Perturbation method for determining the group of invariance of hierarchical models

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Transcription:

Perturbation method for determining the group of invariance of hierarchical models 1 2 1 1 2 2009/11/27 ( ) 2009/11/27 1 / 31

2 3 p 11 p 12 p 13 p 21 p 22 p 23 (p ij 0, i;j p ij = 1). p ij = a i b j log p ij = α i + β j. (log p ij ) (α i ) (β j ) (1, j) (2, j) ( ) 2009/11/27 2 / 31

( ) 2009/11/27 3 / 31

m (p i1 i m ) R I 1ˆ ˆI m. (log p i1 i m ) L, L L = {(α i + β j )}. S I1ˆ ˆI m. Problem (Aoki & Takemura 2008, J. Symbolic Comput.) L G S I1ˆ ˆI m Sei, T., Aoki, S. and Takemura, A. (2009). Perturbation method for determining the group of invariance of hierarchical models, Advances in Applied Mathematics, 43 (4), 375 389. ( ) 2009/11/27 4 / 31

1 2 (setwise stabilizer) 3 4 5 6 ( ) 2009/11/27 5 / 31

I = m j=1 I j, I j = {1,..., I j }. (p i ) i2i I p i = K k=1 qa k(i) k ( (log p i ) span(a k )) F(t) = {x i A k(i)x i = t k } Remark: K[p] p x i i ( P ) i q A k(i)x i k K[q]. i k (Diaconis & Strumfels 1998). ( ) 2009/11/27 6 / 31

2 = x i+ x +j F({r i }, {c j }) = {{x ij } x i+ = r i, x +j = c j }. move: 2 F({r i }, {c j }) move j j M(i, i 0, j, j 0 ) = i +1 1 i 1 +1 {M(i, i 0, j, j 0 )} 1 M(1, 2, 1, 2) ( ) 2009/11/27 7 / 31

{{1, 2}, {2, 3}} log p ijk = α ij + β jk M = (z ijk ) j 0 (z ijk ) j=j0 = k i 1 1 1 1, OK. (Dobra 2003 ) (Aoki & Takemura 2003 ) ( ) 2009/11/27 8 / 31

3 log p ijk = α ij + β jk + γ ik. Table: 3 3 K (Aoki & Takemura 2003) K 3 4 5 6 7 # unique minimal MB 81 450 2670 10665 31815 # reduced GB 110 622 3240 12085 34790 # orbits in the MB 4 5 6 6 6 6 ( ) 2009/11/27 9 / 31

(setwise stabilizer) I = m j=1 I j, I j = {1,..., I j }. (log p i ) i2i L. S I S I = {g : I I one to one}. g S I R I (gx) i = x g`1 (i). g(hx) = (gh)x. L g gl := {gx x L}. L (setwise stabilizer) G L := {g S I gl = L}. ( ) 2009/11/27 10 / 31

(setwise stabilizer) G L = {g S I gl = L} L Problem (Aoki & Takemura 2008 ) L G L Remark G L = G L? L? = {x i x iy i = 0, y L} L? = ker A L move ker A G L ( ) 2009/11/27 11 / 31

(setwise stabilizer) F = {{1},..., {m}} (Aoki & Takemura, JSC2008) ( ) G L = S I1 S Im, (τ jj 0) Ij =I 0 j τ jj 0 : (, i j,, i j 0, ) (, i j 0,, i j, ) 1 3 F = {{1, 2}, {1, 3}, {2, 3}} 2 Hardy-Weinberg / ( ) 2009/11/27 12 / 31

(setwise stabilizer) 2 log p ij = α i + β j. I 1 = I 2 G r( ) = S I1 S I2, ( ) 2009/11/27 13 / 31

Lauritzen (1996). m: m I j = {1,..., I j }: j (j = 1,..., m) I = m j=1 I j: : [m] = {1,..., m} F := red( ): m = 3, = {, {1}, {2}, {3}, {1, 2}, {2, 3}} F = {{1, 2}, {2, 3}}. 1 2 3 ( ) 2009/11/27 14 / 31

Definition F (log p i ) L F := D2F L D, L D := {i D }. m = 3, F = {{1, 2}, {2, 3}} log p ijk = α ij + β jk. 1 2 3 G F := G LF = {g S I gl F = L F }. ( ) 2009/11/27 15 / 31

1 wreath product G F. (S I1 ) I 2 = {g : I 2 S I1 } = I 2 I 1 (S I1 ) I 2 S I2 wreath product P j2p (S I j ) A(j) generalized wreath product A(ρ) ρ P = {1, 2, 3}, 1 < 2, 3 < 2 (S I1 ) I 2 S I2 (S I3 ) I 2. log p i1 i 2 i 3 = α i1 i 2 + β i2 i 3 ( ) 2009/11/27 16 / 31

2 pseudofactor poset F P. 1 3 5 {3} {4} 2 4 6 {1} {2} {5,6} i j def ( D F (i D j D)). P := [m]/, pseudofactor ρ, ρ 0 P ρ ρ 0 def ( D F (ρ D ρ 0 D)). ( ) 2009/11/27 17 / 31

2 pseudofactor poset Remark. 1 pseudofactor poset P intersection poset Q F P Q φ 1 1 2 3 2 3 {1} {2} {3} {1,2} {1,3} {2,3} P Q 2 pseudofactor poset poset Problem ( ) poset pseudofactor poset ( ) 2009/11/27 18 / 31

Theorem ( ) F G F. G F = (S Ij ) I A(j). j2p ρ pseudofactor poset, A(ρ) ρ I D := j2d I j (D F). j I j 3. ( ) 2009/11/27 19 / 31

1 m = 3, F = {{1, 2}, {2, 3}}. {2} 1 2 3 {1} {3} G F = (S I1 ) I 2 S I2 (S I3 ) I 2. ( ) 2009/11/27 20 / 31

2. m = 5, F = {{1, 3}, {2, 4}, {3, 4, 5}} 1 3 5 {3} {4} 4 2 {1} {2} {5} G F = (S I1 ) I 3 (S I2 ) I 4 S I3 S I4 (S I5 ) I f3;4g ( ) 2009/11/27 21 / 31

2 ( ) m = 5, F = {{1, 3}, {2, 4}, {3, 4, 5}}. G F = (S I1 ) I f3g (S I2 ) I f4g S I3 S I4 (S I5 ) I 3;4. 2 move (indispensable) M 1 = (11111)(12211) (12111)(11211), M 2 = (11112)(12211) (12112)(11211). M 1 M 2, (i 3 i 4 ) = (11) i 5 = 1, 2 M 1 = (11111)(12211) (12111)(11211), M 2 = (11112)(12211) (12112)(11211). ( ) 2009/11/27 22 / 31

3,. m = 6, F = {{1, 4, 5}, {2, 5, 6}, {3, 4, 6}}. ({4, 5, 6} / F). 1 {4} {5} {6} 5 4 2 3 6 {1} {2} {3} G F = (S I1 ) I f4;5g (S I2 ) I f5;6g (S I3 ) I f4;6g S I4 S I5 S I6 ( ) 2009/11/27 23 / 31

9 9 3 3 1 9 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 1 2 3 7 8 9 1 2 3 4 5 6 2 3 1 5 6 4 8 9 7 5 6 4 8 9 7 2 3 1 8 9 7 2 3 1 5 6 4 3 1 2 6 4 5 9 7 8 6 4 5 9 7 8 3 1 2 9 7 8 3 1 2 6 4 5 (Russell & Jarvis 2006) 3. 3. 3. 3.. ( ) 2009/11/27 24 / 31

3 4 9 (x ijklc ) (Lawrence lifting). x ij++c = x i+k+c = x ++klc = x ijkl+ = 1.. F = {{1, 2, 5}, {1, 3, 5}, {3, 4, 5}, {1, 2, 3, 4}}. 2 1 3 5 4, S I1 (S I2 ) I 1 S I3 (S I4 ) I 3 S I5.. ( ) 2009/11/27 25 / 31

Theorem ( ) G F = j2p (S I j ) I A(j). 1 (Bailey et al. 1983 ) G fdg = (S Ij ) I A(j) D2F j2p 2 G F D2F G fdg 3 D 0 = argmin D2F I D G F D2FnfD 0 g G fdg 4 G F G fd0 g 3, 4 perturbation method ( ) 2009/11/27 26 / 31

( 1/3) I 1 I 2 F = {{1}, {2}} G F = S I1 S I2 I 1 < I 2 I 1 = 2, I 2 = 4 perturbation x = 1 0 0 0 1 0 0 0 g G F gx = 1 0 0 0 1 0 0 0, 0 1 0 0 0 1 0 0,, 0 0 0 1 0 0 0 1 ( ) 2009/11/27 27 / 31

( 2/3) gx 1 1 gx L = {(α i + β j )} gx = 1 0. (1) gx = α 1 + β 1 α 1 + β 2 α 1 + β 3 α 1 + β 4 α 2 + β 1 α 2 + β 2 α 2 + β 3 α 2 + β 4 1 2 (gx) 1 (gx) 2 = α 1 α 2 α 1 α 2 α 1 α 2 α 1 α 2 (2) (1), (2) gx 1 1 ( ) 2009/11/27 28 / 31

( 3/3) g G F g : 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0,, 0 0 0 1 0 0 0 1 g G f2g g G f1g g N := L F L > f2g g 1 1 1 1 1 1 1 1 N perturbation g G f1g ( ) 2009/11/27 29 / 31

,. e.g. structural zero, HSM,... pseudofactor poset. ( ) 2009/11/27 30 / 31

Bibliography S. Aoki & A. Takemura (2003). Minimal basis for a connected Markov chain over 3 ˆ 3 ˆ K contingency tables with fixed two-dimensional marginals. Aust. N. Z. J. Stardust., 45 (2), 229 249. S. Aoki & A. Takemura (2008). The largest group of invariance for Markov bases and toric ideals. J. Symbolic Com-put., 43 (5), 342 358. R. A. Bailey, C. E. Praeger, C. A. Rowley & T. P. Speed (1983). Generalized wreath products of permutation groups. Proc. London Math. Soc., 47 (3), 69 82. D. A. Cox. (2007). Gröbner basis tutorial. Part II. A sampler from recent developments. (available from Cox s homepage). A. Dobra (2003). Markov bases for decomposable graphical models, Bernoulli, 9 (6), 1093 1108. R. Fontana & M.-P. Rogantin (2008). Indicator function and sudoku designs, in press. E. Russel & F. Jarvis (2006), Mathematics of Sudoku II, Preprint. T. Sei, S. Aoki & A. Takemura (2008). Perturbation method for determining the group of invariance of hierarchical models, Advances in Applied Mathematics, 43 (4), 375 389. (arxiv:0808.2725v2) ( ) 2009/11/27 31 / 31

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