fi=112 Statistiikka|sv=112 Statistik|en=112 Statistics|
https://www.doria.fi:443/handle/10024/92545
2024-05-14T10:23:56ZStructure learning of context-specific graphical models
https://www.doria.fi:443/handle/10024/124252
Structure learning of context-specific graphical models
Pensar, Johan
Abstract The ultimate problem considered in this thesis is modeling a high-dimensional joint distribution over a set of discrete variables. For this purpose, we consider classes of context-specific graphical models and the main emphasis is on learning the structure of such models from data. Traditional graphical models compactly represent a joint distribution through a factorization justi ed by statements of conditional independence which are encoded by a graph structure. Context-speci c independence is a natural generalization of conditional independence that only holds in a certain context, speci ed by the conditioning variables. We introduce context-speci c generalizations of both Bayesian networks and Markov networks by including statements of context-specific independence which can be encoded as a part of the model structures. For the purpose of learning context-speci c model structures from data, we derive score functions, based on results from Bayesian statistics, by which the plausibility of a structure is assessed. To identify high-scoring structures, we construct stochastic and deterministic search algorithms designed to exploit the structural decomposition of our score functions. Numerical experiments on synthetic and real-world data show that the increased exibility of context-specific structures can more accurately emulate the dependence structure among the variables and thereby improve the predictive accuracy of the models.
2016-06-14T05:12:34ZEconomic capital allocation
https://www.doria.fi:443/handle/10024/120265
Economic capital allocation
Wang, Min
For my Licentiate thesis, I conducted research on risk measures. Continuing with this research, I now focus on capital allocation. In the proportional capital allocation principle, the choice of risk measure plays a very important part.
In the chapters Introduction and Basic concepts, we introduce three definitions of economic capital, discuss the purpose of capital allocation, give different viewpoints of capital allocation and present an overview of relevant literature. Risk measures are defined and the concept of coherent risk measure is introduced. Examples of important risk measures are given, e. g., Value at Risk (VaR), Tail Value at Risk (TVaR). We also discuss the implications of dependence and review some important distributions.
In the following chapter on Capital allocation we introduce different principles for allocating capital. We prefer to work with the proportional allocation method.
In the following chapter, Capital allocation based on tails, we focus on insurance business lines with heavy-tailed loss distribution. To emphasize capital allocation based on tails, we define the following risk measures: Conditional Expectation, Upper Tail Covariance and Tail Covariance Premium Adjusted (TCPA).
In the final chapter, called Illustrative case study, we simulate two sets of data with five insurance business lines using Normal copulas and Cauchy copulas. The proportional capital allocation is calculated using TCPA as risk measure. It is compared with the result when VaR is used as risk measure and with covariance capital allocation.
In this thesis, it is emphasized that no single allocation principle is perfect for all purposes. When focusing on the tail of losses, the allocation based on TCPA is a good one, since TCPA in a sense includes features of TVaR and Tail covariance.
2016-02-18T09:37:14ZContext-specific independence in graphical models
https://www.doria.fi:443/handle/10024/101008
Context-specific independence in graphical models
Nyman, Henrik
The theme of this thesis is context-speci c independence in graphical models. Considering a system of stochastic variables it is often the case that the variables are dependent of each other. This can, for instance, be seen by measuring the covariance between a pair of variables. Using graphical models, it is possible to visualize the dependence structure found in a set of stochastic variables. Using ordinary graphical models, such as Markov networks, Bayesian networks, and Gaussian graphical models, the type of dependencies that can be modeled is limited to marginal and conditional (in)dependencies. The models introduced in this thesis enable the graphical representation of context-speci c independencies, i.e. conditional independencies that hold only in a subset of the outcome space of the conditioning variables.
In the articles included in this thesis, we introduce several types of graphical models that can represent context-speci c independencies. Models for both discrete variables and continuous variables are considered. A wide range of properties are examined for the introduced models, including identi ability, robustness, scoring, and optimization. In one article, a predictive classi er which utilizes context-speci c independence models is introduced. This classi er clearly demonstrates the potential bene ts of the introduced models. The purpose of the material included in the thesis prior to the articles is to provide the basic theory needed to understand the articles.; Temat för avhandlingen är kontextspecifikt oberoende i grafiska modeller. Inom sannolikhetslära och statistik är en stokastisk variabel en variabel som påverkas av slumpen. Till skillnad från vanliga matematiska variabler antar en stokastisk variabel ett givet värde med en viss sannolikhet. För en mängd stokastiska variabler gäller det i regel att variablerna är beroende av varandra. Graden av beroende kan t.ex. mätas med kovariansen mellan två variabler. Med hjälp av grafiska modeller är det möjligt att visualisera beroendestrukturen för ett system av stokastiska variabler. Med hjälp av traditionella grafiska modeller såsom Markov nätverk, Bayesianska nätverk och Gaussiska grafiska modeller är det möjligt att visualisera marginellt och betingat oberoende. De modeller som introduceras i denna avhandling möjliggör en grafisk representation av kontextspecifikt oberoende, d.v.s. betingat oberoende som endast håller i en delmängd av de betingande variablernas utfallsrum.
I artiklarna som inkluderats i avhandlingen introduceras flera typer av grafiska modeller som kan representera kontextspecifika oberoende. Både diskreta och kontinuerliga system behandlas. För dessa modeller undersöks många egenskaper inklusive identifierbarhet, stabilitet, modelljämförelse och optimering. I en artikel introduceras en prediktiv klassificerare som utnyttjar kontextspecifikt oberoende i grafiska modeller. Denna klassificerare visar tydligt hur användningen av kontextspecifika oberoende kan leda till förbättrade resultat i praktiska tillämpningar.
2014-11-03T07:11:06ZExcessive functions, Appell polynomials and optimal stopping
https://www.doria.fi:443/handle/10024/96492
Excessive functions, Appell polynomials and optimal stopping
Ta, Bao Quoc
The main topic of the thesis is optimal stopping. This is treated in two
research articles. In the first article we introduce a new approach to optimal stopping of general strong Markov processes. The approach is based on the representation of excessive functions as expected suprema. We present a variety of examples, in particular, the Novikov-Shiryaev problem for Lévy processes. In the second article on optimal stopping we focus on differentiability of excessive functions of diffusions and apply these results to study the validity of the principle of smooth fit. As an example we discuss optimal stopping of sticky Brownian motion. The third research article offers a survey like discussion on Appell polynomials. The crucial role of Appell polynomials in optimal stopping of Lévy processes was noticed by Novikov and Shiryaev. They described the optimal rule in a large class of problems via these polynomials. We exploit the probabilistic approach to Appell polynomials and show that many classical results are obtained with ease in this framework. In the fourth article we derive a new relationship between the generalized Bernoulli polynomials and the generalized Euler polynomials.
2014-04-28T08:41:58Z