This thesis is concerned, loosely, with time series analysis. It is also, loosely, concerned with smoothers and Generalized Additive Models. And, finally, it is also concerned with the estimation of health risk due to air pollution. In the field of time series analysis, we develop two data-driven interpolation algorithms for interpolation of mixed time series data; that is, data which has a stationary or “almost” stationary background with embedded deterministic trend and sinusoidal components. These interpolators are developed to deal with the problem of estimating power spectra under the condition that some observations of the series are unavailable. We examine the structure of time-based cubic regression spline smoothers in Generalized Additive Models and demonstrate several interpretation problems with the resultant models. We propose, implement, and test a replacement smoother and show dramatic improvement. We further demonstrate a new, spectrally motivated way of examining residuals in Generalized Additive Models which drives many of the findings of this thesis. Finally, we create and analyze a large-scale Canadian air pollution and mortality database. In the course of analyzing the data we rebuild the standard risk estimation model and demonstrate several improvements. We conclude with a comparison of the original model and the updated model and show that the new model gives consistently more positive risk estimates. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-10-26 14:32:00.678