In this paper, we develop a method for estimating and clustering two-dimensional spectral density functions (2D-SDFs) for spatial data from multiple subregions. We use a common set of adaptive basis functions to explain the similarities among the 2D-SDFs in a low-dimensional space and estimate the basis coefficients by maximizing the Whittle likelihood with two penalties. We apply these penalties to impose the smoothness of the estimated 2D-SDFs and the spatial dependence of the spatially-correlated subregions. The proposed technique provides a score matrix, that is comprised of the estimated coefficients associated with the common set of basis functions representing the 2D-SDFs. Instead of clustering the estimated SDFs directly, we propose to employ the score matrix for clustering purposes, taking advantage of its low-dimensional property. In a simulation study, we demonstrate that our proposed method outperforms other competing estimation procedures used for clustering. Finally, to validate the described clustering method, we apply the procedure to soil moisture data from the Mississippi basin to produce homogeneous spatial clusters. We produce animations to dynamically show the estimation procedure, including the estimated 2D-SDFs and the score matrix, which provide an intuitive illustration of the proposed method.

在本文中，我们开发了一种用于估计和聚类来自多个子区域的空间数据的二维光谱密度函数（2D-SDF）的方法。我们使用一组通用的自适应基函数来解释低维空间中2D-SDF之间的相似性，并通过使用两个惩罚最大化Whittle可能性来估计基系数。我们应用这些惩罚来施加估计的2D-SDF的平滑度以及与空间相关的子区域的空间依赖性。所提出的技术提供了分数矩阵，该分数矩阵由与代表2D-SDF的通用基础函数集相关的估计系数组成。我们建议使用分数矩阵的低维特性，而不是直接对估计的SDF进行聚类，而是将分数矩阵用于聚类目的。在仿真研究中，我们证明了我们提出的方法优于用于聚类的其他竞争性估计程序。最后，为了验证所描述的聚类方法，我们将该程序应用于密西西比盆地的土壤湿度数据，以产生均匀的空间聚类。我们制作动画来动态显示估计过程，包括估计的2D-SDF和得分矩阵，它们为所提出的方法提供了直观的说明。