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Bottom-to-top decomposition of time series by smoothness-controlled cubic splines: Uncovering distinct freezing-melting dynamics between the Arctic and the Antarctic

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Jánosi,  Imre M.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Beims,  Marcus W.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Citation

Jánosi, I. M., Baki, A., Beims, M. W., & Gallas, J. A. C. (2020). Bottom-to-top decomposition of time series by smoothness-controlled cubic splines: Uncovering distinct freezing-melting dynamics between the Arctic and the Antarctic. Physical Review Research, 2(4): 043040. doi:10.1103/PhysRevResearch.2.043040.


Cite as: http://hdl.handle.net/21.11116/0000-0008-E569-0
Abstract
The classical methods of identifying significant slow components (modes) in a strongly fluctuating signal usually require strict stationarity. A notable exception is the procedure called empirical mode decomposition (EMD), which is designed to work well for nonstationary and nonlinear (quasiperiodic) time series. However, EMD has some well-known limitations such as the end divergence effect, mode mixing, and the general problem of interpreting the modes. Methods to overcome these limitations, such as ensemble EMD or complete ensemble EMD with adaptive noise, promise an exact reconstruction of the original signal and a better spectral separation of the intrinsic mode functions (IMFs). All these variants share the feature that the decomposition runs from the top to the bottom: The first few IMFs represent the noise contribution and the last is a long-term trend. Here we propose a decomposition from the bottom to the top, by the introduction of smoothness-controlled cubic spline fits. The key tool is a systematic scan by cubic spline fits with an input parameter controlling the smoothness, essentially the number of knots. Regression qualities are evaluated by the usual coefficient of determination R-2, which grows monotonically when the number of knots increases. In contrast, the growth rate of R-2 is not monotonic: When an essential slow mode is approached, the growth rate exhibits a local minimum. We demonstrate that this behavior provides an optimal tool to identify strongly quasiperiodic slow modes in nonstationary signals. We illustrate the capability of our method by reconstruction of a synthetic signal composed of a chirp, a strong nonlinear background, and a large-amplitude additive noise, where all EMD-based algorithms fail spectacularly. As a practical application, we identify essential slow modes in daily ice extent anomalies at both the Arctic and the Antarctic. Our analysis demonstrates the distinct freezing-melting dynamics on the two poles, where apparently different factors are determining the time evolution of ice sheets. Thus, we believe that our methodology offers a competitive tool to identify modes in strongly fluctuating data and advances significantly the state of the art regarding the decomposition of nonlinear time series.