In statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to... Show moreIn statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to perform exact testing for the p1 random graph model. By comparing it to the simple switch Markov chain, we prove that it mixes rapidly on many classes of degree sequences, and we discuss why it is sometimes better suited than the simple switch chain, and try to easily introduce the concepts from the general theory along the way. M.S. in Applied Mathematics, May 2016 Show less

The generic identification problem is to decide whether a stochastic process (X_t) is a hidden Markov process and if yes to infer its... Show moreThe generic identification problem is to decide whether a stochastic process (X_t) is a hidden Markov process and if yes to infer its parameters for all but a subset of parametrizations that form a lower-dimensional subvariety in parameter space. Partial answers so far available depend on extra assumptions on the processes, which are usually centered around stationarity. Here we present a general solution for binary-valued hidden Markov processes. Our approach is rooted in algebraic statistics hence it is geometric in nature. We find that the algebraic varieties associated with the probability distributions of binary-valued hidden Markov processes are zero sets of determinantal equations which draws a connection to well-studied objects from algebra. As a consequence, our solution allows for algorithmic implementation based on elementary (linear) algebraic routines. Show less

Methods from Commutative Algebra and Numerical Analysis are combined to address a problem common to many disciplines: the estimation of the... Show moreMethods from Commutative Algebra and Numerical Analysis are combined to address a problem common to many disciplines: the estimation of the expected value of a polynomial of a random vector using a linear combination of a finite number of its values. In this work we remark on the error estimation in cubature formulæ for polynomial functions and introduce the notion of a precision space for a cubature rule. Show less

In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model... Show moreIn this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the use of tehse models in different fields of applications. Show less

We consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is... Show moreWe consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is bounded from above by the Markov complexity of the base configuration. As important examples of base configurations we consider incidence matrices of graphs and study the maximum Markov degree of configurations defined by fibers of the incidence matrices. In particular we give a proof that the Markov degree for two-way transportation polytopes is three. Show less