We propose a novel class of multiscale spatiotemporal models for multivariate Gaussian data. First, we decompose the multivariate data and the underlying latent process with a novel multivariate multiscale decomposition. This decomposition results in multiscale coefficient matrices with elements that are multiscale approximations of spatial directional derivatives. We then assume that the associated latent multiscale coefficient matrices evolve through time with matrix-variate state-space equations. We allow for different speeds of change through time for each latent multiscale coefficient matrix, which induces distinct spatiotemporal dynamics for the mean process in different regions at multiple spatial scales of resolution. This flexible model framework allows for both stationary and nonstationary latent multiscale spatiotemporal processes. Further, we develop a singular matrix-variate forward filter backward sampler for efficient posterior exploration. Importantly for practical purposes, our proposed multiscale spatiotemporal algorithm scales linearly with dataset size and is fully parallelizable. To illustrate the usefulness and flexibility of our dynamic multivariate multiscale framework, we present an application to the spatiotemporal NCEP/NCAR Reanalysis-I dataset on stratospheric temperatures over North America from 1951 to 2016. Our analysis indicates substantial long-term trends in stratospheric temperatures.

我们为多元高斯数据提出了一类新颖的多尺度时空模型。首先，我们用新颖的多元多尺度分解方法分解多元数据和潜在的潜在过程。这种分解导致具有元素的多尺度系数矩阵，这些元素是空间方向导数的多尺度近似。然后，我们假定关联的潜在多尺度系数矩阵会随着矩阵变量状态空间方程随时间演化。我们为每个潜在的多尺度系数矩阵允许随时间变化的不同速度，从而在分辨率的多个空间尺度上为不同区域的平均过程引入了独特的时空动力学。这种灵活的模型框架允许静态和非静态潜在多尺度时空过程。此外，我们开发了一种奇异的矩阵变量正向滤波器后向采样器，以进行高效的后向探测。重要的是出于实际目的，我们提出的多尺度时空算法与数据集大小成线性比例，并且是完全可并行化的。为了说明我们的动态多元多尺度框架的有用性和灵活性，我们将其应用于1951年至2016年北美平流层温度的时空NCEP / NCAR Reanalysis-I数据集。我们的分析表明平流层温度存在长期的长期趋势。