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

Tässä tutkielmassa esitetään logistisen regressiomallin teoriaa sekä havainnollistetaan sen soveltuvuutta terveystieteelliseen tutkimukseen. Tutkielman tarkoituksena on tarkastella logistisen regressiomallin parametrin estimaattien tulkintaa. Mallin estimaatteja voidaan tulkita kolmen eri metriikan avulla mutta usein tarkastelut rajoittuu vain yhteen. Tutkielmassa käydään läpi kaikki kolme metriikkaa, eli todennäköisyys-, logit- sekä ristisuhdemetriikka ja tarkastelaan näitä teorian ja empiirisen esimerkin avulla. Esimerkissä käytetty aineisto koostuu THL:n Kouluterveyskyselyyn vastanneiden vantaalaisten 8. ja 9. luokan oppilaiden vastauksista ja on tehty yhteistyössä Vantaan kaupungin kanssa. Tutkielman analyysit on tehty Stata ohjelmistolla minkä käytöstä esitetään muutama esimerkki. Tutkielman alussa käydään läpi logistisen regressiomallin teoriaa kuten yleistettyjen lineaaristen mallien teoriaa sekä mallin sovitus suurimman uskottavuuden menetelmällä. Tämän jälkeen käydään läpi metriikan valintaa ja tulkintaa sekä nostetaan esiin myös mallin yhteisvaikutustermin tulkintaan liittyviä huomioita. Tutkielman lopussa havainnollistetaan logistisen regressiomallin soveltuvuutta laadullisiin tutkimuskysymyksiin. Analyyseissä keskitytään tarkastelemaan ilmeneekö terveyden kokemuksessa eroja ulkomaalaistaustaisten ja suomalaistaustaisten nuorten välillä ja muuttaako perheen resursseihin ja elintapoihin liittyvien muuttujien lisääminen malleihin näitä havaintoja. Mallin kolmen eri metriikan teoreettinen sekä empiirinen tarkastelu osoittavat, että tulkinta on riippuvainen metriikan valinnasta mutta tehtävät johtopäätökset eivät välttämättä ole riippuvaisia metriikasta. Erityisesti laadullisen tulkinnan kannalta on haastavaa muuttujien yhteyksien suuruuden tulkinta sekä tilastollisen merkitsevyyden toteamisessa ilmenee eroja. Vaikka tulkinta on riippuvainen metriikan valinnasta oli tutkielmassa laadulliset johtopäätökset kuitenkin lopulta samankaltaiset. Logistisen regressiomallin analyysit toivat siis esiin samankaltaiset päätelmät, riippumatta käytettävästä metriikasta. Analyysit osoittavat, että Vantaalla nuoren ulkomaalaistausta ei ole vahva selittävä tekijä nuoren terveyden kokemukselle. Kuitenkin sukupolvien välillä ilmenee merkitseviä eroja suomalaistaustaisiin nuoriin verratuna. Nuorten kokemus perheen huonosta taloudellisesta tilanteesta sekä arkeen kuuluvien terveyteen positiivisesti vaikuttavien elintapojen puuttuminen selittivät merkitsevän osan nuorten terveyden kokemuksesta.

In this thesis, we prove the existence of a generalization of the matrix product state (MPS) decomposition in infinite-dimensional separable Hilbert spaces. Matrix product states, as a specific type of tensor network, are typically applied in the context of finite-dimensional spaces. However, as quantum mechanics regularly makes use of infinite-dimensional Hilbert spaces, it is an interesting mathematical question whether certain tensor network methods can be extended to infinite dimensions. It is a well-known result that an arbitrary vector in a tensor product of finite-dimensional Hilbert spaces can be written in MPS form by applying repeated singular value or Schmidt decompositions. In this thesis, we use an analogous method in the infinitedimensional context based on the singular value decomposition of compact operators. In order to acquire sufficient theoretical background for proving the main result, we first discuss compact operators and their spectral theory, and introduce Hilbert-Schmidt operators. We also provide a brief overview of the mathematical formulation of quantum mechanics. Additionally, we introduce the reader to tensor products of Hilbert spaces, in both finite- and infinite-dimensional contexts, and discuss their connection to Hilbert-Schmidt operators and quantum mechanics. We also prove a generalization of the Schmidt decomposition in infinite-dimensional Hilbert spaces. After establishing the required mathematical background, we provide an overview of matrix product states in finite-dimensional spaces. The thesis culminates in the proof of the existence of an MPS decomposition in infinite-dimensional Hilbert spaces.

A major innovation in statistical mechanics has been the introduction of conformal field theory in the mid 1980’s. The theory postulates the existence of conformally invariant scaling limits for many critical 2D lattice models, and then uses representation theory of a certain algebraic object that can be associated to these limits to derive exact solvability results. Providing mathematical foundations for the existence of these scaling limits has been a major ongoing project ever since, and lead to the introduction of Schramm-Löwner evolution (or SLE for short) in the early 2000’s. The core insight behind SLE is that if a conformally invariant random planar curve can be described by Löwner evolution and fulfills a condition known as the domain Markov property, it must be driven by a Wiener process with no drift. Furthermore, the variance of the Wiener process can be used to define a family SLE𝜅 of random curves which are simple, self-touching or space-filling depending on 𝜅 ≥ 0. This combination of flexibility and rigidity has allowed the scaling limits of various lattice models, such as the loop-erased random walk, the harmonic explorer, and the critical Ising model with a single interface, to be described by SLE. Once we move (for example) to the critical Ising model with multiple interfaces it turns out that the standard theory of SLE is inadequate. As such we would like establish the existence of multiple SLE to handle these more general situations. However, conformal invariance and the domain Markov property no longer guarantee uniqueness of the object so the situation is more complicated. This has led to two main approaches to the study of multiple SLE, known as the global and local approaches. Global methods are often simpler, but they often do not yield explicit descriptions of the curves. On the other hand, local methods are far more involved but as a result give descriptions of the laws of the curves. Both approaches have lead to distinct proofs that the laws of the driv- ing terms of the critical Ising model on a finitely-connected domain are described by multiple SLE3 . The aim of this thesis is to provide a proof of this result on a simply-connected domain that is simpler than the ones found in the literature. Our idea is to take the proof by local approach as our base, simplify it after restricting to a simply-connected domain, and bypass the hard part of dealing with a martingale observable. We do this by defining a function as a ratio of what are known as SLE3 partition functions, and use it as a Radon-Nikodym derivative with respect to chordal SLE3 to construct a new measure. A convergence theorem for fermionic observables shows that this measure is the scaling limit of the law of the driving term of the critical Ising model with multiple interfaces, and due to our knowledge of the Radon-Nikodym derivative an application of Girsanov’s theorem shows that the measure we constructed is just local multiple SLE3.

Työni aihe on Gaussisten prosessien (Gp) soveltaminen aikasarjojen analysointiin. Erityisesti lähestyn aikasarjojen analysointia verrattain harvinaisen sovellusalan, historiallisten aikasarja-aineistojen analysoinnin näkökulmasta. Bayesilaisuus on tärkeä osa työtä: parametreja itsessään kohdellaan satunnaismuuttujina, mikä vaikuttaa sekä mallinnusongelmien muotoiluun että uusien ennusteiden tekemiseen työssä esitellyillä malleilla. Työni rakentuu paloittain. Ensin esittelen Gp:t yleisellä tasolla, tilastollisen mallinnuksen työkaluna. Gp:eiden keskeinen idea on, että Gp-prosessin äärelliset osajoukot noudattavat multinormaalijakaumaa, ja havaintojen välisiä yhteyksiä mallinnetaan ydinfunktiolla (kernel), joka samaistaa havaintoja niihin liittyvien selittäjien ja parametriensa funktiona. Oikeanlaisen ydinfunktion valinta ja datan suhteen optimoidut parametrit mahdollistavat hyvinkin monimutkaisten ja heikosti ymmärrettyjen ilmiöiden mallintamisen Gp:llä. Esittelen keskeiset tulokset, jotka mahdollistavat sekä GP:n sovittamisen aineistoon että sen käytön ennusteiden tekemiseen ja mallinnetun ilmiön alatrendien erittelyyn. Näiden perusteiden jälkeen siirryn käsittelemään sitä, miten GP-malli formalisoidaan ja sovitetaan, kun lähestymistapa on Bayesilainen. Käsittelen sekä eri sovittamistapojen vahvuuksia ja heikkouksia, että mahdollisuutta liittää Gp osaksi laajempaa tilastollista mallia. Bayesilainen lähestymistapa mahdollistaa mallinnettua ilmiötä koskevan ennakkotiedon syöttämisen osaksi mallin formalismia parametrien priorijakaumien muodossa. Lisäksi se tarjoaa systemaattisen, todennäköisyyksiin perustuvan tavan puhua sekä ennakko-oletuksista että datan jälkeisistä parametreihin ja mallinnetun ilmiön tuleviin arvoihin liittyvistä uskomuksista. Seuraava luku käsittelee aikasarjoihin erityisesti liittyviä Gp-mallintamisen tekniikoita. Erityisesti käsittelen kolmea erilaista mallinnustilannetta: ajassa tapahtuvan Gp:n muutoksen, useammasta eri alaprosessista koostuvan Gp:n ja useamman keskenään korreloivan Gp:n mallintamista. Tämän käsittelyn jälkeen työn teoreettinen osuus on valmis: aikasarjojen konkreettinen analysointi työssä esitellyillä työkaluilla on mahdollista. Viimeinen luku käsittelee historiallisten ilmiöiden mallintamista aiemmissa luvuissa esitellyillä tekniikoilla. Luvun tarkoitus on ensisijaisesti esitellä lyhyesti useampi potentiaalinen sovelluskohde, joita on yhteensä kolme. Ensimmäinen luvussa käsitelty mahdollisuus on usein vain repalaisesti havaintoja sisältävien historiallisten aikasarja-aineistojen täydentäminen GP-malleista saatavilla ennusteilla. Käytännön tulokset korostivat tarvetta vahvoille prioreille, sillä historialliset aikasarjat ovat usein niin harvoja, että mallit ovat valmiita hylkäämän havaintojen merkityksen ennustamisessa. Toinen esimerkki käsittelee historiallisia muutoskohtia, esimerkkitapaus on Englannin sisällissodan aikana äkillisesti räjähtävä painotuotteiden määrä 1640-luvun alussa. Sovitettu malli onnistuu päättelemään sisällissodan alkamisen ajankohdan. Viimeisessä esimerkissä mallinnan painotuotteiden määrää per henkilö varhaismodernissa Englannissa, käyttäen ajan sijaan selittäjinä muita ajassa kehittyviä muuttujia (esim. urbanisaation aste), jotka tulkitaan alaprosesseiksi. Tämänkin esimerkin tekninen toteutus onnistui, mikä kannustaa sekä tilastollisesti että historiallisesti kattavampaan analyysiin. Kokonaisuutena työni sekä esittelee että demonstroi Gp-lähestymistavan mahdollisuuksia aikasarjojen analysoinnissa. Erityisesti viimeinen luku kannustaa jatkokehitykseen historiallisten ilmiöiden mallintamisen uudella sovellusalalla.

This thesis focuses on statistical topics that proved important during a research project involving quality control in chemical forensics. This includes general observations about the goals and challenges a statistician may face when working together with a researcher. The research project involved analyzing a dataset with high dimensionality compared to the sample size in order to figure out if parts of the dataset can be considered distinct from the rest. Principal component analysis and Hotelling's T^2 statistic were used to answer this research question. Because of this the thesis introduces the ideas behind both procedures as well as the general idea behind multivariate analysis of variance. Principal component analysis is a procedure that is used to reduce the dimension of a sample. On the other hand, the Hotelling's T^2 statistic is a method for conducting multivariate hypothesis testing for a dataset consisting of one or two samples. One way of detecting outliers in a sample transformed with principal component analysis involves the use of the Hotelling's T^2 statistic. However, using both procedures together breaks the theory behind the Hotelling's T^2 statistic. Due to this the resulting information is considered more of a guideline than a hard rule for the purposes of outlier detection. To figure out how the different attributes of the transformed sample influence the number of outliers detected according to the Hotelling's T^2 statistic, the thesis includes a simulation experiment. The simulation experiment involves generating a large number of datasets. Each observation in a dataset contains the number of outliers according to the Hotelling's T^2 statistic in a sample that is generated from a specific multivariate normal distribution and transformed with principal component analysis. The attributes that are used to create the transformed samples vary between the datasets, and in some datasets the samples are instead generated from two different multivariate normal distributions. The datasets are observed and compared against each other to find out how the specific attributes affect the frequencies of different numbers of outliers in a dataset, and to see how much the datasets differ when a part of the sample is generated from a different multivariate normal distribution. The results of the experiment indicate that the only attributes that directly influence the number of outliers are the sample size and the number of principal components used in the principal component analysis. The mean number of outliers divided by the sample size is smaller than the significance level used for the outlier detection and approaches the significance level when the sample size increases, implying that the procedure is consistent and conservative. In addition, when some part of the sample is generated from a different multivariate normal distribution than the rest, the frequency of outliers can potentially increase significantly. This indicates that the number of outliers according to Hotelling's T^2 statistic in a sample transformed with principal component analysis can potentially be used to confirm that some part of the sample is distinct from the rest.

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.

This thesis is motivated by the following questions: What can we say about the set of primes p for which the equation f(x) = 0 (mod p) is solvable when f is (i) a polynomial or (ii) of the form a^x - b? Part I focuses on polynomial equations modulo primes. Chapter 2 focuses on the simultaneous solvability of such equations. Chapter 3 discusses classical topics in algebraic number theory, including Galois groups, finite fields and the Artin symbol, from this point of view. Part II focuses on exponential equations modulo primes. Artin's famous primitive root conjecture and Hooley's conditional solution is discussed in Chapter 4. Tools on Kummer-type extensions are given in Chapter 5 and a multivariable generalization of a method of Lenstra is presented in Chapter 6. These are put to use in Chapter 7, where solutions to several applications, including the Schinzel-Wójcik problem on the equality of orders of integers modulo primes, are given.

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.