Abstract The cubic smoothing spline has been a popular method for detrending tree-ring data since the 1980s. The common implementation of this procedure (e.g., ARSTAN, dplR) uses a unique method for determining the smoothing parameter that is widely known as the %n criterion. However, this smoothing parameter selection method carries the assumption that end point effects are ignorable. In this paper, we complete the mathematical derivation and show how the original method differs from the complete version, both in the interpretation of the smoothing parameter and in the spline fit. Frequency response curves (FRC) demonstrate how the smoothing parameter is affected by the original assumption. For example, the FRC results indicate that a tree core of 250-year length has a 14% difference in the cut-off frequency when looking at the 67%n criterion. The FRC analysis shows that the existing approach produces a more flexible fit than anticipated, i.e., it is removing more variance than previously thought. For example, a 67%n spline under the existing approach corresponds to a 53%n spline fit. By using both simulated tree-core sequences and a dataset from a Midwest forest, we discuss which conditions result in greater differences between the spline fits and which conditions will have small differences. Tree-core sequences that have more curvature, such as a large-amplitude growth release, will lead to greater differences. Finally, we provide approximations to the end-point effect procedure. For example, using an 83%n criterion under the original approach produces a spline fit approximating the 67%n fit under the complete approach. These approximations could be easily implemented within existing programs like ARSTAN.

中文翻译：

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端点调整对用于树木年轮标准化的平滑样条的影响

摘要 自 80 年代以来，三次平滑样条一直是一种流行的树轮数据去趋势方法。此过程的常见实现（例如，ARSTAN、dplR）使用一种独特的方法来确定平滑参数，该方法被广泛称为 %n 标准。然而，这种平滑参数选择方法假设终点效应是可忽略的。在本文中，我们完成了数学推导，并展示了原始方法与完整版本的不同之处，包括平滑参数的解释和样条拟合。频率响应曲线 (FRC) 展示了平滑参数如何受到原始假设的影响。例如，FRC 结果表明，当查看 67%n 标准时，长度为 250 年的树核在截止频率上有 14% 的差异。FRC 分析表明，现有方法产生比预期更灵活的拟合，即它消除的方差比以前认为的多。例如，现有方法下的 67%n 样条对应于 53%n 样条拟合。通过使用模拟的树核序列和来自中西部森林的数据集，我们讨论了哪些条件会导致样条拟合之间的差异更大，哪些条件会有很小的差异。具有更大曲率的树核序列，例如大振幅的增长释放，将导致更大的差异。最后，我们提供了终点效应程序的近似值。例如，在原始方法下使用 83%n 标准会产生近似于完全方法下 67%n 拟合的样条拟合。