复杂时间动态下的多尺度跳跃检测与估计
Multiscale Jump Testing and Estimation Under Complex Temporal Dynamics
AI总结：
针对复杂时间动态下趋势中跳跃的检测与估计问题，提出一种多尺度最优跳跃通过滤波器方法，能处理跳跃点数量发散且幅度趋零的情形，计算复杂度为$O(n\log^{1+\epsilon} n)$，数值实验验证了其稳健性和准确性。
摘要：
我们考虑在系统协方差和高阶结构可能随时间发生平滑及突变的情况下，检测原本平滑演化趋势中的跳跃问题。跳跃点的数量允许发散至无穷大，且跳跃幅度可能缩小至零。该方法基于将最优跳跃通过滤波器多尺度应用于时间序列，其中尺度在允许的下限和上限之间密集分布。对于一大类非平稳时间序列模型和趋势函数，所提出的方法在温和条件下渐近地能够以指定概率在近乎最优的范围内检测所有跳跃点。对于长度为$n$的时间序列，所提方法在每个尺度上的计算复杂度为$O(n)$，总体为$O(n\log^{1+\epsilon} n)$，其中$\epsilon$是任意小的正常数。数值研究表明，所提出的跳跃检测与估计方法在复杂时间动态下表现稳健且准确。
英文摘要：
We consider the problem of detecting jumps in an otherwise smoothly evolving trend whilst the covariance and higher-order structures of the system can experience both smooth and abrupt changes over time. The number of jump points is allowed to diverge to infinity with the jump sizes possibly shrinking to zero. The method is based on a multiscale application of an optimal jump-pass filter to the time series, where the scales are dense between admissible lower and upper bounds. For a wide class of non-stationary time series models and trend functions, the proposed method is shown to be able to detect all jump points within a nearly optimal range with a prescribed probability asymptotically under mild conditions. For a time series of length $n$, the computational complexity of the proposed method is $O(n)$ for each scale and $O(n\log^{1+ε} n)$ overall, where $ε$ is an arbitrarily small positive constant. Numerical studies show that the proposed jump testing and estimation method performs robustly and accurately under complex temporal dynamics.
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