|
|
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.
|
|
|
Models for zero-inflated data
Matula, Dominik ; Kulich, Michal (advisor) ; Hlubinka, Daniel (referee)
The aim of this thesis is to provide a comprehensive overview of the main approaches to modeling data loaded with redundant zeros. There are three main subclasses of zero modified models (ZMM) described here - zero inflated models (the main focus lies on models of this subclass), zero truncated models and hurdle models. Models of each subclass are defined and then a construction of maximum likelihood estimates of regression coefficients is described. ZMM models are mostly based on Poisson or negative binomial type 2 distribution (NB2). In this work, author has extended the theory to ZIM models generally based on any discrete distributions of exponential type. There is described a construction of MLE of regression coefficients of theese models, too. Just few of present works are interested in ZIM models based on negative binomial type 1 distribution (NB1). This distribution is not of exponential type therefore a common method of MLE construction in ZIM models cannot be used here. In this work provides modification of this method using quasi-likelihood method. There are two simulation studies concluding the work. 1
|
|
|
Cure-rate models
Drabinová, Adéla ; Kulich, Michal (advisor) ; Omelka, Marek (referee)
In this work we deal with survival models, when we consider that with positive probability some patients never relapse because they are cured. We focus on two-component mixture model and model with biological motivation. For each model, we derive estimate of probability of cure and estimate of survival function of time to relaps of uncured patients by maximum likelihood method. Further we consider, that both probability of cure and survival time can depend on regressors. Models are then compared through simulation study. 1
|
|
|
Estimation in continuous time Markov chains
Nemčovič, Bohuš ; Prokešová, Michaela (advisor) ; Kadlec, Karel (referee)
Title: Estimation in continuous time Markov chains Author: Bohuš Nemčovič Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this work we deal with estimating the intensity matrices of continu- ous Markov chains in the case of complete observation and observation at selected discrete time points. To obtain an estimate we use the maximum likelihood met- hod. In the second chapter we first introduce the general EM algorithm and then adjust it for finding the intensity matrix estimate based on observations at disc- rete time points. In the last chapter we will illustrate the impact of the discrete step size on the quality of intensity matrix estimate. Keywords: Markov chains, intensity matrix, maximum likelihood estimation, EM algorithm 1
|
|
|
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
|
|
|
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)
|
|
|
Finite Mixture Models
Rusý, Tomáš ; Komárek, Arnošt (advisor) ; Omelka, Marek (referee)
This work focuses on finite mixture models and aims to introduce the maximum likelihood method as an approach of fitting finite mixtures. For that purpose the EM algorithm is adopted and derived in detail. Both the E and M steps of the EM algorithm are presented and performed for general finite mixture model. We derive new estimates for some of parameters defining the model. All updated estimates of the iteration of the EM algorithm are derived explicitly for the specific family of normal mixtures. The described theory is then applied to a model of the production of cytokine interleukin 10 in human periodontitis attack, which clearly demonstrates an application of the model in practice. Finally, we discuss the theory of clustering, which is based on our previous results. Like the previous theory, this one is also illustrated in the aforementioned model of cytokine IL10 production. Powered by TCPDF (www.tcpdf.org)
|
|
|
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.
|
|
|
Applications of EM-algorithm
Komora, Antonín ; Omelka, Marek (advisor) ; Kulich, Michal (referee)
EM algorithm is a very valuable tool in solving statistical problems, where the data presented is incomplete. It is an iterative algorithm, which in its first step estimates the missing data based on the parameter estimate from the last iteration and the given data and it does so by using the conditional expectation. In the second step it uses the maximum likelihood estimation to find the value that maximizes the logarithmic likelihood function and passes it along to the next iteration. This is repeated until the point, where the value increment of the logarithmic likelihood function is small enough to stop the algorithm without significant errors. A very important characteristic of this algorithm is its monotone convergence and that it does so under fairly general conditions. However the convergence itself is not very fast, and therefore at times requires a great number of iterations.
|
|
|
Classification of Testing Maneuvers from Flight Data
Funiak, Martin ; Dittrich, Petr (referee) ; Chudý, Peter (advisor)
Zapisovač letových údajů je zařízení určené pro zaznamenávání letových dat z různých senzorů v letadlech. Analýza letových údajů hraje důležitou roli ve vývoji a testování avioniky. Testování a hodnocení charakteristik letadla se často provádí pomocí testovacích manévrů. Naměřená data z jednoho letu jsou uložena v jednom letovém záznamu, který může obsahovat několik testovacích manévrů. Cílem této práce je identi kovat základní testovací manévry s pomocí naměřených letových dat. Teoretická část popisuje letové manévry a formát měřených letových dat. Analytická část popisuje výzkum v oblasti klasi kace založené na statistice a teorii pravděpodobnosti potřebnou pro pochopení složitých Gaussovských směšovacích modelů. Práce uvádí implementaci, kde jsou Gaussovy směšovací modely použité pro klasifi kaci testovacích manévrů. Navržené řešení bylo testováno pro data získána z letového simulátoru a ze skutečného letadla. Ukázalo se, že Gaussovy směšovací modely poskytují vhodné řešení pro tento úkol. Další možný vývoj práce je popsán v závěrečné kapitole.
|
guest ::
