Spectral methods are important for both theory and computation in spatial data analysis. When data lie on a grid, spectral approaches can take advantage of the discrete Fourier transform for fast computation. If data are not on a grid, then low-rank processes with Fourier basis functions may be sufficient approximations. However, deciding which basis functions to use is difficult and can depend on unknown parameters. Here, we introduce Bayesian Random Fourier Frequencies (BRFF), a fully Bayesian extension of the random Fourier features approach. BRFF treats the spectral frequencies as random parameters, which unlike fixed frequency approximations allows the frequencies to be data-adaptive and averages over uncertainty in frequency selection. We apply this method to non-gridded continuous, binary, and count data. We compare BRFF using simulated and observed data to another popular low-rank method, the predictive processes (PP) model. BRFF is faster than PP, and outperforms or matches the predictive performance of the PP model in settings with high numbers of observations.

光谱方法对于空间数据分析的理论和计算都很重要。当数据位于网格上时，谱方法可以利用离散傅里叶变换进行快速计算。如果数据不在网格上，则具有傅里叶基函数的低秩过程可能是足够的近似值。然而，决定使用哪些基函数是困难的，并且可能取决于未知参数。在这里，我们介绍了贝叶斯随机傅里叶频率 (BRFF)，它是随机傅里叶特征方法的完全贝叶斯扩展。BRFF 将频谱频率视为随机参数，与固定频率近似不同，它允许频率具有数据自适应性，并在频率选择的不确定性上进行平均。我们将此方法应用于非网格连续、二进制和计数数据。我们将使用模拟和观察数据的 BRFF 与另一种流行的低秩方法预测过程 (PP) 模型进行比较。BRFF 比 PP 更快，并且在具有大量观察值的设置中优于或匹配 PP 模型的预测性能。