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Advancements in Algebraic Statistics for Statistical Computing: Current Research Areas

Discover the latest research areas in algebraic statistics, from Gröbner bases to sufficient statistics in statistical computing. Explore designs, probability models, toric varieties, and more in this insightful collection.

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Advancements in Algebraic Statistics for Statistical Computing: Current Research Areas

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  1. Advances in Algebraic Statistics General Applications and Statistical Computing Sections 8 June 2005 Henry Wynn

  2. Contents 1. Gröbner bases, varieties, ideals 2. Designs and supports 3. Probability models 5. Toric varieties and saturation 6. Graphical models 7. Moments and cumulants 8. Sufficient statistics and Maximum likelihood 9. Markov bases and simulation 10. Live research areas

  3. 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) [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] [1, b, c, s, t, y, z, sz, tz]

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

  5. gg := [-p3^8*p5+p2^3*p4^6, -p2^3*p5+p1*p4^3, p1*p3^8-p2^6*p4^3, v*p5^4*p3^17-p2^2*p4^14,-p4^8+p2*v*p5^3*p3^9, -p4^2+p2^4*v*p5^2*p3, v*p2^7*p4^4*p5-p3^7, -p1*p3^7+v*p2^10*p5^2*p4, p2^13*v*p5^3-p1^2*p4^2*p3^7, v*p1*p2*p3*p4*p5-1]

  6. Live research areas • More on design: corner cut, inverse problems • Complex probability models: large area • Boundary exponential models, MLE etc • Kernel v Markov v Gröbner bases • Secant varieties and hidden Markov • Design/Probability link: structural zeros etc

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