The stationary higher-order Markov process for circular data is considered. We employ the mixture transition distribution (MTD) model to express the transition density of the process on the circle. The underlying circular transition distribution is based on Wehrly and Johnson's bivariate joint circular models. The structures of the circular autocorrelation function together with the circular partial autocorrelation function are found to be similar to those of the autocorrelation and partial autocorrelation functions of the real-valued autoregressive process when the underlying binding density has zero sine moments. The validity of the model is assessed by applying it to some Monte Carlo simulations and real directional data.

翻译：本文研究了循环数据的平稳高阶马尔可夫过程。我们采用混合转移分布（MTD）模型来表达该过程在圆上的转移密度。基础循环转移分布基于Wehrly和Johnson的双变量联合循环模型。研究发现，当基础绑定密度具有零正弦矩时，循环自相关函数与循环偏自相关函数的结构与实值自回归过程的自相关函数及偏自相关函数具有相似性。通过蒙特卡洛模拟和实际方向性数据的应用，评估了该模型的有效性。