一阶平面自回归模型
First-order planar autoregressive model
AI总结：
本文研究平面一阶自回归模型的平稳解存在条件、因果性及纯非确定性，并利用Yule-Walker方程推导其自协方差函数，刻画了参数不可识别的情形。
摘要：
本文建立了一阶自回归方程在平面上存在平稳解的条件以及平稳解的性质。平面一阶自回归模型由方程 $X_{i,j} = a X_{i-1,j} + b X_{i,j-1} + c X_{i-1,j-1} + ε_{i,j}$ 定义。当且仅当 $(1-a-b-c) (1-a+b+c) (1+a-b+c) (1+a+b-c) > 0$ 时，方程存在平稳解 $X$。当且仅当 $1-a-b-c>0$、$1-a+b+c>0$、$1+a-b+c>0$ 且 $1+a+b-c>0$ 时，平稳解 $X$ 满足关于白噪声 $ε$ 的因果性条件。本文给出了 $X$ 为纯非确定性的充分条件。此外，给出了 $X$ 在某些点处自协方差函数的显式表达式。借助 Yule-Walker 方程，可以计算出所有位置的自协方差函数。同时，描述了不同参数决定 $X$ 相同自协方差函数的所有情形。
英文摘要：
This paper establishes the conditions of existence of a stationary solution to the first order autoregressive equation on a plane as well as properties of the stationarity solution. The first-order autoregressive model on a plane is defined by the equation $X_{i,j} = a X_{i-1,j} + b X_{i,j-1} + c X_{i-1,j-1} + ε_{i,j}.$ A stationary solution $X$ to the equation exists if and only if $(1-a-b-c) (1-a+b+c) (1+a-b+c) (1+a+b-c) > 0$. The stationary solution $X$ satisfies the causality condition with respect to the white noise $ε$ if and only if $1-a-b-c>0$, $1-a+b+c>0$, $1+a-b+c>0$ and $1+a+b-c>0$. A sufficient condition for X to be purely nondeterministic is provided. An explicit expression for the autocovariance function of $X$ at some points is provided. With Yule-Walker equations, this allows to compute the autocovariance function everywhere. In addition, all situations are described where different parameters determine the same autocovariance function of $X$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
