Public defence in Mathematics, M.Sc. Muhammad Ardiyansyah

Title of the doctoral thesis: Algebraic Aspects of Hidden Variable Models

Doctoral student: Muhammad Ardiyansyah
Opponent: Prof. Marta Casanellas Rius, Universitat Politècnica de Catalunya, Spain
Custos: Assistant Professor Kaie Kubjas, Aalto University School of Science, Department of Mathematics and Systems Analysis

Thesis available for public display 10 days prior to the defence at: 
https://aaltodoc.aalto.fi/doc_public/eonly/riiputus/

Statistical Modeling with Hidden Variables

Statistical models have been used to study real phenomena that are rather complex. If we want to build reasonably nice statistical models that could explain the complicated feature of these real phenomena, then we often need to gather a big amount of data and this process could be time-consuming, expensive, or just infeasible. We can overcome these limitations by incorporating hidden random variables corresponding to the unmeasurable variables into our models. In this thesis, we focus on two important examples of hidden variable models: phylogenetic model and factor analysis model. 

Phylogenetics is a field in biology that studies the evolutionary relationships between biological entities. From biological data such as DNA sequences, we can extract some useful information that can be used to infer evolutionary relationships that are presented in terms of networks. Hidden variables in phylogenetic models correspond to ancestors whose DNA samples are unmeasurable. 

In phylogenetics, we focus our study to the embedding and the network distinguishability problems. Roughly speaking, the embedding problem asks how restrictive it is to model the evolutionary processes which regard time as a continuous variable and assume that substitution events always occur at the same rate. On the other hand, the goal of studying the network distinguishability problems is to distinguish the set of probability distributions arising from two phylogenetic network models. 

For the embedding problem, we provide criteria of embeddability of Markov matrices belonging in some phylogenetic models that include the most common models studied in the literature. These results enable us to approximate how large the set of embeddable Markov matrices is inside the set of Markov matrices within the models. Moreover, we provide some conditions under which we can distinguish certain phylogenetic network models. 

Factor analysis models seek to reduce the number of observable variables that is often quite large in terms of fewer number of underlying hidden factors which are thought to better explain the covariance between the observable variables. Dropping the Gaussianity assumption of the classical factor analysis model, we introduce the higher order factor analysis model which takes into account higher order moment or cumulant tensors. Additionally, we compute the dimension of this model which can be used to measure the complexity of the model.

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Doctoral theses in the School of Science: https://aaltodoc.aalto.fi/handle/123456789/52