The Markov chain process for predicting the occurence of a sequence of rainless days, a standard technique, is critically examined in light of the basic underlying assumptions that must be made each time it is used. This is then compared to a simple binomial model wherein an event is defined to be a series of rainless days of desired length. Computer programs to perform the required calculations are then presented and compared as to complexity and operating characteristics. Finally, an example of applying both programs to real data is presented and further comparisons are drawn between the two techniques.

Indiana University-Purdue University Indianapolis (IUPUI) / The essence of Bayesian data analysis is to ascertain posterior distributions. Posteriors generally do not have closed-form expressions for direct computation in practical applications. Analysts, therefore, resort to Markov Chain Monte Carlo (MCMC) methods for the generation of sample observations that approximate the desired posterior distribution. Standard MCMC methods simulate sample values from the desired posterior distribution via random proposals. As a result, the mechanism used to generate the proposals inevitably determines the efficiency of the algorithm. One of the modern MCMC techniques designed to explore the high-dimensional space more efficiently is Hamiltonian Monte Carlo (HMC), based on the Hamiltonian differential equations. Inspired by classical mechanics, these equations incorporate a latent variable to generate MCMC proposals that are likely to be accepted. This dissertation discusses how such a powerful computational approach can be used for implementing statistical models. Along this line, I created a unified computational procedure for using HMC to fit various types of statistical models. The procedure that I proposed can be applied to a broad class of models, including linear models, generalized linear models, mixed-effects models, and various types of semiparametric regression models. To facilitate the fitting of a diverse set of models, I incorporated new parameterization and decomposition schemes to ensure the numerical performance of Bayesian model fitting without sacrificing the procedure’s general applicability. As a concrete application, I demonstrate how to use the proposed procedure to fit a multivariate generalized additive model (GAM), a nonstandard statistical model with a complex covariance structure and numerous parameters. Byproducts of the research include two software packages that all practical data analysts to use the proposed computational method to fit their own models. The research’s main methodological contribution is the unified computational approach that it presents for Bayesian model fitting that can be used for standard and nonstandard statistical models. Availability of such a procedure has greatly enhanced statistical modelers’ toolbox for implementing new and nonstandard statistical models.

This project applies statistical inference methods to energy data from the island of Luzon in the Philippines. The goal of the project is to explore different ways of creating predictive models and to understand the assumptions that are made about reality when a certain model is selected. The main models discussed in the project are Simple Linear Regression and Markov Chain Models. The predictions were used to assess Luzon's progress towards the sustainable development goals. All models considered in this project suggest that they are not on target to meet the sustainability goal.

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Discrete-time Markov chain models can be used to make future predictions in many important fields including education. Government and educational institutions today are concerned about college enrollment and what impacts the number of students enrolling. One challenge is how to make an accurate prediction about student enrollment so institutions can plan appropriately. In this thesis, we model student enrollment at East Tennessee State University (ETSU) with a discrete-time Markov chain model developed using ETSU student data from Fall 2008 to Spring 2017. In this thesis, we focus on the progression from one level to another within the university system including graduation and dropout probabilities as indicated by the data. We further include the probability that a student will leave school for a limited period of time and then return to the institution. We conclude with a simulation of the model and a comparison to the trends seen in the data.

This paper focuses on how to improve strategic asset allocation in practice. Strategic asset allocation is perhaps the most fundamental issue in portfolio management and it has been thoroughly discussed in previous research. We take our starting point in the traditional work of Markowitz within portfolio optimization. We provide a new solution of how to perform portfolio optimization in practice, or more specifically how to estimate the covariance matrix, which is needed to perform conventional portfolio optimization. Many researchers within this field have noted that the return distribution of financial assets seems to vary over time, so called regime switching, which makes it dicult to estimate the covariance matrix. We solve this problem by using a Bayesian approach for developing a Markov chain Monte Carlo algorithm that detects break points in the return distribution of financial assets, thus enabling us to improve the estimation of the covariance matrix. We find that there are two break points during the time period studied and that the main difference between the periods are that the volatility was substantially higher for all assets during the period that corresponds to the financial crisis, whereas correlations were less affected. By evaluating the performance of the algorithm we find that the algorithm can increase the Sharpe ratio of a portfolio, thus that our algorithm can improve strategic asset allocation over time. / Detta examensarbete fokuserar på hur man kan förbättra tillämpningen av strategisk tillgångsslagsallokering i praktiken. Hur man allokerar kapital mellan tillgångsslag är kanske de mest fundamentala beslutet inom kapitalförvaltning och ämnet har diskuterats grundligt i litteraturen. Vårt arbete utgår från Markowitz traditionella teorier inom portföljoptimering och utifrån dessa tar vi fram ett nytt angreppssätt för att genomföra portföljoptimering i praktiken. Mer specifikt utvecklar vi ett nytt sätt att uppskatta kovar-iansmatrisen för avkastningsfördelningen för finansiella tillgångar, något som är essentiellt för att kunna beräkna de optimala portföljvikterna enligt Markowitz. Det påstås ofta att avkastningens fördelning förändras över tid; att det sker så kallade regimskiften, vilket försvårar uppskattningen av kovariansmatrisen. Vi löser detta problem genom att använda ett Bayesiansk angreppssätt där vi utvecklar en Markov chain Monte Carlo-algoritm som upptäcker brytpunkter i avkastningsfördelningen, vilket gör att uppskattningen av kovar-iansmatrisen kan förbättras. Vi finner två brytpunkter i fördelningen under den studerade tidsperioden och den huvudsakliga skillnaden mellan de olika tidsperioderna är att volatiliten var betydligt högre för samtliga tillgångar under den tidsperiod som motsvaras av finanskrisen, medan korrelationerna mellan tillgångsslagen inte påverkades lika mycket. Genom att utvärdera hur algoritmen presterar finner vi att den ökar en portföljs Sharpe ratio och således att den kan förbättra den strategiska allokeringen mellan tillgångsslagen över tid.

One of the key tasks in the study of the recommendation system is to model the dynamics aspect of a person's preference, i.e. to give sequential recommendations. Markov Chain (MC), which is famous for its capability of learning a transition graph, is the most popular approach to address the task. In previous work, the recommendation system attempts to model the short-term dynamics of the personal preference based on the long-term dynamics, which implies the assumption that the personal preference over a set of items remains same over time. However, in the field of food science, the study of Sensory-Specific Satiety (SSS) shows that the personal preference on food changes along time and previous meals. However, whether such changes follow certain patterns remains unclear. In this paper, a recommendation system is built based on Variable Order Markov Chain (VOMC), which is capable of modeling various lengths of sequential patterns using the suffix tree (ST) search. This recommendation system aims to understand and model the short-term dynamics aspect of the personal preference on food. To evaluate the system, a Food Diary survey is carried to collect users’ meals data over seven days. The results show that this recommendation system can give meaningful recommendations. / En av huvuduppgifterna när det kommer till rekommenderingsplatformar är att modellera kortsidiga dynamiska egenskaper, dvs. användares sekventiella beteenden. Markov Chain (MC), som är mest känd för sin förmåga att lära sig övergångsgrafer, är den mest populära metoden för att ge sig på denna uppgift. I föregående arbeten så har rekommenderingsplatformar ofta tenderat att modellera kortsidig dynamik baserat på långsidig dynamik, t.ex. likheter mellan objekt eller användares relativa preferenser givet olika tillfällen. Att använda den här metoden brukar medföra att användares långsiktiga dynamik, i detta fall personliga smakpreferenser, är alltid densamma. Däremot, så har studien av Sensory-Specific Satiety visat att användares preferenser gällande mat varierar. I detta arbete så undersöks ett rekommenderingssystem som baseras på Variable Order Markov Chain (VOMC) som kan anpassa sig efter den observerade realiseringen genom att använda suffix tree (ST) för att extrahera sekventiella mönster. Detta rekommenderingssystem fokuserar på kortsidig dynamik istället för att kombinera kort- och långsidig dynamik. För att evaluera metoden, en undersökning av vilken mat som konsumeras, under loppet av sju dagar, ges ut för att samla data om vilken mat och i vilken ordning användare konsumerar. I resultaten så visas att det föreslagna rekommenderingsystemet kan ge meningsfulla rekommendationer.

Yes / It is shown that the probability generating function of a lifetime random variable T on a finite lattice with polynomial failure rate satisfies a certain differential equation. The interrelationship with Markov chain theory is highlighted. The differential equation gives rise to a system of differential equations which, when inverted, can be used in the limit to express the polynomial coefficients in terms of the factorial moments of T. This then can be used to estimate the polynomial coefficients. Some special cases are worked through symbolically using Computer Algebra. A simulation study is used to validate the approach and to explore its potential in the reliability context.

<p>This dissertation focuses on the simulation efficiency of the Markov process for two scenarios: Stochastic differential equations(SDEs) and simulated weather data. </p> <p><br></p> <p>For SDEs, we propose a novel Gibbs sampling algorithm that allows sampling from a particular class of SDEs without any discretization error and shows the proposed algorithm improves the sampling efficiency by orders of magnitude against the existing popular algorithms.  </p> <p><br></p> <p>In the weather data simulation study, we investigate how representative the simulated data are for three popular stochastic weather generators. Our results suggest the need for more than a single realization when generating weather data to obtain suitable representations of climate. </p>

History matching is a critical process used for calibrating simulation models and assessing subsurface uncertainties. This common technique aims to align the reservoir models with the observed data. However, achieving this goal is often challenging due to the non uniqueness of the solution, underlying subsurface uncertainties, and usually the high computational cost of simulations. The traditional approach is often based on trial and error, which is exhaustive and labor-intensive. Some analytical and numerical proxies combined with Monte Carlo simulations are utilized to reduce the computational time. However, these approaches suffer from low accuracy and may not fully capture subsurface uncertainties. This study proposes a new robust method utilizing Bayesian Markov Chain Monte Carlo (MCMC) to perform assisted history matching under uncertainties. We propose a novel three-step workflow that includes 1) multi-resolution low-fidelity models to guarantee high-quality matching; 2) Long-Short Term Memory (LSTM) network as a low-fidelity model to reproduce continuous time-response based on the simulation model, combined with Bayesian optimization to obtain the optimum low fidelity model; 3) Bayesian MCMC runs to obtain the Bayesian inversion of the uncertainty parameters. We perform sensitivity analysis on the LSTM’s architecture, hyperparameters, training set, number of chains, and chain length to obtain the optimum setup for Bayesian-LSTM history matching. We also compare the performance of predicting the recovery factor using different surrogate methods, including polynomial chaos expansions (PCE), kriging, and support vector machines for regression (SVR). We demonstrate the proposed method using a water flooding problem for the upper Tarbert formation of the tenth SPE comparative model. This study case represents a highly heterogeneous nearshore environment. Results showed that the Bayesian-optimized LSTM has successfully captured the physics in the high-fidelity model. The Bayesian-LSTM MCMC produces an accurate prediction with narrow ranges of uncertainties. The posterior prediction through the high-fidelity model ensures the robustness and accuracy of the workflow. This approach provides an efficient and practical history-matching method for reservoir simulation and subsurface flow modeling with significant uncertainties.