"We investigate the use of 1-dimensional persistence diagrams to determine minimum embedding dimension. In particular, we test the claim that persistence diagrams look qualitatively the same once the correct dimension is reached. In some cases, this appears to not be true so we turn to a quantitative measure, the bottleneck distance, to see if the persistence diagrams are close once the minimum embedding dimension is attained. In some instances, we see that the persistence diagrams fail to converge experimentally under the bottleneck distance. The main issue appears to be that it is difficult to explicitly characterize the persistent homology of delay embeddings of arbitrary time series. Instead we restrict to periodic time series where there exists such an explicit characterization. We apply Fourier analysis to see that that number of peaks in the frequency spectrum of a delay embedded time series is related to the minimum embedding dimension. Moreover, we give a method to filter out less significant peaks while not altering the persistent homology much, with respect to the bottleneck distance."--Page ii.
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