Browsing by master's degree program "Master 's Programme in Mathematics and Statistics"

In this thesis, we develop a Bayesian approach to the inverse problem of inferring the shape of an asteroid from time-series measurements of its brightness. We define a probabilistic model over possibly non-convex asteroid shapes, choosing parameters carefully to avoid potential identifiability issues. Applying this probabilistic model to synthetic observations and sampling from the posterior via Markov Chain Monte Carlo, we show that the model is able to recover the asteroid shape well in the limit of many well-separated observations, and is able to capture posterior uncertainty in the case of limited observations. We greatly accelerate the computation of the forward problem (predicting the measured light curve given the asteroid’s shape parameters) by using a bounding volume hierarchy and by exploiting data parallelism on a graphics processing unit.

Colorectal cancer (CRC) accounts for one in 10 new cancer cases worldwide. CRC risk is determined by a complex interplay of constitutional, behavioral, and environmental factors. Patients with ulcerative colitis (UC) are at increased risk of CRC, but effect estimates are heterogeneous, and many studies are limited by small numbers of events. Furthermore, it has been challenging to distinguish the effects of age at UC diagnosis and duration of UC. Multistate models provide a useful statistical framework for analyses of cancers and premalignant conditions. This thesis has three aims: to review the mathematical and statistical background of multistate models; to study maximum likelihood estimation in the illness-death model with piecewise constant hazards; and to apply the illness-death model to UC and CRC in a population-based cohort study in Finland in 2000–2017, considering UC as a premalignant state that may precede CRC. A likelihood function is derived for multistate models under noninformative censoring. The multistate process is considered as a multivariate counting process, and product integration is reviewed. The likelihood is constructed by partitioning the study time into subintervals and finding the limit as the number of subintervals tends to infinity. Two special cases of the illness-death model with piecewise constant hazards are studied: a simple Markov model and a non-Markov model with multiple time scales. In the latter case, the likelihood is factorized into terms proportional to Poisson likelihoods, which permits estimation with standard software for generalized linear models. The illness-death model was applied to study the relationship between UC and CRC in a population-based sample of 2.5 million individuals in Finland in 2000–2017. Dates of UC and CRC diagnoses were obtained from the Finnish Care Register for Health Care and the Finnish Cancer Registry, respectively. Individuals with prevalent CRC were excluded from the study cohort. Individuals in the study cohort were followed from January 1, 2000, to the date of first CRC diagnosis, death from other cause, emigration, or December 31, 2017, whichever came first. A total of 23,533 incident CRCs were diagnosed during 41 million person-years of follow-up. In addition to 8,630 patients with prevalent UC, there were 19,435 cases of incident UC. Of the 23,533 incident CRCs, 298 (1.3%) were diagnosed in patients with pre-existing UC. In the first year after UC diagnosis, the HR for incident CRC was 4.67 (95% CI: 3.07, 7.09) in females and 7.62 (95% CI: 5.65, 10.3) in males. In patients with UC diagnosed 1–3 or 4–9 years earlier, CRC incidence did not differ from persons without UC. When 10–19 years had passed from UC diagnosis, the HR for incident CRC was 1.63 (95% CI: 1.19, 2.24) in females and 1.29 (95% CI: 0.96, 1.75) in males, and after 20 years, the HR was 1.61 (95% CI: 1.13, 2.31) in females and 1.74 (95% CI: 1.31, 2.31) in males. Early-onset UC (age <40 years) was associated with a markedly increased long-term risk of CRC. The HR for CRC in early-onset UC was 4.13 (95% CI: 2.28, 7.47) between 4–9 years from UC diagnosis, 4.88 (95% CI: 3.46, 6.88) between 10–19 years, and 2.63 (95% CI: 2.01, 3.43) after 20 years. In this large population-based cohort study, we estimated CRC risk in persons with and without UC in Finland in 2000–2017, considering both the duration of UC and age at UC diagnosis. Patients with early-onset UC are at increased risk of CRC, but the risk is likely to depend on disease duration, extent of disease, attained age, and other risk factors. Increased CRC risk in the first year after UC diagnosis may be in part due to detection bias, whereas chronic inflammation may underlie the long-term excess risk of CRC in patients with UC.

Modal inclusion logic is modal logic extended with inclusion atoms. It is the modal variant of first-order inclusion logic, which was introduced by Galliani (2012). Inclusion logic is a main variant of dependence logic (Väänänen 2007). Dependence logic and its variants adopt team semantics, introduced by Hodges (1997). Under team semantics, a modal (inclusion) logic formula is evaluated in a set of states, called a team. The inclusion atom is a type of dependency atom, which describes that the possible values a sequence of formulas can obtain are values of another sequence of formulas. In this thesis, we introduce a sound and complete natural deduction system for modal inclusion logic, which is currently missing in the literature. The thesis consists of an introductory part, in which we recall the definitions and basic properties of modal logic and modal inclusion logic, followed by two main parts. The first part concerns the expressive power of modal inclusion logic. We review the result of Hella and Stumpf (2015) that modal inclusion logic is expressively complete: A class of Kripke models with teams is closed under unions, closed under k-bisimulation for some natural number k, and has the empty team property if and only if the class can be defined with a modal inclusion logic formula. Through the expressive completeness proof, we obtain characteristic formulas for classes with these three properties. This also provides a normal form for formulas in MIL. The proof of this result is due to Hella and Stumpf, and we suggest a simplification to the normal form by making it similar to the normal form introduced by Kontinen et al. (2014). In the second part, we introduce a sound and complete natural deduction proof system for modal inclusion logic. Our proof system builds on the proof systems defined for modal dependence logic and propositional inclusion logic by Yang (2017, 2022). We show the completeness theorem using the normal form of modal inclusion logic.

In the thesis we assess the ability of two different models to predict cash flows in private credit investment funds. Models are a stochastic type and a deterministic type which makes them quite different. The data that has been obtained for the analysis is divided in three subsamples. These subsamples are mature funds, liquidated funds and all funds. The data consists of 62 funds, subsample of mature funds 36 and subsample of liquidated funds 17 funds. Both of our models will be fitted for all subsamples. Parameters of the models are estimated with different techniques. The parameters of the Stochastic model are estimated with the conditional least squares method. The parameters of the Yale model are estimated with the numerical methods. After the estimation of the parameters, the values are explained in detail and their effect on the cash flows are investigated. This helps to understand what properties of the cash flows the models are able to capture. In addition, we assess to both models' ability to predict cash flows in the future. This is done by using the coefficient of determination, QQ-plots and comparison of predicted and observed cumulated cash flows. By using the coefficient of determination we try to explain how well the models explain the variation around the residuals of the observed and predicted values. With QQ-plots we try to determine if the values produced of the process follow the normal distribution. Finally, with the cumulated cash flows of contributions and distributions we try to determine if models are able to predict the cumulated committed capital and returns of the fund in a form of distributions. The results show that the Stochastic model performs better in its prediction of contributions and distributions. However, this is not the case for all the subsamples. The Yale model seems to do better in cumulated contributions of the subsample of the mature funds. Although, the flexibility of the Stochastic model is more suitable for different types of cash flows and subsamples. Therefore, it is suggested that the Stochastic model should be the model to be used in prediction and modelling of the private credit funds. It is harder to implement than the Yale model but it does provide more accurate results in its prediction.

Gaussiset prosessit ovat stokastisia prosesseja, joiden äärelliset osajoukot noudattavat multinormaa- lijakaumaa. Niihin pohjautuvien mallien käyttö on suosittua bayesiläisessä tilastotieteessä, sillä ne mahdollistavat monimutkaisten ajallisten tai avaruudellisten riippuvuuksien joustavan mallintami- sen. Gaussisen latentin muuttujan malleissa havaintojen oletetaan noudattavan ehdollista jakaumaa, joka riippuu priorijakaumaltaan gaussisen latentin prosessin saamista arvoista. Havaintoaineiston koostuessa kategorisista arvoista, ovat gaussisen latentin muuttujan mallit ovat laskennallisesti han- kalia, sillä latenttien muuttujien posteriorijakaumaa ei yleensä voida käsitellä analyyttisesti. Täl- löin posterioripäättelyyn on käytettävä analyyttisiä approksimaatioita tai numeerisia menetelmiä. Laskennalliset hankaluudet korostuvat entisestään, kun latentin gaussisen muuttujan kovarianssi- funktion parametreille asetetaan oma priorijakauma. Tässä työssä käsitellään approksimatiivisia menetelmiä, joita voidaan käyttää posterioripäättelyyn gaussisen latentin muuttujan malleissa. Työssä keskitytään pääasiallisesti usean luokan luokitte- lumalliin, jossa havaintomallina on softmax-funktio, mutta suuri osa esitellyistä ideoista on so- vellettavissa myös muille havaintomalleille. Tutkielmassa käsiteltäviä approksimatiivisia menetel- miä on kolme. Ensimmäinen menetelmä on bayesiläisessä tilastotieteessä usein käytetty, satunnai- sotantaan perustuva Markovin ketju Monte Carlo -menetelmä, joka on asymptoottisesti eksakti, mutta laskennallisesti raskas. Toinen menetelmä käyttää Laplace-approksimaatioksi kutsuttua ana- lyyttista approksimaatiota latentin muuttujan posteriorille, yhdessä Markovin ketju Monte Carlo -menetelmän kanssa. Kolmas menetelmä yhdistää Laplace-approksimaation ja hyperparametrien piste-estimoinnin. Käsiteltävien menetelmien perustana oleva teoria esitellään tutkielmassa mini- maalisesti, jonka jälkeen approksimatiivisten menetelmien suoriutumista vertaillaan usean luokan luokittelumallis- sa simuloidulla havaintoaineistolla. Vertailussa voidaan havaita Laplace-approksimaation vaikutus hyperparametrien, sekä latentin muuttujan posteriorijakaumiin.

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.

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.

This thesis studies equilibrium in a continuous-time overlapping generations (OLG) model. OLG models are used in economics to study the effect of demographics and life-cycle behavior on macroeconomic variables such as the interest rate and aggregate investment. These models are typically set in discrete time but continuous-time versions have also received attention recently for their desirable properties. Competitive equilibrium in a continuous-time OLG model can be represented as a solution to an integral equation. This equation is linear in the special case of logarithmic utility function. This thesis provides the necessary and sufficient conditions under which the linear equation is a convolution type integral equation and derives a distributional solution using Fourier transform. We also show that the operator norm of the integral operator is not generally less than one. Hence, the equation cannot be solved using Neumann series. However, in a special case the distributional solution is characterized by a geometric series on the Fourier side when the operator norm is equal to one.

Often in spatial statistics the modelled domain contains physical barriers that can have impact on how the modelled phenomena behaves. This barrier can be, for example, land in case of modelling a fish population, or road for different animal populations. Common model that is used in spatial statistics is a stationary Gaussian model, because of its computational requirements, relatively easy interpretation of results. The physical barrier does not have an effect on this type of models unless the barrier is transformed into variable, but this can cause issues in the polygon selection. In this thesis I discuss how the non-stationary Gaussian model can be deployed in cases where spatial domain contains physical barriers. This non-stationary model reduces spatial correlation continuously towards zero in areas that are considered as a physical barrier. When the correlation is selected to reduce smoothly to zero, the model is more likely to results similar output with slightly different polygons. The advantage of the barrier model is that it is as fast to train as the stationary model because both models can be trained using finite equation method (FEM). With FEM we can solve stochastic partial differential equations (SPDE). This method interprets continuous random field as a discrete mesh, and the computational requirements increases as the number of nodes in mesh increases. In order to create stationary and non-stationary models, I have described the required methods such as Bayesian statistics, stochastic process, and covariance function in the second chapter. I use these methods to define spatial random effect model, and one commonly used spatial model is the Gaussian latent variable model. At the end of second chapter, I describe how the barrier model is created, and what types of requirements this model has. The barrier model is based on a Matern model that is a Gaussian random field, and it can be represented by using Matern covariance function. The second chapter ends with description of how to create a mesh mentioned above, and how the FEM is used to solve SPDE. The performance of stationary and non-stationary Gaussian models are first tested by training both models with simulated data. This simulated data is a random sample from polygon of Helsinki where the coastline is interpreted as a physical barrier. The results show that the barrier model estimates the true parameters better than the stationary model. The last chapter contains data analysis of the rat populations in Helsinki. The data contains number of rat observations in each zip code, and a set of covariates. Both models, stationary and non-stationary, are trained with and without covariates, and the best model out of these four models was the stationary model with covariates.

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.