Nonparametric methods of uncovering coupling provides a flexible framework for researchers to study network configurations and discover causal graphs. For example, Gaussian Processes have been applied to discover coupling in dynamical systems under the convergent cross-mapping (GP-CCM) framework. Though GP-CCM is a nonparametric method, a function of the data, variations in hyperparameters may impact model results. Maximum marginal likelihood is commonly used to optimize such hyperparameters, however only point estimates are provided from such a routine. In this work, we determine a posteriori the hyperparameter distribution conditioned on the data by the use of variational Bayesian methods with mean field approximations on the posterior distribution of the hyperparameters, thus introducing the variational form of Gaussian process convergent cross-mapping (VGP-CCM). We show that using the a posteriori best approximate model that maximizes evidence of the data for hyperparameters sampling in VGP-CCM, in conjunction with a permutation sampling for the null distribution of uncoupled time series, permits significance statistics that are more robust than mere permutation sampling of the null GP-CCM distribution. We perform tests on synthetic data derived from unidirectionally coupled Lorenz-Rossler systems to demonstrate that the proposed method yields substantially improved specificity over the original GP-CCM.

翻译：在这项工作中,我们通过使用变式巴伊西亚方法,在数据基础上进行超参数分布,在超参数分布时使用数据,在超参数分布表的外近近度上采用平均的实地近似法,从而引入了高斯进程交叉分布图的变异形式。 我们发现,使用最接近性的最佳假设模型,使VGP-CCM的超参数取样数据证据最大化,同时对原GMM-CM的原定时间序列进行不相交分布的整流抽样抽样抽样抽样取样。