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.

我们提出了一种非参数最小熵方法，用于根据位置时间序列估计最佳速度，该时间序列可能包含未知噪声、数据间隙、负载效应、瞬态、异常值和步骤不连续性。尽管是非参数的，但所提出的方法基于最小二乘和最大似然用户熟悉的基本统计概念。它寻求具有最佳可能（现实）方差的恒定速度，而不是适合最接近位置数据的最佳可变速度。我们基于信息论、合成数据和真实数据证明了最小熵速度估计：（1）考虑有色噪声，而不对其分布或其时间相关性程度进行假设； (2)不受初始位置、台阶不连续高度等系列确定性内容的影响，并且对小幅度周期性变化和瞬变不敏感； (3) 对异常值具有鲁棒性，并且对于长时间序列，对阶跃不连续性甚至噪声的轻微非平稳性具有鲁棒性； (4)不涉及协方差矩阵或特征/奇异值分析，因此可以通过简短而高效的软件实现； (5) 在任何情况下都不会导致速度方差以\(1/N\)的形式衰减，其中\(N\)是观测值的数量。该方法基于合成数据进行验证，然后应用于几百个不同特征的NGL（内华达大地测量实验室）位置时间序列，并将结果与​​偏度调整中值年际差（MIDAS）算法的结果进行比较。比较的时间序列包括连续和线性的时间序列，用于在存在未知噪声、数据间隙和负载效应的情况下测试两种方法之间的一致性，选择不连续但线性的序列以包括一些（1-4）不连续性的影响，选择非线性但连续的时间序列以包含瞬态的影响。最小熵和 MIDAS 方法都是非参数的，因为它们仅从位置时间序列中提取速度，几乎没有对其噪声分布或相关结构进行任何明确的假设。否则，这两种方法在每一个可能的技术意义上都不同。除了指出导出的速度之间非常一致之外，比较一致地表明，最小熵速度不确定性表明 NGL 时间序列中的时间相关性程度比 MIDAS 更小。