Vector autoregressions (VARs) have an associated order $p$; conditional on observations at the preceding $p$ time points, the variable at time $t$ is conditionally independent of all the earlier history. Learning the order of the model is therefore vital for its characterisation and subsequent use in forecasting. It is common to assume that a VAR is stationary. This prevents the predictive variance of the process from increasing without bound as the forecast horizon increases and facilitates interpretation of the relationships between variables. A VAR is stable if and only if the roots of its characteristic equation lie outside the unit circle, constraining the autoregressive coefficient matrices to lie in the stationary region. Unfortunately, the geometry of the stationary region is very complicated which impedes specification of a prior. In this work, the autoregressive coefficients are mapped to a set of transformed partial autocorrelation matrices which are unconstrained, allowing for straightforward prior specification, routine computational inference, and meaningful interpretation of the magnitude of the elements in the matrix. The multiplicative gamma process is used to build a prior for the unconstrained matrices, which encourages increasing shrinkage of the partial autocorrelation parameters as the lag increases. Identifying the lag beyond which the partial autocorrelations become equal to zero then determines the order of the process. Posterior inference is performed using Hamiltonian Monte Carlo via Stan. A truncation criterion is used to determine whether a partial autocorrelation matrix has been effectively shrunk to zero. The value of the truncation threshold is motivated by classical theory on the sampling distribution of the partial autocorrelation function. The work is applied to neural activity data in order to investigate ultradian rhythms in the brain.

翻译：向量自回归（VAR）模型具有关联的阶数 $p$；在给定前 $p$ 个时间点观测值的条件下，时间 $t$ 处的变量与所有更早的历史条件独立。因此，学习模型的阶数对于其表征及后续在预测中的应用至关重要。通常假设VAR过程是平稳的。这防止了预测方差随着预测范围增加而无界增长，并便于解释变量之间的关系。当且仅当其特征方程的根位于单位圆之外时VAR是稳定的，这限制了自回归系数矩阵位于平稳区域内。不幸的是，平稳区域的几何结构非常复杂，阻碍了先验分布的设定。在本文中，自回归系数映射到一组无约束的变换偏自相关矩阵，从而允许直接进行先验设定、常规的计算推断，以及对矩阵元素幅度的有意义解释。使用乘法伽马过程为无约束矩阵构建先验分布，这鼓励随着滞后阶数增加对偏自相关参数进行递增的收缩。识别偏自相关系数变得为零的滞后阶数即确定了过程的阶数。使用Stan通过哈密顿蒙特卡洛方法进行后验推断。采用截断标准来判断偏自相关矩阵是否已被有效收缩为零。截断阈值的取值基于偏自相关函数采样分布的经典理论。本文将方法应用于神经活动数据，以研究大脑中的超日节律。