Toric Homogeneous Markov Chains
This page contains supplimentary material for the paper: "Degree bounds for a minimal Markov basis for the three-state toric homogeneous Markov chain model", David Haws, Abraham Martin Del Campo, Ruriko Yoshida. Submitted to the Proceedings of the Second CREST–SBM International Conference “Harmony of Grobner Bases and the Modern Industrial Society.” Available at http://arxiv.org/abs/1108.0481 , 2011.
Quick Background
We study the three state toric homogeneous Markov chain model and three special cases of it, namely: (i) when the initial state parameters are constant, (ii) without self-loops, and (iii) when both cases are satisfied at the same time. Using as a key tool a directed multigraph associated to the model, the state- graph, we give a bound on the number of vertices of the polytope associated to the model which does not depend on the time. Based on our computations, we also conjecture the stabilization of the f-vector of the polytope, analyze the normality of the semigroup, give conjectural bounds on the degree of the Markov bases.
Thus we study four toric statistical models given by four types of matrices (see pdf above). The software https://github.com/dchaws/GenWordsTrans can be used to generate the matrices defining the four models we study. We focus mainly on the loopless case.
Loopless No Initial State
State and Move graphs of the Markov Bases above, sorted by orbits. Drawn with my software https://github.com/dchaws/DrawStateMoveGraphs.
Using my software to generate design matrics and Normaliz, I have confirmed for S=3 and T <= 170 that the design matrices is normal.
Polymake files containing he design matrix, the vertices, facets, and f-vector.
Using Polymake and my software to draw state graphs, I drew all the vertices, non-vertices and non-vertices with no two-cycles. These drawings lead me to the proof that the number of vertices are bounded by a constant as T grows.
Loopless With Initial State
State and Move graphs of the Markov Bases above, sorted by orbits. Drawn with my software https://github.com/dchaws/DrawStateMoveGraphs.