Topics in functional data analysis

© 2012 Dr. Stephen Edward LaneAs the amount of data captured in experimental and observational situations has grown, so too has the need for more sophisticated tools of analysis. The broad area of functional data analysis (FDA) has received a great deal of attention within the statistics community over the past decade, as a way of dealing with the high dimensional data that is becoming more commonplace in areas such as medical science and biology. High dimensional data has classically been analysed by treating the relevant data sets as multivariate data sets. FDA seeks to analyse these data as if they are in fact functions, observed at either sparse or dense sample points. Motivated by problems in forest science and capture/recapture experiments, this thesis explores some of the FDA methodology through modelling data sets that contain functional responses or functional covariates. We show that a nonparametric 'whole of function' approach to predicting conditional probability density functions is a particularly useful alternative to commonly used parametric approaches, especially when density functions have 'non–standard' shapes. The functional prediction approach is further extended by developing functional regression models for functional longitudinal data. We show that these models allow prediction and extrapolation of density functions with arbitrarily (smooth) changing shapes over time, conditional on a functional growth covariate. A semiparametric population size estimator for mark-recapture data with continuous, time–varying covariates is developed from a functional data perspective. This method allows the full use of data resulting from such experiments, circumventing the usual loss of information in current approaches. We demonstrate that an iterative estimation approach in the form of an EM algorithm outperforms methods that ignore the time variation in the covariates