Time management and distance evaluation have long been a difficult task for workers and companies. This thesis studies 6712 pothole coordinates in Gävle municipality, and evaluates the minimal total road distance needed to visit each pothole once, and return to an initial pothole. Road distance is approximated using the flight distance and a simple random sample of 113 road distances from Google Maps. Thereafter, the data from the sample along with a Bayesian approach is used to find a distribution of the ratio between road distance and flight distance. Lastly, a solution to the shortest distance is devised using the Nearest Neighbor algorithm (NNA) and Simulated Annealing (SA). Computational work is performed with Markov Chain Monte Carlo (MCMC). The results provide a minimal road distance of 717 km. / Tidshantering och distansutvärdering är som regel en svår uppgift för arbetare och företag. Den här uppsatsen studerar 6712 potthål i Gävle kommun, och utvärderar den bilväg som på kortast sträcka besöker varje potthål och återgår till den ursprungliga startpunkten. Bilvägsavståndet mellan potthålen uppskattas med hjälp av flygavståndet, där ett obundet slumpmässigt urval av 113 bilvägsavstånd mellan potthålens koordinatpunkter dras. Bilvägsdistanser hittas med hjälp av Google Maps. Därefter används data från urvalet tillsammans med en bayesiansk modell för att hitta en fördelning för förhållandet mellan bilvägsavstånd och flygavstånd. Slutligen framförs en lösning på det kortaste bilvägsavståndet med hjälp av en Nearest Neighbour algoritm (NNA) samt Simulated Annealing (SA). Statistiskt beräkningsarbete utförs med Markov Chain Monte Carlo (MCMC). Resultaten ger en kortaste bilvägssträcka på 717 km.

This thesis attempts to evaluate the Markov Chain Monte Carlo (MCMC) methods Metropolis-Hastings (MH) and No-U-Turn Sampler (NUTS) to estimate systemic risk measures. The subject of analysis is an equity portfolio provided by a Nordic asset management firm, which is modelled using a vine copula. The evaluation considers three different crisis outcomes on a portfolio level, and the results are compared with a Monte Carlo (MC) benchmark. The MCMC samplers attempt to increase sampling efficiency by sampling from these crisis events directly, which is impossible for an MC sampler. The resulting systemic risk measures are evaluated both on the portfolio level as well as marginal level.  The results are divided. In part, the MCMC samplers proved to be efficient in terms of accepted samples, where NUTS outperformed MH. However, due to the practical implementation of the MCMC samplers and the vine copula model, the computational time required outweighed the gains in sampler efficiency - causing the MC sampler to outperform both MCMC samplers in certain settings. For NUTS, there seems to be great potential in the context of estimating systemic risk measures as it explores high-dimensional and multimodal joint distributions efficiently with low autocorrelation. It is concluded that asset management companies can benefit from both using vine copulas to model portfolio risk, as well as using MC or MCMC methods for evaluating systemic risk. However, for the MCMC samplers to be of practical relevance, it is recommended to further investigate efficient implementations of vine copulas in the context of MCMC sampling. / Detta examensarbete utvärderar Markov Chain Monte Carlo (MCMC)-metoderna No-U-Turn Sampler (NUTS) och Metropolis-Hastings (MH) vid uppskattning av systemiska riskmått. För att göra detta används en vine copula för att modellera en portfölj, baserad på empirisk data från ett nordiskt kapitalförvaltningsföretag. Metoderna utvärderas givet tre olika krishändelser och jämförs därefter med ett Monte Carlo (MC) benchmark. MCMC-metoderna försöker öka samplingseffektiviteten genom att simulera direkt från dessa krishändelser, vilket är omöjligt för en klassisk MC-metod. De resulterande systemiska riskmåtten utvärderas både på portföljnivå och på marginalnivå. Resultaten är delade. Dels visade sig MCMC-metoderna vara effektiva när det gäller accepterade samples där NUTS överträffade MH. Dock, med anledning av av den praktiska implementationen av MCMC-metoderna och vine copula modellen var beräkningstiden för hög trots effektiviteten hos metoden - vilket fick MC-metoden att överträffa de andra metoderna i givet dessa särskilda kontexter. När det kommer till att uppskatta systemiska riskmått finns det dock stor potential för NUTS eftersom metoden utforskar högdimensionella och multimodala sannolikhetsfördelningar effektivt med låg autokorrelation. Vi drar även slutsatsen att kapitalförvaltare kan dra nytta av att både använda riskmodeller baserade på vine copulas, samt använda MC- eller MCMC-metoder för att utvärdera systemisk risk. För att MCMC-metoderna ska vara av praktisk relevans rekommenderas det dock att framtida forskning görs där mer effektiva implementeringar av vine copula-baserade modeller görs i samband med MCMC-sampling.

我們提供一個新的可靠度模型，DwACM，並提供一個模式選擇準則CCP，我們利用DwACM和CCP來選擇衰變量。 / We propose a new reliability model, DwACM (Dichotomous-data with Auxiliary Continuous Measurements model) to describe a data set which consists of classical dichotomous response (Go or No Go) associated with a set of continuous auxiliary measurement. In this model, the lifetime of each individual is considered as a latent variable. Given the value of the latent variable, the dichotomous response is either 0 or 1 depending on if it fails or not at the measuring time. The continuous measurement can be regarded as observations of an underlying possible degradation candidate of which descending process is a function of the lifetime. Under the assumption that the failure of products is defined as the time at which the continuous measurement reaches a threshold, these two measurements can be linked in the proposed model. Statistical inference under this model are both in frequentist and Bayesian frameworks. To evaluate the continuous measurements, we provide a criterion, CCP (correct classification probability), to select the best degradation measurement. We also report our simulation studies of the performances of parameters estimators and CCP.

本篇論文以Regime Switching Stochastic Volatility（RSV）作為外匯選擇權市場的波動度模型，採用馬可夫鏈蒙地卡羅法（Markov Chain Monte Carlo）中的GibbS Sampling演算法估計RSV模型的參數，並預測外匯選擇權在RSV模型下的價格。 數值結果方面首先對GibbS Sampling參數估計的結果做討論，再對預測出的選擇權價格與Black and Scholes作比較，最後並提出笑狀波幅與隱含波動度平面的結果。 本研究所得到之結論： 1. RSV模型與MCMC模擬法的組合，具備產生笑狀波幅的能力，提供足夠證據顯示，RSV模型與MCMC演算法所計算出來的選擇權價格，確實反應且捕捉到了市場上選擇權價格所應具備的特色。 2. 本模型能有效解釋期限結構 （Term Stucture of Volatility）、笑狀波幅（Volatility Smile）的現象。 關鍵字：馬可夫鏈蒙地卡羅法、外匯選擇權、貝氏選擇權評價、MCMC、Regime switching Regine change、Gibbs Sampling、currency option、Markov Chain Montec Carlo

函數型資料分析為近年發展的統計方法。函數型資料是在一段特定時間上，我們只在離散的時間點上收集觀測值。例如:氣象觀測站所收集到的每月氣溫、雨量資料，即是一種常見的函數型資料。函數型資料主要有三種特色，共同趨勢性、觀測個體反應強度不同，觀測個體時間特色上的差異。本文研究主要是使用，Brumback與Lindstrom在2004所提出的自模型迴歸族(self-modeling)當作模型架構來處理函數型資料的趨勢性與個體反應強度。而為了處理函數型資料的時間差異性，我們在模型中加入時間轉換函數(time transformation function)，處理函數型資料的時間差異性步驟，這個過程稱為同步化。經過同步化的處理後，能幫助研究者更清楚資料的特性。模型中除了時間轉換函數的部份，其餘模型中的參數我們是利用馬可夫鏈蒙地卡羅法中的Gibbs Sampling來進行參數的抽樣，並以取出的抽樣值來估計參數。時間轉換函數的部份，我們使用概似懲罰函數(penalized likelihood function)來估計時間轉換函數的參數部份。由於函數型資料擁有趨勢性，我們預期不同類別的資料，會呈現不同的趨勢性，我們將利用此一特色當做分類上的標準。 關鍵詞:函數型資料分析、曲線同步化、曲線區別分析、馬可夫鏈蒙地卡羅法。 / Functional data are random curves observed in a period of time at discrete time points.They often exhibit a common shape, but with variations in amplitude and phase across curves.To estimate the common shape,some adjustment for synchronization is often made,which is also known as time warping or curve registration.In this thesis,splines are used to model the warping functions and the common shape. Certain parameters are allowed to be random.For the estimation of the random parameters,priors are proposed so that samples from the posteriors can be obtained using Markov chain Monte Carlo methods.For the estimation of non-random parameters, a penalized likelihood approach is used. It is found via simulation studies that for a set of random curves with a common shape,the estimated common shape function looks like the true function up to a location-scale transform,and the curve alignment based on estimated time warping functions looks reasonable.For two groups of random curves which differ in the group common shape functions,synchronization also improves the discrimination between groups in some cases. Key words: functional data analysis,curve registration,curve discrimination,markov chain monte carlo method.

The availability of ultra high-frequency (UHF) data on transactions has revolutionised data processing and statistical modelling techniques in finance. The unique characteristics of such data, e.g. discrete structure of price change, unequally spaced time intervals and multiple transactions have introduced new theoretical and computational challenges. In this study, we develop a Bayesian framework for modelling integer-valued variables to capture the fundamental properties of price change. We propose the application of the zero inflated Poisson difference (ZPD) distribution for modelling UHF data and assess the effect of covariates on the behaviour of price change. For this purpose, we present two modelling schemes; the first one is based on the analysis of the data after the market closes for the day and is referred to as off-line data processing. In this case, the Bayesian interpretation and analysis are undertaken using Markov chain Monte Carlo methods. The second modelling scheme introduces the dynamic ZPD model which is implemented through Sequential Monte Carlo methods (also known as particle filters). This procedure enables us to update our inference from data as new transactions take place and is known as online data processing. We apply our models to a set of FTSE100 index changes. Based on the probability integral transform, modified for the case of integer-valued random variables, we show that our models are capable of explaining well the observed distribution of price change. We then apply the deviance information criterion and introduce its sequential version for the purpose of model comparison for off-line and online modelling, respectively. Moreover, in order to add more flexibility to the tails of the ZPD distribution, we introduce the zero inflated generalised Poisson difference distribution and outline its possible application for modelling UHF data.

This thesis consists of three chapters on economic and econometric applications of Bayesian parameter estimation and model comparison. The first two chapters study the incidental parameter problem mainly under a linear autoregressive (AR) panel data model with fixed effect. The first chapter investigates the problem from a model comparison perspective. The major finding in the first chapter is that consistency in parameter estimation and model selection are interrelated. The reparameterization of the fixed effect parameter proposed by Lancaster (2002) may not provide a valid solution to the incidental parameter problem if the wrong set of exogenous regressors are included. To estimate the model consistently and to measure its goodness of fit, the Bayes factor is found to be more preferable for model comparson than the Bayesian information criterion based on the biased maximum likelihood estimates. When the model uncertainty is substantial, Bayesian model averaging is recommended. The method is applied to study the relationship between financial development and economic growth. The second chapter proposes a correction function approach to solve the incidental parameter problem. It is discovered that the correction function exists for the linear AR panel model of order p when the model is stationary with strictly exogenous regressors. MCMC algorithms are developed for parameter estimation and to calculate the Bayes factor for model comparison. The last chapter studies how stock return's predictability and model uncertainty affect a rational buy-and-hold investor's decision to allocate her wealth for different lengths of investment horizons in the UK market. The FTSE All-Share Index is treated as the risky asset, and the UK Treasury bill as the riskless asset in forming the investor's portfolio. Bayesian methods are employed to identify the most powerful predictors by accounting for model uncertainty. It is found that though stock return predictability is weak, it can still affect the investor's optimal portfolio decisions over different investment horizons.

Dans ce mémoire, nous cherchons à modéliser des tables à deux entrées monotones en lignes et/ou en colonnes, pour une éventuelle application sur les tables de mortalité. Nous adoptons une approche bayésienne non paramétrique et représentons la forme fonctionnelle des données par splines bidimensionnelles. L’objectif consiste à condenser une table de mortalité, c’est-à-dire de réduire l’espace d’entreposage de la table en minimisant la perte d’information. De même, nous désirons étudier le temps nécessaire pour reconstituer la table. L’approximation doit conserver les mêmes propriétés que la table de référence, en particulier la monotonie des données. Nous travaillons avec une base de fonctions splines monotones afin d’imposer plus facilement la monotonie au modèle. En effet, la structure flexible des splines et leurs dérivées faciles à manipuler favorisent l’imposition de contraintes sur le modèle désiré. Après un rappel sur la modélisation unidimensionnelle de fonctions monotones, nous généralisons l’approche au cas bidimensionnel. Nous décrivons l’intégration des contraintes de monotonie dans le modèle a priori sous l’approche hiérarchique bayésienne. Ensuite, nous indiquons comment obtenir un estimateur a posteriori à l’aide des méthodes de Monte Carlo par chaînes de Markov. Finalement, nous étudions le comportement de notre estimateur en modélisant une table de la loi normale ainsi qu’une table t de distribution de Student. L’estimation de nos données d’intérêt, soit la table de mortalité, s’ensuit afin d’évaluer l’amélioration de leur accessibilité. / This master’s thesis is about the estimation of bivariate tables which are monotone within the rows and/or the columns, with a special interest in the approximation of life tables. This problem is approached through a nonparametric Bayesian regression model, in particular linear combinations of regression splines. By condensing a life table, our goal is to reduce its storage space without losing the entries’ accuracy. We will also study the reconstruction time of the table with our estimators. The properties of the reference table, specifically its monotonicity, must be preserved in the estimation. We are working with a monotone spline basis since splines are flexible and their derivatives can easily be manipulated. Those properties enable the imposition of constraints of monotonicity on our model. A brief review on univariate approximations of monotone functions is then extended to bivariate estimations. We use hierarchical Bayesian modeling to include the constraints in the prior distributions. We then explain the Markov chain Monte Carlo algorithm to obtain a posterior estimator. Finally, we study the estimator’s behaviour by applying our model on the Standard Normal table and the Student’s t table. We estimate our data of interest, the life table, to establish the improvement in data accessibility.

In my PhD, I advanced the scientific exploration of the Next Generation Transit Survey (NGTS), a ground-based wide-field survey operating at ESO’s Paranal Observatory in Chile since 2016. My original contribution to knowledge is the development of novel methods to 1) estimate NGTS’ yield of planets and false positives; 2) disentangle planets from false positives; and 3) accurately characterise planets. If an exoplanet passes (transits) in front of its host star, we can measure a periodic decrease in brightness. The study of transiting exoplanets gives insight into their size, formation, bulk composition and atmospheric properties. Transit surveys are limited by their ability to identify false positives, which can mimic planets and out-number them by a hundredfold. First, I designed a novel yield simulator to optimise NGTS’ observing strategy and identification of false positives (published in Günther et al., 2017a). This showed that NGTS’ prime targets, Neptune- and Earth-sized signals, are frequently mimicked by blended eclipsing binaries, allowing me to quantify and prepare strategies for candidate vetting and follow-up. Second, I developed a centroiding algorithm for NGTS, achieving a precision of 0.25 milli-pixel in a CCD image (published in Günther et al., 2017b). With this, one can measure a shift of light during an eclipse, readily identifying unresolved blended objects. Third, I innovated a joint Bayesian fitting framework for photometry, centroids, and radial velocity cross-correlation function profiles. This allows to disentangle which object (target or blend) is causing the signal and to characterise the system. My method has already unmasked numerous false positives. Most importantly, I confirmed that a signal which was almost erroneously rejected, is in fact an exoplanet (published in Günther et al., 2018). The presented achievements minimise the contamination with blended false positives in NGTS candidates by 80%, and show a new approach for unmasking hidden exoplanets. This research enhanced the success of NGTS, and can provide guidance for future missions.

L'apprentissage de dictionnaire pour la représentation parcimonieuse est bien connu dans le cadre de la résolution de problèmes inverses. Les méthodes d'optimisation et les approches paramétriques ont été particulièrement explorées. Ces méthodes rencontrent certaines limitations, notamment liées au choix de paramètres. En général, la taille de dictionnaire doit être fixée à l'avance et une connaissance des niveaux de bruit et éventuellement de parcimonie sont aussi nécessaires. Les contributions méthodologies de cette thèse concernent l'apprentissage conjoint du dictionnaire et de ces paramètres, notamment pour les problèmes inverses en traitement d'image. Nous étudions et proposons la méthode IBP-DL (Indien Buffet Process for Dictionary Learning) en utilisant une approche bayésienne non paramétrique. Une introduction sur les approches bayésiennes non paramétriques est présentée. Le processus de Dirichlet et son dérivé, le processus du restaurant chinois, ainsi que le processus Bêta et son dérivé, le processus du buffet indien, sont décrits. Le modèle proposé pour l'apprentissage de dictionnaire s'appuie sur un a priori de type Buffet Indien qui permet d'apprendre un dictionnaire de taille adaptative. Nous détaillons la méthode de Monte-Carlo proposée pour l'inférence. Le niveau de bruit et celui de la parcimonie sont aussi échantillonnés, de sorte qu'aucun réglage de paramètres n'est nécessaire en pratique. Des expériences numériques illustrent les performances de l'approche pour les problèmes du débruitage, de l'inpainting et de l'acquisition compressée. Les résultats sont comparés avec l'état de l'art.Le code source en Matlab et en C est mis à disposition. / Dictionary learning for sparse representation has been widely advocated for solving inverse problems. Optimization methods and parametric approaches towards dictionary learning have been particularly explored. These methods meet some limitations, particularly related to the choice of parameters. In general, the dictionary size is fixed in advance, and sparsity or noise level may also be needed. In this thesis, we show how to perform jointly dictionary and parameter learning, with an emphasis on image processing. We propose and study the Indian Buffet Process for Dictionary Learning (IBP-DL) method, using a bayesian nonparametric approach.A primer on bayesian nonparametrics is first presented. Dirichlet and Beta processes and their respective derivatives, the Chinese restaurant and Indian Buffet processes are described. The proposed model for dictionary learning relies on an Indian Buffet prior, which permits to learn an adaptive size dictionary. The Monte-Carlo method for inference is detailed. Noise and sparsity levels are also inferred, so that in practice no parameter tuning is required. Numerical experiments illustrate the performances of the approach in different settings: image denoising, inpainting and compressed sensing. Results are compared with state-of-the art methods is made. Matlab and C sources are available for sake of reproducibility.