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，本文研究了平稳高阶马尔可夫过程。我们采用混合转移分布（MTD）模型，在圆上对该过程的转移密度进行表达。其基础圆形转移分布基于Wehrly和Johnson的二元联合圆形模型。研究发现，当底层绑定密度的正弦矩为零时，圆形自相关函数与圆形偏自相关函数的结构，与实值自回归过程的自相关及偏自相关函数的结构相似。通过蒙特卡罗模拟及实际方向数据的应用，验证了模型的有效性。