Browsing by master's degree program "Magisterprogrammet i matematik och statistik"

In statistics, data can often be high-dimensional with a very large number of variables, often larger than the number of samples themselves. In such cases, selection of a relevant configuration of significant variables is often needed. One such case is in genetics, especially genome-wide association studies (GWAS). To select the relevant variables from high-dimensional data, there exists various statistical methods, with many of them relating to Bayesian statistics. This thesis aims to review and compare two such methods, FINEMAP and Sum of Single Effects (SuSiE). The methods are reviewed according to their accuracy of identifying the relevant configurations of variables and their computational efficiency, especially in the case where there exists high inter-variable correlations within the dataset. The methods were also compared to more conventional variable selection methods, such as LASSO. The results show that both FINEMAP and SuSiE outperform LASSO in terms of selection accuracy and efficiency, with FINEMAP producing sligthly more accurate results with the expense of computation time compared to SuSiE. These results can be used as guidelines in selecting an appropriate variable selection method based on the study and data.

In this thesis we consider the inverse problem for the one-dimensional wave equation. That is, we would like to recover the velocity function, the wave speed, from the equation given Neumann and Dirichlet boundary conditions, when the solution to the equation is known. It has been shown that an operator Λ corresponding to the boundary conditions determines the volumes of the domain of influence, which is the set where the travel time for the wave is limited. These volumes then in turn determine the velocity function. We present some theorems and propositions about determining the wave speed and present proofs for a few of them. Artificial neural networks are a form of machine learning widely used in various applications. It has been previously proven that a one-layer feedforward neural network with a non-polynomial activation function with some additional constraints on the activation function can approximate any continuous real valued functions. In this thesis we present proof of this result for a continuous non-polynomial activation function. Furthermore, in this thesis we apply two neural network architectures to the volume inversion problem, which means that we train the networks to approximate a single volume when the operator Λ is given. The neural networks in question are the feedforward neural network and the operator recurrent neural network. Before the volume inversion problem, we consider a simpler problem of finding an inverse matrix of a small invertible matrix. Finally, we compare the performances of these two neural networks for both the volume and matrix inversion problems.

A continuous-time Markov chain is a stochastic process which has the Markov property. The Markov property states that the transition to a next state of the process only depends on the current state, that is, it does not depend on the process’ preceding states. Continuous-time Markov Chains are fundamental tools to model stochastic systems in finance and insurance such as option pricing and modelling insurance claim processes. This thesis examines continuous-time Markov chains and their most important concepts and typical properties. For instance, we introduce and investigate the Kolmogorov forward and backward equations, which are essential for continuous-time systems. However, the main aim of the thesis is to present a method and proof for constructing a Markov process from continuous transition intensity matrix. This is achieved by generating a transition probability matrix from given transition intensity matrix. When the transition intensities are known, the challenge is to determine the transition probabilities since the calculations can easily become difficult to solve analytically. Through the introduced theorem it becomes possible to simplify the calculations by approximations. In this thesis, we also make applications of the theory. We demonstrate how determining transition probabilities using Kolmogorov’s forward equations can become challenging in a simple setup. Furthermore, we will compare the approximations of transition probabilities derived from the main theorem to the actual transition probabilities. We make observations about the theorem’s transition probability function; the approximations derived from the main theorem provides quite satisfactory estimates of the actual transition probabilities.

Sileät monistot laajentavat matemaattisen analyysin keinoja euklidisista avaruuksista yleisemmille topologisille avaruuksille. De Rhamin lause lisää tähän vielä yhteyden algebralliseen topologiaan näyttämällä, että tietyt monistojen topologiset invariantit voidaan karakterisoida joko analyysin tai topologian keinoin. Toisin sanottuna moniston analyyttiset ominaisuudet paljastavat jotain sen topologisista ominaisuuksista ja päinvastoin. Tässä gradussa esitetään De Rhamin lauseelle kaksi todistusta. Ensimmäinen niistä todistaa lauseen sen klassisessa muodossa, joka vaatii vain monistojen ja singulaarihomologian perusteorian ymmärtämisen. Toinen todistus on muotoiltu hyvin yleisesti lyhteiden avulla; tarvittava lyhteiden teoria esitellään lähes kokonaan tekstissä. Tämä rakenne jakaa tekstin luontevasti kahtia. Ensimmäisessä osassa kerrataan ensin lyhyesti de Rhamin kohomologian ja singulaarihomologian perusteet. Seuraavaksi esitellään singulaarikohomologia sekä ketjujen integrointi monistoilla, jotka johtavat klassisen de Rhamin lauseen todistukseen. Toisessa osassa tutustutaan aluksi esilyhteiden ja lyhteiden teoriaan. Sitten esitellään lyhdekoho- mologiateoriat ja niiden yhteys ensimmäisen osan kohomologiaryhmiin. Lopulta näytetään, että kaikki lyhdekohomologiateoriat ovat yksikäsitteisesti isomorfisia keskenään. De Rhamin kohomologian ja singulaarikohomologian tapauksessa tälle isomorfismille annetaan lisäksi suoraviivainen konstruktio.

Children’s height and weight development remains a subject of interest especially due to increasing prevalence of overweight and obesity in the children. With statistical modeling, height and weight development can be examined as separate or connected outcomes, aiding with understanding of the phenomenon of growth. As biological connection between height and weight development can be assumed, their joint modeling is expected to be beneficial. One more advantage of joint modeling is its convenience of the Body Mass Index (BMI) prediction. In the thesis, we modeled longitudinal data of children’s heights and weights of the dataset obtained from Finlapset register of the Institute of Health and Welfare (THL). The research aims were to predict the modeled quantities together with the BMI, interpret the obtained parameters with relation to the phenomenon of growth, as well as to investigate the impact of municipalities on to the growth of children. The dataset’s irregular, register-based nature together with positively skewed, heteroschedastic weight distributions and within- and between-subject variability suggested Hierarchical Linear Models (HLMs) as the modeling method of choice. We used HLMs in Bayesian setting with the benefits of incorporating existing knowledge, and obtaining full posterior predictive distribution for the outcome variables. HLMs were compared with the less suitable classical linear regression model, and bivariate and univariate HLMs with or without area as a covariate were compared in terms of their posterior predictive precision and accuracy. One of the main research questions was the model’s ability to predict the BMI of the child, which we assessed with various posterior predictive checks (PPC). The most suitable model was used to estimate growth parameters of 2-6 year old males and females in Vihti, Kirkkonummi and Tuusula. With the parameter estimates, we could compare growth of males and females, assess the differences of within-subject and between-subject variability on growth and examine correlation between height and weight development. Based on the work, we could conclude that the bivariate HLM constructed provided the most accurate and precise predictions especially for the BMI. The area covariates did not provide additional advantage to the models. Overall, Bayesian HLMs are a suitable tool for the register-based dataset of the work, and together with log-transformation of height and weight they can be used to model skewed and heteroschedastic longitudinal data. However, the modeling would ideally require more observations per individual than we had, and proper out-of-sample predictive evaluation would ensure that current models are not over-fitted with regards to the data. Nevertheless, the built models can already provide insight into contemporary Finnish childhood growth and to simulate and create predictions for the future BMI population distributions.

Bonus-malus systems are used globally to determine insurance premiums of motor liability policy-holders by observing past accident behavior. In these systems, policy-holders move between classes that represent different premiums. The number of accidents is used as an indicator of driving skills or risk. The aim of bonus-malus systems is to assign premiums that correspond to risks by increasing premiums of policy-holders that have reported accidents and awarding discounts to those who have not. Many types of bonus-malus systems are used and there is no consensus about what the optimal system looks like. Different tools can be utilized to measure the optimality, which is defined differently according to each tool. The purpose of this thesis is to examine one of these tools, elasticity. Elasticity aims to evaluate how well a given bonus-malus system achieves its goal of assigning premiums fairly according to the policy-holders’ risks by measuring the response of the premiums to changes in the number of accidents. Bonus-malus systems can be mathematically modeled using stochastic processes called Markov chains, and accident behavior can be modeled using Poisson distributions. These two concepts of probability theory and their properties are introduced and applied to bonus-malus systems in the beginning of this thesis. Two types of elasticities are then discussed. Asymptotic elasticity is defined using Markov chain properties, while transient elasticity is based on a concept called the discounted expectation of payments. It is shown how elasticity can be interpreted as a measure of optimality. We will observe that it is typically impossible to have an optimal bonus-malus system for all policy-holders when optimality is measured using elasticity. Some policy-holders will inevitably subsidize other policy-holders by paying premiums that are unfairly large. More specifically, it will be shown that, for bonus-malus systems with certain elasticity values, lower-risk policy-holders will subsidize the higher-risk ones. Lastly, a method is devised to calculate the elasticity of a given bonus-malus system using programming language R. This method is then used to find the elasticities of five Finnish bonus-malus systems in order to evaluate and compare them.

The Finnish Customs collects and maintains the statistics of the Finnish intra-EU trade with the Intrastat system. Companies with significant intra-EU trade are obligated to give monthly Intrastat declarations, and the statistics of the Finnish intra-EU trade are compiled based on the information collected with the declarations. In case of a company not giving the declaration in time, there needs to exist an estimation method for the missing values. In this thesis we propose an automatic multivariate time series forecasting process for the estimation of the missing Intrastat import and export values. The forecasting is done separately for each company with missing values. For forecasting we use two dimensional time series models, where the other component is the import or export value of the company to be forecasted, and the other component is the import or export value of the industrial group of the company. To complement the time series forecasting we use forecast combining. Combined forecasts, for example the averages of the obtained forecasts, have been found to perform well in terms of forecast accuracy compared to the forecasts created by individual methods. In the forecasting process we use two multivariate time series models, the Vector Autoregressive (VAR) model, and a specific VAR model called the Vector Error Correction (VEC) model. The choice of the model is based on the stationary properties of the time series to be modelled. An alternative option for the VEC model is the so-called augmented VAR model, which is an over-fitted VAR model. We use the VEC model and the augmented VAR model together by using the average of the forecasts created with them as the forecast for the missing value. When the usual VAR model is used, only the forecast created by the single model is used. The forecasting process is created as automatic and as fast as possible, therefore the estimation of a time series model for a single company is made as simple as possible. Thus, only statistical tests which can be applied automatically are used in the model building. We compare the forecast accuracy of the forecasts created with the automatic forecasting process to the forecast accuracy of forecasts created with two simple forecasting methods. In the non-stationary-deemed time series the Naïve forecast performs well in terms of forecast accuracy compared to the time series model based forecasts. On the other hand, in the stationary-deemed time series the average over the past 12 months performs well as a forecast in terms of forecast accuracy compared to the time series model based forecasts. We also consider forecast combinations where the forecast combinations are created by calculating the average of the time series model based forecasts and the simple forecasts. In line with the literature, the forecast combinations perform overall better in terms of the forecast accuracy than the forecasts based on the individual models.

The fields of insurance and financial mathematics require increasingly intricate descriptors of dependency. In the realm of financial mathematics, this demand arises from globalisation effects over the past decade, which have caused financial asset returns to exhibit increasingly intricate dependencies between each other. Of particular interest are measurements describing the probabilities of simultaneous occurrences between unusually negative stock returns. In insurance mathematics, the ability to evaluate probabilities associated with the simultaneous occurrence of unusually large claim amounts can be crucial for both the solvency and the competitiveness of an insurance company. These sorts of dependencies are referred to by the term tail dependence. In this thesis, we introduce the concept of tail dependence and the tail dependence coefficient, a tool for determining the amount of tail dependence between random variables. We also present statistical estimators for the tail dependence coefficient. Favourable properties of these estimators are investigated and a simulation study is executed in order to evaluate and compare estimator performance under a variety of distributions. Some necessary stochastics concepts are presented. Mathematical models of dependence are introduced. Elementary notions of extreme value theory and empirical processes are touched on. These motivate the presented estimators and facilitate the proofs of their favourable properties.

In this thesis we study extension results related to compact bilinear operators in the setting of interpolation theory and more specifically the complex interpolation method, as introduced by Calderón. We say that: 1. the bilinear operator T is compact if it maps bounded sets to sets of compact closure. 2.\bar{ A} = (A_0,A_1) is a Banach couple if A_0,A_1 are Banach spaces that are continuously embedded in the same Hausdorff topological vector space. Moreover, if (Ω,\mathcal{A}, μ) is a σ-finite measure space, we say that: 3. E is a Banach function space if E is a Banach space of scalar-valued functions defined on Ω that are finite μ-a.e. and so that the norm of E is related to the measure μ in an appropriate way. 4. the Banach function space E has absolutely continuous norm if for any function f ∈ E and for any sequence (Γ_n)_{n=1}^{+∞}⊂ \mathcal{A} satisfying χ_{Γn} → 0 μ-a.e. we have that ∥f · χ_{Γ_n}∥_E → 0. Assume that \bar{A} and \bar{B} are Banach couples, \bar{E} is a couple of Banach function spaces on Ω, θ ∈ (0, 1) and E_0 has absolutely continuous norm. If the bilinear operator T : (A_0 ∩ A_1) × (B_0 ∩ B_1) → E_0 ∩ E_1 satisfies a certain boundedness assumption and T : \tilde{A_0} × \tilde{B_0} → E_0 compactly, we show that T may be uniquely extended to a compact bilinear operator T : [A_0,A_1]_θ × [B_0,B_1]_θ → [E_0,E_1]_θ where \tilde{A_j} denotes the closure of A_0 ∩ A_1 in A_j and [A_0,A_1]_θ denotes the complex interpolation space generated by \bar{A}. The proof of this result comes after we study the case where the couple of Banach function spaces is replaced by a single Banach space.

Tiivistelmä – Referat – Abstract Tässä työssä on tarkoituksena esittää Fukushiman hajotelma, jota voidaan käyttää yleistyksenä Itôn lemmalle. Ensimmäisessä luvussa käydään läpi perusteita stokastiselle analyysille. Työ etenee stokastisen analyysin perusteista Markovin prosesseihin ja tähän liittyviin käsitteisiin. Käydään läpi additiivisen funktionaalin käsite ja miten se liittyy käsiteltäviin prosesseihin. Martingaalien kohdalla käydään läpi peruskäsitteet. Tämän jälkeen siirrytään käsittelemään Itôn lemmaan ja tämän todistukseen. Itôn lemma on tärkeä työkalu taloustieteessä, etenkin kun työskennellään varallisuushintojen ja osakemarkkinoiden parissa. Itôn lemma luo pohja sille, kuinka varallisuushinnat voidaan määritellä Brownin liikkeen avulla. Samassa luvussa käsitellään myös muita hyödyllisiä stokastisen analyysin työkaluja. Yksi tällainen työkalu on Doobin-Meyer’n hajotelma martingaaleille ja ennustettavissa oleville prosesseille. Hajotelma on tärkeä työkalu, kun siirrytään korkeammalle tasolle stokastisten yhtälöiden kanssa. Ensimmäisen luvun lopussa käsitellään Sobolevin avaruutta, Dirichlet’n avaruutta ja Dirichlet’n muotoja. Näiden tarkoituksena on valmistaa lukijaa pohjatiedoiltaan seuraavaan lukuun, jossa käsitellään yhtä työn päälauseista Toisessa luvussa käsitellään additiivisen funktionaalin ja martingaaliadditiivisen funktionaalin energiaa ja Radon mittaa. Näiden käsittelyn jälkeen, siirrytään Itôn lemman yleistyksen pariin. Lopulta käsitellään yleistystä Itôn lemmalle. Yleistyksen pohjalla on mahdollisuus ottaa lauseesta “heikompi” versio, jolloin kaikkein vahvimpien ehtojen ja oletusten ei välttämättä tarvitse olla voimassa. Tämä on tärkeää, sillä Itôn lemman ehtona on jatkuvasti kahdesti differentioituvuus, joka ei läheskään aina toteudu stokastisissa prosesseissa. Näin ollen voidaan saavuttaa Itôn lemman edut kevyemmillä ehdoilla. Lopulta käsitellään Fukushiman hajotelmaa, joka on käytännöllinen prosesseille, jotka ovat semimartingaaleja. Fukushiman hajotelman avulla voidaan käsitellä tapauksia, joissa aiemmin käsiteltyjen lauseiden oletukset eivät täyty. Fukushiman hajotelma saadaan rakennettua aiemmin esitellyn lauseen avulla.