LTMS magistritööd -- Master's theses
Selle kollektsiooni püsiv URIhttps://hdl.handle.net/10062/50402
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Kirje Predicting annual returns of individual stocks on the Baltic Stock Exchange(Tartu Ülikool, 2025) Künnapas, Indrek; Kangro, Raul, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThe objective of this master’s thesis is to investigate possibilities of deploying statistical models in predicting annual returns of stocks on the Baltic Stock Exchange. Both, theoretical and practical aspects of linear regression, fixed effects, random effects, mixed effects and auto regressive model, are illustrated.Kirje Kaskokindlustuse kindlustuskahjude hindamine(Tartu Ülikool, 2025) Tomson, Mikk; Kangro, Raul, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutKäesoleva magistritöö eesmärk on modelleerida kaskokindlustuse kahjusid. Töös antakse esmalt ülevaade üldistatud lineaarsetest ja üldistatud aditiivsetest mudelitest ning potentsiaalsetest jaotustest. Seejärel sobitatakse tutvustatud meetodide põhjal mudelid LHV Kindlustuse kaskokindlustuse andmetele. Kahjude modelleerimine toimub kaheastmeliselt: esmalt modelleeritakse kahju sagedust ning seejärel hüvitise suurust tingimusel, et kahju on juba juhtunud.Kirje Model selection and AIC(Tartu Ülikool, 2025) Guliyev, Farhad; Sova, Joonas, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThis master thesis investigates the role of Akaike Information Criterion (AIC) in choosing statistical models, combining theoretical foundations with practical simulations to evaluate its effectiveness. The theoretical part of the thesis shows that AIC is based on Kullback-Leibler divergence and cross-entropy, emphasizing its role in minimizing information loss while maintaining a balance between model fit and complexity. In practical analysis, samples from parametric distributions supported on unit interval are simulated to evaluate the effectiveness of AIC in correctly identifying the true model among various competing alternatives. The results demonstrate that AIC successfully prevents overfitting by penalizing excessive parameters. Moreover, the asymptotic behavior of AIC is also analyzed to see if and how the probability of choosing the correct model converges as the sample size increases. In addition, AIC bias-corrected estimate of the cross-entropy is analyzed and compared with other bias-corrected estimates in order to evaluate its effectiveness in reducing bias. Overall, this thesis demonstrates the importance of AIC in model selection by optimizing the trade-off between model fit and complexity.Kirje Structural time series models in GDP analysis(Tartu Ülikool, 2025) Kippar, Kätlin; Raus, Toomas, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThe aim of this thesis is to use structural time series models to estimate the business cycles of Estonia and its five key trading partners, with additional objectives of examining potential economic dependencies between the estimated cycles and evaluating the forecast accuracy of structural models. To address these research questions, structural models with both trigonometric and ARMA cycle formulations were applied to GDP time series data, alongside an alternative cycle estimation method of the Hodrick-Prescott (HP) filter. The resulting cycle estimates were further used to test Granger causality. Additionally, ARIMA models were estimated for comparative purposes in forecasting evaluation. The thesis includes a brief overview of gross domestic product (GDP) and business cycles, a comprehensive overview of the applied methods, a summary of relevant previous research, and the results of the analysis. As a result of this thesis, business cycles were successfully estimated for all the countries considered in the analysis, economic dependencies between Estonia and its trading partners were identifies, and the forecast accuracy of the models was assessed.Kirje Volatility modeling of asset returns(Tartu Ülikool, 2025) Michael, Alex Chiwete; Raus, Toomas, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThe research investigates financial market volatility modeling through an analysis of daily stock price data from Tallink Grupp together with OMX Baltic Index data spanning from 01 February 2007 until 10 October 2023. Financial econometric theory guides the analysis through the combination of Autoregressive Moving Average (ARMA) models with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) frameworks to properly model heteroskedasticity. This research evaluates asymmetric GARCH extensions including TGARCH, EGARCH and other asymmetric variants to account for the leverage effect and examine how market shocks affect volatility differently based on their positive or negative nature. The ‘rugarch‘ package in R serves as a tool and provides a robust and flexible framework for specifying, fitting, and comparing various volatility models. The research further investigates heteroskedasticity and asymmetry characteristics in the volatility dynamics of Tallink Grupp stock prices and OMX Baltic Index data across three economic periods: the global financial crisis (2007–2010), the stable market phase (2011–2019), and the COVID-19 pandemic (2020–2023). The research provides both theoretical and practical value by advancing knowledge about risk-trade-off and the relationship between expected returns and associated risk. Advanced GARCH and asymmetric models which further support the performance of the models in capturing market shocks and volatility effects.Kirje Claims frequency modeling and model comparison for travel insurance(Tartu Ülikool, 2025) Paarmets, Minni-Marii; Kangro, Raul, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutTravel insurance is a widely used insurance product covering various risks that might occur while traveling. The aim of this thesis is to model the frequency of travel insurance claims using generalized linear models and different types of metrics to determine which type of model performs best. The thesis is split into three parts, where the first gives an overview of the background, the data, and travel insurance. The second part describes the theory of the models and model comparison techniques. The third part details the analysis of the models and their comparisons.Kirje Forecasting loan default rates using macroeconomic variables(Tartu Ülikool, 2025) Pool, Stein-Marten; Käärik, Meelis, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThe aim of this thesis is to model and forecast Estonian loan default rates using macroeconomic variables. Debt portfolio analysis and forecasting are important parts of credit risk management and are used in capital planning, risk assessment and scenario analysis. The selection of the final model and its suitability with the financial supervisors are discussed, highlighting the pros and cons of machine learning and simpler statistical models. The thesis also highlights the real-life value of default rate forecasting and describes its usage from the perspective of an Estonian bank - Coop Pank.Kirje Lyme disease: modeling and analyzing long-term costs related to the infection based on Estonian Biobank data(Tartu Ülikool, 2025) Raak, Sten; Abner, Erik, juhendaja; Fischer, Krista, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutLyme disease is a prevalent vector-borne illness with significant public health implications due to its potential for multisystem effects and persistent symptoms. Understanding the associated economic burden is crucial for healthcare planning. This thesis investigates the temporal dynamics of healthcare costs surrounding a Lyme disease diagnosis, aiming to quantify whether cost increases are primarily acute or persist over a longer period, which contributes to understanding the extended healthcare services utilization potentially linked to the condition. Using longitudinal health data from the Estonian Biobank, this study uses a relative time scale indexed to the year of first diagnosis. Linear Mixed-Effects (LME) models serve as the primary analytical framework to handle correlated repeated measures and model cost trajectories. The analysis compares diagnosed individuals to a reference group over a defined time window surrounding diagnosis. The thesis includes a background on the disease and methods, details the analysis, and presents results within the Estonian context.Kirje Arithmetic Asian options pricing using general lattice method(Tartu Ülikool, 2025) Sulojeva, Anastasija; Raus, Toomas, juhendaja; Lätt, Kaido, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThe aim of this thesis is to study the pricing of path-dependent discrete arithmetic Asian options. In the case of arithmetic Asian options, the option price cannot be found analytically and various numerical methods must be used to find the price. The theoretical part of this thesis focuses on the general lattice method proposed in 2020 by Gambaro, Kyriakou, and Fusai for pricing European and American style Asian options with fixed and floating strike prices. The novelty of this approach is that by a change of numeraire, lattice becomes one dimensional, while previous lattice methods were two dimensional. In addition, this thesis also examines the pricing of Asian options using Monte Carlo simulations and Hull-White method. Numerical examples compare the accuracy and computation speed of option prices obtained by general lattice method, the Hull-White method and the Monte Carlo method.Kirje Comparative analysis of traditional time series, machine learning, deep learning and hybrid models for profit forecasting in financial markets(Tartu Ülikool, 2025) Vaask, William; Raus, Toomas, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThis thesis compared the forecasting performance of traditional time series, machine learning, deep learning and hybrid models on daily banking profit data in financial markets area aggregated on three different levels. To evaluate different methods, this thesis used a novel performance metric - corrected mean average scaled error (cMASE), which improves interpretability of MASE by using T one-step naive forecasts instead of T −1, which results in naive method always having a score of cMASE = 1. Despite advancements in computational power, traditional time series method SARIMA still outperformed other models, also showing the most consistent results between average cross-validation cMASE and testing cMASE. For best hybrid models, gradient boosting methods complemented SARIMA by correcting forecasts using long lags, rolling means and standard deviations. While SARIMA models required refitting after every forecast, the machine learning, deep learning and non-linear parts of hybrid models performed best when refit only on average once every two weeks, which reduced the overall computing cost significantly.Kirje Parameter estimation in progressive multistate models(Tartu Ülikool, 2025) Xu, Zhen; Lember, Jüri, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutMultistate models are widely used to analyze systems or entities that transition between different states over time. These models have applications in various fields, including survival analysis, reliability engineering, and financial mathematics. This thesis will focus on parameter estimation for time-homogeneous progressive Markov chains, where transitions are limited to higher-numbered states, and the final state is absorbing. The thesis can be divided into two parts, the first part focusing on normal multistate models and the other part focusing on hidden multistate models. In the first part, we study progressive, time-homogeneous multistate Markov chains. During this part, maximum likelihood estimators (MLEs) for transition probabilities are derived for uncensored, fixed-censored, and random-censored scenarios. Consistency proofs for these estimators are provided using theoretical tools such as the Strong Law of Large Numbers and the Continuous Mapping Theorem. The second part focuses on hidden multistate models, specifically the twostate hidden Markov chains. In this case, the underlying states are not fully observable: In each observation, each position of the chain has a constant probability of being observed. We study the block structure of the observations and derive statistical properties of block lengths under different censoring mechanisms. Here we only have one transition probability to estimate, moment-based estimators and maximum likelihood estimators are developed for the transition probability, and their consistency is validated through simulations. Simulation experiments are also conducted for both parts to evaluate the performance of the derived estimators under various parameter configurations and censoring mechanisms. The results confirm the consistency of all proposed estimators.Kirje Absoluutriski prognoosimine konkureerivate riskide korral: meetodite võrdlus teise tüübi diabeedi näitel(Tartu Ülikool, 2025) Teder, Karmel; Fischer, Krista, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutKäesoleva magistritöö eesmärk on anda ülevaade konkureerivaid riske arvesse võtvatest haigusriski hindamise meetoditest ning võrrelda neist peamiseid Tartu Ülikooli Eesti geenivaramu andmetel teise tüübi diabeedi riski hindamisel. Peamiselt soovitakse näha, millise mudeliga saaks kõige paremini hinnata vanemaealiste inimeste teist tüüpi diabeeti haigestumise riski. Töös võrreldakse omavahel Coxi võrdeliste riskide mudelit, Fine-Gray võrdeliste alamriskide mudelit ning pseudovaatluste abil hinnatud lineaarset riskimudelit. Mudelite abil hinnatakse teise tüübi diabeeti haigestumise 10 aasta absoluutriski, võttes seejuures arvesse võimalust enne haigestumist surra. Töö tulemusena selgub, et kõige täpsemalt võimaldab riski hinnata konkureerivaid riske arvesse võttev Coxi võrdeliste riskide mudel.Kirje Optimization of parameters of a portfolio reinsurance in non-life insurance(Tartu Ülikool, 2025) Miriyeva, Arzu; Chepizhko, Oleksandr, juhendaja; Tverdostup, Maryna, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThis thesis assesses the optimization of reinsurance parameters within a non-life insurance portfolio, focusing specifically on determining the optimal retention levels for two insurance products. The research aims to balance the dual objectives of risk reduction and profitability enhancement, which are crucial for the financial stability and growth of insurance companies. From a broader perspective, it was noted that this trade-off would determine an optimal level of retention which can be applied to both products. To achieve this objective, it was employed two methodologies, utility optimization theory and value at risk. The first one helps determine how much should be kept from what is earned. In contrast, the second one identifies potential maximum loss. It was shown that an increased level of retention may bring about higher profits but at the same time expose the company to significant financial loss calling for a balanced approach in reinsurance.Kirje Real Estate Indexation Model based on Estonian Land Board summary statistics(Tartu Ülikool, 2024) Kodasma, Margus; Puusepp, Johannes, juhendaja; Möls, Märt, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThe objective of this thesis is to develop a real estate indexation model that accurately predicts the value of real estate collateral. The resulting real estate indexation model is essential for adjusting collateral valuations in response to market changes. The model developed during this thesis will be widely used within the bank for portfolio credit risk assessment, serving as an input for internal Loss Given Default estimates or capital requirement calculations. The thesis is divided into two parts. The first part gives background on topics necessary for understanding the subsequent chapters, describing the market price index used for real estate indexation and the principles of clustering. The second part presents the results of time series clustering and develops indexation models based on these results to describe the value of collateral in relation to the current market conditions.Kirje Predicting loan default with XGBoost: an examination of strength and application(Tartu Ülikool, 2024) Felt, Grayson Steven; Kangro, Raul, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutThis thesis explores the application of the XGBoost algorithm for predicting loan defaults, a vital aspect of credit risk management. By leveraging advanced machine learning techniques, the study aims to improve the accuracy and reliability of default predictions over traditional methods. We begin with an overview of fundamental machine learning concepts, including loss functions and tree-based models, which sets the stage for a detailed examination of gradient boosting and its implementations. The focus then shifts to XGBoost, where we delve into its objective function, optimization process, and hyperparameters. Using a publicly available dataset from Bondora, we conduct thorough data preprocessing, followed by careful hyperparameter tuning using grid search and cross-validation. Our results highlight XGBoost’s ability to handle complex, real-world data effectively, resulting in significant improvements in prediction performance. This study illustrates the importance of sophisticated algorithms in advancing the field of financial predictive analytics.Kirje Osaliste moodulite tensorkorrutis(Tartu Ülikool, 2024) Saarse, Heleen; Laan, Valdis, juhendaja; Väljako, Kristo, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutKäesolevas magistritöös uuritakse osalisi mooduleid üle ringide ja nende tensorkorrutist ning selle omadusi. Töös defineeritakse nii vasak- ja parempoolsed osalised moodulid kui ka osalised bimoodulid ning nende homomorfismid. Järgmiseks defineeritakse osaliste moodulite osalised faktormoodulid ning sõnastatakse ja tõestatakse homomorfismiteoreem osaliste moodulite jaoks. Edasi defineeritakse osaliste moodulite tensorkorrutis ja osaliste moodulite homomorfismide tensorkorrutis. Tõestatakse tensorkorrutisega seotud omadused, osaliste moodulite hom-funktori ja tensorfunktori olemasolu ning lõpuks viimaste adjunktsioon. Tulemused on üldistused tulemustele K. Väljako raamatus „Idempotentsete ringide Morita ekvivalentsus“ ning analoogid tulemustele H. Saarse ja K. Väljako artiklis „Tensor product of partial acts“.Kirje Caputo murrulist tuletist sisaldava algväärtusülesande lahendamisest(Tartu Ülikool, 2024) Määrits, Erik-Jürgen; Pedas, Arvet, juhendaja; Vikerpuur, Mikk, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutKäesolevas magistritöös vaadeldakse Caputo murrulist tuletist sisaldava algväärtusülesande lahendamist tükiti polünomiaalse kollokatsioonimeetodi abil. Põhjendatakse selle meetodi koondumist ja koondumise järku. Tuuakse sisse itereeritud meetod lähislahendi leidmiseks ja uuritakse selle koondumise järku. Vaadeldakse ka ühe konkreetse näidisülesande lahendamist.Kirje Globalizations of strong partial acts over monoids(Tartu Ülikool, 2024) Luhaäär, Urmas; Laan, Valdis, juhendaja; Kudryavtseva, Ganna, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutOlgu S monoid. Tugevaks osaliseks S-polügooniks nimetame sellist osalist S-polügooni, mis tekib kui jätame mingist globaalsest S-polügoonist mingid elemendid ära. Kui A on osaline polügoon, B mingi A globalisatsioon ja B on moodustatud S-polügoonina A elementide poolt, siis ütleme, et B on A-genereeritud. Kellendonk ja Lawson on näidanud, et kui S on rühm, siis igal tugeval osalisel S-polügoonil leidub ühene A-genereeritud globalisatsioon. See aga ei kehti monoidide juhul. Laan ja Kudryavtseva on andnud kaks konstruktsiooni osaliste polügoonide globaliseerimiseks üle poolrühmade, mis ei pruugi olla isomorfsed: tensor-globalisatsioon A ⊗ S ja hom-hulkglobalisatisoon AS. Selles magistritöös defineerime hom-hulk-globalisatsiooni osaliste S-polügoonide morfismidel nii, et saame kovariantse täpse funktori. See funktor aga ei ole reflektor ega koreflektor. Siis näitame, et A-genereeritud globalisatsioonide isomorfismiklassid moodustavad täieliku võre, mis on duaalselt isomorfne võre Con A⊗S alamvõrega. Lõpuks näitame, et ainsad monoidid, mille puhul kõik tugevad osalised polügoonid on üheselt globaliseeritavad on rühmad.Kirje Troopiline algebra ja maatriksid üle järjestatud Abeli rühmade(Tartu Ülikool, 2024-06) Kutti, Marilyn; Laan, Valdis, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutTöös vaadeldakse maatrikseid üle lineaarselt järjestatud Abeli rühma A ja üle kommutatiivse monoidi A?, mis on saadud rühmast A välise vähima elemendi ? lisamisel. Analüüsitakse teist järku ruutmaatriksite poolrühmade (M_2(A); ·) ja (M_2(A?); ·) struktuuri, kus maatriksite korrutamine on defineeritud sarnaselt troopiliste maatriksite korrutamisega. Antakse poolrühma (M_2(A), ·) Greeni seoste R, L ja H kirjeldus ning poolrühma (M_2(A?), ·) idempotentide kirjeldus. Näidatakse, et poolringis M_2(A?) on kõik nullist erinevad idempotendid täisidempotendid.Kirje Banachi ruumi ühikkera plastilisus(Tartu Ülikool, 2024-07) Leo, Nikita; Haller, Rainis, juhendaja; Tartu Ülikool. Loodus- ja täppisteaduste valdkond; Tartu Ülikool. Matemaatika ja statistika instituutSelles magistritöös tõestatakse mõned Banachi ruumi ühikkera plastilisusega seonduvad tulemused (nimetame meetrilist ruumi M plastiliseks, kui iga kaugusi mittesuurendav bijektsioon f : M → M on tegelikult isomeetria). Esimeses peatükis loetletakse vajalikke eelteadmisi. Teises peatükis tõestatakse ruumi ℓ_1 ⊕_p R ühikkera plastilisus p ∈ (1, ∞) korral. Kolmandas peatükis vaadeldakse lõpliku arvu rangelt kumerate Banachi ruumide ℓ∞-summa ühikkera plastilisust. Tõestatakse, et kahe rangelt kumera Banachi ruumi ℓ∞-summal on plastiline ühikkera ning et suvalise lõpliku arvu liidetavate korral on mittelaiendava bijektsiooni F : B_X → B_X isomeetrilisus tagatud lisaeeldusega F(S_X) ⊂ S_X või F(ext B_X) ⊂ ext B_X.