  We develop in this paper a theoretical framework for the topological study of
time series data. Broadly speaking, we describe geometrical and topological
properties of sliding window (or time-delay) embeddings, as seen through the
lens of persistent homology. In particular, we show that maximum persistence at
the point-cloud level can be used to quantify periodicity at the signal level,
prove structural and convergence theorems for the resulting persistence
diagrams, and derive estimates for their dependency on window size and
embedding dimension. We apply this methodology to quantifying periodicity in
synthetic data sets, and compare the results with those obtained using
state-of-the-art methods in gene expression analysis. We call this new method
SW1PerS which stands for Sliding Windows and 1-dimensional Persistence Scoring.
