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Variable selection based on penalized likelihood
Chlubnová, Tereza ; Kulich, Michal (advisor) ; Maciak, Matúš (referee)
Selection of variables and estimation of regression coefficients in datasets with the number of variables exceeding the number of observations consti- tutes an often discussed topic in modern statistics. Today the maximum penalized likelihood method with an appropriately selected function of the parameter as the penalty is used for solving this problem. The penalty should evaluate the benefit of the variable and possibly mitigate or nullify the re- spective regression coefficient. The SCAD and LASSO penalty functions are popular for their ability to choose appropriate regressors and at the same time estimate the parameters in a model. This thesis presents an overview of up to date results in the area of characteristics of estimates obtained by using these two methods for both small number of regressors and multidimensional datasets in a normal linear model. Due to the fact that the amount of pe- nalty and therefore also the choice of the model is heavily influenced by the tuning parameter, this thesis further discusses its selection. The behavior of the LASSO and SCAD penalty functions for different values and possibili- ties for selection of the tuning parameter is tested with various numbers of regressors on simulated datasets.
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Big data - extraction of key information combining methods of mathematical statistics and machine learning
Masák, Tomáš ; Antoch, Jaromír (advisor)
This thesis is concerned with data analysis, especially with principal component analysis and its sparse modi cation (SPCA), which is NP-hard-to- solve. SPCA problem can be recast into the regression framework in which spar- sity is usually induced with ℓ1-penalty. In the thesis, we propose to use iteratively reweighted ℓ2-penalty instead of the aforementioned ℓ1-approach. We compare the resulting algorithm with several well-known approaches to SPCA using both simulation study and interesting practical example in which we analyze voting re- cords of the Parliament of the Czech Republic. We show experimentally that the proposed algorithm outperforms the other considered algorithms. We also prove convergence of both the proposed algorithm and the original regression-based approach to PCA. vi
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Big data - extraction of key information combining methods of mathematical statistics and machine learning
Masák, Tomáš ; Antoch, Jaromír (advisor)
This thesis is concerned with data analysis, especially with principal component analysis and its sparse modi cation (SPCA), which is NP-hard-to- solve. SPCA problem can be recast into the regression framework in which spar- sity is usually induced with ℓ1-penalty. In the thesis, we propose to use iteratively reweighted ℓ2-penalty instead of the aforementioned ℓ1-approach. We compare the resulting algorithm with several well-known approaches to SPCA using both simulation study and interesting practical example in which we analyze voting re- cords of the Parliament of the Czech Republic. We show experimentally that the proposed algorithm outperforms the other considered algorithms. We also prove convergence of both the proposed algorithm and the original regression-based approach to PCA. vi
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Big data - extraction of key information combining methods of mathematical statistics and machine learning
Masák, Tomáš ; Antoch, Jaromír (advisor) ; Maciak, Matúš (referee)
This thesis is concerned with data analysis, especially with principal component analysis and its sparse modi cation (SPCA), which is NP-hard-to- solve. SPCA problem can be recast into the regression framework in which spar- sity is usually induced with ℓ1-penalty. In the thesis, we propose to use iteratively reweighted ℓ2-penalty instead of the aforementioned ℓ1-approach. We compare the resulting algorithm with several well-known approaches to SPCA using both simulation study and interesting practical example in which we analyze voting re- cords of the Parliament of the Czech Republic. We show experimentally that the proposed algorithm outperforms the other considered algorithms. We also prove convergence of both the proposed algorithm and the original regression-based approach to PCA. vi
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Variable selection based on penalized likelihood
Chlubnová, Tereza ; Kulich, Michal (advisor) ; Maciak, Matúš (referee)
Selection of variables and estimation of regression coefficients in datasets with the number of variables exceeding the number of observations consti- tutes an often discussed topic in modern statistics. Today the maximum penalized likelihood method with an appropriately selected function of the parameter as the penalty is used for solving this problem. The penalty should evaluate the benefit of the variable and possibly mitigate or nullify the re- spective regression coefficient. The SCAD and LASSO penalty functions are popular for their ability to choose appropriate regressors and at the same time estimate the parameters in a model. This thesis presents an overview of up to date results in the area of characteristics of estimates obtained by using these two methods for both small number of regressors and multidimensional datasets in a normal linear model. Due to the fact that the amount of pe- nalty and therefore also the choice of the model is heavily influenced by the tuning parameter, this thesis further discusses its selection. The behavior of the LASSO and SCAD penalty functions for different values and possibili- ties for selection of the tuning parameter is tested with various numbers of regressors on simulated datasets.
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Analýza regionálních cen nemovitostí ve Spojených státech pomocí vysokodimenzionálního VAR modelu
Krčál, Adam ; Čížek, Ondřej (advisor) ; Zouhar, Jan (referee)
In this thesis the heterogeneity of regional real estate prices in United States is investigated. A high dimensional VAR model with additional exogenous predictors, originally introduced by \cite{fan11}, is adopted. In this framework, the common factor in regional house prices dynamics is explained by exogenous predictors and the spatial dependencies are captured by lagged house prices in other regions. For the purpose of estimation and variable selection under high-dimensional setting the concept of Penalized Least Squares (PLS) with different penalty functions (e.g. LASSO penalty) is studied in detail and implemented. Moreover, clustering methods are employed to identify subsets of statistical regions with similar house prices dynamics. It is demonstrated that these clusters are well geographically defined and contribute to a better interpretation of the VAR model. Next, we make use of the LASSO variable selection property in order to construct the impulse response functions and to simulate the prices behavior when a shock occurs. And last but not least, one-period-ahead forecasts from VAR model are compared to those from the Diffusion Index Factor Model by \cite{stock02}, a commonly used model for forecasts.
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