[论文解读] Change point analysis of second order characteristics in non-stationary time series

Holger Dette, Weichi Wu+1 more|arXiv (Cornell University)|Jan 1, 2015

Probabilistic and Robust Engineering Design参考文献 32被引用 21

一句话总结

本文提出了一种基于CUSUM的变更点检验方法，用于非平稳时间序列中的二阶特征（如方差和滞后-k自相关）的检测，放宽了原假设下完全平稳性的限制性假设。该文推导了检验统计量的渐近分布，并提出了一种自助法以获取临界值，从而在均值或高阶矩非恒定的情况下也能检测结构性突变。该方法应用于英格兰中部气温数据，揭示了19世纪末冬季气温方差和自相关性的显著变更点。

ABSTRACT

A restrictive assumption in the work on testing for structural breaks in time series consists in the fact that the model is formulated such that the stochastic process under the null hypothesis of "no change-point" is stationary. This assumption is crucial to derive (asymptotic) critical values for the corresponding testing procedures using an elegant and powerful mathematical theory, but it might be not very realistic from a practical point of view. For example, if change point analysis for a particular parameter of the process (such as the variance) is performed, it is not necessary clear why other parameters (such as the mean or higher order moments) have to stay constant under the hypothesis that there is no change point in the parameter of interest. This paper develops change point analysis under less restrictive assumptions and deals with the problem of detecting change points in the marginal variance and correlation structures of a non-stationary time series. A CUSUM approach is proposed, which is used to test the "classical" hypothesis of the form H₀ : θ₁ = θ₂ vs. H₁ : θ₀ ≠ θ₂, where θ₁ and θ₂ denote second order parameters (such as the variance or the lag k-correlation) of the process before and after a change point. The asymptotic distribution of the CUSUM test statistic is derived under the null hypothesis. This distribution depends in a complicated way on the dependency structure of the nonlinear non-stationary time series and a bootstrap approach is developed to generate critical values. The results are then extended to test the hypothesis of a non relevant change point, i.e. H₀ : | θ₀ - θ₂ | ≤ δ , which reflects the fact that inference should not be changed, if the difference between the parameters before and after the change-point is small. In contrast to previous work, our approach does neither require the mean to be constant nor - in the case of testing for lag k-correlation - that the mean, variance and fourth order joint cumulants are constant under the null hypothesis. In particular, we allow that the variance has a change point at a different location than the auto-covariance. The results are illustrated by means of a simulation study, which shows that the new procedures have nice finite sample properties. The central England monthly temperature series are analyzed and significant change points in the variance and lag 1-correlation are found in the winter monthly temperature at the late 19th century.

研究动机与目标

• 解决现有变更点检验方法在原假设下假设完全平稳性的局限性。
• 开发一种检测二阶结构（方差与滞后-k自相关）变化的方法，而无需要求均值或高阶矩保持恒定。
• 允许方差与自协方差的变更点出现在不同位置，以反映现实中的非平稳行为。
• 将框架扩展至检测非显著变更点，即原假设为 |θ₀ − θ₂| ≤ δ，其中参数差异被限定在小阈值δ以内。
• 提供一种实用的渐近有效检验程序，并通过模拟研究与真实数据分析验证其有限样本性能。

提出的方法

• 构建一个CUSUM检验统计量，用于比较潜在变更点前后二阶参数（如方差、滞后-k自相关）的差异。
• 在无变更点的原假设下，推导CUSUM统计量的渐近分布，考虑非线性非平稳时间序列中复杂的依赖结构。
• 开发一种自助程序以生成临界值，因为渐近分布依赖于未知的依赖参数，且在分析上难以处理。
• 通过原假设 H₀: |θ₀ − θ₂| ≤ δ 将方法扩展至检测非显著变更点，允许在参数值上实现实际等价性。
• 该方法在原假设下不要求均值恒定，也无需方差或四阶累积量保持不变，尤其在检验滞后-k自相关时更为适用。
• 通过模拟研究验证该方法，并将其应用于英格兰中部月度气温序列，以检测冬季气温方差与滞后-1自相关性的变更点。

实验结果

研究问题

• RQ1 是否可以在不假设原假设下均值恒定的前提下，对二阶特征的变更点进行检测？
• RQ2 如何为具有复杂依赖结构的非线性非平稳时间序列推导CUSUM检验统计量的渐近分布？
• RQ3 何种自助方法可在渐近分布分析上不可行时确保临界值的准确性？
• RQ4 该框架是否能够检测到方差与自协方差的变更点，即使它们出现在不同位置？
• RQ5 所提出的方法在有限样本中是否具有鲁棒性与有效性，特别是对非显著变更点的检测能力如何？

主要发现

• CUSUM检验统计量的渐近分布依赖于时间序列的依赖结构，无法以简洁的闭式表达。
• 自助方法成功生成了检验统计量的临界值，使得在非平稳条件下仍能进行有效推断。
• 模拟研究证实，所提出程序具有良好的有限样本性质，保持了正确的尺寸与合理的功效。
• 对英格兰中部月度气温序列的分析揭示了19世纪末前后冬季气温方差与滞后-1自相关性的显著变更点。
• 该方法即使在均值非恒定的情况下，也能成功检测出二阶结构的结构性变化，表现出对现实非平稳性的强鲁棒性。
• 对非显著变更点的扩展使得可通过检验参数差异是否在可忽略的阈值δ内，实现实际推断。