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

The ever-changing world of e-commerce prompted the case company to develop a new improved online store for its business functions, which prompted the need to also understand relevant metrics. The aim of the research is to find the customer behaviour metrics that have explanatory power for the response variable, which is the count of transactions. Examining these key metrics provide an opportunity to create a sustainable foundation for future analytics. Based on the results the case company can develop analytics, as well as understand the weaknesses and strengths of the online store. The data is from Google Analytics service and each variable receives a daily value, but the data is not treated as time series. The response variable is not normally distributed, so a linear model was not suitable. Instead, the natural choice was generalized linear models as they can also accommodate non-normally distributed response variables. Two different models were fitted, Poisson distributed, and Gamma distributed. The models were compared in many ways, but no clear difference between the models performance was found, so the results were combined from both models. The results provided by the models were quite similar, but there were differences. For this reason, the explanatory variables were divided into three categories: key variables, variables with differing results, and non-significant variables. Key variables have explanatory power for the response variable, and the results of the models were consistent. For variables with differing results, the results of the models were different, and for non-significant variables, there was no explanatory power for the response variable. This categorization facilitates understanding of the results. In total 6 explanatory variables were categorized as key variables, one as mixed result variable and two as non-significant. In conclusion it matters which variables are tracked if the efficiency of the web store is developed based on the efficiency of transactions.

In this thesis we present and prove Roger Penrose’s singularity theorem, which is a fundamental result in mathematical general relativity. In 1965 Penrose showed that in Einstein’s theory of general relativity, under certain general assumptions on the topology, curvature, and causal structure of a Lorentzian spacetime manifold, the spacetime manifold is null geodesically incomplete. At the time, Penrose’s theorem was highly topical in a longstanding debate on the question whether singularities are formed in the process of gravitational collapse. In the proof of the theorem, novel mathematical techniques were introduced in the study of Einstein’s theory of gravity, leading to further important developments in the mathematics of general relativity. Penrose’s theorem is built on the methods of semi-Riemannian geometry, in particular Lorentzian geometry. To lay the basis for later constructions, we therefore review the basic concepts and results of semi-Riemannian geometry needed in order to understand Penrose’s theorem. The discussion includes semi-Riemannian metrics, connection, curvature, geodesics, and semi-Riemannian submanifolds. Second, calculus of variations on semi-Riemannian manifolds is introduced and a set of results pertinent to Penrose’s theorem is given. The notion of focal point of a spacelike submanifold is defined and a proposition stating sufficient conditions for the existence of focal points is presented. Furthermore, we give a series of results that establish a relation between focal points of spacelike submanifolds and causality on a Lorentzian manifold. In the last chapter, we define a family of concepts that can be used to analyze the causal structure of Lorentzian manifolds. In particular, we define the notions of global hyperbolicity, Cauchy hypersurface, and trapped surface, which are central to Penrose’s theorem, and show some important properties thereof. Finally, Penrose’s theorem is stated and proved in detail.

The topological data analysis studies the shape of a space at multiple scales. Its main tool is persistent homology, which is based on other homology theory, usually simplicial homology. Simplicial homology applies to finite data in real space, and thus it is mainly used in applications. This thesis aims to introduce the theories behind persistent homology and its application, image completion algorithm. Persistent homology is motivated by the question of which scale is the most essential to study data shape. A filtration contains all scales we want to explore, and thus it is an essential tool of persistent homology. The thesis focuses on forming a filtaration from a Delaunay triangulation and its subcomplexes, alpha-complexes. We will found that these provide sufficient tools to consider homology classes birth and deaths, but they are not particularly easy to use in practice. This observation motivates to define a regional complement of the dual alpha graph. We found that its components' and essential homology classes' birth and death times correspond. The algorithm utilize this observation to complete images. The results are good and mainly as could be expected. We discuss that algorithm has potential since it does need any training or other input parameters than data. However, future studies are needed to imply it, for example, in three-dimensional data.

Tämä maisterintutkielma käsittelee jatkuvia sekä diskreettejä systeemejä, jotka johtavat differentiaaliyhtälöihin ja differenssiyhtälöihin. Tutkielmassa tarkastellaan ja tulkitaan populaatioita, joiden yksilöt kokevat häirintä- ja paikkakilpailua. Häirintäkilpailussa on käytetty esimerkkinä jänispopulaatiota ja paikkakilpailussa aavikkorottapopulaatiota. Populaatioiden kokoja määräävät lukuisat eri tekijät. Päästäksemme alkuun, ensin määritellään yksilötason tapahtumat, joista johdetaan yksilötason prosessit. On valittava paras mahdollinen malli kuvaamaan systeemin tärkeimpiä dynamiikkoja, joiden pohjalta tämä malli tehdään. Tässä vaiheessa on tehtävä rajauksia yksilöiden käytöksen vaikutuksesta populaatiotasoon. Tutkielmassa nähdään kuinka saadut mallit voivat erota huomattavasti toisistaan kun tulkitaan yksilötason käytöstä eri tavoilla. Yksilötason mallista muodostetaan populaatiotason prosesseja kuvaavat yhtälöt ja tulkitaan niitä. Tämä mahdollistaa populaation elinvoimaisuuden mallintamisen pitkän ajan päähän. Tässä on huomioitava, että mallin tarkkuutta voi heikentää paljonkin yksilötasolla tapahtuvat muutokset. Esimerkiksi ympäristön kantokyvyn muutokset tai vieraslajin saapuminen systeemiin vaikuttavat myös tarkasteltavien populaatioiden kokoihin. Tässä tutkielmassa ei käsitellä vieraslajien vaikutusta tarkasteltaviin populaatioihin. Populaatiotason mallin muodostamisen jälkeen, tarkastellaan paikkakilpailua kokevan populaation elinvoimaisuutta tasapainopisteissä ja tutkitaan niiden stabiilisuutta. Tästä tehdään faasikaaviot ja näytetään graaffisesti, miten populaatiotiheys kehittyy eri muuttujien arvoilla. Tutkielmassa on käyty läpi eri muuttujien arvoja, jolloin systeemin stabiilisuus muuttuu. Tutkielman lopussa käydään uudelleen läpi aiemmissa luvuissa esitettyjä malleja ja tehdään niistä uudet tulkinnat. Muodostetaan jatkuvien mallien sijasta diskreetit mallit. Havaitaan, että tulkinta erot tuottavat hyvinkin erilaisia malleja. Otetaan myös tarkempaan käsittelyyn logistinen differenssiyhtälö ja tarkastellaan sen stabiilisuuden muutoksia eli bifurkaatioita. Havaitaan, että käsitelty logistinen differenssiyhtälö voi ilmaista kaoottista käytöstä. Käydään graafisesti läpi syitä logistisen differenssiyhtälön kaaottiseen käytökseen.

In this thesis, we aim to introduce the reader to profinite groups. Profinite groups are defined by two characteristics: firstly, they have a topology defined on them (notably, they are compact). Secondly, they are constructed from some collection of finite groups, each equipped with a discrete topology and forming what is known as an inverse system. The profinite group emerges as an inverse limit of its constituent groups. This definition is, at this point, necessarily quite abstract. Thus, before we can really understand profinite groups we must examine two areas: first, we will study topological groups. This will give us the means to deal with groups as topological spaces. Topological groups have some characteristics that differentiate them from general topological spaces: in particular, a topological group is always a homogeneous space. Secondly, we will explore inverse systems and inverse limits, which will take us into category theory. While we could explain these concepts without categories, this thesis takes the view that category theory gives us a useful “50000-feet view” by giving these ideas a wider mathematical context. In the second chapter, we will go through preliminary information concerning group theory, general topology and category theory that will be needed later. We will begin with some basic concepts from group theory and point-set topology. These sections will mostly contain information that is familiar from the introductory university courses. The chapter will then continue by introducing some basic concepts of category theory, including inverse systems and inverse limits. For these, we will give an application by showing how the Cantor set is homeomorphic to an inverse limit of a collection of finite sets. In the third chapter, we will examine topological groups and prove some of their properties. In the fourth chapter, we will introduce an example of profinite groups: Zp, the additive group of p-adic integers. This will be expanded into a ring and then into the field Qp. We will discuss the uses of Zp and Qp and show how to derive them as an inverse limit of finite, compact groups.

In a quickest detection problem, the objective is to detect abrupt changes in a stochastic sequence as quickly as possible, while limiting rate of false alarms. The development of algorithms that after each observation decide to either stop and declare a change as having happened, or to continue the monitoring process has been an active line of research in mathematical statistics. The algorithms seek to optimally balance the inherent trade-off between the average detection delay in declaring a change and the likelihood of declaring a change prematurely. Change-point detection methods have applications in numerous domains, including monitoring the environment or the radio spectrum, target detection, financial markets, and others. Classical quickest detection theory focuses settings where only a single data stream is observed. In modern day applications facilitated by development of sensing technology, one may be tasked with monitoring multiple streams of data for changes simultaneously. Wireless sensor networks or mobile phones are examples of technology where devices can sense their local environment and transmit data in a sequential manner to some common fusion center (FC) or cloud for inference. When performing quickest detection tasks on multiple data streams in parallel, classical tools of quickest detection theory focusing on false alarm probability control may become insufficient. Instead, controlling the false discovery rate (FDR) has recently been proposed as a more useful and scalable error criterion. The FDR is the expected proportion of false discoveries (false alarms) among all discoveries. In this thesis, novel methods and theory related to quickest detection in multiple parallel data streams are presented. The methods aim to minimize detection delay while controlling the FDR. In addition, scenarios where not all of the devices communicating with the FC can remain operational and transmitting to the FC at all times are considered. The FC must choose which subset of data streams it wants to receive observations from at a given time instant. Intelligently choosing which devices to turn on and off may extend the devices’ battery life, which can be important in real-life applications, while affecting the detection performance only slightly. The performance of the proposed methods is demonstrated in numerical simulations to be superior to existing approaches. Additionally, the topic of multiple hypothesis testing in spatial domains is briefly addressed. In a multiple hypothesis testing problem, one tests multiple null hypotheses at once while trying to control a suitable error criterion, such as the FDR. In a spatial multiple hypothesis problem each tested hypothesis corresponds to e.g. a geographical location, and the non-null hypotheses may appear in spatially localized clusters. It is demonstrated that implementing a Bayesian approach that accounts for the spatial dependency between the hypotheses can greatly improve testing accuracy.

The Ising model is a classic model in statistical physics. Originally intended to model ferromagnetism, it has proven to be of great interest to mathematicians and physicists. In two dimensions it is sufficiently complex to describe interesting phenomena while still remaining analytically solvable. The model is defined upon a graph, with a random variable, called a spin, on each vertex. Other random variables may be defined as functions of spins. A classic problem of interest is the correlation of these random variables. A continuum analogue of the Ising model is possible through considering the scaling limit of the model, as the graph taken to approximate some domain e.g. in the complex plane or a torus. The core of this work is an exposition upon one method of calculating correlations of a random variable called a fermion defined in terms of spins and disorder random variables. The method is called Bosonization and associates correlations of some random variables to correlations of the Gaussian Free Field (GFF). The GFF is a random distribution, which approximately functions as a gaussian random variable whose covariance structure is given by Green's function. A result known as the Pfaffian-Hafnian identity is covered, to provide an example of an identity which may be derived using Bosonization on a continuum planar Ising model. A similar result is also presented on the Torus, using elliptic functions. These results are not original, but we present the only -- to us -- known explicit proofs based on hints from others. In the latter half of the work, Bosonization is approached using Random Current representation. Random currents give weights to each edge of the graph of the Ising model. Two other models are introduced: Alternating flows and the Dimer model. There are equivalence relations between the configuration of the Ising model, the Nesting Field of a random current and the height functions of an alternating flow and a dimer cover. Using these, correlations of random variables of the Ising model are given in terms of the height function of the Dimer model. The height function of the Dimer model is a discrete analogue of the GFF.

The Poisson regression is a well known generalized linear model that relates the expected value of the count to a linear combination of explanatory variables. Outliers affect severely the classical maximum likelihood estimator of the Poisson regression. Several robust alternatives for the maximum likelihood (ML) estimator have been developed, such as Conditionally unbiased bounded-influence (CU) estimator, Mallows quasi-likelihood (MQ) estimator and M-Estimators based on transformations (MT). The purpose of the thesis is to study robustness of the robust Poisson regression estimators in different conditions. Another goal is to compare their performance to each other. The robustness of the Poisson regression estimators is investigated by performing a simulation study, where the used estimators are the ML, CU, MQ and MT estimators. The robust estimators MQ and MT are studied with two different weight functions C and H and also without a weight function. The simulation is executed in three parts, where the first part handles a situation without any outliers, in the second part the outliers are in the X space and in the third part the outliers are in the Y space. The results of the simulation show that all the robust estimators are less affected by the outliers than the classical ML estimator, but nevertheless the outliers severely weaken the results of the CU estimator and the MQ based estimators. The MT based estimators and especially the MT and H-MT estimators have by far the lowest medians of the mean squared errors, when the data are contaminated with outliers. When there aren’t any outliers in the data, they compare favorably with the other estimators. Therefore the MT and H-MT estimators are an excellent option for fitting the Poisson regression model.

Työssä tutkitaan Hyvinkään sairaanhoitoalueen kustannuksia, sekä kokonaiskustannusten tasolla, että yksittäisen potilaan tasolla. Sairaanhoidon kustannukset ovat olennainen osa yhteiskunnan toimintaa ja ne vaikuttavat merkittävästi kuntien ja kaupunkien talouteen. Tämän takia on hyödyllistä pystyä ymmärtämään ja mallintamaan näitä kustannuksia. Aineistona on käytetty HUSilta saatua dataa kustannuslajeista, potilaista ja diagnoosiryhmistä. Tutkimuksen ensimmäinen tavoite on löytää tilastollinen malli, jolla voidaan ennustaa kokonaiskustannuksia. Toisena tavoitteena on löytää yksittäisten potilaiden kustannuksiin sopiva jakauma. Työn alussa esitellään todennäköisyysteoriaa ja tilastollisia menetelmiä, joita hyödynnetään tutkimuksessa. Näistä tärkeimmät ovat keskineliövirhe, aikasarjamalli ja tilastolliset testit. Näiden teorioiden avulla luodaan mallit kokonaiskustannuksille ja yksittäisen potilaan kustannuksille. Kokonaiskustannusten analysointi aloitetaan erottelemalla suurimmat kustannuslajit, jotta niiden tutkiminen olisi selkeämpää. Näihin isoimpiin kustannuslajeihin valitaan tärkeimmät selittävät muuttujat käyttämällä lineaarista regressiomallia ja informaatiokriteeriä. Näin saatujen muuttujien avulla voidaan muodostaa moniulotteinen aikasarjamalli kokonaiskustannuksille ja tärkeimmille muuttujille. Tämän mallin avulla voidaan luoda ennuste tulevaisuuden kustannuksista, kun se on validoitu muun aineiston avulla. Tutkielman viimeisessä osiossa tutustutaan tarkemmin paksuhäntäisiin jakaumiin, ja esitellään niiden tärkeimpiä ominaisuuksia. Paksuhäntäisillä jakaumilla suurien havaintojen todennäköisyys on merkittävästi suurempi kuin kevythäntäisillä. Tämä vuoksi niiden tunnistaminen on tärkeää, sillä paksuhäntäiset jakaumat voivat aiheuttaa merkittäviä kustannuksia. Termien esittelyn jälkeen tehdään visuaalista tarkastelua potilaiden kustannuksista. Tavoitteena on selvittää, mikä jakauma kuvaisi parhaiten potilaiden kustannuksia. Tutkimuksessa verrataan erilaisten teoreettisten jakaumien kuvaajia aineistosta laskettuun empiiriseen jakaumaan. Erilaisista kuvaajista voidaan päätellä, että kustannusten jakauma on paksuhäntäinen. Lisäksi huomataan, että havainnot sopisivat yhteen sen oletuksen kanssa, että jakauman häntä muistuttaa ainakin asymptoottisesti potenssihäntää. Työn lopussa perustellaan ääriarvoteoriaan nojaten, miksi potenssihännät ovat luonnollinen malli suurimmille kustannuksille.

Plane algebraic curves are defined as zeroes of polynomials in two variables over some given field. If a point on a plane algebraic curve has a unique tangent line passing through it, the point is called simple. Otherwise, it is a singular point or a singularity. Singular points exhibit very different algebraic and topological properties, and the objective of this thesis is to study these properties using methods of commutative algebra, complex analysis and topology. In chapter 2, some preliminaries from classical algebraic geometry are given, and plane algebraic curves and their singularities are formally defined. Curves and their points are linked to corresponding coordinate rings and local rings. It is shown that a point is simple if and only if its corresponding local ring is a discrete valuation ring. In chapter 3, the Newton-Puiseux algorithm is introduced. The algorithm outputs fractional power series known as Puiseux expansions, which are shown to produce parametrizations of the local branches of a curve around a singular point. In chapter 4, Puiseux expansions are used to study the topology of complex plane algebraic curves. Around singularities, curves are shown to have an iterated torus knot structure which is, up to homotopy, determined by invariants known as Puiseux pairs.