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Outliers
Kudrnáč, Vojtěch ; Zvára, Karel (advisor) ; Anděl, Jiří (referee)
This paper concerns itself with the methods of identifying outliers in an otherwise normally distributed data set. Several significant tests and criteria designed for this purpose are described here, Peirce's criterion, Chauvenet's criterion, Grubbs' test, Dixon's test and Cochran's test. Deriving of the tests and criteria is indicated and finally the results of the use of the test and criteria on simulated data with normal distribution and inserted outlier are looked into. Codes in programming language R with the implementation of these test and criteria using existing functions are included. Powered by TCPDF (www.tcpdf.org)
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Durbin-Watson test
Lipták, Patrik ; Zvára, Karel (advisor) ; Anděl, Jiří (referee)
The Bachelor Thesis deals with Durbin-Watson test which is used to test an inde- pendence of residuals in a normal linear regression model. The test is applicable in a case of collecting data gradually and if values of a dependent variable form time series. In the first part, thesis provides detailed derivation of a distribution of test statistic (or its bounds), as well as conclusion describing how to make a right decision in testing a hypothesis that the value of correlation coefficient is equal to 0. In the second part, three practical examples with real data are used to demonstrate this theoretical basis. Moreover, calculations are supplemented by illustrative graphs and they are made in computing environment R for com- parison. 1
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Random triangles
Matula, Dominik ; Anděl, Jiří (advisor) ; Dvořák, Jiří (referee)
The author summarizes some previous results concerning random triangles. He describes the Gaussian triangle and random triangles whose vertices lie in a unit n-dimensional ball, in a rectangle or in a general bounded convex set. In the second part, the author deals with an inscribed triangle in a triangle - let ABC be an equilateral triangle and let M, N, O be three points, each laying on one side of the ABC. We call MNO inscribed triangle (in an equi- laterral triangle). The median triangle is a special case of that triangle. Author starts with the median triangle and one by one replaces it's vertices by random points with uniform distribution on the corresponding sides. He proves that propability of such inscribed triangle to be an obtuse triangle increases with number of randomly chosen points while the expected area reminds constant. The whole thesis is concluded with a simulation study. 1
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Multiple Testing Problem
Waczulík, Oliver ; Komárek, Arnošt (advisor) ; Anděl, Jiří (referee)
Title: Multiple Testing Problem Author: Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Arnošt Komárek, Ph.D., Department of Probability and Mat- hematical Statistics Abstract: This thesis looks into the different approaches to control the type-1 error inflation, more specifically, it looks into different procedures that control familywise error rate (FWER), when testing more than one elementary hypothesis simultaneously. It focuses on two basic types of problems: comparing the mean of more than one independently normally distributed random sample with the control and com- paring the means of more than one independently normally distributed random sample between each other. To further illustrate the mechanics of the multiple comparison procedures stated in this thesis we use a real measurements of the brain volume of four independent groups of males, where we anticipate normal distribution with the same variance. Global null hypothesis is always tested as first, followed by testing of elementary hypothesis in the case of rejection. To tackle the first problem, we use Bonfferoni and Holm multiple comparison pro- cedures along with Dunnett method and closure principle. Unlike Dunnett with Bonfferoni, Holm and Bonfferoni's closed test provide different conclusion...
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Chebyshev inequality and some its modifications
Drabinová, Adéla ; Anděl, Jiří (advisor) ; Nagy, Stanislav (referee)
In the presented thesis we describe some improvements of Chebyshev inequa- lity. In the first chapter we introduce inequalities for random variables with uni- modal distributions. We prove Gauss and Camp-Meidell inequality and we deduce Vysochanskii-Petunin inequality. We describe inequalities for variables with mode 0 and with unspecified mode. In the second chapter we consider constants C(r), for which the approximations are the best. We are interested in finding optimal parameter r or its approximation. In the third chapter we state inequalities from the first chapter for specific distributions, calculation of their constants, appli- cations and graphic presentations of the results. 1
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Nonlinearity in time series models
Kalibán, František ; Anděl, Jiří (advisor) ; Zvára, Karel (referee)
The thesis concentrates on property of linearity in time series models, its definitions and possibilities of testing. Presented tests focus mainly on the time domain; these are based on various statistical methods such as regression, neural networks and random fields. Their implementation in R software is described. Advantages and disadvantages for tests, which are implemented in more than one package, are discussed. Second topic of the thesis is additivity in nonlinear models. The definition is introduced as well as tests developed for testing its presence. Several test (both linearity and additivity) have been implemented in R for purposes of simulations. The last chapter deals with application of tests to real data. 1
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Statistical analysis of historical temperature series
Gergelits, Václav ; Antoch, Jaromír (advisor) ; Anděl, Jiří (referee)
Title: Statistical analysis of historical temperature series Author: Václav Gergelits Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jaromír Antoch CSc. Supervisor's e-mail address: antoch@karlin.mff.cuni.cz Abstract: In the present work we deal with the statistical analysis of time-series of a mean-temperature obtained from seven European cities from the Europe Union project "IMPROVE". Properties of the time series are analyzed by means of descriptive statistics, being assessing their homoscedasticity, autocorrelation and normality. We report the ways in which the data has been adjusted, including consideration of the impact of the urban heat island and we discuss the availability of additional data. The theoretical part presents a theory of change point detection for a one change model as well as more than one change model taking an autocorrelation into account. In the practical part we analyze the data using change point detection method. The significant increase was not detected for time series of Cadiz and Uppsala. The significant increase was rather detected for the rest of the time series. The increase of temperature could be in a relation to the adjustment for the urban heat island. Keywords: change point detection, temperature time series 1
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Some functions of ARMA processes
Štufka, Miroslav ; Anděl, Jiří (advisor) ; Prášková, Zuzana (referee)
This study provides a comprehensive overview of changes in the autoregressive-moving- average model (ARMA) when applied to various functions. First, the necessary and sufficient condition for a weakly stationary stochastic process described by ARMA is given. Next, some particular transformations of ARMA processes are presented: first, non- correlated and generic sums of ARMA processes; next, products of independent and dependent Gaussian ARMA processes; and finally, time aggregations, namely, systematic sampling and temporal aggregations. Tables are included to clearly summarize special cases of particular transformations. Some of these cases are then demonstrated through concrete examples. In addition to theoretical results, extensive numerical simulation in statistical software R is also given, which systematically covers the obtained results.
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Birthday problem
Drápal, Lukáš ; Anděl, Jiří (advisor) ; Dostál, Petr (referee)
In the presented work we discuss the birthday problem with unlike probabilities. First, we introduce the concept of majorization of vectors, Schur convexity of functions and Bell polynomials. Using these concepts we show the results from papers [6] and [8]. We also discuss the paper [7] and we point out its error. Then we present a program in language R that is simulating the problem. We use this program to calculate the probability for the true birthday problem in the Czech Republic and the effect of leap years. Finally, we show some applications of the birthday problem, especially the true surname problem in Japan [8].
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Weighted Data Depth and Depth Based Discrimination
Vencálek, Ondřej ; Hlubinka, Daniel (advisor) ; Anděl, Jiří (referee) ; Malý, Marek (referee)
The concept of data depth provides a powerful nonparametric tool for multivariate data analysis. We propose a generalization of the well-known halfspace depth called weighted data depth. The weighted data depth is not affine invariant in general, but it has some useful properties as possible nonconvex central areas. We further discuss application of data depth methodology to solve discrimination problem. Several classifiers based on data depth are reviewed and one new classifier is proposed. The new classifier is a modification of k-nearest- neighbour classifier. Classifiers are compared in a short simulation study. Advantage gained from use of the weighted data depth for discrimination purposes is shown.
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