1 / 24

The Algebraic Methods in the Design of Experiments

This book by Henry Wynn covers Gröbner bases, state polytopes, linear aberrations, and Hilbert series in experimental design. It explores algebraic methods using Q[a,b,c,r,s,t,x,y,z], ideal equations, leading terms, and references.

sileas
Télécharger la présentation

The Algebraic Methods in the Design of Experiments

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related