dc.contributor.author Banos, Hector dc.date.accessioned 2016-06-13T20:19:15Z dc.date.available 2016-06-13T20:19:15Z dc.date.issued 2016-05 dc.identifier.uri http://hdl.handle.net/11122/6600 dc.description Thesis (M.S.) University of Alaska Fairbanks, 2016 en_US dc.description.abstract Latent tree models are tree structured graphical models where some random variables are observable while others are latent. These models are used to model data in many areas, such as bioinformatics, phylogenetics, computer vision among others. This work contains some background on latent tree models and algebraic geometry with the goal of estimating the volume of the latent tree model known as the 3-leaf model M₂ (where the root is a hidden variable with 2 states, and is the parent of three observable variables with 2 states) in the probability simplex Δ₇, and to estimate the volume of the latent tree model known as the 3-leaf model M₃ (where the root is a hidden variable with 3 states, and is the parent of two observable variables with 3 states and one observable variable with 2 states) in the probability simplex Δ₁₇. For the model M₃, we estimate that the rough percentage of distributions that arise from stochastic parameters is 0:015%, the rough percentage of distributions that arise from real parameters is 64:742% and the rough percentage of distributions that arise from complex parameters is 35:206%. We will also discuss the algebraic boundary of these models and we observe the behavior of the estimates of the Expectation Maximization algorithm (EM algorithm), an iterative method typically used to try to find a maximum likelihood estimator. en_US dc.description.tableofcontents Chapter 1: Introduction -- 1.1 Chapter Overview -- Chapter 2: Basic Concepts -- 2.1 A statistical model as a geometric object -- 2.2 Varieties -- 2.2.1 Zariski closure -- 2.2.2 Semialgebraic sets -- 2.3 Tensors -- 2.4 Latent tree models -- 2.4.1 Basic graph theory -- 2.4.2 Latent tree model -- Chapter 3: The 3-leaf model -- 3.0.3 Parameter identiability -- 3.1 Volume of the model in the probability simplex -- 3.1.1 The volume of M₂ in Δ₇ -- 3.1.2 The volume of M₃ in Δ₁₇ -- Chapter 4: The algebraic boundary of M -- 4.1 Algebraic boundary -- 4.2 Boundary strata of M₂ -- 4.3 Boundary strata of M₃ -- Chapter 5: The EM algorithm -- 5.1 Maximum likelihood estimator -- 5.2 The EM algorithm -- 5.3 E-step -- 5.4 M-step -- 5.5 EM estimates -- Chapter 6: Conclusions -- Appendix -- References. en_US dc.language.iso en_US en_US dc.title Expectation maximization and latent class models en_US dc.type Thesis en_US dc.type.degree ms en_US dc.identifier.department Department of Mathematics and Statistics en_US dc.contributor.chair Allman, Elizabeth dc.contributor.committee Faudree, Jill dc.contributor.committee Barry, Ron dc.contributor.committee Bueler, Edward refterms.dateFOA 2020-03-05T13:38:44Z
﻿

### Files in this item

Name:
Banos_uaf_0006N_10510.pdf
Size:
2.363Mb
Format:
PDF