Abstract: This paper proposes a joint semi-parameter probability graphical model to learn the conditional dependence relationship of non-normal distribution heterogeneous data. Considering the smoothly varying heterogeneous data, this paper further proposes a joint dynamic semi-parameter probability graphical model. Moreover, combining non-parametric rank-based correlation matrix estimator with fused graphical Lasso, this paper proposes a semi-parametric fused graphical Lasso to estimate the joint models. In particular, it proposes a novel kernel smoothing Kendall's tau correlation coefficient matrix to estimate the dynamic graphical models. Due to relaxing the normal distribution assumption, the proposed models are more flexible than the existing joint Gaussian graphical models. At the same time, due to using non-parametric rank-based correlation matrix estimator, the proposed models are also more robust. Moreover, this paper uses an efficient alternating direction method of multipliers (ADMM) to optimize the proposed models. Finally, some numerical results on simulations and real datasets such as brain imaging data and stock trading data demonstrate the effectiveness of the proposed models.

Key words: semi-parameter graphical models, dynamic graphical models, joint learning, alternating direction method of multipliers, brain network

摘要： 提出了联合的半参数概率图模型用于学习非正态分布异构数据的条件相关性。进一步针对光滑变化的异构数据，给出了联合的动态半参数概率图模型。将基于非参排序的相关矩阵估计方法与融合图套索方法相结合，提出了半参数融合图套索方法估计上述两类联合图模型。特别针对动态半参数图模型，提出了一种新的核光滑Kendall’s tau相关矩阵。由于放宽了正态分布假设，使得该模型比当前联合的高斯图模型更灵活。由于采用了基于非参排序的相关矩阵估计方法，使得该模型更加鲁棒。利用有效的交替方向乘子法（alter-nating direction method of multipliers，ADMM）对这些模型进行求解。最后，在一些人工数据与真实数据上，如脑影像数据及股票交易数据，验证了该模型的有效性。

关键词: 半参数图模型, 动态图模型, 联合学习, 交替方向乘子法, 脑网络