This dissertation is concerned with time series analysis and signal processing. Two related topics are addressed. The first topic involves the problem of data analysis for very large collections of time series curves. The primary result is an analytic procedure that is computationally efficient, and that is interpretable from several different perspectives. The method involves viewing each of the observed time series curves as a randomly weighted sum of independent draws from a collection of stationary sources. This probability model is shown to have a natural correspondence to the geometry of the set of autocovariance functions of the time series in the collection, viewed as points in Euclidean space. The estimation algorithm consists of a PCA-like procedure for performing the source identification, followed by a posterior restoration of the hidden components. We apply the method to various types of simulated data, as well as to functional brain imaging data, and to U.S. unemployment rates. Results from these applications can be viewed at http://www.stat.ucla.edu/∼kshedden. We also draw connections to basis-construction methods (ICA, PCA, FDA), and to ARIMA models. The second topic of the dissertation involves signal restoration for a single observed time series. We introduce two sampling procedures for carrying out Monte Carlo signal restoration that are shown to significantly reduce the computational burden relative to other Monte Carlo restoration procedures. These procedures can be incorporated into the data analysis described in the first section of the dissertation as a method for restoring the hidden signal components.