A Novel Method for Topological Embedding of Time-Series Data

In this paper, we propose a novel method for embedding one-dimensional, periodic time-series data into higher-dimensional topological spaces to support robust recovery of signal features via topological data analysis under noisy sampling conditions. Our method can be considered an extension of the popular time delay embedding method to a larger class of linear operators. To provide evidence for the viability of this method, we analyze the simple case of sinusoidal data in three steps. First, we discuss some of the drawbacks of the time delay embedding framework in the context of periodic, sinusoidal data. Next, we show analytically that using the Hilbert transform as an alternative embedding function for sinusoidal data overcomes these drawbacks. Finally, we provide empirical evidence of the viability of the Hilbert transform as an embedding function when the parameters of the sinusoidal data vary over time.