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3 I δ Web Web BILOG-MG i
4 II EM IRT EM IRT EM IRT EM IRT EM IRT ii
5 PC e-learning IRT: item response theory [9, 10, 27] 1
6 3,000 2,500 Population of 18 years old , population 1,500 Higher Education University & Junior college rate 1,000 University Junior college Graduate school year 1.1: 18 ( [41] IRT e-learning IRT 20 IRT IRT MMLE: marginal maximum likelihood estimation [4, 5, 17] EM [18] [22] PC 2
7 20 IRT BILOG-MG [3] IRT e-learning 1.2 IRT SSI BILOG- MG [3] IRT IRT Microsoft Office Excel Web IRT IRT EM [4, 5, 17] e-learning 3
8 e-learning IRT 1.3 II I IRT IRT Web II IRT 2 IRT IRT IRT [4,5,17] MBE: marginal baysian estimation [18] MCMC: Malcov chain Monte Carlo IRT [22] 3 IRT Web [1] PC Microsoft Office Excel
9 IRT BILOG-MG [3] 4 IRT e-learning 5 IRT IRT
10 I
11 2 [9, 10, 27] 1 0 0/ δ δ N n δ N n 0 1 δ i,j = 1 i j
12 2.1: T i j w j T i = n w j δ i,j j= [29]
13 2.2: 4 [29] Q δ P i,j j Q 1 δ i,j j θ P 1 (θ) P 2 (θ) P 3 (θ) P 4 (θ) δ i,j θ 1 9
14 i θ i j φ j i j P (θ i, φ j ) Q(θ i, φ j ) = 1 P (θ i, φ j ) n j = 1, 2,, n i j δ i,j δ i,j = 1 = 0 n 1 1 θ i i n P (δ i,j θ i ) = P (θ i, φ j ) δ i,j Q(θ i, φ j ) 1 δ i,j (2.1) j=1 δ i,j θ i likelihood n N i = 1, 2,..., N δ N n L = P (δ i,j θ i, φ j ) = P (θ i, φ j ) δ i,j Q(θ i, φ j ) 1 δ i,j (2.2) i=1 j=1 δ i,j φ j i θ i φ j 2.3 φ j θ i P (θ, φ j ) θ ICC θ 10
15 φ(z) = 1 exp ( 12 ) 2π z2 (2.3) Φ(f(θ)) = f(θ) φ(z)dz (2.4) ICC f(θ) θ Φ(f(θ)) θ 1 1 Rash 1 φ j = {b j } P (θ, φ j ) = exp ( Da(θ b j )) (2.5) D D = 1.7 b j j item difficulty a ICC ICC 11
16 1 0.8 probability ability θ 2.1: 1 ICC 2 2 φ j = {a j, b j } P (θ, φ j ) = exp ( Da j (θ b j )) (2.6) a j j descriminating parameter 2 1 ICC ICC ICC 2.2 δ (2.2) δ δ = 1 δ = 0 2 m l 12
17 1 0.8 probability ability θ 2.2: 2 ICC δ = l/m δ m = 5 l l = {0, 1, 2, 3, 4, 5} δ δ = l { m } 0 = 5, 1 5, 2 5, 3 5, 4 5, 5 5 = {0, 0.2, 0.4, 0.6, 0.8, 1} 2.3 (2.2) 13
18 [21] MMLE: marginal maximum likelihood estimation [4, 5, 17] MBE: marginal Bayesian estimation [18] BMAP: Bayesian maximum a posteriori BEAP: Bayesian expectation a posteriori [18] MBE BEAP [18] MCMC: malcov chain Monte Carlo method [22] MBE BMAP BEAP MCMC ICC {θ, a, b} g(θ), g(a), g(b) {θ, a, b} L(δ θ, a, b) {θ, a, b} g{θ, a, b δ} g{θ, a, b δ} L{δ θ, a, b}g(a)g(b)g(θ) (2.7) δ {θ, a, b} 14
19 {θ, a, b} θ i θ i N(µ θ, σ θ 2 ) (2.8) < θ i < b j b j N(µ b, σ b 2 ) (2.9) < b j < a j a j log-normal(µ α, σ α 2 ) (2.10) 0 < a j < (2.7) 1 0 (2.7) l log l = log L{δ θ, a, b} + log g(a)g(b) + log g(θ) (2.11) (2.11) 1 log L{δ θ, a, b} + a j log L{δ θ, a, b} + b j a j log g(a)g(b) + b j log g(a)g(b) + log g(θ) = 0 a j (2.12) log g(θ) = 0 b j (2.13) log g(θ) 0 0 log l = log L{δ θ, a, b} + a j a j log l b j = b j log L{δ θ, a, b} + 15 log g(a) = 0 a j (2.14) log g(b) = 0 b j (2.15)
20 (2.2) L = L{δ θ, a, b} = = N P (δ i θ i, a, b) i=1 N i=1 P (δ i a, b)g(θ)dθ (2.16) (2.14) 1 (2.16) log L = N { } log P (δ i a, b)g(θ)dθ a j a j i=1 N = D (δ i,j P (θ i, a j, b j ))(θ i b j ) i=1 P (δ a j, b j )g(θ) dθ (2.17) P (δ a j, b j )g(θ)dθ (2.15) 1 (2.16) b j log L = Da j N (δ i,j P (θ i, a j, b j )) i=1 P (δ a j, b j )g(θ) dθ (2.18) P (δ a j, b j )g(θ)dθ (2.14) (2.15) 2 (2.9), (2.10) [ 1 g(a) = exp 1 ( ) ] 2 log aj µ a 2πσa a j 2 σ a [ 1 g(b) = exp 1 ( ) ] 2 bj µ b 2πσb 2 σ b (2.19) (2.20) 16
21 0.5! density! 0.4! 0.3! 0.2! g(!) A(X k ) 0.1! X 1 X 2 X 3 X 4 X 5 X 6 X 7 0.0! -4! -3! -2! -1! 0! 1! 2! 3! 4! Ability 2.3: log g(a) = 1 log a j µ a a j a j a j σa 2 log g(b) = b j µ b b j σb 2 (2.21) (2.22) (2.17) (2.18) θ X k, (k = 1, 2,..., q) A(X k ) θ X k g(θ) A(X k ) 2.3 q = (2.1) P (X k, a j, b j ) = exp{ Da j (X k b j )} 17
22 2.3: q = 15 k X k A(X k ) (2.16) N q n log L = log A(X k ) P (X k, a j, b j ) δ i,j (1 P (X k, a j, b j )) 1 δ i,j (2.23) i=1 k=1 j=1 (2.17) (2.18) log L a j N q = D [δ i,j P (X k, a j, b j )](X k b j ) i=1 k=1 n A(X k ) P (X k, a j, b j ) δ i,j (1 P (X k, a j, b j )) 1 δ i,j j=1 q n A(X k ) P (X k, a j, b j ) δ i,j (1 P (X k, a j, b j )) 1 δ i,j k=1 j=1 (2.24) 18
23 log L b j = Da j k=1 N i=1 k=1 q [δ i,j P (X k, a j, b j )] n A(X k ) P (X k, a j, b j ) δ i,j (1 P (X k, a j, b j )) 1 δ i,j j=1 q n A(X k ) P (X k, a j, b j ) δ i,j (1 P (X k, a j, b j )) 1 δ i,j j=1 (2.25) EM E-step θ = X k n L(X k ) = P (X k, a j, b j ) δ i,j (1 P (X k, a j, b j )) 1 δ i,j (2.26) j=1 X k f jk f jk = N i=1 L(X k )A(X k ) q L(X k )A(X k ) k=1 r jk r jk = N δ i,j L(X k )A(X k ) q L(X k )A(X k ) i=1 k=1 (2.27) (2.28) M-step Newton-Raphson-Fisher [ a j b j ] [ = t+1 a j b j ] + t ( ) 2 log l E a 2 j ( 2 ) log l E b j a j ( 2 ) log l E a j b ( j ) 2 log l E b 2 j 1 t log l a j log l b j t 19
24 log l 1 (2.21) (2.22) (2.24) (2.25) (2.27) (2.28) log l a j log l b j = D (X k b j )[ r jk f jk P (X k, a j, b j )] 1 log a j µ a a j a j σ 2 k a = Da j [ r jk f jk P (X k, a j, b j )] b j µ b k σ 2 b 2 E[ r jk ] = f jk P j (X k ) ( ) 2 log l E a 2 = D 2 (X k b j ) 2 fjk P (X k, a j, b j )(1 P (X k, a j, b j )) j k + 1 a 2 1 log a j + µ a j a 2 j σ2 a ( ) 2 log l E b 2 = D 2 a 2 j f jk P (X k, a j, b j )(1 P (X k, a j, b j )) 1 j σ 2 k b ( 2 ) log l E = D 2 a j (X k b j ) b j a f jk P (X k, a j, b j )(1 P (X k, a j, b j )) j k E-step E-step M-step BMAP: bayesian maximum a posteriori g{θ i δ i, a, b} L{δ i θ i, a, b}g(θ) (2.29) g(θ) N(µ θ, σ θ 2 ) (2.29) log g{δ i θ i, a, b} log L{δ i θ i, a, b} + log g(θ) (2.30) 20
25 n L{δ i θ i, a, b} = P (θ i, a j, b j ) δ i,j (1 P (θ i, a j, b j )) 1 δ i,j (2.31) j=1 (2.30) log l θ i log l θ i = D n j=1 {a j (δ i,j P (θ i, a j, b j ))} θ i µ θ σ θ 2 (2.32) 2 ( 2 ) log l n E = D 2 { 2 aj P (θ i, a j, b j )(1 P (θ i, a j, b j )) } 1 (2.33) σ 2 θ θ i 2 j=1 Newton-Raphson-Fisher ˆθ i [ ˆθ i ] t+1 = [ ˆθ i ] t + [ E ( 2 log l θ i 2 )] 1 t [ ] log l θ i t (2.34) BEAP: bayesian expect a posteriori g{θ i δ i, a, b} = n g(θ) P (θ i, a j, b j ) δ i,j (1 P (θ i, a j, b j )) 1 δ i,j g(θ) j=1 n P (θ i, a j, b j ) δ i,j (1 P (θ i, a j, b j )) 1 δ i,j dθ j=1 (2.35) E{θ i δ i, a, b} = ˆθ i = q X k L(X k )A(X k ) k=1 q L(X k )A(X k ) k=1 (2.36) 21
26 θ i V ar{θ i δ i, a, b} = q (X k ˆθ i ) 2 L(X k )A(X k ) k=1 q L(X k )A(X k ) k=1 (2.37) IRT IRT MCMC: malcov chain Monte Carlo method M-H Gibbs-sampler IRT M-H within Gibbs β = [a, b] 2 MCMC 1 k θ k 2 k β k 2 K 1. θ (k) p(θ δ, a, b) a θi N(θ(k 1) i, σθ 2 ), (i = 1, 2,..., N) u U(0, 1) b u α(θ (k 1) i, θi ) θ(k) i = θi 22
27 α(θ (k 1) i, θ i ) = min{r θ, 1} R θ = p(θ i δ, a, b) p(θ (k 1) i δ, a, b) p(δ θi, a, b)p(θi ) c θ (k) i p(δ θ (k 1) i, a, b)p(θ (k 1) p(δ i,j θ i, a j, b j ) = p(θ i ) = 1 2πσθ exp = θ (k 1) i 2. (a (k), b (k) ) p(a, b δ, θ (k) ) i ) N i=1 j=1 [ n P j (θ i ) δ i,j Q j (θ i ) 1 δ i,j 1 2 ( ) ] 2 θi µ θ a a j lognormal(a(k 1) j, σa), 2 b j N(b(k 1) j, σb 2 ), (j = 1, 2,..., n) u U(0, 1) b u α((a (k 1) j, b (k 1) j ), (a j, b j )) (a(k) j α((a (k 1) j, b (k 1) j ), (a j, b j )) = min{r a,b, 1} R a,b = c (a (k) j p(a j, b j δ, θ) p(a (k 1) j, b (k 1) j δ, θ) p(δ θ, a i, b i )p(a i, b i ) p(δ θ, a (k 1) i, b (k 1) i p(δ i,j θ i, a j, b j ) = p(a j ) = p(b j ) =, b (k) j N )p(a (k 1) i i=1 j=1 1 2πσa a j exp σ θ, b (k) j, b (k 1) i ) ) = (a ( ) j n P j (θ i ) δ i,j Q j (θ i ) 1 δ i,j [ [ 1 exp 1 2πσb 2 ) = (a (k 1) j ( ) ] 2 log aj µ a 1 2 σ a ( ) ] 2 bj µ b σ b, b (k 1) j ), b ( ) j ) K 23
28 Burn-In MBE BEAP MCMC MCMC i θ i j a j, b j MCMC 2.4 k = 5000 i θ i j a j b j MCMC MCMC θ : MCMC ˆθ [31] µ s.d. bias RMSE RMSE 24
29 θ θ value a density a b b iterate value 2.4: MCMC k = 5000 [31] Burn-In= 0 RMSE MCMC a : MCMC â [31] µ s.d. bias RMSE
30 2.6: MCMC ˆb [31] µ s.d. bias RMSE ! 0.80! 0.70! RMSE 0.60! 0.50! 0.40! 0.30! 0.20! 0.10!! b a 0.00! 0! 5000! 10000! 15000! 20000! 25000! 30000! iterate 2.5: MCMC RMSE [31] θ RMSE MCMC b 2.6 θ a RMSE RMSE
31 MBE BEAP MCMC MBE BEAP MCMC θ 2.6 MBE BEAP MMLE MCMC 2.6(a) 2.6(b) MBE BEAP MCMC a 2.7 MBE BEAP MMLE MCMC 2.7(a) 2.7(b) MBE BEAP MCMC b 2.8 MBE BEAP MMLE MCMC 2.8(a) 2.8(b) RMSE 2.7 θ a MCMC RMSE b MBE RMSE RMSE MCMC 27
32 3! MMLE! MCMC! 2.5! 2! 1.5! 1! 0.5! 0! 0! -0.5! 5! 10! 15! 20! 25! -1! -1.5! -2! -2.5! ID (a) 2.5! MMLE! 2! MCMC! 1.5! 1! 0.5! 0! -4! -3! -2! -1! 0! -0.5! 1! 2! 3! 4! -1! -1.5! -2! -2.5! (b) 2.6: θ [31] 28
33 4! 3.5! MMLE! MCMC! 3! a 2.5! 2! 1.5! 1! 0.5! 0! 0! 5! 10! 15! 20! 25! ID (a) 3! 2.5! MMLE! MCMC! 2! 1.5! 1! 0.5! 0! 0! 0.5! 1! 1.5! 2! 2.5! 3! 3.5! 4! (b) 2.7: a [31] 29
34 1.5! 1! MMLE! MCMC! 0.5! b 0! 0! 5! 10! 15! 20! 25! -0.5! -1! -1.5! -2! ID (a) MMLE! MCMC! 1.5! 1! 0.5! 0! -2! -1.5! -1! -0.5! 0! 0.5! 1! 1.5! -0.5! -1! -1.5! -2! (b) 2.8: b [31] 30
35 2.7: MBE BEAP MCMC RMSE [31] RMSE MCMC MBE BEAP θ a b MBE BEAP MBE BEAP 2.4 IRT 0/1 [0,1] IRT 3 IRT TOEFL IRT EM 31
36 EM 32
37 3 Web IRT IRT IRT BILOG-MG [3] Web Excel [0,1] Excel IRT 1) 2) 33
38 3.1 IRT: item response theory IRT [33 35] e-learning Moodle IRT [37, 38] IRT 1 BILOG-MG [3] IRT 2 EM 3) 0/1 IRT Web IRT [23] 3.2 Web [0,1] Excel 2 Web Excel Excel Java OS Web PC GUI GUI 34
39 Excel Response pattern ID Item 1 Item 2 Item 3 Item Drag & Drop input sheet1 Examinee s Ability BME: Bayesian Modal Estimation BEAP: Bayesian Expected a Posteriori Estimation Item Parameter MMLE/EM: Marginal Maximum Likelihood Estimation and EM algorithm MBE/EM: Marginalized Bayesian Estimation and EM algorithm computing output Excel sheet1,2,3 Item Parameter Estimates Label a (slope) b (threshold) Item Item Item Item sheet2 Examinee's Ability Estimates ID θ (Ability) sheet3 3.1: IRT Web [23] [0,1] Excel Run Excel Excel pattern 1 1 i + 1 j + 1 i j δ i,j 1 0 δ
![3.2: [23] pattern examinee item info summary](/docs-images/102/155072267/images/40-0.jpg "pattern δ examinee item info summary 3.")
40 3.2: [23] pattern examinee item info summary pattern δ examinee item info summary 3.4 item 36
41 3.3: [23]
![3.4: [23] IRT 3.](/docs-images/102/155072267/images/42-0.jpg "3 BILOG-MG IRT BILOG-MG [3] Web [40] 1 2")
42 3.4: [23] IRT 3.3 BILOG-MG IRT BILOG-MG [3] Web [40] BEAP 38
43 a BILOG-MG b BILOG-MG ID -7 ID (a) a j (b) b j θ 0-1 BILOG-MG -2-3 ID (c) θ i 3.5: 1 BILOG-MG ID BILOG-MG BILOG-MG 3.5(a) 3.5(b) 3.5(c) 39
44 a BILOG-MG b BILOG-MG ID ID (a) a j (b) b j θ BILOG-MG ID (c) θ i 3.6: 2 BILOG-MG ID BILOG-MG BILOG-MG 3.6(a) 3.6(b) 3.6(c) 0 40
45 3.1: BILOG-MG RMSE a j b j θ i RMSE a j, b j, θ i RMSE 3.1 RMSE BILOG-MG Web 3.4 IRT Web BILOG-MG IRT BILOG-MG IRT IRT Web IRT PC 0/1 Microsoft Office Excel IRT BILOG-MG 41
46 IRT IRT 42
47 II
48 4 IRT; item response theory IRT IRT [7,9,10,27] IRT [3] IRT 44
49 [23] [0,1] EXCEL IRT [13] IRT [16, 24] [11] IRT IRT 45
50 IRT id id δ item a j b j st 2nd 3rd 4th 5th 4.1: (2.2)
51 I(θ) (2.2) (2.6) [ 2 ] log L I(θ) = E θ 2 = n D 2 a 2 jp (θ i, a j, b j )Q(θ i, a j, b j ) (4.1) j=1 I j (θ) I j (θ) = D 2 a 2 jp (θ i, a j, b j )Q(θ i, a j, b j ) (4.2) P (θ i, a j, b j ) = Q(θ i, a j, b j ) = 0.5 a j (2.6) P (θ i, a j, b j ) = exp( Da j (θ i b j )) = 0.5 (4.3) θ i = b j θ i θ i = b j a j a j θ i b j P (θ i, a j, b j ) 0.5 P (θ i, a j, b j ) θ i b j
52 θi 1st item 2nd Item 3rd Item 4th Item 5th Item 4.2: [6] 48
53 ) 1 3 2) 3) 4.3: IRT 49
54 ) IRT 2) 3) 4) 4 1) 3 Web BEAP Java 2) MySQL 3) PHP 4) Web HTML HTML PHP 4.4 Web Web PHP MySQL HTML Web 50
55 4.4: Web PHP Java Web 4.5 Web 3 51
56 4.5: [24] CSV
57 10 18 i j : [30] IRT 4.6 IRT 53
58 (a) 4.7(b) PC Web PC
59 (a) (b) 4.7: [32] 55
60 4.8:
61 4.9: 57
62 4.10: [30] IRT 58
63 4.11: 5 [30] id id 4.12: 2013 [30] 59
64 5 EM IRT EM IRT [14, 15, 24] 60
65 5.1 EM IRT EM IRT j a j, b j i θ i P (θ i, a j, b j ) 2 δ EM (expectation-maximization algorithm [8]) δ 0 i,j [0, 1] δ i,j = 0, 1 0 δi,j 0 1 δ0 i,j j µ j i µ i a 0 j, b0 j, θ0 i L 0 (2) {δi,j 0 } (2) L a1 j, b1 j, θ 1 i L1 2 MCMC EM maximization (2.6) P j (θ i ) [0, 1] ˆδ i,j = P j (θ i ) P j (θ i ) δi,j 1 EM expectation 2 L k, δ k i,j, ak j, bk j, θk i (k = 0,... ) k L, δ i,j, a j, b j, θ i EM EM EM IRT limiting IRT (LIRT) [15] [25,28] [14] 5.1 EM IRT 61
66 δ (0) δ (1) 1 0 µ13 1 µ15 µ21 µ µ32 1 µ35 0 µ42 1 µ P13 1 P15 (1) (1) P21 P P33 1 P35 0 P42 1 P44 0 E[ˆδ] =P j (ˆθ i ) (1) (1) (1) (1) (1) (1) IRT a j (1), b j (1), θ i (1) (1) (1) (1) (1) (1) P11 P12 P13 P14 P15 (1) (1) (1) (1) (1) P21 P22 P23 P24 P25 (1) (1) (1) (1) (1) P31 P32 P33 P34 P35 (1) (1) (1) (1) (1) P41 P42 P43 P44 P45 IRT a j (2), b j (2), θ i (2) EM IRT a j ( ), b j ( ), θ i ( ) ( ) ( ) ( ) ( ) ( ) P11 P12 P13 P14 P15 ( ) ( ) ( ) ( ) ( ) P21 P22 P23 P24 P25 ( ) ( ) ( ) ( ) ( ) P31 P32 P33 P34 P35 ( ) ( ) ( ) ( ) ( ) P41 P42 P43 P44 P45 5.1: EM IRT S k ˆδ i,j k 2 S k = 1 (ˆδ i,j k δ i,j) 2 (5.1) (i,j) S k S k S k 1 < δ [0, 1] 62
67 * * 0 * * * * 1 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 * * 0 0 * * * 0 1 * * * * * * * * * * * * * * * * * * * 0 1 * * 1 1 * * * 0 * * * * * * * * * * * * * 1 * 1 * 0 * 0 * * * * * * * * 0 * * * * * * * * * * * * * * 1 0 * * * * 1 * * * * * * * 0 1 * * * * * * * * * * * * * IRT a j, b j, θ i EM IRT a j, b j, θ i EM IRT a j, b j, θ i new user175 EM IRT a j, b j, θ i adaptive test 5.2: EM IRT EM IRT EM IRT 5.2 EM IRT 63
68 受験者 id 正答 誤答 問題 id 5.3: [30] 5.2 EM IRT EM IRT [36] IRT IRT 64
69 問題 id 正答 誤答 ****** * ************** ** ****** * ********** * * **** *** ******* * *************** * ******** **** ********** * * **** * * ****** * * * ***** **** ** ***** ******** ***** * ************** ******* ******* **** **** **** * * * ***** * * * * ***** **** * * * * ** ** *** * *** * * * **** 受験者 id 未回答 予備テストオンライン適応型テスト 5.4:
70 id i id j 5.5: 2013 [30] EM IRT IRT θ i (i = 1,..., N) a j, b j (j = 1,..., n) p i,j 2. θ i, a j, b j 2009 A m (m = 1, 2,..., M) 3. M a A m Training K Test k b Training EM IRT ˆθ i â j, ˆb j Test t i,j c ˆθ i, â j, ˆb j, t i,j bias mse 66
71 bias = 1 n x (ˆx x), mse = 1 n x (ˆx x) 2 (x, ˆx, n x ) {(p i,j, t i,j, k), (θ i, ˆθ i, N), (a j, â j, n), (b j, ˆb j, n)} [39] M = 10 K + k = N n = = 3480 K = % K = % bias mse [0, 1] K = 2784 bias mse 5.1 random 5.1 EM IRT bias mse K = 696 bias mse 5.2 K = 2784 EM IRT bias mse bias mse 5.1: bias mse K = 2784 [30] EM IRT random m bias mse bias mse
72 5.2: bias mse K = 696 [30] EM IRT random m bias mse bias mse bias mse K = 2784 θ i,a j,b j bias mse 5.3 bias mse 5.3: θ i,a j,b j bias mse K = 2784 [30] θ a b m bias mse bias mse bias mse
73 K = 696 θ i,a j,b j bias mse 5.4 K = 2784 mse 5.4: θ i,a j,b j bias mse K = 696 [30] θ a b m bias mse bias mse bias mse bias mse EM IRT [24] EM IRT [0,1] 0 1 EM IRT
![0 5.6: 2011 EM IRT [24] start 1st](/docs-images/102/155072267/images/74-2.jpg "2nd 3rd 4th 10th 30th 536th 5.")
74 solved failed incomplete matrix problem id j EM IRT problem id j student id i student id i not tackled estimated complete matrix : 2011 EM IRT [24] start 1st 2nd 3rd 4th 10th 30th 536th 5.7: EM IRT
75 5.8: EM IRT [24] 0 1 EM IRT EM IRT 5.8 EM IRT 71
76 5.9: EM IRT [24] EM IRT 5.10 EM IRT 5 EM IRT EM IRT 72
77 5.10: EM IRT [24] EM IRT RMSE 5.11 RMSE 3 EM IRT 5.11 µ j log L 5.11 RMSE EM IRT 73
![5.11: EM IRT [14] EM IRT 5.](/docs-images/102/155072267/images/78-0.jpg "8 5.9 175 EM IRT EM IRT 5.")
78 5.11: EM IRT [14] EM IRT EM IRT EM IRT
79 5.12: EM IRT a j b j [14]
80 5.13: EM IRT sd(a j ) sd(b j ) [14] EM IRT
![5.14: EM IRT θ i [14]](/docs-images/102/155072267/images/81-0.jpg "5.15 5.")
81 5.14: EM IRT θ i [14] EM IRT IRT 2 77
82 5.15: EM IRT sd(θ i [14] ) 78
83 6 EM IRT 1 MD; matrix decomposition method [12, 26] EM IRT MD EM IRT MD [26] matrix decomposition method; MD [2, 19, 20] 79
84 V 2 U M P = U T M U, M 2 [26] 2 2 f(u, V ) = 1 2 m i=1 j=1 + k u 2 n I(i, j) {V (i, j) P (i, j)} 2 m i=1 U i 2 + k m 2 n M j 2 (6.1) j=1 I(i, j) k u k m f U i = f M i = n I(i, j) {V (i, j) P (i, j)} M j + k u U i (6.2) j=1 m I(i, j) {V (i, j) P (i, j)} U i + k m M j (6.3) i=1 U M µ U (t+1) i M (t+1) j U (t) i M (t) j µ f U i (6.4) µ f M j (6.5) Training Test Training Test (5.1) RMSE 6.1 Training Test Training Test T 80
85 2 MCMC maximization (1) P j (θ i ) [0, 1] ˆδ i,j = P j (θ i ) P j (θ i ) δ 1 i,j expectation L k, δi,j k, ak j,bk j, θk i Training (k =0,...) k 1 1 L, δi,j, a j, b j, 1 θ i limiting P12 P13 1 IRT P15 (LIRT) [11] 1 0 P21 P P25 [19], [22] 1 1 [10] 1 1 P33 P34 P RMSE P41 P42 1 P44 0 (root mean squared error) S k Test ˆδ i,j k 0 RMSE S k 1 S k = 1 S k = 1 (ˆδ i,j k (ˆδ i,j k 1 δ i,j) 2, δ i,j) 2, (3) (i,j) (i,j) 0 c 2012 Information Processing Society of Japan 6.1: Training Test [30] c 2012 Information Processing Society of Japan EM IRT MD % 80% 2 Test Training Training 81
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87 6.1: RMSE [30] EM IRT MD Training Test Training Test Training 80% Test 20% Training 20% Test 80% Test RMSE EM IRT RMSE EM IRT MD Training Test Training Test A 2013B 6.2 RMSE 6.2 Test RMSE B2 EM IRT 2013B EM IRT 2013A 2013B Test RMSE MD EM IRT Test RMSE 2013A 2013B 95.1% 85.7% EM IRT 83
88 6.2: :1 Training Test 30 RMSE [30] EM IRT MD Training Test Training Test 2013A B EM IRT RMSE Training 6.3(a) 2013A 6.3(b) 2013B 30 Training RMSE Training log L 2013A 6.4(a) 2013B 6.4(b) log L 30 EM IRT RMSE log L [14] 84
89 6.3: MD RMSE [30] MD MD 2013A B MD EM IRT MD RMSE EM IRT T 0 T 1 MD 1 0 MD 0 T 1 T / / x i,j ˆx i,j (x i,j = 0) (0 ˆx i,j < 0.5) T = 0 (x i,j = 0) (0.5 ˆx i,j 1) T = 1 (x i,j = 1) (0 ˆx i,j < 0.5) T = 1 85
90 6.4: Test [30] EM IRT MD 2013A B (x i,j = 1) (0.5 ˆx i,j 1) T = 0 T 2 T = 1 T = 0 30 Test i,j T /#T 6.4 MD EM IRT 0/1 MD 6.5 IRT IRT IRT EM IRT 1 MD; matrix decomposition method [12, 26] EM IRT 86
91 EM IRT MD EM IRT EM IRT 87
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93 -240 logl iterate (a) 2013A logl iterate (b) 2013B 6.4: 9:1 Training Test 30 EM IRT Training log L [30] 89
94 7 I IRT IRT Web II IRT 2 IRT [0,1] IRT IRT TOEFL IRT i j P (θ i φ j ) (i j) 2 θ i φ j i θ i 90
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100 journal/ html. [21] J. Neyman and E.L. Scott. Consistent estimates based on partially consistent observations. Econometrica: Journal of the Econometric Society, Vol. 16, No. 1, pp. 1 32, [22] R.J. Patz and B.W. Junker. A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, Vol. 24, No. 2, p. 146, [23] T. Sakumura and H. Hirose. Test evaluation system via the web using the item response theory. Information, Vol. 13, No. 3, pp , May [24] T. Sakumura, T. Kuwahata, and H. Hirose. An adaptive online ability evaluation system using the item response theory. In in Education and e-learning (EeL2011)., pp Global Science and Technology Forum (GSTF), [25] H.K. Suen and P.S.C. Lee. Constraint optimization: An alternative perspective of IRT parameter estimation, chapter 17, pp Norwood, NJ: Ablex., [26] S. Takimoto and H. Hirose. Recommendation systems and their preference prediction algorithms in a large-scale database. Information, Vol. 12, No. 5, pp , [27] W.J. van der Linden and R.K. Hambleton. Handbook of modern item response theory. Springer, [28] W.M. Yen, G.R. Burket, and R.C. Sykes. Nonunique solutions to the likelihood equation for the three-parameter logistic model. Psychometrika, Vol. 56, No. 1, pp , [29].., [30],,. EM IRT.., Vol. 7, No. 2, pp , Nov [31],.. 96
101 OR 2010, Oct [32],. IRT adaptive online system. 27, pp , Nov [33],,. IRT. 35, pp , September [34],,. IRT. 2007, p. 280, September [35],,. IRT. 2008, p. 106, Sep [36],,. IRT e-learning. PC Conference, pp , Aug [37],,. IRT e-learning :. 60, Vol. 10-2A-06, p. 371, September [38],,. IRT e-learning :. 2008, Vol. D-15-43, p. 237, March [39],,. e-learning. CIEC, Vol. 24, pp , Jun [40].., [41]. 97
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