Algebraic Statistics

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Volume Title
School of Science | Doctoral thesis (article-based) | Defence date: 2013-04-15
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Date
2013
Department
Matematiikan ja systeemianalyysin laitos
Department of Mathematics and Systems Analysis
Major/Subject
Mcode
Degree programme
Language
en
Pages
34 + app. 106
Series
Aalto University publication series DOCTORAL DISSERTATIONS, 38/2013
Abstract
This thesis on algebraic statistics contains five papers. In paper I we define ideals of graph homomorphisms. These ideals generalize many of the toric ideals defined in terms of graphs that are important in algebraic statistics and commutative algebra.   In paper II we study polytopes from subgraph statistics. Polytopes from subgraph statistics are important for statistical models for large graphs and many problems in extremal graph theory can be stated in terms of them. We find easily described semi-algebraic sets that are contained in these polytopes, and using them we compute dimensions and get volume bounds for the polytopes.  In paper III we study the topological Tverberg theorem and its generalizations. We develop a toolbox for complexes from graphs using vertex decomposability to bound the connectivity.  In paper IV we prove a conjecture by Haws, Martin del Campo, Takemura and Yoshida. It states that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means that a certain class of toric ideals are generated by quadratic binomials.  In paper V we produce cellular resolutions for a large class of edge ideals and their powers. Using algebraic discrete Morse theory it is then possible to make many of these resolutions minimal, for example explicit minimal resolutions for powers of edge ideals of paths are constructed this way.

Denna avhandling om algebraisk statistik innehåller fem artiklar. I artikel I definieras ideal av grafhomomorfier. Dessa ideal generaliserar ett flertal konstruktioner av ideal från grafer som är viktiga i algebraisk statistik samt kommutativ algebra. I artikel II behandlas polytoper från delgrafsstatistik. Dessa är viktiga för att förstå statistiska modeller som beskriver stora grafer och många problem om ytterlighetsgrafer kan formuleras med dem. Bland verktygen som används är att beskriva semi-algebraiska mängder i polytoperna och genom detta bestämma deras dimension samt begränsa volymen. I artikel III behandlas den topologiska tverbergssatsen med generaliseringar. Grafkomplexen förstås genom att begränsa sammanhängandegraden medelst hörnnedbrytbarhet. I artikel IV bevisas att ideal tillhörande markovkedjor med tre tillstånd är genererade i grad två, vilket förmodats av Haws, Martin del Campo, Takemura och Yoshida. I artikel V skapas cellulära upplösningar för en stor klass av kantideal samt deras potenser. Med algebraisk diskret morseteori görs dessa upplösningar minimala för kantideal från stigar. 
Description
Supervising professor
Engström, Alexander, Prof., Aalto University, Finland
Thesis advisor
Engström, Alexander, Prof., Aalto University, Finland
Keywords
algebra, statistics
Other note
Parts
  • [Publication 1]: Alexander Engstrom and Patrik Noren. Ideals of graph homomorphisms. Annals of Combinatorics, 17, 2013.
  • [Publication 2]: Alexander Engstrom and Patrik Noren. Polytopes from subgraph statistics. arxiv:1011.3552, 2010.
  • [Publication 3]: Alexander Engstrom and Patrik Noren. Tverberg’s theorem and graph coloring. arxiv:1105.1455, 2011.
  • [Publication 4]: Patrik Noren. The three-state torics homogenous Markov chain model has Markov degree two. arxiv:1207.0077, 2012.
  • [Publication 5]: Alexander Engstrom and Patrik Noren. Cellular resolutions of powers of monomial ideals. arxiv:1212.2146, 2012.
Citation