确定性混沌时间序列参数估计的统计方法研究
On Statistical Methods of Parameter Estimation for Deterministically Chaotic Time-Series
摘要：
讨论了将一些标准统计方法（最小二乘法、极大似然法、统计矩估计参数法）应用于含有观测噪声的确定性混沌低维动力系统（Logistic映射）的可能性。提出了一种“纯”极大似然（ML）方法来估计Logistic映射的结构参数，并将初始值$x_1$作为一个额外的未知参数.与以前提出的技术模拟数值例子的比较给出了有利的结果（至少，对于样本量$N$和噪声水平的调查组合）。此外，与一些建议的技术，我们的方法不需要噪声方差的先验知识。我们还澄清了确定性混沌时间序列的统计分析和以前提出的贝叶斯方法的状态的固有困难的性质。我们注意到需要在ML分析中使用大量数据点来减少偏差（以保证估计的一致性）与初始条件的记忆指数快速丢失的动态轨迹的不稳定性之间的权衡。讨论了Logistic映射参数估计的统计矩方法。这种方法似乎是唯一的方法，其一致性的确定性混沌时间序列证明到目前为止，理论上（不仅数值）。
英文摘要：
We discuss the possibility of applying some standard statistical methods (the least square method, the maximum likelihood method, the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A ``pure'' Maximum Likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value $x_1$ considered as an additional unknown parameter. Comparisons with previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size $N$ and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade-off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).
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