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：针对圆形数据的平稳高阶马尔可夫过程进行了研究。我们采用混合转移分布（MTD）模型来刻画该过程在圆上的转移密度。底层圆形转移分布基于Wehrly与Johnson的二元联合圆形模型。当底层绑定密度具有零正弦矩时，圆形自相关函数及圆形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结构相似。通过蒙特卡洛模拟及真实方向数据的应用，验证了该模型的有效性。