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Stochastic Equations with Correlated Noise and Their Applications
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Peszat, Szymon (referee) ; Hlubinka, Daniel (referee)
Stochastic Equations with Correlated Noise and Their Applications Ondřej Týbl Doctoral Thesis Abstract Properties of stochastic differential equations with jumps are stud- ied. Lyapunov-type methods are derived to assess long-time behavior of solu- tions and general results are applied in specific cases. In the first case, conditions in terms of the geometric properties of the coefficients for stability in terms of boundedness in probability in the mean are obtained. By means of Krylov Bogolyubov Theorem criterion for existence of invariant measures is given sub- sequentely. In the second case, the long-time behavior refers to existence of an almost sure single-point limit not depending on the initial condition. This result is then applied to get a continuous-time Robbins-Monro type stochastic approximation procedure for finding roots of a given function. 1
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Conformal prediction
Krynická, Michaela ; Maciak, Matúš (advisor) ; Týbl, Ondřej (referee)
The main objective of this work is to formalize the concept of conformal prediction. This robust, nonparametric method allows the construction of an accurate prediction interval at a specified level, for which it is sufficient to assume that the input data are independent, equally distributed. In the context of random sampling from a one- dimensional continuous distribution, we expose the theoretical foundations of the method. Subsequently, we define the key concept of the degree of nonconformance and present the algorithmic design, first for random sampling and then in the context of regression ana- lysis. At the end of the work, we compare the reliability and effectiveness of conformal prediction with a specific frequency method on randomly generated data. 1
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Uniform law of large numbers, VC dimension and machine learning
Kossumov, Aibat ; Omelka, Marek (advisor) ; Týbl, Ondřej (referee)
In this thesis we study the generalized Glivenko-Cantelli theorem and its application in mathematical foundations of machine learning. Firstly, we prove the generalized Glivenko-Cantelli's theorem using covering numbers and lemma of symmetrization. Next we show the uniform law of large numbers. Then, we deal with Vapnik-Chervonenkis classes of functions (VC classes). We show that for VC classes covering numbers are uniformly bounded. Finally, we describe the task of machine learning and give an example of one specific task that can be "learned". The main application will be to prove the fundamental theorem of statistical learning. Usually this theorem is proved for classes of predictors that are Probably Approximately Correct learnable (PAC learnable). In this work we strengthen the property of PAC learnable and for it we prove the basic theorem of statistical learning. 1
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Kalman-Bucy Filter in Continuous Time
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1
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Kalman-Bucy Filter in Continuous Time
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1
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Stochastic Integration
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Dostál, Petr (referee)
The object of this thesis is a theory of stochastic integration, i.e., an inte- gration of a stochastic process with respect to a stochastic process. First, the Ito integral with respect to processes with finite quadratic variation is presented. This integral is then used to define the Stratonovich integral and both integrals are subsequently compared in terms of a martingale property and so-called chain rule. The core of this work is then a comparison of these two integrals as limits of aproximating sums. A third variant of an integral, first introduced in Strato- novich (1966), is then defined as a limit of sums of a different type. The resulting integral is equivalent to the original Stratonovich integral when the integrand is the Wiener process, however, it may differ if even when integrating with respect to a continuous process (a counterexample Yor (1977) is provided). A sufficient condition for an equivalence of these two integrals from Protter (2004) is presen- ted. 1
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