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Robust estimation of autocorrelation function
Lain, Michal ; Hudecová, Šárka (advisor) ; Hlávka, Zdeněk (referee)
The autocorrelation function is a basic tool for time series analysis. The clas- sical estimation is very sensitive to outliers and can lead to misleading results. This thesis deals with robust estimations of the autocorrelation function, which is more resistant to the outliers than the classical estimation. There are presen- ted following approaches: leaving out the outliers from the data, replacement the average with the median, data transformation, the estimation of another coeffici- ent, robust estimation of the partial autocorrelation function or linear regression. The thesis describes the applicability of the presented methods, their advantages and disadvantages and necessary assumptions. All the approaches are compared in simulation study and applied to real financial data. 1
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Comparison of Models for Probabilities in Football Betting
Kožnar, František ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of the thesis is to compare different statistical models for football betting odds and determine the best performing once based on the historical performance of sport teams. There are at least three possible approaches for computing the odds, namely logistic regression, Poisson regression and methods based on statistical machine learning. The idea is that the historical performance of teams is a good predictor of the future performance. Thus we can take the past performances, say all matches in the full season of the English Premier League (380 matches), and use these data for predicting the odds for the following season. The resulting odds should be compared with the actual results using the scoring rules, which will identify the best performing model.
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Statistical machine learning with applications in music
Janásková, Eliška ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of this thesis is to train a computer on Beatles' songs using the re- search project Magenta from the Google Brain Team to produce its own music, to derive backpropagation formulas for recurrent neural networks with LSTM cells used in the Magenta music composing model, to overview machine learning techniques and discuss its similarities with methods of mathematical statistics. In order to explore the qualities of the artificially composed music more thor- oughly, we restrict ourselves to monophonic melodies only. We train three deep learning models with three different configurations (Basic, Lookback, and At- tention) and compare generated results. Even though the artificially composed music is not as interesting as the original Beatles, it is quite likeable. According to our analysis based on musically informed metrics, artificial melodies differ from the original ones especially in lengths of notes and in pitch differences be- tween consecutive notes. The artificially composed melodies tend to use shorter notes and higher pitch differences. 1
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Yield Curves
Korbel, Michal ; Hurt, Jan (advisor) ; Hlávka, Zdeněk (referee)
The master thesis is looking into the estimation of yield curve using two ap- proaches. The first one is searching for parametric model which is able to describe the behavior of yield curve well and estimate its parameters. The parametric mo- dels used in the thesis are derived from the class of models introduced by Nelson and Siegel. The second approach is nonparametric estimation of yield curves using spline smoothing and kernel smoothing. All used methods are then compared on real observed data and their suitability for various tasks and concrete available observations is considered. 1
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Zero inflated Poisson model
Veselý, Martin ; Komárek, Arnošt (advisor) ; Hlávka, Zdeněk (referee)
This paper deals with the zero-inflated Poisson distribution. First the Poisson model is defined and generalized to a zero-inflated model. The basic properties of this generalized model are derived. After- wards the basics of the method of moments and the maximum likelihood method are described. Both of these are used to derive parameter estimates of such distribution. The feasibility of calculating the distribution of moment method estimates is analyzed. Then the asymptotic distribution of maximum likelihood estimates is derived and used to create confidence intervals. In the last chapter a numeric si- mulation of the derived asymptotic properties is performed. Special attention is paid to situations where regularity conditions are not met. 1
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Boxplot for multivariate data
Brabenec, Tomáš ; Nagy, Stanislav (advisor) ; Hlávka, Zdeněk (referee)
We will introduce three methods of extension of the classical Tukey's Boxplot for multivariate data. These are the Rangefinder, the Relplot and the Bagplot. To implement the methods, we will need the notions like Mahalanobis distance, elliptically symmetric distributions and halfspace depth. A big part of the thesis is focused on the construction of the Relplot and the Bagplot. We will also discuss, how do these methods detect outliers and what are their advantages and disadvantages. This work contains many examples and illustrating images. 1
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Statistical machine learning with applications in music
Janásková, Eliška ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of this thesis is to review the current state of machine learning in music composition and to train a computer on Beatles' songs using research project Magenta from the Google Brain Team to produce its own music. In order to explore the qualities of the generated music more thoroughly, we restrict our- selves to monophonic melodies only. We train three deep learning models with three different configurations (Basic, Lookback, and Attention) and compare generated results. Even though the generated music is not as interesting as the original Beatles, it is quite likable. According to our analysis based on musically informed metrics, generated melodies differ from the original ones especially in lengths of notes and in pitch differences between consecutive notes. Generated melodies tend to use shorter notes and higher pitch differences. In theoretical background, we cover the most commonly used machine learning algorithms, introduce neural networks and review related work of music generation. 1
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Principal components
Zavadilová, Anna ; Hlávka, Zdeněk (advisor) ; Nagy, Stanislav (referee)
This thesis presents principal components as a useful tool for data dimensio- nality reduction. In the first part, the basic terminology and theoretical properties of principal components are described and a biplot construction is derived there as well. Besides, heuristic methods for a choice of the optimum number of prin- cipal components are summarised there. Subsequently, asymptotical properties of sample eigenvalues of covariance and white Wishart matrices are described and cases of equality of some eigenvalues are distinguished at the same time. In the second part of the thesis, asymptotic distribution of the largest eigenva- lue of white Wishart matrices is described, completed with graphic illustrations. A test of the number of significant eigenvalues is suggested on the basis of this limiting distribution, and the connection of this test to the number of suitable principal components is presented. The final part of the thesis provides an over- view of advanced computational methods for the choice of an adequate number of principal components. The thesis is completed with graphical illustrations and a simulation study using Wolfram Mathematica and R.
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k-sample problem with ordered alternative
Nováková, Martina ; Hlávka, Zdeněk (advisor) ; Pešta, Michal (referee)
In this thesis we deal with k-sample problem with ordered alternative. At the beginning of the thesis isotonic regression is introduced. We use isotonic regression for maximum likelihood estimation of ordered parameters. In the second chapter, we describe the χ2 and E 2 tests that use the knowledge of isotonic regression and are based on the likelihood ratio. The exact null hypothesis distributions of their test statistics are derived in detail. The one-sided studentized range test is also further described. At the end of the thesis, we show the use of the E 2 test on the real data. 1
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Ratio estimators
Klyuchevskiy, Iakov ; Hlávka, Zdeněk (advisor) ; Antoch, Jaromír (referee)
The aim of the bachelor thesis is to estimate the incidence of fractures in women from 0 to 20 years in the Czech Republic. In the introductory chapter we will introduce the concept of incidence and show the statistical data that we will continue to work with. In the second and third chapters we define statistical models for estimating the incidence and also the unit estimation by which we estimate the incidence, we will examine its properties. In the fourth chapter, we will show the real data to estimate the incidence of fractures in women for each age category.
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