Abstract
We propose a nonparametric minimum entropy method for estimating an optimal velocity from position time series, which may contain unknown noise, data gaps, loading effects, transients, outliers and step discontinuities. Although nonparametric, the proposed method is based on elementary statistical concepts familiar to least-squares and maximum-likelihood users. It seeks a constant velocity with a best possible (realistic) variance rather than a best variable velocity fit to the closest position data. We show, based on information theory, synthetic and real data, that minimum-entropy velocity estimation: (1) accounts for colored noise without assumptions about its distribution or the extent of its temporal correlations; (2) is unaffected by the series deterministic content such as an initial position and the heights of step discontinuities and insensitive to small-amplitude periodic variations and transients; (3) is robust against outliers and, for long time series, against step discontinuities and even slight non-stationarity of the noise; (4) does not involve covariance matrices or eigen/singular value analysis, thus can be implemented by a short and efficient software; (5) under no circumstances results in a velocity variance that decays as \(1/N\), where \(N\) is the number of observations. The proposed method is verified based on synthetic data and then applied to a few hundred NGL (Nevada Geodetic Lab) position time series of different characteristics, and the results are compared to those of the Median Interannual Difference Adjusted for Skewness (MIDAS) algorithm. The compared time series include continuous and linear ones used to test the agreement between the two methods in the presence of unknown noise, data gaps and loading effects, discontinuous but linear series selected to include the effect of a few (1–4) discontinuities, and nonlinear but continuous time series selected for including the effects of transients. Both the minimum-entropy and MIDAS methods are nonparametric in the sense that they only extract the velocity from a position time series with hardly any explicit assumptions about its noise distribution or correlation structure. Otherwise, the two methods differ in every single possible technical sense. Other than pointing to a close agreement between the derived velocities, the comparisons consistently revealed that minimum-entropy velocity uncertainties suggest a smaller degree of temporal correlations in the NGL time series than the MIDAS does.
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Data availability
We downloaded the NGL time series from http://geodesy.unr.edu/gps_timeseries/tenv/IGS14. Section S5 in the supplementary information lists the 4-character ID of all downloaded stations. The MIDAS software was downloaded from the NGL website http://geodesy.unr.edu/MIDAS_release and compiled on NGS computers. Our little software and its short documentation are included with the supplementary material.
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Acknowledgements
We thank NGS for supporting this work and John Galetzka for allowing its completion. We thank professor Duncan Agnew and two anonymous reviewers for comments that focused this paper and improved its presentation. Thanks to NGS colleagues who reviewed this paper in-house and to colleagues who continued to answer our questions over the years.
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Contributions
SW provided simulated noise series and RB facilitated the extraction of the NGL data and the MIDAS software and edited initial versions of the manuscript. As long-time experts on GNSS time series analysis, SW and RB oversaw the soundness of ideas, logic and results. JS did the rest.
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Supplementary Information
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Fig S1: Examples of simulated noise time series (Top) Stationary colored noise consisting of 2.7 mm of white noise and 5 mm of power-law noise with spectral index of -0.9. This series (out of 100 similar ones) was chosen for display because its limited length (of 10,000 days) makes it appear to be non-stationary. The green curve connects the simulated daily noise values and the smoother red curve represents a low-pass (LP) filtered values of the noise using an edge detection filter called “the weak elastic string”, added only to bring out the seemingly non-stationary shape of the series; (Middle) Mildly non-stationary noise series structured as above except that the spectral index is -1.1. (Bottom) Mildly non-stationary noise as in the top plot but with the addition of 0.1 mm of random walk (PNG 319 KB)
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Fig S2: (Top) Simulated nonlinear transient; (Middle) The transient plus a sequence of simulated colored noise (green curve) and its smoothed values (red curve); (Bottom) The difference between the noisy and smoothed transient. The smoothing was done by the “weak elastic string (PNG 382 KB)
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Fig S3: Minimum-entropy (red) and MIDAS (blue) estimated horizontal velocities at 171 stations with continuous and linear NGL time series (PNG 381 KB)
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Fig S4: Horizontal velocity differences: minimum-entropy minus MIDAS estimated at the 171 stations with continuous and linear position time series (PNG 358 KB)
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Fig S5: Minimal-entropy- and MIDAS-derived velocity uncertainty versus lifespan of station in years, for 55 NGL time series which exhibit visible nonlinearities (mostly in the Up component) but no discontinuities (PNG 125 KB)
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Fig S6: Histograms (counts versus bins in mm/year) of velocity differences (minimal-entropy minus MIDAS) for 55 NGL time series which exhibit visible nonlinearities (mostly in the Up components) but no discontinuities. The 4-character IDs of these stations are listed in section S5 (PNG 56 KB)
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Fig S7: Histograms of velocity differences (minimal-entropy minus MIDAS) for time series which experienced one to four discontinuities. Only components which experienced discontinuities are shown. (PNG 56 KB)
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Fig S8: Velocity standard deviations (SDs) versus time series lifespan derived by minimal-entropy and the MIDAS method for 50 stations with discontinuities. Only components which experienced discontinuities are shown (PNG 121 KB)
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Saleh, J., Bennett, R.A. & Williams, S.D.P. Minimum-entropy velocity estimation from GPS position time series. J Geod 98, 11 (2024). https://doi.org/10.1007/s00190-023-01820-3
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DOI: https://doi.org/10.1007/s00190-023-01820-3