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In this Bachelor thesis, a novel algorithm for sampling from bandlimited circular probability distributions is presented. The algorithm leverages results from Fourier analysis concerning the Fejér kernel to simulate data with some desired probability distribution, realized as a sum of data sampled from a discrete distribution and a small continuous perturbation sampled from the Fejér kernel distribution. Relevant theory is presented before formally proving exact simulation using the algorithm. Experimental results confirm the validity of the theoretical results, and the efficiency of the algorithm is then compared with that of other sampling methods such as rejection sampling with a uniform envelope function.

The present licentiate thesis is devoted to the rigorous derivation of the equations governing thin-film flow of incompressible Newtonian and non-Newtonian fluids. More precisely, we consider flow in a generalized Hele-Shaw cell, which is a thin three-dimensional domain confined between two surfaces connected by cylindrical obstacles of various shapes. Thin-film flows arise naturally in several applications. For instance, it is commonly used when the domain itself has different characteristic lengths in different directions, i.e. when the domain is a thin layer or a slender tube. Mathematically, the flow is described by a set of partial differential equations in a thin domain which depends on a small parameterε, e.g. the ratio of two characteristic lengths. By letting ε tend to zero, one can obtain a better understanding of the properties of solutions of such equations. In this limit process, all variables involved (e.g. velocity and pressure) depend on ε and the resulting limit problem yields a simplified model of the flow. There exist several mathematical techniques that have been developed to deal with such problems, e.g. asymptotic expansions, two-scale convergence for thin domains, etc. The scientific results in this thesis are presented in two papers (I and II) and a complimentary appendix. The results are discussed in a more general context in an introduction which also gives an overview of the subject. In both papers, we assume that the flow is governed by the Stokes system posed in a generalized Hele-Shaw cell satisfying a mixed boundary condition. The so-called no-slip and no-penetration conditions require that the velocity vanishes on the solid surfaces of the cell. This condition is complemented by the normal stress condition on the lateral boundary which is defined by an external pressure. Physically this means that the motion of the fluid is caused by the external pressure gradient, which acts in a direction parallel to the surfaces. One of the main objectives of this thesis is to develop a rigorous mathematical description of pressure-driven flow in thin domains. In paper I, we consider Hele-Shaw flow of an incompressible Newtonian fluid. The results are based on the formal asymptotic expansion method, i.e. by introducing a small parameter ε representing the thickness of the domain, rescaling the problem to a fixed domain, and considering solutions in the form of power series of ε. It is shown that the leading term of the velocity satisfies the so-called Poiseuille-law, i.e. the velocity is a linear function of the pressure gradient, whereas the leading pressure term satisfies the generalized Hele-Shaw equation. The main result is the construction of an approximate solution, which is justified by estimating the L2-norm of the error, i.e. the difference between the exact solution and the approximation. In paper II, the situation is similar to that of paper I, but the fluid obeys a more general constitutive relationship between the stress and the shear rate. More precisely, the functional relationship between the viscosity and the symmetrical part of the velocity gradient is given by a power-law. We develop techniques of functional analysis and calculus of variations in order to justify theorems concerning the existence and uniqueness of weak solutions of the corresponding Stokes system. The nonlinear Poiseuille-law, i.e. the limit velocity and the limit pressure gradient follow a power-law, is derived by using a two-scale convergence procedure and monotonicity arguments. Finally, uniqueness and regularity results for the solution of the limit problem are proved.

This thesis consists of an introduction, and one research paper with results related to potential theory both in the classical Euclidean setting, as well as in quite general metric spaces. The introduction contains a theoretical and historical background of some basic concepts, and their more modern generalisations to metric spaces developed in the last 30 years. By using upper gradients it is possible to define such notions as first order Sobolev spaces, p-harmonic functions and capacity on metric spaces. When generalising classical results to metric spaces, one often needs to impose some structure on the space by making additional assumptions, such as a doubling condition and a Poincaré inequality. In the included research paper, we study a certain type of metric spaces called bow-ties, which consist of two metric spaces glued together at a single designated point. For a doubling measure μ, we characterise when μ supports a Poincar´e inequality on the bow-tie, in terms of Poincaré inequalities on the separate parts together with a variational p-capacity condition and a quasiconvexity-type condition. The variational p-capacity condition is then characterised by a sharp measure decay condition at the designated point. We also study the special case when the bow-tie consists of the positive and negative hyperquadrants in Rn, equipped with a radial doubling measure. In this setting, we characterise the validity of the p-Poincaré inequality in various ways, and then provide a formula for the variational p-capacity of annuli centred at the origin.i

This Master Thesis focuses on denoising of Monte Carlo calculated dose distributions of radiosurgery treatment plans. The objective of this project is to implement a Denoising Autoencoder (DAE) and investigate its denoising performance when it has been trained on Monte Carlo calculated dose distributions generated with lower number of photon showers. The DAE is trained in a supervised setting to learn the mapping between corrupted observations and clean ones. The questions this thesis aims to answer are: (i) Can a DAE be used to denoise Monte Carlo calculated dose distributions, and thus predict the dose prior to a full simulation? Additionally, (ii) does incorporating prior knowledge of shot position increase the denoising performance? The results in this investigation have shown that the network successfully predicts the dose for low number of photon showers. In very heavy noise inputs the network denoising was in general successful, and the network could fill in missing data. The results indicated that the DAE could reduce the level of noise with an amount comparable with simulations that were done with 102 times more samples. / Denna masteruppsats fokus är på att brusreducera Monte Carlo-beräknade dosdis-tributioner för behandlingsplaner i hjärnstereotaktisk radiokirurgi. Projektets avsikt är att implementera en brusreducerande Autoencoder (DAE) samt undersöka dess brusreducerande egenskaper, när nätverket har tränats på Monte Carlo-beräknade dosdistributioner genererade med få fotonsimulationer. Den brusreducerande Au-toencodern har genomgått övervakad träning, där den lär sig en avbildning mellan brusiga till brusfria distributioner. Frågorna som denna uppsats ämnar besvara är;(i) Kan en brusreducerande Autoencoder användas för att brusreducera Monte Carlo-beräknade dosdistributioner, och därmed förutspå dosen på förhand? Dessutom, (ii) förbättras nätverkets brusreducerande prestanda när ytterligare information angående skottpositionerna tillförs till nätverket? Resultaten i denna undersökning pekade på att nätverket framgångsrikt förutspår dosdistributionerna, baserat på dosdistribu-tioner som simulerats med få fotonsimulationer. I de fall då bruset i indata är väldigt kraftigt lyckas fortfarande nätverket att brusreducera, samt lyckat fylla i data som saknas. Resultaten indikerade att den brusreducerande Autoencodern kunde reduc-era brus i en mängd som kan jämföras med en simulation som gjorts med en faktor 102 fler fotonsimulationer.

This thesis investigates if a model can be created to map misspelled answers from open-ended questions to a finite set of brands. The data used for the paper comes from the company Nepa that uses open-questions to measure brand-awareness and consists of misspelled answers and brands to be mapped to. A data structure called match candidate was created and consists of a misspelled answer and brand that it poten-tially be mapped to. Features for the match candidates were engineered and based on the edited distances, posterior probability and common misspellings among other. Multiple machine learning models were tested for classifying the match candidates as positive if the mapping was correct and negative otherwise. The model was tested in two scenarios, one when the answers in the training and testing data came from the same questions and secondly when they came from different ones. Among the classifiers tested, the random forest model performed best in terms of PPV as well as sensitivity. The resulting mapping identified on average 92% of the misspelled answers and map then with 98% accuracy in the first scenario. While in the second scenario 70% of the answers were identified with 95% confidence in the mapping on average. / Detta examensarbete undersöker huruvida en modell kan skapas för att kartlägga fel-stavade svar till öppna frågor till ett finit set av företagsnamn. Datan till denna uppsats kommer ifrån företaget Nepa som använder öppna frågor för att mäta märkesmedvetenhet. Denna data består av öppna svar samt företagsnamn som dessa kan matchas till. En datastruktur skapades som kallas för match candidate och består av ett felstavat svar samt ett företagsnamn som svaret kan matchas med. Attribut skapades till match candidate och bygger bland annat på sträng likhet, aposteriorisan-nolikhet samt vanliga fel stavningar med mera. Ett flertal maskininlärningsmodeller testades för att klassifiera match candidates som korrekt om och endast om svaret och företagsnamnet matchade och inkorrekt annars. Modellen testades i två olika scenarior. I det första kom datan som modellen tränade och testade på ifrån samma frågor. I det andra scenariot var det olika frågor som tränings och test data byggdes på. Av de maskininlärningsmodeller som testades så presterade radom forest modellen bäst i avseende på PPV och sensitivity. Den resulterande kartläggningen lyckades i genomsnitt identifiera 92% av alla felstavade svar och matchades i 98% till korrekt företagsnamn i det första scenariot. I det andra scenariot identifiera 70% av alla felstavade svar och matchades i 95% till korrekt företagsnamn i genomsnitt.

With more and more digital text-valued data available, the need to be able to cluster, classify and study them arises. We develop in this thesis statistical tools to perform null hypothesis testing and clustering or classification on text-valued data in the framework of Object-Oriented Data Analysis. The project includes research on semantic methods to represent texts, comparisons between representations, distances for such representations and performance of permutation tests. Main methods compared are Vector Space Model and topic model. More precisely, this thesis will provide an algorithm to compute permutation tests at document or sentence level to study the equality in terms of distribution of two texts for different representations and distances. Lastly, we describe the study of texts regarding a syntactic point of view and its structure with a tree representation. / Med ständigt ökande tillgänglighet av textvärd data ökar behovet att kunna klustra och klassificera denna data. I detta arbete utvecklar vi statistiska verktyg för hypotestestning, klustring och klassificering av textvärd data inom ramen för objektorienterad dataanalys. Projektet inkluderar forskning på semantiska metoder för att representera texter, jämförelser mellan representationer, avstånd för sådana representationer och prestanda hos permutationstest. De viktigaste metoderna som jämförs är vektorrumsmodeller och ämnesmodeller. Mer specifikt tillhandahåller detta arbete en algoritm för permutationstest, på dokument- eller meningsnivå, i syfte att pröva hypotesen att två texter har samma fördelning med avseende på olika representationer och avstånd. Till sist används en trädrepresentation för att beskriva studiet av texter ur en syntaktisk synvinkel.

The modeling of non-maturity deposits is a topic that is highlyimportant to many banks because of the large amount of funding that comes fromthese products. It is also a topic that currently is in the focus oflegislators. Although a non-maturity deposit may seem to be a trivial product,it has several characteristics that make it rather complex. One of the twopurposes of this thesis is to compare different models for the deposit rate ofnon-maturity deposits and to investigate the strengths and weaknesses of themodels. The other purpose is to find a new model for the deposit rate ofnon-maturity deposits. Several different models that are suggested in the literatureare described and evaluated based on the four aspects; goodness of fit,stability, negative interest rate environment and simplicity. Three new modelsfor the deposit rate are suggested in this thesis, one of which shows a verygood performance compared to the models that can be found in the literature. / Modellering av icke tidsbunden inlåning är ett ämne som är mycket viktigt för många banker på grund av den stora andel finansiering som kommer från dessa produkter. Det är också ett ämne som för närvarande väcker lagstiftares intresse. Även om icke tidsbunden inlåning kan tyckas vara en trivial produkt, har den flera egenskaper som gör den komplex. Ett av de två syftena med detta arbete är att jämföra olika modeller för kundräntan i icke tidsbunden inlåning och att undersöka modellernas styrkor och svagheter. Det andra syftet är att introducera en ny modell för kundräntan i icke tidsbunden inlåning. Flera olika modeller från litteraturen beskrivs och utvärderas baserat på de fyra utgångspunkterna passform, stabilitet, negativ räntemiljö och enkelhet. Tre nya modeller av kundräntor för icke tidsbunden inlåning föreslås i detta arbete, varav en visar ett mycket bra resultat jämfört med de modeller som föreslås i litteraturen.

This study is an attempt to model Swedish CPI inflation using ARIMA and variations of distributed lag model with market data as explanatory variables. The model will be constructed on the CPI subcomponents level and the results are aggregated to the CPI. Three approaches are tested in this report. In the first approach, only ARIMA model is used to model each of the subcomponents. In the second approach we use a distributed lag model (DLM) on subcomponents with significant correlation to the market data, the residual of the DLM is then modelled using ARIMA. In the third approach we use an restricted finite distributed lag model (RFDLM) instead of DLM. The study found that RFDLM was the best approach to model inflation with 20% RMSE compared to 32% of the naive forecast. However, there is little forecast potential using this approach due to the short lag of market data used as input. The model would be most useful in testing CPI inflation scenarios using predictions or assumptions of market data as input. / Denna studie är ett försök att modellera svensk inflation genom att använda ARIMA-modell och variationer av distributed lag modell med marknadsdata som förklarande variabler. Modellen är konstrukterad på underkomponents nivå av KPI och sedan aggregerad till KPI. Tre metoder prövas i denna studie. I första metoden modelleras underkomponenterna direkt med ARIMA-modeller. I andra metoden används distributed lag modell (DLM) på underkomponenter med signifikant korrelation till marknadsdata, residualen från DLM modelleras i sin tur med ARIMA-modeller. I den tredje metoden ersätter vi DLM med restricted finite distributed lag modell (RFDLM). Resultaten från studien visar att RFDLM är den bästa metoden att modellera inflationen och hade ett RMSE på 20 %. Detta jämfört med den naiva prognosen som hade en RMSE på 32 %. Däremot har RFDLM inte särskilt mycket praktiskt nytta i prognostisering av inflationen då man behöver marknadsdata för prognosperiod i förhand på grund att modellen använder sig av väldigt korta lagg. Däremot skulle modellen kunna ha nytta i scenario byggande med prognoser and antagande på marknadsdata som input.

A model of credit spreads variations, based on macroeconomic and market variables, has been developed and presented in this paper. Credit spreads of speculative and investment grade bonds have been investigated, leading us to a linear relationship between their quarterly variations. Thanks to their risk contribution we clearly identify government bond rates and a financial conditions index as the most significant variables. Hence, based on macroeconomic views on the market in 2017, we perform some predictions on future variations on spreads based on this model, displaying the flattening of high yield credit spreads and the widening of investment grade spreads in the long run. In addition, a cointegration relationship between spreads, rates and the ISM has been found, meaning that there exists a mean-reverting process representing the spread between credit spreads and a linear combination of these factors. As a consequence, thanks to this process we can conclude about the potential immediate tightening of credit spreads. / Vi studerar en modell för variationer i kreditspreadar, baserad på makroekonomiska och marknadsvariabler,  undersökningen av kreditspreadar av spekulativa och investment grade (dvs BBB klassade)  obligationer gav upphov till  ett linjärt förhållande mellan deras kvartalsvisa variationer. Tack vare deras riskbidrag identifierar vi tydligt Statsobligationsräntor och ett finansiellt förhållningsindex som de viktigaste variablerna. Därför baseras på en makroekonomiska syn på marknaden år 2017 utför vi vissa prediktioner om framtida variationer i spreadarna.