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### Functional Component Analysis and Regression Using Elastic Methods

 Title: Name(s): Functional Component Analysis and Regression Using Elastic Methods. 65 views 28 downloads Tucker, J. Derek, authorSrivastava, Anuj, professor co-directing dissertationWu, Wei, professor co-directing dissertationKlassen, Eric, university representativeHuﬀer, Fred, committee member Department of Statistics, degree granting department Florida State University, degree granting institution text Text monographic 2014 Florida State University Tallahassee, Florida computeronline resource 1 online resource English Constructing generative models for functional observations is an important task in statistical function analysis. In general, functional data contains both phase (or x or horizontal) and amplitude (or y or vertical) variability. Traditional methods often ignore the phase variability and focus solely on the amplitude variation, using cross-sectional techniques such as functional principal component analysis for dimensional reduction and regression for data modeling. Ignoring phase variability leads to a loss of structure in the data, and inefficiency in data models. Moreover, most methods use a "pre-processing'' alignment step to remove the phase-variability; without considering a more natural joint solution. This dissertation presents three approaches to this problem. The first relies on separating the phase (x-axis) and amplitude (y-axis), then modeling these components using joint distributions. This separation in turn, is performed using a technique called elastic alignment of functions that involves a new mathematical representation of functional data. Then, using individual principal components, one for each phase and amplitude components, it imposes joint probability models on principal coefficients of these components while respecting the nonlinear geometry of the phase representation space. The second combines the phase-variability into the objective function for two component analysis methods, functional principal component analysis and functional principal least squares. This creates a more complete solution, as the phase-variability is removed while simultaneously extracting the components. The third approach combines the phase-variability into the functional linear regression model and then extends the model to logistic and multinomial logistic regression. Through incorporating the phase-variability a more parsimonious regression model is obtained and therefore, more accurate prediction of observations is achieved. These models then are easily extended from functional data to curves (which are essentially functions in R2) to perform regression with curves as predictors. These ideas are demonstrated using random sampling for models estimated from simulated and real datasets, and show their superiority over models that ignore phase-amplitude separation. Furthermore, the models are applied to classification of functional data and achieve high performance in applications involving SONAR signals of underwater objects, handwritten signatures, periodic body movements recorded by smart phones, and physiological data. FSU_migr_etd-9106 (IID) A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Summer Semester, 2014. May 20, 2014. Amplitude Variability, Functional Data Analysis, Function Alignment, Functional Regression, Function Principal Component Analysis, Phase Variability Includes bibliographical references. Anuj Srivastava, Professor Co-Directing Dissertation; Wei Wu, Professor Co-Directing Dissertation; Eric Klassen, University Representative; Fred Huﬀer, Committee Member. Statistics http://purl.flvc.org/fsu/fd/FSU_migr_etd-9106 FSU

Tucker, J. D. (2014). Functional Component Analysis and Regression Using Elastic Methods. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-9106