Posterior contraction and credible sets for filaments of regression functions
Posterior contraction and credible sets for filaments of regression functions
复制标题
回归函数细丝的后收缩和可信集
DOI：
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发表时间：
2018
影响因子：
1.1
通讯作者：
S. Ghosal
中科院分区：
文献类型：
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作者：
Wei Li;S. Ghosal
一个细丝是由一个光滑函数f在某个方向上运动时的局部极大值组成的。丝状结构是表征物体形状的重要特征，也被认为是多元数据的重要低维表征。最近在非参数核密度估计上下文中有一些关于细丝的理论研究。本文从两个方面对现有文献进行了补充。首先，我们提供了一个贝叶斯方法回归背景下的细丝估计和研究后验收缩率使用有限的随机序列的B样条的基础上。与核估计方法相比，这具有理论优势，因为当函数更平滑时，可以更好地控制偏差，从而可以获得更好的速率。假设$f：mathbb{R}^2映射mathbb{R}$属于α geq 4阶的迷向H”{o}lder类，在最佳光滑参数的选择下，某些适当定义的积分曲线上的丝点的后验收缩率和丝的Hausdorff距离的后验收缩率都是（n/log n）^{（2-α）/（2（1+ α））}$.其次，我们提供了一种方法来构造一个可信的集合，具有足够的频率覆盖的细丝。我们的有效可信区域由具有频率论解释的后丝组成。我们证明了我们所提出的方法在模拟和应用地震数据的成功。
A filament consists of local maximizers of a smooth function $f$ when moving in a certain direction. Filamentary structures are important features of the shape of objects and are also considered as important lower dimensional characterization of multivariate data. There have been some recent theoretical studies of filaments in the nonparametric kernel density estimation context. This paper supplements the current literature in two ways. First, we provide a Bayesian approach to the filament estimation in regression context and study the posterior contraction rates using a finite random series of B-splines basis. Compared with the kernel-estimation method, this has theoretical advantage as the bias can be better controlled when the function is smoother, which allows obtaining better rates. Assuming that $f: mathbb{R}^2 mapsto mathbb{R}$ belongs to an isotropic H"{o}lder class of order $alpha geq 4$, with the optimal choice of smoothing parameters, the posterior contraction rates for the filament points on some appropriately defined integral curves and for the Hausdorff distance of the filament are both $(n/log n)^{(2-alpha)/(2(1+alpha))}$. Secondly, we provide a way to construct a credible set with sufficient frequentist coverage for the filaments. Our valid credible region consists of posterior filaments that have frequentist interpretation. We demonstrate the success of our proposed method in simulations and application to earthquake data.