UTINAM, Observatory Terre-Homme-Environnement-Temps-Astronomie (THETA) of Franche-Comté, University of Franche-Comté/Centre National de la Recherche Scientifique (CNRS), Besançon, France.
IEEE Trans Ultrason Ferroelectr Freq Control. 2012 Mar;59(3):523-30. doi: 10.1109/TUFFC.2012.2223.
The prediction of very-long-term time stability is a key issue in various fields, such as time keeping, obviously, but also navigation and spatial applications. This is usually performed by extrapolating the measurement data obtained by estimators such as the Allan variance, modified Allan variance, Hadamard variance, etc. This extrapolation may be assessed from a fit over the variance estimates. However, this fit should be performed on the log-log graph of the estimates, which corresponds to a least-squares minimization of the relative difference between the variance estimates and the fitting curve. However, a bias exists between the average of the log of the estimates and the log of the true value of the estimated variance. This paper presents the theoretical calculation of this log-log bias based on the number of equivalent degrees of freedom of the estimates, shows simulations over a large number of realizations, and provides a reliable method of unbiased logarithmic fit. Extrapolating this fit yields a more confident assessment of the very-long-term time stability.
长期时间稳定性的预测是各种领域的关键问题，例如计时，显然还有导航和空间应用。这通常是通过外推诸如 Allan 方差、修正 Allan 方差、Hadamard 方差等估计器获得的测量数据来完成的。这种外推可以从对方差估计的拟合来评估。然而，这种拟合应该在估计值的对数-对数图上进行，这对应于方差估计值和拟合曲线之间的相对差的最小二乘最小化。然而，估计值的对数的平均值与估计方差的真实值的对数之间存在偏差。本文基于估计值的等效自由度的数量，给出了这种对数-对数偏差的理论计算，展示了大量实现的模拟，并提供了一种可靠的无偏对数拟合方法。对这种拟合的外推可以更有信心地评估长期时间稳定性。
