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thesis.toc
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thesis.toc
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\contentsline {chapter}{\numberline {1}Introduction and Related Work}{11}{chapter.1}
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\contentsline {chapter}{\numberline {2}Theory}{13}{chapter.2}
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\contentsline {section}{\numberline {2.1}Gaussian Process}{13}{section.2.1}
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\contentsline {subsection}{\numberline {2.1.1}Linear Regression and Linear Basis Function Model}{13}{subsection.2.1.1}
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\contentsline {subsection}{\numberline {2.1.2}Gaussian Process for Regression}{14}{subsection.2.1.2}
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\contentsline {subsection}{\numberline {2.1.3}Learning the Hyperparameters in Gaussian Process for Regression}{17}{subsection.2.1.3}
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\contentsline {section}{\numberline {2.2}Covariance Functions}{18}{section.2.2}
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\contentsline {subsection}{\numberline {2.2.1}Preliminaries}{18}{subsection.2.2.1}
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\contentsline {subsection}{\numberline {2.2.2}Examples of Covariance Functions}{19}{subsection.2.2.2}
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\contentsline {section}{\numberline {2.3}Sparse Approximation of Gaussian Process}{22}{section.2.3}
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\contentsline {subsection}{\numberline {2.3.1}Sparse Input Gaussian Process (SPGP)}{23}{subsection.2.3.1}
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\contentsline {subsection}{\numberline {2.3.2}Sparse Input Gaussian Process with Variable Noise (SPGP+HS)}{25}{subsection.2.3.2}
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\contentsline {subsection}{\numberline {2.3.3}Sparse Input Gaussian Process with Functional Variable Noise (SPGP+FUNC-HS)}{27}{subsection.2.3.3}
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\contentsline {section}{\numberline {2.4}Poisson Processes}{29}{section.2.4}
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\contentsline {section}{\numberline {2.5}Clustering Time Series of User Behavior}{30}{section.2.5}
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\contentsline {subsection}{\numberline {2.5.1}Problem Definition}{31}{subsection.2.5.1}
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\contentsline {subsection}{\numberline {2.5.2}Dynamic Piecewise Time Series Similarity Measure}{31}{subsection.2.5.2}
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\contentsline {subsection}{\numberline {2.5.3}K-Piece Wise Spectral Centroid}{34}{subsection.2.5.3}
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\contentsline {chapter}{\numberline {3}Coarse Grained Analysis of Population}{37}{chapter.3}
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\contentsline {section}{\numberline {3.1}Experimental Setup}{37}{section.3.1}
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\contentsline {section}{\numberline {3.2}Results}{37}{section.3.2}
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\contentsline {section}{\numberline {3.3}Analysis of the Learned Kernels Parameters}{40}{section.3.3}
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\contentsline {chapter}{\numberline {4}Fine Grained Analysis of Population}{43}{chapter.4}
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\contentsline {section}{\numberline {4.1}User Behavior Models Results}{43}{section.4.1}
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\contentsline {section}{\numberline {4.2}Common Patterns in the Users Behavior}{45}{section.4.2}
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\contentsline {chapter}{\numberline {5}Conclusion and Feature Work}{48}{chapter.5}
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\contentsline {chapter}{\numberline {A}Mathematical Background}{50}{appendix.A}
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\contentsline {section}{\numberline {A.1}Matrix Properties}{50}{section.A.1}
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\contentsline {section}{\numberline {A.2}Gaussian Distribution}{50}{section.A.2}
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\contentsline {chapter}{\numberline {B}Gaussian Process Derivations}{52}{appendix.B}
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\contentsline {section}{\numberline {B.1}Derivation of the Sparse Input Gaussian Process with Functional Variable Noise}{52}{section.B.1}
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\contentsline {section}{\numberline {B.2}Gradient Calculation of the Negative Log Marginal Likelihood of the Sparse Input Gaussian Process with Functional Variable Noise}{54}{section.B.2}
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\contentsline {section}{\numberline {B.3}Kernels Derivatives}{55}{section.B.3}
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\contentsline {chapter}{\numberline {C}Fine Grained Analysis Clusters}{58}{appendix.C}