Improved efficiency of maximum likelihood analysis of time series with temporally correlated errors
Improved efficiency of maximum likelihood analysis of time series with temporally correlated errors
复制标题
DOI：
10.1007/s00190-017-1002-5
复制
发表时间：
2017-08-01
影响因子：
4.4
通讯作者：
Langbein, John
中科院分区：
文献类型：
--
作者：
Langbein, John
大多数地球物理现象的时间序列具有时间相关误差。根据这些测量，估计各种参数。例如，通过对位置的大地测量，经常可以估计出速率和速率变化，并用于模拟构造过程。沿着参数大小的估计，需要评估这些参数的误差。如果不考虑时间相关性，或者假设每个观测值是独立的，则这些参数的误差的任何估计值都可能过低，并且参数的估计值将有偏差。纳入更好的不确定性估计值受到几个因素的限制，包括为背景噪声选择正确的模型，以及在有许多观测值的情况下估计所选噪声模型参数的计算要求。在这里，我使用最大似然估计（MLE）解决计算效率的第二个问题。大多数地球物理时间序列具有背景噪声过程，其可以表示为频率为f的白色和幂律噪声的组合。对于缺失数据，涉及FFT的标准光谱技术是不合适的。相反，时域技术，涉及建设和大数据协方差矩阵的逆。Bos等人（J Geod，2013. doi：10.1007/s 00190 -012-0605-0）展示了一种技术，该技术大大提高了MLE方法的效率，但仅是幂律指数> 1.0的近似解，因为它们要求数据协方差矩阵是Toeplitz。该限制可以通过简单地形成添加噪声过程而不是以正交方式组合它们的数据滤波器来去除。因此，简化了数据协方差矩阵的求逆，但为更宽范围的幂律指数提供了稳健的结果。
Most time series of geophysical phenomena have temporally correlated errors. From these measurements, various parameters are estimated. For instance, from geodetic measurements of positions, the rates and changes in rates are often estimated and are used to model tectonic processes. Along with the estimates of the size of the parameters, the error in these parameters needs to be assessed. If temporal correlations are not taken into account, or each observation is assumed to be independent, it is likely that any estimate of the error of these parameters will be too low and the estimated value of the parameter will be biased. Inclusion of better estimates of uncertainties is limited by several factors, including selection of the correct model for the background noise and the computational requirements to estimate the parameters of the selected noise model for cases where there are numerous observations. Here, I address the second problem of computational efficiency using maximum likelihood estimates (MLE). Most geophysical time series have background noise processes that can be represented as a combination of white and power-law noise, with frequency, f. With missing data, standard spectral techniques involving FFTs are not appropriate. Instead, time domain techniques involving construction and inversion of large data covariance matrices are employed. Bos et al. (J Geod, 2013. doi:10.1007/s00190-012-0605-0) demonstrate one technique that substantially increases the efficiency of the MLE methods, yet is only an approximate solution for power-law indices > 1.0 since they require the data covariance matrix to be Toeplitz. That restriction can be removed by simply forming a data filter that adds noise processes rather than combining them in quadrature. Consequently, the inversion of the data covariance matrix is simplified yet provides robust results for a wider range of power-law indices.