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The Algebraic Methods in the Design of Experiments

DEMA2006 . The Algebraic Methods in the Design of Experiments. Henry Wynn (Joint with Eva Riccomagno, Shmuel Onn, Hugo Maruri-Aguilar, Yael Berstein). Contents. 1. Gr öbner bases 2. Designs and interpolation 3. Generic designs 4. The average degree vector 5. The state polytope

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The Algebraic Methods in the Design of Experiments

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  1. DEMA2006 The Algebraic Methods in the Design of Experiments Henry Wynn (Joint with Eva Riccomagno, Shmuel Onn, Hugo Maruri-Aguilar, Yael Berstein)

  2. Contents 1. Gröbner bases 2. Designs and interpolation 3. Generic designs 4. The average degree vector 5. The state polytope 6. Linear “aberrations” 7. Hilbert series 8. Current work

  3. 2. Designs and interpolation

  4. Use Q[a,b,c,r,s,t,x,y,z]; Points:= [[1,0,0,1,0,0,1,0,0],[1,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,1], [0,1,0,1,0,0,0,0,1],[0,1,0,0,1,0,1,0,0],[0,1,0,0,0,1,0,1,0], [0,0,1,1,0,0,0,1,0],[0,0,1,0,1,0,0,0,1],[0,0,1,0,0,1,1,0,0]]; Ideal(x + y + z - 1, r + s + t - 1, a + b + c - 1, z^2 - z, yz, cz - sz, bz + sz + tz - z, y^2 - y, ty - sz - 1/3b + 1/3c + 1/3s - 1/3t - 1/3y + 1/3z, sy + sz + tz + 2/3b + 1/3c - 2/3s - 1/3t - 1/3y - 2/3z, cy - tz - 1/3b - 2/3c + 1/3s + 2/3t - 1/3y + 1/3z, by - sz - 1/3b + 1/3c + 1/3s - 1/3t - 1/3y + 1/3z, t^2 - t, st, ct + sz + tz + 1/3b - 1/3c - 1/3s - 2/3t + 1/3y - 1/3z, bt - sz - 1/3b + 1/3c + 1/3s - 1/3t - 1/3y + 1/3z, s^2 - s, cs - sz, bs - tz - 2/3b - 1/3c - 1/3s + 1/3t + 1/3y + 2/3z, c^2 - c, bc, b^2 - b) Leading terms [x, r, a, z^2, yz, cz, bz, y^2, ty, sy, cy, by, t^2, st, ct, bt, s^2, cs, bs, c^2, bc, b^2] Basis [1, b, c, s, t, y, z, sz, tz]

  5. Ideal(x^2y + 1/3y^3 - 4/3y, x^3 + 3xy^2 - 4x, xy^3 - xy, y^5 - 5y^3 + 4y) Leading terms: [x^2y, x^3, xy^3, y^5] Basis: [1, x, y, x^2, xy, y^2, xy^2, y^3, y^4]

  6. 3. Generic designs

  7. 4. The state polytope

  8. 5. Linear aberrations

  9. 6. Hilbert series

  10. 8. Current work

  11. References • Basic book on G-bases: Cox, Little, O’Shea • Monograph: “Algebraic Statistics”: Pistone, Riccomagno and W • Book by Sturmfels: Algebraic Statistics… • PhD Thesis: Fabio Rapallo, Torino (supervisor P) • Assorted notes PRW and Rapallo • Lecture notes by Serken Hosken • Work by Onn and Sturmfels • Key papers: Sturmfels and co-authors, particularly on graphical models: edition of J Symb. Comp • PhD: Hugo Maruri-Aguilar

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