5 % 0.25 % 2 1.0 0.8 0.6 0.4 0.2 0.1 200 300 400 500 600 700 800 1000 1200 14001600
/ *1 *1 c Mike Gonzalez, October 14, Wikimedia Commons.
2010 05 22 1/ 35 2010 2010 05 22 *1 kubo@ees.hokudai.ac.jp *1 c Mike Gonzalez, October 14, 2007. Wikimedia Commons. 2010 05 22 2/ 35 1. 2. 3. 2010 05 22 3/ 35 : 1.? 2. 2010 05 22 4/ 35 1. 2010 05 22 5/
s = 1.15 (s = 1.07), R = 0.786, R = 0.679, DW =.03 5 Y = 0.3 (0.095) (.708) X, R = 0.786, R = 0.679, s = 1.07, DW =.03, t û Y = 0.3 (3.163) + 0
7 DW 7.1 DW u 1, u,, u (DW ) u u 1 = u 1, u,, u + + + - - - - + + - - - + + u 1, u,, u + - + - + - + - + u 1, u,, u u 1, u,, u u +1 = u 1, u,, u Y = α + βx + u, u = ρu 1 + ɛ, H 0 : ρ = 0, H 1 : ρ 0 ɛ 1,