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 structure of the circular autocorrelation function together with the circular partial autocorrelation function is found to be similar to 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：考虑圆形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。底层圆形转移分布基于Wehrly和Johnson的二元联合圆形模型。研究发现，当底层绑定密度具有零正弦矩时，圆形自相关函数与圆形偏自相关函数的结构类似于实值自回归过程的自相关函数和偏自相关函数。通过将该模型应用于蒙特卡洛模拟和真实方向数据，验证了其有效性。