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Bayesian and Maximum Likelihood Nonparametric Estimation in Monotone Aalen Model
Timková, Jana
This work is devoted to seeking methods for analysis of survival data with the Aalen model under special circumstances. We supposed, that all regres- sion functions and all covariates of the observed individuals were nonnegative and we named this class of models monotone Aalen models. To find estimators of the unknown regression functions we considered three maximum likelihood based approaches, namely the nonparametric maximum likelihood method, the Bayesian analysis using Beta processes as the priors for the unknown cumulative regression functions and the Bayesian analysis using a correlated prior approach, where the regression functions were supposed to be jump processes with a martingale structure. Powered by TCPDF (www.tcpdf.org)
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EM algorithm
Vacula, Ondřej ; Komárek, Arnošt (advisor) ; Antoch, Jaromír (referee)
This paper discusses the EM algorithm. This algorithm is used, for example, to calculate maximum likelihood estimate of unknown parameter. The algorithm is based on repeated calculations of certain expected value and maximizing specific function. We begin with parameter estimation problem, describe the maximum likelihood method and concept of incomplete data. Then we formulate the EM algorithm and its properties. In the next chapter we apply this knowledge to three selected statistical problems. At first we examine standard mixture model, then the linear mixed model and finally we analyze censored data. Powered by TCPDF (www.tcpdf.org)
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Expectation-Maximization Algorithm
Vichr, Jaroslav ; Pešta, Michal (advisor) ; Zvára, Karel (referee)
EM (Expectation-Maximization) algorithm is an iterative method for finding maximum likelihood estimates in cases, when either complete data include missing values or assuming the existence of additional unobserved data points can lead to more simple formulation of the model. Each of its iterations consists of two parts. During the E step (expectation) we calculate the expected value of the log-likelihood function of the complete data, with respect to the observed data and the current estimate of the parameter. The M step (maximization) then finds new estimate, which will maximize the function obtained in the previous step and which will be used in the next iteration in step E. EM algorithm has important use in e.g. price and manage risk of the portfolio.
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Gini coefficient maximization in binary logistic regression
Říha, Samuel ; Hanzák, Tomáš (advisor) ; Hlávka, Zdeněk (referee)
This Bachelor thesis describes a binary logistic regression model. By means of the term loss function a parameter estimation for the model is derived. A "rich" set of "proper" loss functions - beta family of Fisher-consistent loss functions - is defined. In the second part of the thesis, four basic goodness-of-fit criteria - Gini coefficient, C-statistics, Kolmogorov-Smirnov statistics and coefficient of determination R2 are defined. Further on, a possibility of parameter estimation by maximizing the Gini coefficient is analysed. Several algorithms are designed for this purpose. They are compared with so far existing methods in one simulated data set and three real ones. 1
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Evaluation of Long-term Land Cover Changes in the Region of Svitavy
Drahošová, Stanislava ; Štych, Přemysl (advisor) ; Slačíková, Jana (referee)
Land Cover changes in region Svitavy Abstract The aim of this thesis is to evaluate land cover changes of remote sensing data in region Svitavy after 1990. Landsat data (1994 and 2001) and IRS-P6 data (2006) were used. The literature search concerns the spectral characteristics of surface, the image classification and evaluating land cover changes in the Czech Republic. For image classification was used Maximum Likelihood algorithm of supervised method. The parts of image classification are the training stage, the classification stage, postclassification smoothing and classification accuracy assessment. Land cover changes were evaluated by compare area and share of land cover category and overlap of vector layers for 1994 - 2001 and 2001 - 2006. Two programmes were used: PCI Geomatica and the ArcGIS. In my classification were found some errors based on evaluation error matrix. For the whole time period was confirm relatively assumed tendency as growth of grasslands and forests and spreading built-up area in region Svitavy. Keywords: land cover, classification, region Svitavy, Maximum Likelihood
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Estimation and goodness-of-fit criteria in logistic regression model
Ondrušková, Markéta ; Hanzák, Tomáš (advisor) ; Zvára, Karel (referee)
In this bachelor thesis we describe binary logistic regression model and estimation of model's parameters by maximum likelihood method. Then we propose algorithm for the least squares method. In the goodness-of-fit criteria part we define Lorenz curve, Gini coefficient, C-statistics, Kolmogorov-Smirnov statistics and coefficient of determination R2 . We derive their relation to different sample coefficients of correlation. We derive typical relation between Gini coeffi- cient, Kolmogorov-Smirnov statistics and newly also coefficient of determination R2 via model of normally distributed score of bad and good clients. These derived teoretical results are verified on three real data sets. Keywords: Binary logistic regression, maximum likelihood, ordinary least squa- res, Gini coefficient, coefficient of determination. 1
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Evolution of brain size in bats (Chiroptera)
Králová, Zuzana ; Němec, Pavel (advisor) ; Kratochvíl, Lukáš (referee)
According to the prevailing doctrine, brain size has mainly increased throughout the evolution of mammals and reductions in brain size were rare. On the other hand, energetic costs of developing and maintaining big brain are high, so brain size reduction should occur every time when the respective selective pressure is present. Modern phylogenetic methods make it possible to test the presence of evolutionary trend and to infer the ancestral values of the trait in question based on knowledge of phylogeny and trait values for recent species. However, this approach has been rarely applied to study brain evolution so far. In this thesis, I focus on bats (Chiroptera). Bats are a suitable group for demonstrating the importance of brain size reductions. Considering their energetically demanding mode of locomotion, they are likely to have been under selection pressure for brain reduction. Furthermore, there is a large amount of data on body and brain mass of recent species available. Finally, phylogenetic relationships among bats are relatively well resolved. My present study is based on body masses and brain masses of 334 recent bat species (Baron et al., 1996) and on a phylogeny obtained by adjusting existing bat supertree (Jones et al., 2002) according to recent molecular studies. Analysing the data for...
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Extreme Value Distributions with Applications
Fusek, Michal ; Skalská,, Hana (referee) ; Karpíšek, Zdeněk (referee) ; Michálek, Jaroslav (advisor)
The thesis is focused on extreme value distributions and their applications. Firstly, basics of the extreme value theory for one-dimensional observations are summarized. Using the limit theorem for distribution of maximum, three extreme value distributions (Gumbel, Fréchet, Weibull) are introduced and their domains of attraction are described. Two models for parametric functions estimation based on the generalized extreme value distribution (block maxima model) and the generalized Pareto distribution (threshold model) are introduced. Parameters estimates of these distributions are derived using the method of maximum likelihood and the probability weighted moment method. Described methods are used for analysis of the rainfall data in the Brno Region. Further attention is paid to Gumbel class of distributions, which is frequently used in practice. Methods for statistical inference of multiply left-censored samples from exponential and Weibull distribution considering the type I censoring are developed and subsequently used in the analysis of synthetic musk compounds concentrations. The last part of the thesis deals with the extreme value theory for two-dimensional observations. Demonstrational software for the extreme value distributions was developed as a part of this thesis.
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Channel estimation in CDMA systems
Kadlec, Petr ; Prokopec, Jan (referee) ; Kejík, Petr (advisor)
The subject of this work deals with the problem of channel estimation for CDMA systems. This method of multiple access when individual users share the same full bandwidth simultaneously and are differentiated with any of pseudorandom sequences, is now the most perspective method. That is proved by its wide implementation in mobile networks of the third generation and higher systems. This work describes basic theory principles of spread spectrum, above all DS-CDMA (Direct Sequence-CDMA) and furthermore some phenomena of radio wireless channel that affect changes in transmitted signal in its way from transmitter to receiver. Terms of fading, multipath propagation, loss, refraction, scattering of the wave and Rice and Rayleigh probability density functions are mentioned. The third chapter deals with yet known and used capabilities of channel estimation. Differences, advantages and disadvantages of so-called blind estimation or training-based estimation are discussed. Two algorithms: LS method and sliding correlator are analyzed in more detail. There is also description of their simulations in Matlab and some results of these simulations are discussed. The last chapter deals with comparison of main characteristics and achievable accuracy of wireless channel impulse response estimation by both methods, and their possible utilization in real live.
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Probabilistic model for textile concrete reinforcement and comparison with experiments
Lomič, Jiří ; Rypl, Rostislav (referee) ; Vořechovský, Miroslav (advisor)
The scope of the presented bachelor’s thesis was the establishment of a probabilistic model for material strength of textile reinforcement used for textile reinforced concrete. This reinforcement is composed of AR-glass multi-filament yarns. The goal of this thesis was to determine the potential weak spot of the textile yarn and evaluate its strength in overall. The weak spot could have been a lateral cross-connection, which narrowed the textile yarn at several locations. Another thing of interest was the observation of statistical size effect with the increasing length of textile yarn. In order to properly fit the numerical model to real behavior of multi-filament yarns, five series of experimental tensile testing has been executed in laboratory. Each series consisted of 8-10 specimens and had a different yarn length. Maximum tensile force and maximum deformation have been measured to obtain L-D diagrams for each specimen. Measured data were statistically analyzed and gave the information necessary for the identification of probabilistic model parameters. This parameter estimation has been carried out with the help of numerical and optimization methods included in Python programming algorithms. The problem statement resulted in a combination of model parameters describing the textile yarn behavior. The statistical size effect was observed corresponding to the Weibull theory. The performed study showed that the failure of the textile yarn depends on material strength of its filaments. There are no load concentrators at the location of lateral cross-connections affecting the yarn failure.
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