In this study we investigate a data-driven stochastic methodology to parametrize small-scale features in a prototype multiscale dynamical system, the Lorenz ’96 (L96) model. We propose to model the small-scale features using a vector autoregressive process with exogenous variables (VARX), estimated from given sample data. To reduce the number of parameters of the VARX we impose a diagonal structure on its coefficient matrices. We apply the VARX to two different configurations of the 2-layer L96 model, one with common parameter choices giving unimodal invariant probability distributions for the L96 model variables, and one with nonstandard parameters giving trimodal distributions. We show through various statistical criteria that the proposed VARX performs very well for the unimodal configuration, while keeping the number of parameters linear in the number of model variables. We also show that the parametrization performs accurately for the very challenging trimodal L96 configuration by allowing for a dense (nondiagonal) VARX covariance matrix.

在这项研究中，我们研究了一种数据驱动的随机方法，以对原型多尺度动力学系统Lorenz '96（L96）模型中的小尺度特征进行参数化。我们建议使用带有自定义样本数据估算的外生变量（VARX）的向量自回归过程对小尺度特征进行建模。为了减少VARX的参数数量，我们在其系数矩阵上添加了对角线结构。我们将VARX应用于2层L96模型的两种不同配置，一种具有公共参数选择，可为L96模型变量提供单峰不变概率分布，另一种具有非标准参数，可提供三峰分布。我们通过各种统计标准表明，建议的VARX对于单峰配置的效果非常好，同时保持参数数量与模型变量数量成线性关系。我们还表明，通过允许密集（非对角线）VARX协方差矩阵，对于非常具有挑战性的三峰L96配置，参数化可以准确执行。