This paper investigates how Markov chain modelling can be applied to the Swedish stock index OMXS30. The investigation is two-fold. Firstly, a Markov chain is based on index data from recent years, where properties such as transition matrix, stationary distribution and hitting time are studied. This first investigation shows, for example, signs of volatility clustering and a steady-state distribution that indicates a positive long term trend for OMXS30. Secondly, the predictive ability of a Markov model, consisting of a voting ensemble of ten Markov chains, is evaluated. The results show that a Markov model with six states has a prediction accuracy slightly above the performance of random chance. Moreover, there was no considerable difference in prediction accuracy between models with Markov chains of first and second order. / I denna studie undersöks hur modellering med markovkedjor kan appliceras på det svenska börsindexet OMXS30. Studien består av två delar. Först baseras en markovkedja på de senaste årens indexutveckling där egenskaper såsom övergångsmatris, stationär fördelning och ``hitting time" undersöks. Den här första undersökningen visar bland annat tecken på volatilitetsklustering och en stationärfördelning som indikerar en långsiktig positiv trend för OMXS30. Vidare evalueras prediktionsförmågan hos en markovmodell bestående av en ensemble med tio markovkedjor. Resultatet visar att en markovmodell med sex tillstånd har en prediktionsförmåga som är något bättre än slumpen. Vidare fanns det ingen betydande skillnad i prediktionsförmåga mellan modeller med markovkedjor av första och andra ordningen.

An important task in brain modeling is that of estimating model parameters and quantifying their uncertainty. In this thesis we tackle this problem from a Bayesian perspective: we use experimental data to update the prior information about model parameters, in order to obtain their posterior distribution. Uncertainty quantification via a direct computation of the posterior has a prohibitive computational cost in high dimensions. An alternative to a direct computation is offered by Markov chain Monte Carlo (MCMC) methods. The aim of this project is to analyse some of the methods within this class and improve their convergence. In this thesis we describe the following MCMC methods: Metropolis-Hastings (MH) algorithm, Metropolis adjusted Langevin algorithm (MALA), simplified manifold MALA (smMALA) and Approximate Bayesian Computation MCMC (ABCMCMC). SmMALA is further analysed in Paper A, where we propose an algorithm to approximate a key component of this algorithm (the Fisher Information) when applied to ODE models, with the purpose of reducing the computational cost of the method. A theoretical analysis of MCMC methods is carried out in Paper B and relies on tools from the theory of large deviations. In particular, we analyse the convergence of the MH algorithm by stating and proving a large deviation principle (LDP) for the empirical measures produced by the algorithm. Some of the methods analysed in this thesis are implemented in an R package, available on GitHub as “icpm-kth/uqsa” and presented in Paper C, and are applied to subcellular pathway models within neurons in the context of uncertainty quantification of the model parameters. / En viktig uppgift inom hjärnmodellering är att uppskatta parametrar i modellen och kvantifiera deras osäkerhet. I denna avhandling hanterar vi detta problem från ett Bayesianskt perspektiv: vi använder experimentell data för att uppdatera a priori kunskap av modellparametrar, för att erhålla deras posteriori-fördelning. Osäkerhetskvantifiering (UQ) via direkt beräkning av posteriorfördelningen har en hög beräkningskostnad vid höga dimensioner. Ett alternativ till direkt beräkning ges av Markov chain Monte Carlo (MCMC) metoder. Syftet med det här projektet är att analysera några MCMC metoder och förbättra deras konvergens. I denna avhandling beskriver vi följande MCMC algoritmer: “Metropolis-Hastings” (MH), “Metropolis adjusted Langevin” (MALA), “Simplified Manifold MALA” (smMALA) och “Approximate Bayesian Computation MCMC” (ABCMCMC). SmMALA analyseras i artikel A. Där presenterar vi en algoritm för att approximera en nyckelkomponent av denna algoritm (Fisher informationen) när den tillämpas på ODE modeller i syfte att minska metodens beräkningskostnad. En teoretisk analys av MCMC metoder behandlas i artikel B och bygger på verktyg från teorin av stora avvikelser. Mer specifikt, vi analyserar MH algoritmens konvergens genom att formulera och bevisa en stora avvikelser princip (LDP) för de empiriska mått som produceras av algoritmen. Några av metoderna analyserade i den här avhandlingen har implementerats i ett R paket som finns på GitHub som “icpm-kth/uqsa” och presenteras i artikel C. Metoderna tillämpas på subcellulära vägmodeller inom neuroner i sammanhanget av osäkerhetskvantifieringen av modellparametrar. / <p>QC 2023-08-21</p>

The Rasch sampler is an efficient algorithm to sample binary matrices with given marginal sums. It is a Markov chain Monte Carlo (MCMC) algorithm. The program can handle matrices of up to 1024 rows and 64 columns. A special option allows to sample square matrices with given marginals and fixed main diagonal, a problem prominent in social network analysis. In all cases the stationary distribution is uniform. The user has control on the serial dependency. (authors' abstract)

The first section of this thesis addresses the problem of simultaneously identifying multiple loci that are associated with a trait, using a Bayesian Markov Chain Monte Carlo method. It is applicable to both case/control and quantitative data. I present simulations comparing the methods to standard frequentist methods in human case/control and mouse QTL datasets, and show that in the case/control simulations the standard frequentist method out performs my model for all but the highest effect simulations and that for the mouse QTL simulations my method performs as well as the frequentist method in some cases and worse in others. I also present analysis of real data and simulations applying my method to a simulated epistasis data set. The next section was inspired by the challenges involved in applying a Markov Chain Monte Carlo method to genetic data. It is an investigation into the performance and benefits of the Matlab parallel computing toolbox, specifically its implementation of the Cuda programing language to Matlab's higher level language. Cuda is a language which allows computational calculations to be carried out on the computer's graphics processing unit (GPU) rather than its central processing unit (CPU). The appeal of this tool box is its ease of use as few code adaptions are needed. The final project of this thesis was to develop an HMM for reconstructing the founders of sparsely sequenced inbred populations. The motivation here, that whilst sequencing costs are rapidly decreasing, it is still prohibitively expensive to fully sequence a large number of individuals. It was proposed that, for populations descended from a known number of founders, it would be possible to sequence these individuals with a very low coverage, use a hidden Markov model (HMM) to represent the chromosomes as mosaics of the founders, then use these states to impute the missing data. For this I developed a Viterbi algorithm with a transition probability matrix based on recombination rate which changes for each observed state.

The spectrum has become one of the most important and scarce resources for future wireless communications. However, the current static spectrum policy cannot meet the increasing demands for spectrum access. To improve spectrum efficiency, dynamic spectrum access (DSA) attempts to allocate the spectrum to users in an intelligent manner. Cognitive radio (CR) is an enabling technology for DSA, and can maximize spectrum utilization by introducing unlicensed or secondary users (SUs) to the primary system. The key component of DSA is dynamic spectrum sharing (DSS), which is responsible for providing efficient and fair spectrum allocation or scheduling solutions among licensed or primary users (PUs) and SUs. This thesis focuses on the design of efficient DSS schemes for the future wireless communication networks. Firstly, based on the coordinated DSS model, this thesis proposes a heterogeneous-prioritized spectrum sharing policy for coordinated dynamic spectrum access networks. Secondly, based on the uncoordinated DSS model, a novel partial spectrum sharing strategy and the cross-layer optimization method have been proposed to achieve efficient spectrum sharing between two licensed networks. Then, a hybrid strategy which combines the overlay and underlay schemes is proposed under uncoordinated DSS model. The proposed analytical methods can provide efficient and accurate modeling to predict the behaviors of the PUs and SUs in DSS systems. This thesis presents the performance prediction of the proposed novel DSS schemes that achieve efficient spectrum sharing for coordinated and uncoordinated future wireless networks.

The study on the longest stretch of consecutive successes in \random" trials dates back to 1916 when the German philosopher Karl Marbe wrote a paper concerning the longest stretch of consecutive births of children of the same sex as appearing in the birth register of a Bavarian town. The result was actually used by parents to \predict" the sex of their children. The longest stretch of same-sex births during that time in 200 thousand birth registrations was actually 17 t log2(200 103): During the past century, the research of longest stretch of consecutive successes (longest runs) has found applications in various areas, especially in the theory of reliability. The aim of this thesis is to study large deviations on longest runs in the setting of Markov chains. More precisely, we establish a general large deviation principle for the longest success run in a two-state (success or failure) Markov chain. Our tool is based on a recent result regarding a general large deviation for the longest success run in Bernoulli trails. It turns out that the main ingredient in the proof is to implement several global and local estimates of the cumulative distribution function of the longest success run.

Perfect Gibbs sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random vectors. We show that a perfect Gibbs sampling algorithm suggested in the literature is not always generating from the correct distribution. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Dementia is increasingly recognized as a major and growing threat to public health worldwide, and there is a critical need for prevention and treatment strategies. However, it is necessary that appropriate methodologies are used in the identification of risk factors. The purpose of this dissertation research was to develop further the body of literature featuring Markov chains as an analytic tool for data derived from longitudinal studies of aging and dementia. Data drawn from 649 participants in the University of Kentucky’s Alzheimer’s Disease Center’s (UK ADC) Biologically Resilient Adults in Neurological Studies (BRAiNS) cohort, which was established in 1989 and follows adults age 60 years and older who are cognitively normal at baseline to death, were used to conduct three studies. The first study, “Mild cognitive impairment: Statistical models of transition using longitudinal clinical data,” shows that mild cognitive impairment is a stable clinical entity when a rigorous definition is applied. The second study, “Self-reported head injury and risk of cognitive impairment and Alzheimer’s-type pathology in a longitudinal study of aging and dementia,” shows that when the competing risk of death is properly accounted for, self-reported head injury is a clear risk factor for late-life dementia and is associated with increased beta-amyloid deposition in the brain. The third study, “Incorporating prior-state dependence among random effects and beta coefficients improves multistate Markov chain model fit,” shows that the effect of risk factors, like age, may not be constant over time and may be altered based on the subject’s cognitive state and that model fit is significantly improved when this is taken into account.

Proteins are the fundamental unit of computation and signal processing in biological systems. A quantitative understanding of protein biophysics is of paramount importance, since even slight malfunction of proteins can lead to diverse and severe disease states. However, developing accurate and useful mechanistic models of protein function can be strikingly elusive. I demonstrate that the adoption of Bayesian statistical methods can greatly aid in modeling protein systems. I first discuss the pitfall of parameter non-identifiability and how a Bayesian approach to modeling can yield reliable and meaningful models of molecular systems. I then delve into a particular case of non-identifiability within the context of an emerging experimental technique called single molecule photobleaching. I show that the interpretation of this data is non-trivial and provide a rigorous inference model for the analysis of this pervasive experimental tool. Finally, I introduce the use of nonparametric Bayesian inference for the analysis of single molecule time series. These methods aim to circumvent problems of model selection and parameter identifiability and are demonstrated with diverse applications in single molecule biophysics. The adoption of sophisticated inference methods will lead to a more detailed understanding of biophysical systems. / text

This report describes a method to control the density distribution of a large number of autonomous agents. The approach is based on the fact that there are a large number of agents in the system, and hence the time evolution of the probabilistic density distribution of agents can be described as a Markov chain. The main contribution of this paper is the synthesis of a Markov matrix which will guide the multi-agent system density to a desired steady-state density distribution, in a probabilistic sense, while satisfying some motion and safety constraints. Also, an adaptive density control method based on real time density feedback is introduced to synthesize a time-varying Markov ma- trix, which leads to better convergence to the desired density distribution. Finally, a decentralized density computation method is described. This method guarantees that all agents will have a best, and common, density estimate in a finite, with an explicit bound, number of communication updates. / text