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Statistical methods for sparse functional object data: elastic curves, shapes and densities
Statistical methods for sparse functional object data: elastic curves, shapes and densities
Many applications naturally yield data that can be viewed as elements in non-linear spaces. Consequently, there is a need for non-standard statistical methods capable of handling such data. The work presented here deals with the analysis of data in complex spaces derived from functional L2-spaces as quotient spaces (or subsets of such spaces). These data types include elastic curves represented as d-dimensional functions modulo re-parametrization, planar shapes represented as 2-dimensional functions modulo rotation, scaling and translation, and elastic planar shapes combining all of these invariances. Moreover, also probability densities can be thought of as non-negative functions modulo scaling. Since these functional object data spaces lack a natural Hilbert space structure, this work proposes specialized methods that integrate techniques from functional data analysis with those for metric and manifold data. In particular, but not exclusively, novel regression methods for specific metric quotient spaces are discussed. Special attention is given to handling discrete observations, since in practice curves and shapes are typically observed only as a discrete (often sparse or irregular) set of points. Similarly, density functions are usually not directly observed, but a (small) sample from the corresponding probability distribution is available. Overall, this work comprises six contributions that propose new methods for sparse functional object data and apply them to relevant real-world datasets, predominantly in a biomedical context.
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Steyer, Lisa
2023
English
Universitätsbibliothek der Ludwig-Maximilians-Universität München
Steyer, Lisa (2023): Statistical methods for sparse functional object data: elastic curves, shapes and densities = Statistische Methoden für spärlich beobachtete funktionale Objektdaten: elastische Kurven, Formen und Dichten. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

Many applications naturally yield data that can be viewed as elements in non-linear spaces. Consequently, there is a need for non-standard statistical methods capable of handling such data. The work presented here deals with the analysis of data in complex spaces derived from functional L2-spaces as quotient spaces (or subsets of such spaces). These data types include elastic curves represented as d-dimensional functions modulo re-parametrization, planar shapes represented as 2-dimensional functions modulo rotation, scaling and translation, and elastic planar shapes combining all of these invariances. Moreover, also probability densities can be thought of as non-negative functions modulo scaling. Since these functional object data spaces lack a natural Hilbert space structure, this work proposes specialized methods that integrate techniques from functional data analysis with those for metric and manifold data. In particular, but not exclusively, novel regression methods for specific metric quotient spaces are discussed. Special attention is given to handling discrete observations, since in practice curves and shapes are typically observed only as a discrete (often sparse or irregular) set of points. Similarly, density functions are usually not directly observed, but a (small) sample from the corresponding probability distribution is available. Overall, this work comprises six contributions that propose new methods for sparse functional object data and apply them to relevant real-world datasets, predominantly in a biomedical context.