Vamoş Călin, Crăciun Maria
T. Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68, 400110 Cluj-Napoca, Romania.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Sep;78(3 Pt 2):036707. doi: 10.1103/PhysRevE.78.036707. Epub 2008 Sep 23.
A preliminary essential procedure in time series analysis is the separation of the deterministic component from the random one. If the signal is the result of superposing a noise over a deterministic trend, then the first one must estimate and remove the trend from the signal to obtain an estimation of the stationary random component. The errors accompanying the estimated trend are conveyed as well to the estimated noise, taking the form of detrending errors. Therefore the statistical errors of the estimators of the noise parameters obtained after detrending are larger than the statistical errors characteristic to the noise considered separately. In this paper we study the detrending errors by means of a Monte Carlo method based on automatic numerical algorithms for nonmonotonic trends generation and for construction of estimated polynomial trends alike to those obtained by subjective methods. For a first order autoregressive noise we show that in average the detrending errors of the noise parameters evaluated by means of the autocovariance and autocorrelation function are almost uncorrelated to the statistical errors intrinsic to the noise and they have comparable magnitude. For a real time series with significant trend we discuss a recursive method for computing the errors of the estimated parameters after detrending and we show that the detrending error is larger than the half of the total error.
时间序列分析中的一个初步基本步骤是将确定性成分与随机成分分离。如果信号是在确定性趋势上叠加噪声的结果，那么首先必须估计并从信号中去除趋势，以获得平稳随机成分的估计值。伴随估计趋势的误差也会传递到估计的噪声中，形成去趋势误差。因此，去趋势后获得的噪声参数估计量的统计误差大于单独考虑噪声时的特征统计误差。在本文中，我们通过蒙特卡罗方法研究去趋势误差，该方法基于用于生成非单调趋势和构建与主观方法获得的估计多项式趋势相似的自动数值算法。对于一阶自回归噪声，我们表明，平均而言，通过自协方差和自相关函数评估的噪声参数的去趋势误差与噪声固有的统计误差几乎不相关，且它们的大小相当。对于具有显著趋势的实际时间序列，我们讨论了一种用于计算去趋势后估计参数误差的递归方法，并表明去趋势误差大于总误差的一半。
