We construct a high order conditional distance covariance, which generalizes the notation of conditional distance covariance. The joint conditional distance covariance is defined as a linear combination of conditional distance covariances, which can capture the joint relation of many random vectors given one vector. Furthermore, we develop a new method of conditional independent test based on the joint conditional distance covariance. Simulation results indicate that the proposed method is very effective. We also apply our method to analyze the relationships of $PM_{2.5}$ in five Chinese cities: Beijing, Tianjin, Jinan, Tangshan and Qinhuangdao by Gaussian graphical model.
