Constrained functional data models with environmental applications

Measurements made on a continuous process are increasingly becoming available due to technological advances. The resulting data can be considered to be samples from curves. Such data are known as functional data. In this thesis, we make some contributions to functional data analysis methods. After providing a review of functional data, we introduce the functional linear model (FLM) with a scalar response. Then we propose methods to estimate the parameter curve in a FLM with a scalar response, subject to monotonicity constraints. Methodology for estimation of smoothing parameters for our model is provided. In addition, we provide connections between linear and non-linear mixed models for our functional linear model without and with monotonicity constraints, respectively. Methods for assessing the monotonicity assumption and simulation results are discussed.;We also investigate the variations (amplitude and phase variation) in the snowpack levels of the Sierra Nevada using daily snow water equivalent (SWE) measurements. Estimation of the amplitude and phase variations is done through scaling and curve registration. We investigate how these variations are associated with atmospheric indices and spatial location. To model the amplitude variation we incorporate a generalized additive mixed model, and to model the phase variation we employ a FLM with a functional response.