Browsing by discipline "Soveltava matematiikka"

We show that the embedding of a matrix game in a mechanistic population dynamical model can have important consequences for the evolution of how the game is played. In particular, we show that because of this embedding evolutionary branching and multiple evolutionary singular strategies can occur, which is not possible in the conventional theory of matrix games. We show that by means of the example of the hawk-dove game.

The subject of this Master's Thesis is Shannon-McMillan-Breiman theorem, a famous and important result in information theory. Since the theorem is a statement about ergodic stochastic processes and its proof utilises Birkho 's ergodic theorem, a whole chapter has been devoted to ergodic theory. Ergodic theory has developed into a large branch of mathematics, and so the Chapter 1 is only a brief glance at the subject. Nevertheless, we will prove one of the most important theorems in ergodic theory, the before-mentioned Birkho 's ergodic theorem. This theorem is a strong statement about the average behaviour of certain stochastic processes (or dynamical systems), and it can be seen as a generalisation of the Strong Law of Large Numbers. Chapter 2 discusses information theory and the Shannon-McMillan-Breiman theorem. No previous knowledge about information theory is assumed, and therefore the chapter starts with an introduction to information theory. All fundamental de nitions and theorems concerning the entropy of discrete random variables are provided. After this introduction, we study the entropy of stochastic processes, which in turn leads us to the Asymptotic Equipartition Property (the AEP). Informally, a stochastic process has the AEP if almost all sample paths belong to a rather thin set, called the set of typical sequences, which despite having few elements contains most of the probability mass. Then we prove that independent and identically distributed processes have the AEP, and consider its consequences and applications such as data compression. After this, we present the Shannon-McMillan-Theorem which states that stationary, ergodic processes with infite state space have the AEP. The rest of the thesis is then devoted to the rather long, but interesting proof of the theorem.

The past decade has brought about two key changes to the pricing of interest rate products in the European fixed income markets. In 2007, the Euribor-EONIA spread widened, which called into question the use of Euribor rates as a proxy for the risk-free rate. Nine years later, all of the main Euribor rates had fallen below zero. These changes forced market practitioners to reassess their assumptions, which resulted in the development of new models. The Heath-Jarrow-Morton (HJM) and Brace-Gatarek-Musiela (BGM) frameworks, which had gained popularity before the crisis, have served as a foundation for these models. In this thesis, we present two applications of Malliavin calculus to the pricing of interest rate derivatives within a multicurve BGM framework. Although the framework simplifies the pricing of interest rate derivatives, as it takes market rates as primitives, the pricing of exotic interest rate derivatives can pose a number of problems. The complexity of these products can lead to situations that require the use of computational techniques such as Monte Carlo simulation. This, in turn, provides an opening for the application of Malliavin calculus. We end this thesis by presenting a Malliavin-based formula for the price and the delta-sensitivity of a snowball, and discuss the merits of these representations in the context of Monte Carlo simulation. With reference to advances within the field during the last 5 years, we discuss the possible developments within the field that might garner further interest towards Malliavin calculus in the near future.

Gaussian processes can be used through Bayesian models so that they are formed through a multidimensional Gaussian prior with a special covariance matrix structure and an arbitrary likelihood model. They often include a latent variable structure between the features and the response variable. Bayesian modeling's drawbacks are usually related to the normalizing constants that normalize the product of a prior probability density function and a likelihood function to a proper probability distribution. These integrals are hard or even impossible to calculate analytically and hence some approximations are required. One popular approximation is the Laplace approximation, which is a Gaussian approximation for the unnormalized log-posterior distribution. Reparametrization of the observation model can lead to changes in properties of the posterior distribution such as shape and convergence. The performance of approximations made for the posterior distribution also change along with the parametrization. The changes are often related to either computational complexity or the predictive performance of the approximation. This thesis presents the Gaussian processes starting from Bayes' formula and moves quickly towards key concepts in Bayesian modeling such as predictive distributions and hierarchy. An approximation of interest for the posterior distribution, the Laplace approximation, is derived. Traditional optimization algorithm for the Laplace approximation is the Newton method, which is replaced by an algorithm called natural gradient adaptation in this thesis. Then the focus is turned from general introduction of Gaussian processes to more specific treatment of them by choosing the Weibull distribution as an observation model. Two different parametrizations for the Weibull model are studied, one which acts as a baseline and can be thought as traditional parametrization for the model, and another one for which the parameters are orthogonal. The predictive performance of the Laplace approximation is then compared within the two parametrizations in two different kind of data sets. Finally the results show decrease in computation time required for the Laplace approximation but no improvement in the predictive performance for orthogonal parametrization.

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.

In this thesis we present a new likelihood-free inference method for simulator-based models. A simulator-based model is a stochastic mechanism that specifies how data are generated. Simulator-based models can be as complex as needed, but they must allow exact sampling. One common difficulty with simulator-based models is that learning model parameters from observed data is generally challenging, because the likelihood function is typically intractable. Thus, traditional likelihood-based Bayesian inference is not applicable. Several likelihood-free inference methods have been developed to perform inference when a likelihood function is not available. One popular approach is approximate Bayesian computation (ABC), which relies on the fundamental principle of identifying parameter values for which summary statistics of simulated data are close to those of observed data. However, traditional ABC methods tend have high computational cost. The cost is largely due to the need to repeatedly simulate data sets, and the absence of knowledge of how to specify the discrepancy between the simulated and observed data. We consider speeding up the earlier method likelihood-free inference by ratio estimation (LFIRE) by replacing the computationally intensive grid evaluation with Bayesian optimization. The earlier method is an alternative to ABC that relies on transforming the original likelihood-free inference problem into a classification problem that can be solved using machine learning. This method is able to overcome two traditional difficulties with ABC: it avoids using a threshold value that controls the trade-off between computational and statistical efficiency, and combats the curse of dimensionality by offering an automatic selection of relevant summary statistics when using a large number of candidates. Finally, we measure the computational and statistical efficiency of the new method by applying it to three different real-world time series models with intractable likelihood functions. We demonstrate that the proposed method can reduce the computational cost by some orders of magnitude while the statistical efficiency remains comparable to the earlier method.

Tutkielma kartoittaa bayesiläisen lähestymistavan soveltamista joukkoliikenteen lipunmyynnin dynaamiseen hinnoitteluun. Dynaamisessa hinnoittelussa muutetaan tuotteen hintaa taajaan pyrkimyksenä hiljaisempina aikoina houkutella lisää asiakkaita alemmilla hinnoilla ja hyödyntää suuremman kysynnän jaksot nostamalla hintoja. Tutkittavassa tilanteessa pyritään maksimoimaan pitkän matkan linja-autovuoron lipunmyynnistä syntyvä liikevaihto. Oletetaan lippujen myynnin odotusarvon määräytyvän jonkin kysyntäfunktion perusteella, ja että kysyntä riippuu myyntihinnasta sekä muista muuttujista. Hinnoittelija valitsee kullekin myyntijaksolle hinnan, joka tuottaa jonkin myyntitulon, ja tavoite on siis tietylle linja-autolähdölle maksimoida sen lipunmyynnin kokonaisliikevaihdon odotusarvo yli myyntijaksojen etsimällä parhaat hinnat. Hinnoittelussa esitetään noudatettavaksi seuraavaa lähestymistapaa. Oletetaan kysynnän noudattavan log-lineaarista mallia. Käytetään sen parametreille priorijakaumaa, jonka hyperparametreille lasketaan suurimman uskottavuuden estimaatit aiemmin kerätyn datan perusteella. Kussakin yksittäisessä hinnoittelujaksossa kysyntäfunktion parametreille muodostetaan myyntikauden edellisten hinnoittelujaksojen toteutuneiden myyntien perusteella uskottavuus. Sitten parametrien uskottavuuden ja priorijakauman perusteella muodostetaan parametrien posteriorijakauma, jota arvioidaan Markovin ketju -- Monte Carlo -menetelmin. Viimein posteriorijakauman antamaa tietoa kysynnästä käytetään myyntijakson hinnan optimoimiseen varmuutta vastaavan hinnoittelustrategian mukaisesti, eli olettaen parametriestimaatit virheettömiksi.

Tutkielmassani käsitellään Burgersin yhtälön nimellä tunnettua kvasilineaarista osittaisdifferentiaaliyhtälöä, sekä paneudutaan osittaisdifferentiaaliyhtälöiden teoriaan yleisemmin. Fyysikko Johannes Martinus Burgersin mukaan nimetyllä yhtälöllä voidaan kuvata häiriöiden etenemistä fluideissa, ja sitä voidaan soveltaa myös vaikkapa liikenneruuhkien kehittymisen analysointiin. Burgersin yhtälö on esimerkki yleisemmästä säilymislaista. Matemaattisessa fysiikassa säilymislakien mukaan eristetyssä systeemissä vuorovaikutustapahtumissa tiettyjen suureiden kokonaismäärät pysyvät muuttumattomina. Tunnetuin esimerkki säilymislakeihin liittyen on Noetherin lause, jonka mukaan suurella on vastaavuus tietyn systeemin symmetriaominaisuuden kanssa. Esimerkiksi nestedynamiikan Navier-Stokesin yhtälö on esimerkki epähomogeenisesta versiosta säilymislakia. Tutkimukseni alussa esitellään Hamilton-Jakobin yhtälö, sekä sivutaan variaatiolaskentaa, Legendren muunnosta ja Euler-Lagrangen yhtälöitä. Näytetään, miten annettu osittaisdifferentiaaliyhtälö voidaan samaistaa karakteristiseen yhtälöryhmään, joka koostuu tavallisista differentiaaliyhtälöistä. Karakteristinen yhtälöryhmä johdetaan kvasilineaarisen osittaisdifferentiaaliyhtälön tapauksessa ja sille annetaan geometrinen tulkinta. Ensimmäisen luvun lopussa Hamilton-Jacobin yhtälö ratkaistaan Hopf-Lax kaavan avulla. Toisessa ja kolmannessa luvussa esitellään Burgersin yhtälö ja ratkaistaan se karakteristisen yhtälöryhmän avulla. Saatu ratkaisu ei kuitenkaan päde kaikkialla, vaan tapauksissa, joissa karakteristiset käyrät kohtaavat ('shokkikäyrä'), Burgersin yhtälön ratkaisu vaatii ratkaisufunktion ehtojen heikentämistä ja ingraaliratkaisun määrittelemistä. Rankine-Hugoniot-ehto johdetaan ja sen avulla voidaan löytää ratkaisuja tilanteessa, jossa karakteristiset käyrät leikkaavat. Esittelen myös entropia-ehdon, jonka avulla karsitaan 'epäfysikaaliset' ratkaisut pois ja täten saadaan yksikäsitteinen ja yleinen ratkaisu Burgersin yhtälölle. Lopuksi todistan Lax-Oleinikin kaavan, joka antaa ratkaisun yleisemmälle ongelmalle. Lopuksi tälle ratkaisulle räätälöidään entropia-ehto, jotta siitä saadaan yksikäsitteinen.

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.