This paper considers a filter for smoothing spatial data. It can be used to smooth data on the vertices of arbitrary undirected graphs with arbitrary non-negative spatial weights. It consists of a quantity analogous to Geary’s , which is one of the most prominent measures of spatial autocorrelation. In addition, the quantity can be represented by a matrix called the graph Laplacian in spectral graph theory. We show mathematically how spatial data becomes smoother as a parameter, called the smoothing parameter, increases from 0 and is fully smoothed as the parameter goes to infinity, except for the case where the spatial data is originally fully smoothed. We also illustrate the results numerically and apply the spatial filter to climatological/meteorological data. In addition, as supplementary investigations, we examine how the sum of squared residuals and the effective degrees of freedom vary with the smoothing parameter. Finally, we review two closely related literatures to the spatial filter. One is the intrinsic conditional autoregressive model and the other is the eigenvector spatial filter. We clarify how the spatial filter considered in this paper relates to them. We then mention future research.

中文翻译：

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使用图拉普拉斯惩罚滤波器进行空间平滑

本文考虑了一种用于平滑空间数据的滤波器。它可用于平滑具有任意非负空间权重的任意无向图的顶点上的数据。它由类似于 Geary 的量组成，这是空间自相关最突出的度量之一。此外，该量可以用谱图论中称为图拉普拉斯的矩阵来表示。我们以数学方式展示了空间数据如何变得更加平滑，因为参数（称为平滑参数）从 0 开始增加，并随着参数变为无穷大而完全平滑，但空间数据最初完全平滑的情况除外。我们还以数字方式说明结果，并将空间过滤器应用于气候/气象数据。此外，作为补充研究，我们研究了残差平方和和有效自由度如何随平滑参数变化。最后，我们回顾了两篇与空间滤波器密切相关的文献。一种是内在条件自回归模型，另一种是特征向量空间滤波器。我们阐明了本文中考虑的空间滤波器如何与它们相关。然后我们提到未来的研究。