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

Contents1. Grbner bases2. Designs and interpolation3. Generic designs4. The average degree vector5. The state polytope 6. Linear aberrations7. Hilbert series8. Current work

2. Designs and interpolation

ABCCABBCA

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]

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]

3. Generic designs

4. The state polytope

5. Linear aberrations

6. Hilbert series

8. Current work

ReferencesBasic book on G-bases: Cox, Little, OSheaMonograph: Algebraic Statistics: Pistone, Riccomagno and WBook by Sturmfels: Algebraic StatisticsPhD Thesis: Fabio Rapallo, Torino (supervisor P)Assorted notes PRW and RapalloLecture notes by Serken HoskenWork by Onn and SturmfelsKey papers: Sturmfels and co-authors, particularly on graphical models: edition of J Symb. CompPhD: Hugo Maruri-Aguilar