利用曲率选择时延重构中的时延
Using Curvature to Select the Time Lag for Delay Reconstruction
摘要：
我们提出了一种基于曲率的方法来选择延迟重建中的时间延迟参数$τ$的好值。这个想法是基于延迟对重建的几何形状的影响。如果延迟选择得太小，重构的动态沿着嵌入空间的主对角线变平;另一方面，太大的延迟会使动态重叠。计算二维延迟重建的曲率是识别这些极端并找到它们之间的中间地带的有效方法：在未充分展开的重建的末端处的急剧反转和过度折叠的重建中的折叠都在曲率中产生尖峰。我们通过计算不同时间延迟的2D重建的门格尔曲率的平均值来实现这一观察。我们表明，这些值的平均值给出了一个有效的启发式选择的时间延迟。此外，我们表明，即使在使用平均互信息的传统方法失败的情况下，这种基于曲率的启发式也是有用的-例如，噪声或过滤数据。
英文摘要：
We propose a curvature-based approach for choosing good values for the time-delay parameter $τ$ in delay reconstructions. The idea is based on the effects of the delay on the geometry of the reconstructions. If the delay is chosen too small, the reconstructed dynamics are flattened along the main diagonal of the embedding space; too-large delays, on the other hand, can overfold the dynamics. Calculating the curvature of a two-dimensional delay reconstruction is an effective way to identify these extremes and to find a middle ground between them: both the sharp reversals at the ends of an insufficiently unfolded reconstruction and the folds in an overfolded one create spikes in the curvature. We operationalize this observation by computing the mean over the Menger curvature of 2D reconstructions for different time delays. We show that the mean of these values gives an effective heuristic for choosing the time delay. In addition, we show that this curvature-based heuristic is useful even in cases where the customary approach, which uses average mutual information, fails --- e.g., noisy or filtered data.
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