Browsing by discipline "Tillämpad matematik"

Työssä tarkastellaan ääriarvoteorian perusteiden pohjalta kahta tunnettua ääriarvojakaumaa ja niiden antamia häntätodennäköisyyksiä havaintoaineistolle, joka esittää osaketuottojen kuukausitappioita. Lukijalla oletetaan olevan hallussa todennäköisyysteorian -ja laskennan perustiedot. Ääriarvoteoriassa ollaan kiinnostuneita jonkin havaintoaineiston otosmaksimien käyttäytymisestä sekä niiden jakaumasta. Kiinnostuksen kohteena on siis harvoin sattuvien tapahtumien todennäköisyydet eli havaintojen häntätodennäköisyydet ja tarkoitus on analysoida tiettyyn hetkeen mennessä havaittuja tapahtumia suurempien tapahtumien todennäköisyyksiä. Tutkielman alussa käydään lävitse ääriarvoteorian muutamia olennaisia tuloksia. Käsiteltävä teoria on nimenomaan klassista ääriarvoteoriaa, jossa havaintojen oletetaan olevan riippumattomia ja samoin jakautuneita. Olennainen tulos ääriarvoteoriassa on se, että mikäli sopivilla vakioilla normeerattu otosmaksimi suppenee jakaumaltaan kohti jotain ei-degeneroitunutta jakaumaa, kun otoskoko kasvaa rajatta, niin tällöin tämän jakauman täytyy olla tyypiltään yksi kolmesta standardista ääriarvojakaumasta Fréchet, Weibull tai Gumbel. Tällöin sanotaan, että otosmaksimin jakauma kuuluu ääriarvojakauman vaikutuspiiriin maksimin suhteen. Teorian käsittelyn jälkeen esitellään ääriarvoteorian kaksi tunnetuinta ääriarvojakaumaa. Ensimmäinen niistä on standardi yleistetty ääriarvojakauma eli ns. GEV-jakauma, joka pitää sisällään nuo kolme edellä mainittua standardia ääriarvojakaumaa. Toinen esiteltävä jakauma on yleistetty Pareto-jakauma eli ns. GP-jakauma, jonka jakaumaperheen jäsenet niinikään kuuluvat GEV-jakauman antamien ääriarvojakaumien vaikutuspiiriin maksimin suhteen. Molempien jakaumien avulla pystytään vähän eri menetelmin tutkimaan ääriarvojen tapahtumista jonkin tietyn havaintoaineiston pohjalta ja ekstrapoloimaan havaintoaineiston alueelle, jota ei paljon tunneta eli ääriarvoalueelle. Teorian ja jakaumien konkretisoimiseksi tutkielmassa käydään esimerkin avulla läpi minkälaisia tuloksia ääriarvojakaumilla voidaan saavuttaa. GEV-jakauman sovitus havaintoaineistoon tapahtuu ns. blokkimaksimimenetelmällä. Siinä aineisto jaetaan vuoden blokkeihin ja kustakin blokista poimitaan suurin osaketappio. Tämän jälkeen GEV-jakauma sovitetaan ns. suurimman uskottavuuden menetelmällä havaintoihin. GP-jakauman sovitus aineistoon tapahtuu ns. ylitemenetelmällä, jossa havaintoihin otetaan mukaan tietyn korkean tason ylittävät havainnot. Tämän jälkeen myös GP-jakauma sovitetaan havaintoihin suurimman uskottavuuden menetelmällä. Tuloksista käy ilmi, että molemmat jakaumat vaikuttavat sopivan suhteellisen hyvin havaintoihin joskin GP-jakauma antaa monipuolisempia tuloksia. Lopuksi kerrataan vielä käsiteltyjä asioita sekä kurotetaan esitellyn teorian ohi kohti yleisempää teoriaa. Klassinen ääriarvoteoria ei riippumattomuus oletuksineen nimittäin sellaisenaan sovi reaalimaailman havaintoaineistoon. Asia on tutkielmassa kuitenkin pääosin sivuutettu esityksen helpottamiseksi.

In this thesis we present an algorithm for doing mixture modeling for heterogeneous data collections. Our model supports using both Gaussian- and Bernoulli distributions, creating possibilities for analysis of many kinds of different data. A major focus is spent to developing scalable inference for the proposed model, so that the algorithm can be used to analyze even a large amount of data relatively fast. In the beginning of the thesis we review some required concepts from probability theory and then proceed to present the basic theory of an approximate inference framework called variational inference. We then move on to present the mixture modeling framework with examples of the Gaussian- and Bernoulli mixture models. These models are then combined to a joint model which we call GBMM for Gaussian and Bernoulli Mixture Model. We develop scalable and efficient variational inference for the proposed model using state-of-the-art results in Bayesian inference. More specifically, we use a novel data augmentation scheme for the Bernoulli part of the model coupled with overall algorithmic improvements such as incremental variational inference and multicore implementation. The efficiency of the proposed algorithm over standard variational inference is highlighted in a simple toy data experiment. Additionally, we demonstrate a scalable initialization for the main inference algorithm using a state-of-the-art random projection algorithm coupled with k-means++ clustering. The quality of the initialization is studied in an experiment with two separate datasets. As an extension to the GBMM model, we also develop inference for categorical features. This proves to be rather difficult and our presentation covers only the derivation of the required inference algorithm without a concrete implementation. We apply the developed mixture model to analyze a dataset consisting of electronic patient records collected in a major Finnish hospital. We cluster the patients based on their usage of the hospital's services over 28-day time intervals over 7 years to find patterns that help in understanding the data better. This is done by running the GBMM algorithm on a big feature matrix with 269 columns and more than 1.7 million rows. We show that the proposed model is able to extract useful insights from the complex data, and that the results can be used as a guideline and/or preprocessing step for possible further, more detailed analysis that is left for future work.

Often it would be useful to be able to calculate the 'shape' of a set of points even though it is not clear what is formally meant by the word 'shape.' The definitions of shapes presented in this thesis are generalizations of the convex hull of a set of points P which is the smallest convex set that contains all points of P. The k-hull of a point set P is a generalization of the convex hull. It is the complement of the union of all such halfplanes that contain at most k points of P. One application of the k-hull is measuring the data depth of an arbitrary point of the plane with respect to P. Another generalization of the convex hull is the α-hull of a point set P. It is the complement of the union of all α-disks (disks of radius α) that contain no points of the set P in their interior. The value of α controls the detail of the α-hull: 0-hull is P and ∞-hull is the convex hull of P. The α-shape is the 'straight-line' version of the α-hull. The α-hull and the α-shape reconstruct the shape in a more intuitive manner than the convex hull, recognizing that the set might have multiple distinct data clusters. However, α-hull and α-shape are very prone to outlier data that is often present in real-life datasets. A single outlier can drastically change the output shape. The k-order α-hull is a generalization of both the k-hull and the α-hull and as such it is a link between statistical data depth and shape reconstruction. The k-order α-hull of a point set P is the complement of the union of all such α-disks that contain at most k points of P. The k-order α-shape is the α-shape of those points of P that are not included in any of the α-disks. The k-order α-hull and the k-order α-shape can ignore a certain amount of the outlier data which the α-hull and the α-shape cannot. The detail of the shape can be controlled with the parameter α and the amount of outliers ignored with the parameter k. For example, the 0-order α-hull is the α-hull and the k-order ∞-hull is the k-hull. One possible application of the k-order α-shape is a visual representation of spatial data. Multiple k-order α-shapes can be drawn on a map so that shapes that are deeper in the dataset (larger values of k) are drawn with more intensive colors. Example datasets for geospatial visualization in this thesis include motor vehicle collisions in New York and unplanned stops of public transportation vehicles in Helsinki. Another application presented in this thesis is noise reduction from seismic time series using k-order α-disks. For each time tick, two disks of radius α are put above and below the data points. The upper disk is pulled downwards and the lower disk upwards until they contain exactly k data points inside. The algorithm's output for each time tick is the average of the centres of these two α-disks. With a good choice of parameters α and k, the algorithm successfully restores a good estimate of the original noiseless time series.

Tässä työssä tutustumme adaptiiviseen optiikkaan ja ilmakehätomografiaan liittyviin matemaattisiin ongelmiin. Adaptiivinen optiikka on menetelmä, joka pyrkii vähentämään ilmakehän turbulenssista aiheutuvia valon vaiheen vääristymiä ja näin parantamaan suurten optisten teleskooppien suorituskykyä. Adaptiivinen optiikka on tieteenala, joka yhdistää matematiikkaa, fysiikkaa ja insinööritieteitä. Ilmakehätomografialla taas tarkoitamme tiettyä matemaattista kuvantamisongelmaa, joka esiintyy seuraavan sukupolven teleskooppien adaptiivista optiikkaa käyttävissä systeemeissä. Tässä työssä esittelemme tavan mallintaa ilmakehätomografiaa matemaattisesti. Käsittelemme ilmakehän turbulenssia satunnaisfunktioiden avulla sekä esittelemme algoritmin, joka kattaa ilmakehätomografian sekä optisen systeemin kontrolloinnin. Lisäksi tutkimme tämän algoritmin toimintaa yhdessä alan tarkimmista simulointiympäristöistä. Erityisesti testasimme algoritmin vakautta olosuhteissa, joissa kohinataso datassa on korkea. Tutkimustulokset osoittavat, että työssä johdetut regularisointia käyttävät ratkaisumenetelmät parantavat systeemin suorituskykyä verrattuna yksinkertaisempaan ei-regularisoituun menetelmään.

Tutkielma perehdyttää lukijan Steinin menetelmään normaaliapproksimaatiolle sekä esittää tämän avulla todistuksen Berry-Esseen-lauseelle. Steinin menetelmä on todennäköisyysteorian piiriin kuuluva nykyaikainen ja tehokas tapa tuottaa ylärajoja kahden eri todennäköisyysjakauman väliselle etäisyydelle. Tutkielmassa esitetään todennäköisyysjakaumien etäisyydelle kolme eniten käytettyä mittaa, jotka ovat Total variation, Kolmogorov sekä Wasserstein-mitat. Tämän jälkeen käydään läpi Steinin menetelmä aloittaen Steinin lemmasta, joka karakterisoi normaalijakauman Steinin operaattorin avulla siten, että operaattorin arvon ollessa nolla, on tarkasteltava jakauma normaali. Seuraavaksi esitetään Steinin yhtälöt, joiden ratkaisujen avulla saadaan Steinin rajoitukset jokaiselle käytetylle kolmelle mitalle. Näiden rajoitusten avulla voidaan päätellä asymptoottinen normaalijakautuneisuus myös silloin, kun Steinin operaattorin arvo on lähellä nollaa. Berry-Esseen-lause on keskeinen raja-arvolause, johon on erityisesti lisätty suppenemisnopeus Kolmogorov-etäisyyden suhteen. Tämä suppenemisnopeus todistetaan tutkielmassa käyttäen hyväksi Steinin menetelmää. Lopuksi käsitellään vielä ylimalkaisesti Steinin menetelmää moniulotteisen jakauman tapauksessa. Huomataan sen olevan hyvin paljon samankaltaista kuin yksiulotteisessa tapauksessa.

Forecasting of solar power energy production would benefit from accurate sky condition predictions since the presence of clouds is a primary variable effecting the amount of radiation reaching the ground. Unfortunately the spatial and temporal resolution of often used satellite images and numerical weather prediction models can be too small for local, intra-hour estimations. Instead, digital sky images taken from the ground are used as data in this thesis. The two main building blocks needed to make sky condition forecasts are reliable cloud segmentation and cloud movement detection. The cloud segmentation problem is solved using neural networks, a double exposure imaging scheme, automatic sun locationing and a novel method to study the circumsolar region directly without the use of a sun occluder. Two different methods are studied for motion detection. Namely, a block matching method using cross-correlation as the similarity measure and the Lukas-Kanade method. The results chapter shows how neural networks overcome many of the situations labelled as difficult for other methods in the literature. Also, results by the two motion detection methods are presented and analysed. The use of neural networks and the Lukas-Kanade method show much promise for forming the cornerstone of local, intra-hour sky condition now-casting and prediction.

Clustering is one of the core problems of unsupervised machine learning. In a clustering problem we are given a set of data points and asked to partition them into smaller subgroups, known as clusters, such that each point is assigned to exactly one cluster. The quality of the obtained partitioning (clustering) is then evaluated according to some objective measure dependent on the specific clustering paradigm. A traditional approach within the machine learning community to solving clustering problems has been focused on approximative, local search algorithms that in general can not provide optimality guarantees of the clusterings produced. However, recent advances in the field of constraint optimization has allowed for an alternative view on clustering, and many other data analysis problems. The alternative view is based on stating the problem at hand in some declarative language and then using generic solvers for that language in order to solve the problem optimally. This thesis contributes to this approach to clustering by providing a first study on the applicability of state-of-the-art Boolean optimization procedures to cost-optimal correlation clustering under constraints in a general similarity-based setting. The correlation clustering paradigm is geared towards classifying data based on qualitative--- as opposed to quantitative similarity information of pairs of data points. Furthermore, correlation clustering does not require the number of clusters as input. This makes it especially well suited to problem domains in which the true number of clusters is unknown. In this thesis we formulate correlation clustering within the language of propositional logic. As is often done within computational logic, we focus only on formulas in conjunctive normal form (CNF), a limitation which can be done without loss of generality. When encoded as a CNF-formula the correlation clustering problem becomes an instance of partial Maximum Satisfiability (MaxSAT), the optimization version of the Boolean satisfiability (SAT) problem. We present three different encodings of correlation clustering into CNF-formulas and provide proofs of the correctness of each encoding. We also experimentally evaluate them by applying a state-of-the-art MaxSAT solver for solving the resulting MaxSAT instances. The experiments demonstrate both the scalability of our method and the quality of the clusterings obtained. As a more theoretical result we prove that the assumption of the input graph being undirected can be done without loss of generality, this justifies our encodings being applicable to all variants of correlation clustering known to us. This thesis also addresses another clustering paradigm, namely constrained correlation clustering. In constrained correlation clustering additional constraints are used in order to restrict the acceptable solutions to the correlation clustering problem, for example according to some domain specific knowledge provided by an expert. We demonstrate how our MaxSAT-based approach to correlation clustering naturally extends to constrained correlation clustering. Furthermore we show experimentally that added user knowledge allows clustering larger datasets, decreases the running time of our approach, and steers the obtained clusterings fast towards a predefined ground-truth clustering.

Speech is the most common form of human communication. An understanding of the speech production mechanism and the perception of speech is therefore an important topic when studying human communication. This understanding is also of great importance both in medical treatment regarding a patient's voice and in human-computer interaction via speech. In this thesis we will present a model for digital speech called the source-filter model. In this model speech is represented with two independent components, the glottal excitation signal and the vocal tract filter. The glottal excitation signal models the airflow created at the vocal folds, which works as the source for the created speech sound. The vocal tract filter describes how the airflow is filtered as it travels through the vocal tract, creating the sound radiated to the surrounding space from the lips, which we recognize as speech. We will also present two different parametrized models for the glottal excitation signal, the Rosenberg-Klatt model (RK-model) and the Liljencrants-Fant model (LF-model). The RK-model is quite simple, being parametrized with only one parameter in addition to the fundamental frequency of the signal, while the LF-model is more complex, taking in four parameters to define the shape of the signal. A transfer function for vocal tract filter is also derived from a simplified model of the vocal tract. Additionally, relevant parts of the theory of signal processing are presented before the presentation of the source-filter model. A relatively new model for glottal inverse filtering (GIF), called the Markov chain Monte Carlo method for glottal inverse filtering (MCMC-GIF) is also presented in this thesis. Glottal inverse filtering is a technique for estimating the glottal excitation signal from a recorded speech sample. It is a widely used technique for example in phoniatrics, when inspecting the condition of a patient's vocal folds. In practice the aim is to separate the measured signal into the glottal excitation signal and the vocal tract filter. The first method for solving glottal inverse filtering was proposed in the 1950s and since then many different methods have been proposed, but so far none of the methods have been able to yield robust estimates for the glottal excitation signal from recordings with a high fundamental frequency, such as women's and children's voices. Recently, using synthetic vowels, MCMC-GIF has been shown to produce better estimates for these kind of signals compared to other state of the art methods. The MCMC-GIF method requires an initial estimate for the vocal tract filter. This is obtained from the measurements with the iterative adaptive inverse filtering (IAIF) method. A synthetic vowel is then created with the RK-model and the vocal tract filter, and compared to the measurements. The MCMC method is then used to adjust the RK excitation parameter and the parameters for the vocal tract filter to minimize the error between the synthetic vowel and the measurements, and ultimately receive a new estimate for the vocal tract filter. The filter can then be used to calculate the glottal excitation signal from the measurements. We will explain this process in detail, and give numerical examples of the results of the MCMC-GIF method compared against the IAIF method.

In this thesis we consider dynamic X-ray computed tomography (CT) for a two dimensional case. In X-ray CT we take X-ray projection images from many different directions and compute a reconstruction from those measurements. Sometimes the change over time in the imaged object needs to be taken into account, for example in cardiac imaging or in angiography. This is why we're looking at the dynamic (something changing in time, while taking the measurements) case. At the beginning of the thesis in chapter 2 we present some necessary theory on the subject. We first go through some general theory about inverse problems and the concentrate on X-ray CT. We talk about ill-posedness of inverse problems, regularization and the measurement proses in CT. Different measurement settings and the discretization of the continuous case are introduced. In chapter 3 we introduce a solution method for the problem: total variation regularization with Barzilai-Borwein minimization method. The Barzilai-Borwein minimization method is an iterative method and well suited for large scale problems. We also explain two different methods, the multi-resolution parameter choice method and the S-curve method, for choosing the regularization parameter needed in the minimization process. The 4th chapter shows the materials used in the thesis. We have both simulated and real measured data. The simulated data was created using a rendering software and for the real data we took X-ray projection images of a Lego robot. The results of the tests done on the data are shown in chapter 5. We did tests on both the simulated and the measured data with two di erent measurement settings. First assuming we have 9 xed source-detector pairs and then that we only one source-detector pair. For the case where we have only one pair, we tested the implemented regularization method we begin by considering the change in the imaged object to be periodic. Then we assume can only use some number of consecutive moments, based on the rate the object is chancing, to collect the data. Here we only get one X-ray projection image at each moment and we combine measurements from multiple di erent moments. In the last chapter, chapter 6, we discuss the results. We noticed that the regularization method is quite slow, at least partly because of the functions used in the implementation. The results obtained were quite good, especially for the simulated data. The simulated data had less details than the measured data, so it makes sense that we got better results with less data. Already with only four angles, we cold some details with the simulated data, and for the measured data with 8 angles and with 16 angles details were also visible in the case of measured data.