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Cluster analysis is commonly described as the classification of unlabeled observations into groups such that they are more similar to one another than to observations in other groups. Model-based clustering assumes that the data arise from a statistical (mixture) model and typically a group of many models are fit to the data, from which the `best' model is selected by a model selection criterion (often the BIC in mixture model applications). This chosen model is then the only model that is used for making inferences on the data. Although this is common practice, proceeding in this way ignores a large component of model selection uncertainty, especially for situations where the difference between the model selection criterion for two competing models is relatively insignificant. For this reason, recent interest has been placed on selecting a subset of models that are close to the selected best model and using a weighted averaging approach to incorporate information from multiple models in this set. Model averaging is not a novel approach, yet its presence in a clustering framework is minimal. Here, we use Occam's window to select a subset of models eligible for two types of averaging techniques: averaging a posteriori probabilities, and direct averaging of model parameters. The efficacy of these model-based averaging approaches is demonstrated for a family of generalized hyperbolic mixture models using real and simulated data. / Thesis / Master of Science (MSc)

We propose a method to deal simultaneously with model uncertainty and correlated regressors in linear regression models by combining elastic net specifications with a spike and slab prior. The estimation method nests ridge regression and the LASSO estimator and thus allows for a more flexible modelling framework than existing model averaging procedures. In particular, the proposed technique has clear advantages when dealing with datasets of (potentially highly) correlated regressors, a pervasive characteristic of the model averaging datasets used hitherto in the econometric literature. We apply our method to the dataset of economic growth determinants by Sala-i-Martin et al. (Sala-i-Martin, X., Doppelhofer, G., and Miller, R. I. (2004). Determinants of Long-Term Growth: A Bayesian Averaging of Classical Estimates (BACE) Approach. American Economic Review, 94: 813-835) and show that our procedure has superior out-of-sample predictive abilities as compared to the standard Bayesian model averaging methods currently used in the literature. (authors' abstract) / Series: Research Report Series / Department of Statistics and Mathematics

The exchange rate forecasting has been an interesting topic for a long time. Beating the random walk model has been the goal of many researchers, who applied various techniques and used various datasets. We tried to beat it using bayesian model averaging technique, which pools a large amount of models and the final forecast is the average of forecasts of these models. We used quarterly data from 1980 to 2013 and attempted to predict the value of exchange rate return of five currency pairs. The novelty was the fact that none of these currency pairs included U.S. Dollar. The forecasting horizon was one, two, four and eight quarters. In addition to random walk, we also compared our results to historical average return model using several benchmarks, such as root mean squared error, mean absolute error or direction of change statistic. We found out that bayesian model averaging can not generally outperform random walk or historical average return, but in specific setting it can produce forecasts with low error and with high percentage of correctly predicted signs of change.

Performance of stock markets is determined by three classes of variables: macroeconomic indicators, industry & firm heterogeneity and third country effects. When assessing performance of a stock market index, impact of industry & firm heterogeneity is marginal as it is already embedded in the index through its constituent companies. This paper will therefore focus on the other two. Chinese stock market was selected as an application as their performance compared to other domestic indicators (mainly GDP growth) is considered inferior by many researchers. Using econometric framework for panel data and a Bayesian extension, the paper estimates multiple models of Chinese stock market performance examining individual determinants of it. Subsequently, it predicts development of theoretical prices of two main Chinese stock indices on two time samples until 2013. The paper then demonstrates underperformance of Chinese stock market by comparing the modeled prices to actual prices realized on the market. JEL Classification C23, C51, C53, G15, G17 Keywords underperformance, panel data, fixed effects model, Bayesian Model Averaging Author's e-mail roman_kovac@ymail.com Supervisor's e-mail karel.bata@seznam.cz

MASTER THESIS Determinants of Economic Growth: A Bayesian Model Averaging Author: Bc. Nikoloz Kudashvili Abstract The paper estimates the economic growth determinants across 72 countries using a Bayesian Model Averaging. Unlike the other studies we include debt to GDP ratio as an explanatory variable among 29 growth determinants. For given values of the other variables debt to GDP ratio up to the threshold level is positively related with the growth rate. The coefficient on the ratio has nearly 0.8 posterior inclusion probability suggesting that debt to GDP ratio is an important long term growth determinant. We find that the initial level of GDP, life expectancy and equipment investments have a strong effect on the GDP per capita growth rate together with the debt to GDP ratio.

The main aim of this thesis is to design efficient and novel numerical algorithms for a class of deterministic and stochastic dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become, therefore, inefficient when the much longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. This dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three different papers. The framework of the heterogeneous multiscale method (HMM) is considered as a general strategy both for the design and the analysis of multiscale methods. In Chapter 2, we consider a new class of multiscale methods that use a technique related to the construction of a Poincaré map. The main idea is to construct effective paths in the state space whose projection onto the slow subspace shows the correct dynamics. More precisely, we trace the evolution of the invariant manifold M(t), identified by the level sets of slow variables, by introducing a slowly evolving effective path which crosses M(t). The path is locally constructed through interpolation of neighboring points generated from our developed map. This map is qualitatively similar to a Poincaré map, but its construction is based on the procedure which solves two split equations successively backward and forward in time only over a short period. This algorithm does not require an explicit form of any slow variables. In Chapter 3, we present efficient techniques for numerical averaging over the invariant torus defined by ergodic dynamical systems which may not be mixing. These techniques are necessary, for example, in the numerical approximation of the effective slow behavior of highly oscillatory ordinary differential equations in weak near-resonance. In this case, the torus is embedded in a higher dimensional space and is given implicitly as the intersection of level sets of some slow variables, e.g. action variables. In particular, a parametrization of the torus may not be available. Our method constructs an appropriate coordinate system on lifted copies of the torus and uses an iterated convolution with respect to one-dimensional averaging kernels. Non-uniform invariant measures are approximated using a discretization of the Frobenius-Perron operator. These two numerical averaging strategies play a central role in designing multiscale algorithms for dynamical systems, whose fast dynamics is restricted not to a circle, but to the tori. The efficiency of these methods is illustrated by numerical examples. In Chapter 4, we generalize the classical two-scale averaging theory to multiple time scale problems. When more than two time scales are considered, the effective behavior may be described by the new type of slow variables which do not have formally bounded derivatives. Therefore, it is necessary to develop a theory to understand them. Such theory should be applied in the design of multiscale algorithms. In this context, we develop an iterated averaging theory for highly oscillatory dynamical systems involving three separated time scales. The relevant multiscale algorithm is constructed as a family of multilevel solvers which resolve the different time scales and efficiently computes the effective behavior of the slowest time scale. / text

In this paper we investigate the economic, political, institutional, and societal factors that encourage entrepreneurial activity. We do so by applying Bayesian Model Averaging, which controls for model uncertainty, to a panel data set for 33 countries. Our results indicate that the general state of macroeconomic activity, the availability of financing, the level of human capital, fiscal policies implemented and the type of economic system are the main determinants of the level of entrepreneurship. We also document a non-linear, U-shaped relation between distortionary taxation and entrepreneurial activity. Keywords: Entrepreneurship, Entrepreneurial Activity, Total Early-Stage Activity (TEA), Global Entrepreneurial Monitor (GEM), Bayesian Model Averaging (BMA), Panel Estimation. JEL Classification: B30, B53, C11, C23, J20, M13, O10, O40

This thesis presents a modification of existing entropy-based splitting criteria for classification and regression trees. Trees are typically grown using splitting criteria that choose optimal splits without taking future splits into account. This thesis examines localised splitting criteria that are based on local averaging in regression trees or local proportions in classification trees. The use of a localised criterion is motivated by the fact that future splits result in leaves that contain local observations, and hence local deviances provide a better approximation of the deviance of the fully grown tree. While most recent research has focussed on tree-averaging techniques that are aimed at taking a moderately successful splitting criterion and improving its predictive power, this thesis concentrates on improving the splitting criterion. Use of a localised splitting criterion captures local structures and enables later splits to capitalise on the placement of earlier splits when growing a tree. Using the localised splitting criterion results in much simpler trees for pure interaction data (data with no main effects) and can produce trees with fewer errors and lower residual mean deviances than those produced using a global splitting criterion when applied to real data sets with strong interaction effects. The superiority of the localised splitting criterion can persist when multiple trees are grown and averaged using simple methods. Although a single tree grown using the localised splitting criterion can outperform tree averaging using the global criterion, generally improvements in predictive performance are achieved by utilising the localised splitting criterion's property of detecting local discontinuities and averaging over sets of trees grown by placing splits where the deviance is locally minimal. Predictive performance improves further when the degree of localisation of the splitting criterion is randomly selected and weighted randomisation is used with locally minimal deviances to produce sets of trees to average over. Although state of the art methods quickly average very large numbers of trees, thus making the performance of the splitting criterion less critical, predictive performance when the localised criterion is used in bagging indicates that different splitting methods warrant investigation. The localised splitting criterion is most useful for growing one tree or a small number of trees to examine structure in the data. Structurally different trees can be obtained by simply splitting the data where the localised splitting criterion is locally optimal.

Abstract The topic of this thesis is superconducting electric circuits. Technical advances have made possible the experimental study of Josephson junction based circuit elements which sustain quantum mechanical properties long enough to be denoted as quantum devices. The quantum state can be controlled with electronic variables and measured using standard electrical setups. The research is motivated by the possibility to examine quantum phenomena in circumstances that can be customized, prospects of new quantum devices, and the development of quantum information processing. This thesis presents theoretical studies on the nonlinear and stochastic driving of a superconducting quantum two-level system (qubit). We first investigate the energy level shifts a single-Cooper-pair transistor under large amplitude driving realized via the inherently nonlinear Josephson energy by using an external magnetic flux. The effective driving field substantially deviates from a circular polarization and linear coupling. The energy level shifts are compared to the cases of a vanishing and a weak driving field, measured as the Stark shift and the generalized Bloch-Siegert shift, respectively. We describe criteria for the natural basis of the analytical and the numerical calculations. In addition to that, we develop a formalism based on the Floquet method for the weak probe measurement of the strongly driven qubit. In the latter part of the thesis research, we study utilization of a stochastic driving field whose time evolution is not regular but follows probabilistic laws. We concentrate on the motional averaging phenomenon and show that it can be measured with an unparalleled accuracy by employing a flux-modulated transmon qubit. As the stochastically modulated qubit is simultaneously measured with a moderate driving field, we develop a theoretical description accounting the possible interference effects between the modulation and the drive. The comparison with experimental results shows good agreement. Motional averaging phenomenon can be applied to estimate the properties of fluctuation processes occurring in qubits, e.g., the quasiparticle tunneling or the photon shot noise. Resting on the motional averaging, we anticipate that the qubit dephasing times can be improved if one can accelerate the dynamics of two-level fluctuators. We apply a semiclassical formalism where the qubit is treated with quantum mechanical concepts whereas the driving fields are classical. In the solution procedure, the numerical results support the main analytical understanding. As the theoretical results are extensively compared to reflection measurements, we construct an explicit connection between the dynamics of the studied quantum devices and the measured reflection coefficient.