The subject of this thesis is the maximization of the information divergence from an exponential family on a finite set, a problem first formulated by Nihat Ay. A special case is the maximization of the mutual information or the multiinformation between different parts of a composite system. My thesis contributes mainly to the mathematical aspects of the optimization problem. A reformulation is found that relates the maximization of the information divergence with the maximization of an entropic quantity, defined on the normal space of the exponential family. This reformulation simplifies calculations in concrete cases and gives theoretical insight about the general problem. A second emphasis of the thesis is on examples that demonstrate how the theoretical results can be applied in particular cases. Third, my thesis contain first results on the characterization of exponential families with a small maximum value of the information divergence.:1. Introduction 2. Exponential families 2.1. Exponential families, the convex support and the moment map 2.2. The closure of an exponential family 2.3. Algebraic exponential families 2.4. Hierarchical models 3. Maximizing the information divergence from an exponential family 3.1. The directional derivatives of D(*|E ) 3.2. Projection points and kernel distributions 3.3. The function DE 3.4. The first order optimality conditions of DE 3.5. The relation between D(*|E) and DE 3.6. Computing the critical points 3.7. Computing the projection points 4. Examples 4.1. Low-dimensional exponential families 4.1.1. Zero-dimensional exponential families 4.1.2. One-dimensional exponential families 4.1.3. One-dimensional exponential families on four states 4.1.4. Other low-dimensional exponential families 4.2. Partition models 4.3. Exponential families with max D(*|E ) = log(2) 4.4. Binary i.i.d. models and binomial models 5. Applications and Outlook 5.1. Principles of learning, complexity measures and constraints 5.2. Optimally approximating exponential families 5.3. Asymptotic behaviour of the empirical information divergence A. Polytopes and oriented matroids A.1. Polytopes A.2. Oriented matroids Bibliography Index Glossary of notations

In this dissertation, we study the complicated dynamics of two classes of systems: Anosov systems driven by an external force and partially hyperbolic systems driven by an external force. For smooth Anosov systems driven by an external force, we first study the random specification property, which is on the approximation of an N−spaced arbitrary long finite random orbit segments within given precision by a random periodic point. We prove that if such system is topological mixing on fibers, then it has the random specification property. Furthermore, we prove that the homeomorphism induced by such a system on the space of random probability measures also has the specification property. We note that the random specification property implies the positivity of topological fiber entropy. Secondly, we show that if the system is topological mixing on fibers, then its past and future random correlation for Hölder observable functions decay exponentially with respect to the system and the unique random SRB measure. For smooth partially hyperbolic systems driven by an external force, we prove the existence of the random Gibbs u−state, which has absolutely continuous conditional measure on the strong unstable manifolds.

In this thesis, we treat the computation of transforms with asymptotically smooth and oscillatory kernels. We introduce the discrete Laplace transform in a modern form including a generalization to more general kernel functions. These more general kernels lead to specific function transforms. Moreover, we treat the butterfly fast Fourier transform. Based on a local error analysis, we develop a rigorous error analysis for the whole butterfly scheme. In the final part of the thesis, the Laplace and Fourier transform are combined to a fast Fourier transform for nonequispaced complex evaluation nodes. All theoretical results on accuracy and computational complexity are illustrated by numerical experiments.

This paper examines childbearing among Polish migrant women and their descendants in Sweden. While many studies have focused on immigrants' childbearing in relation to women in the destination country, this study uses a country-of-origin and a country-of-destination approach in order to more thoroughly examine the socialization, selection and adaptation hypotheses. Using a piecewise-exponential model, the transitions to first and second births are analyzed using Swedish register data and the Polish Generations and Gender survey (GGS). The results show that the Polish stayers and the first-generation have relatively similar fertility behavior in the transition to first birth but not in the transition to second birth. However, parts of the similarity in the transition to first birth can be attributed to marital status selection. By examining the 1.5-generation and the second-generation in relation to Swedish natives, it is possible to see fertility convergence across generations, both when it comes to timing and quantum. This study also shows that family migrants have higher risk of having a first child compared to migrants moving for other reasons. However, in the transition to second birth, there is no difference.

It the thesis, we study retarded functional differential equations. As a result of the Banach fixed point theorem, it is easy to show that there exists a unique solution to such problems. Alas, this theorem gives us no information on the form of the solution. Therefore, we are particularly interested in expressing it. We achieve that by applying Laplace transform to both sides of the equation, we get a solution to this modified problem and subsequently claim that we can apply the inverse Laplace transform to express the solution of the former problem. At the end of the thesis, we formulate and prove the exponential estimate of the solution. 1

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels'' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics.

The stock market is a non-linear field, but many of the best-known portfolio optimization algorithms are based on linear models. In recent years, the rapid development of machine learning has produced flexible models capable of complex pattern recognition. In this paper, we propose two different methods of portfolio optimization; one based on the development of a multivariate time-dependent neural network,thelongshort-termmemory(LSTM),capable of finding lon gshort-term price trends. The other is the linear Markowitz model, where we add an exponential moving average to the input price data to capture underlying trends. The input data to our neural network are daily prices, volumes and market indicators such as the volatility index (VIX).The output variables are the prices predicted for each asset the following day, which are then further processed to produce metrics such as expected returns, volatilities and prediction error to design a portfolio allocation that optimizes a custom utility function like the Sharpe Ratio. The LSTM model produced a portfolio with a return and risk that was close to the actual market conditions for the date in question, but with a high error value, indicating that our LSTM model is insufficient as a sole forecasting tool. However,the ability to predict upward and downward trends was somewhat better than expected and therefore we conclude that multiple neural network can be used as indicators, each responsible for some specific aspect of what is to be analysed, to draw a conclusion from the result. The findings also suggest that the input data should be more thoroughly considered, as the prediction accuracy is enhanced by the choice of variables and the external information used for training. / Aktiemarknaden är en icke-linjär marknad, men många av de mest kända portföljoptimerings algoritmerna är baserad på linjära modeller. Under de senaste åren har den snabba utvecklingen inom maskininlärning skapat flexibla modeller som kan extrahera information ur komplexa mönster. I det här examensarbetet föreslår vi två sätt att optimera en portfölj, ett där ett neuralt nätverk utvecklas med avseende på multivariata tidsserier och ett annat där vi använder den linjära Markowitz modellen, där vi även lägger ett exponentiellt rörligt medelvärde på prisdatan. Ingångsdatan till vårt neurala nätverk är de dagliga slutpriserna, volymerna och marknadsindikatorer som t.ex. volatilitetsindexet VIX. Utgångsvariablerna kommer vara de predikterade priserna för nästa dag, som sedan bearbetas ytterligare för att producera mätvärden såsom förväntad avkastning, volatilitet och Sharpe ratio. LSTM-modellen producerar en portfölj med avkastning och risk som ligger närmre de verkliga marknadsförhållandena, men däremot gav resultatet ett högt felvärde och det visar att vår LSTM-modell är otillräckligt för att använda som ensamt predikteringssverktyg. Med det sagt så gav det ändå en bättre prediktion när det gäller trender än vad vi antog den skulle göra. Vår slutsats är därför att man bör använda flera neurala nätverk som indikatorer, där var och en är ansvarig för någon specifikt aspekt man vill analysera, och baserat på dessa dra en slutsats. Vårt resultat tyder också på att inmatningsdatan bör övervägas mera noggrant, eftersom predikteringsnoggrannheten.

Queueing theory is the mathematical study of queues or waiting lines, which are formed whenever demand for service exceeds the capacity to provide service. A queueing system is composed of customers, packets or calls that need some kind of service. These entities arrive at queueing system, join a queue if service is not immediately available and leave system after receiving service. There are also cases when customers, packets or calls leave system without joining queue or drop out without receiving service even after waiting for some time. Queueing network models with finite capacity have facilitated the analysis of discrete flow systems, such as computer systems, transportation networks, manufacturing systems and telecommunication networks, by providing powerful and realistic tools for performance evaluation and prediction. In wireless cellular systems mobility is the most important feature and continuous service is achieved by supporting handoff from one cell to another. Hand off is the process of changing channel associated with the current connection while a call is in progress. A handoff is required when a mobile terminal moves from one cell to another or the signal quality deteriorates in current cell. Since neighbouring cells use disjoint subset of frequency bands therefore negotiation must take place between mobile terminal, the current base station and next potential base station. A poorly designed handoff scheme significantly decreases quality of service (QOS). Different schemes have been devised and in these schemes handoff calls are prioritize. Also most of the performance evaluation techniques consider the case where the arrival process is poisson and service is exponential i.e. there is single arrival and single departure. Whereas in practice there is burstiness in cellular traffic i.e. there can be bulk arrivals and bulk departures. Other issue is that, assumptions made by stochastic process models are not satisfied. Most of the effort is concentrated on providing different interpretations of M/M queues rather than attempting to provide a new methodology. In this thesis performance evaluation of multi traffic cellular models i.e. non priority, priority and hand off calling scheme for bursty traffic are devised. Moreover extensions are carried out towards the analysis of a multi-traffic M/M queueing system and state probabilities are calculated analytically.

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