Stationary subspace analysis (SSA) is a blind source separation framework that decomposes linearly mixed multivariate data into stationary and nonstationary components. We extend SSA to spatially indexed data by introducing spatial stationary subspace analysis (spSSA), which explicitly accounts for spatial dependence. We propose three estimation procedures for the unmixing matrix based on first- and second-order spatial statistics. Each procedure targets a different type of nonstationarity and can be formulated as the solution to a generalized eigenvalue problem. To address situations where multiple forms of nonstationarity are present simultaneously, we combine the three procedures using approximate joint diagonalization. Simulation studies demonstrate that this combined approach yields superior separation performance. When the dimension of the nonstationary subspace is known, the proposed methods reliably recover the latent stationary and nonstationary components. However, determining this dimension remains a fundamental challenge in SSA, for which no generally accepted solution currently exists. Building on our estimation procedures, we propose a novel data augmentation approach to estimate the dimension of the nonstationary subspace and demonstrate its effectiveness through simulation studies. The proposed methodology is easily transferable to time series settings, making it of broader methodological interest.

翻译：平稳子空间分析（SSA）是一种盲源分离框架，它将线性混合的多变量数据分解为平稳和非平稳成分。我们通过引入空间平稳子空间分析（spSSA），将SSA扩展到空间索引数据，该分析明确考虑了空间依赖性。我们基于一阶和二阶空间统计量，提出了三种用于混合矩阵估计的方法。每种方法针对不同类型的非平稳性，并可表述为广义特征值问题的解。为了处理同时存在多种非平稳性的情况，我们使用近似联合对角化将三种方法结合起来。模拟研究表明，这种组合方法具有优越的分离性能。当非平稳子空间的维度已知时，所提出的方法能够可靠地恢复潜在平稳和非平稳成分。然而，确定该维度仍是SSA中的一个基本挑战，目前尚无普遍接受的解决方案。基于我们的估计方法，我们提出了一种新颖的数据增强方法来估计非平稳子空间的维度，并通过模拟研究证明了其有效性。所提出的方法易于迁移到时间序列设定中，因此具有更广泛的方法论意义。