  Analyzing data from paleoclimate archives such as tree rings or lake
sediments offers the opportunity of inferring information on past climate
variability. Often, such data sets are univariate and a proper reconstruction
of the system's higher-dimensional phase space can be crucial for further
analyses. In this study, we systematically compare the methods of time delay
embedding and differential embedding for phase space reconstruction.
Differential embedding relates the system's higher-dimensional coordinates to
the derivatives of the measured time series. For implementation, this requires
robust and efficient algorithms to estimate derivatives from noisy and possibly
non-uniformly sampled data. For this purpose, we consider several approaches:
(i) central differences adapted to irregular sampling, (ii) a generalized
version of discrete Legendre coordinates and (iii) the concept of Moving Taylor
Bayesian Regression. We evaluate the performance of differential and time delay
embedding by studying two paradigmatic model systems - the Lorenz and the
R\"ossler system. More precisely, we compare geometric properties of the
reconstructed attractors to those of the original attractors by applying
recurrence network analysis. Finally, we demonstrate the potential and the
limitations of using the different phase space reconstruction methods in
combination with windowed recurrence network analysis for inferring information
about past climate variability. This is done by analyzing two well-studied
paleoclimate data sets from Ecuador and Mexico. We find that studying the
robustness of the results when varying the analysis parameters is an
unavoidable step in order to make well-grounded statements on climate
variability and to judge whether a data set is suitable for this kind of
analysis.
