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Discrete sequential games with random payoffs
Račko, Lukáš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
In our thesis we consider games with random payoff as a generalizations of the stan- dard concept of games in the game theory. We discuss possible optimality conditions for these types of games. In one of these approaches by the concept of a α-Nash equilibria we manage to prove the existence of this generalization of Nash equilibria for the case when the payoff has only finite number of realizations. We then apply those concepts to the case when the game is considered in multiple stages. In the practical part of this thesis we consider an application to a competition of internet providers which we model by a generalized version of the Cornout model of duopoly. We compare results of our optimal strategy with the deterministic approaches to this problem. 1
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Limit theorems for dependent random variables
Švarcová, Anna ; Hlubinka, Daniel (advisor) ; Lachout, Petr (referee)
In presented work we will discuss the central limit theorem for dependent random variables. First of all, we brush up the basic version of the theorem and we illustrate it by an example of its use. Then we introduce a definition of the strong mixing condition that allows us to prove the theorem even for dependent random variables. Next, we focus on the assumptions which are essential for the validity of the theorem. The biggest part of the work we deal with its proof. Last of all, we illustrate this theorem with an example which helps us to better understand the main idea of the proof. We simulate this example for specific value of sequences that we define in the wording of the theorem. 1
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Causality, causality measures
Borák, Daniel ; Lachout, Petr (advisor) ; Pawlas, Zbyněk (referee)
Causality measures are useful tools when looking for causality in time series. This thesis does not only describe the theory behind the definition of measures of causality but also gives comprehensive instructions on how to use measures of causality to search for possible causal conditions. It is important to note that although measures of causality can point to possible causal relationships, they cannot confirm them. Causality is a complex relationship that cannot be captured by data alone - experiments and physical experience must also be considered. 1
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ROC curve
Zatkalíková, Zuzana ; Lachout, Petr (advisor) ; Antoch, Jaromír (referee)
This bachelor's thesis firstly defines the basic terms and then describes the ROC curve. Thesis deals with meaning of the ROC curve, its properties and construction with a graphic representation. Subsequently, the expression of the ROC curve and its area for normal, exponential and uniform distribution is derived in the work, also with a graphical representation. Then, it is related to statistical testing. At the end, there are described the empirical expression of the ROC curve and its application to real data processed in the Python programming language. 1
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Concentration inequalities for sums
Blatská, Tereza ; Hlubinka, Daniel (advisor) ; Lachout, Petr (referee)
In this bachelor thesis we focus on concentration inequalities for sums of indepen- dent random variables, which are bounded and not necessarily identically distributed. The main pillar of the thesis is Hoeffding's inequality, finding its improvement and other similar inequalities. Inequalities are completed with examples for various probability dis- tributions. In each example there is a theoretical calculation, a simulation for specifically selected parameters and a graphical representation of all the obtained estimates, which was created using the R programming language. 1
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Anderson's theorem
Bočinec, Filip ; Nagy, Stanislav (advisor) ; Lachout, Petr (referee)
In this thesis we explore a theorem from real analysis and geometry called Anderson's theorem. It concerns an integral inequality for symmetric quasi-concave functions, where the integration is done over a symmetric convex set. A thorough proof of Anderson's theorem is given. In addition, we investigate cases in which equality or strict inequality occurs. While studying this topic, we come across some problems in published papers and we try to clarify them. Furthermore, we explore possible extensions of the theorem. In particular, results involving group invariance and theory of s-concave functions are mentioned. As outlined in the final part of the thesis, Anderson's theorem is a useful and widely used tool in multivariate statistics. 1
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Scheduling problems under uncertainty with nonidentical machines
Matoušková, Monika ; Branda, Martin (advisor) ; Lachout, Petr (referee)
This thesis deals with operational fixed interval scheduling problems under uncer- tainty. The topic has been covered for identical machines and the theory is summarized in this thesis. The stochastic programming problem has a deterministic reformulation based on network flow under the assumption that the multivariate distribution of ran- dom delays follows an Archimedean copula. In this thesis, we focus on the problem with heterogeneous machines. When there are more possible types of machines, this problem somewhat complicates. The deterministic problem with no delays and more than one machine type is NP-hard. We generalized the deterministic reformulation of the stochas- tic problem with possible random delays for nonidentical machines. This formulation for more than one machine type loses an important property that holds for identical machines reformulation using network flow. Then an algorithm based on Lagrangean relaxation is proposed, implemented and compared with a solution obtained by MIP solver. 1
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Sparse solutions in labeling optimization problems
Komora, Ondřej ; Branda, Martin (advisor) ; Lachout, Petr (referee)
The main goal of this work is to give self-contained description of proximal and ordi- nary stochastic subgradient descent methods, which are used in finding sparse solutions of labeling optimization problems. We will define and interpret necessary concepts lea- ding to the definition of those methods and we will discuss in detail conditions, under which we show convergence of these methods to critical point. At the end, we will pre- sent a numerical experiment on concrete optimization task where we demonstrate use of these methods. In this experiment we will also show how a suitable choice of so called regularization can influence sparsity of solution of this particular task. 1
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Stochastic optimization models for energy trading
Klyuchevskiy, Iakov ; Lachout, Petr (advisor) ; Branda, Martin (referee)
Energy market modelling is one of the most pressing topics. There are several ways how to create the models depending on what features to expect and require. The thesis will be devoted to studying stochastic models of the energy market. We explain first the mechanics and the characteristics of the energy market. We then move on to two commonly used stochastic power price models with different options for volatility and the distribution of spikes. A comparison of the models then shows that some are significantly better than others. Further, two other models (and their modifications) were proposed based on the first four. One of them has shown good performance and can be advised for use in ELIX price modelling. 1
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Statistical inference for Markov processes with continuous time
Křepinská, Dana ; Prokešová, Michaela (advisor) ; Lachout, Petr (referee)
Tato diplomová práce se zabývá odhadováním matice intenzit Markovova pro- cesu se spojitým časem na základě diskrétně pozorovaných dat. Začátek práce je věnován jednoduššímu odhadu ze spojité trajektorie pomocí metody maximální věrohodnosti. Dále je zde popsán odhad z diskrétní trajektorie přes výpočet ma- tice pravděpodobností přechodu. Následně je velmi podrobně rozebrán EM al- goritmus, který předchozí odhad zpřesňuje. Na závěr teoretické části je uvedena metoda odhadu zvaná Monte Carlo Markov Chain. Všechny postupy jsou zároveň implementovány v počítačovém softwaru a prezentace jejich výsledk· je obsahem druhé části práce. V té jsou porovnané odhady pro denní, týdenní a měsíční po- zorování a také pro pětiletou a desetiletou pozorovanou trajektorii. K výsledk·m jsou připojeny odhady rozptyl· a intervaly spolehlivosti. 1
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