Autoregressive-moving-average (ARMA) models seek to express a system function of a discretely sampled process as a rational function in the z-domain. Treating an ARMA model as a complex rational function, we discuss a metric defined on the set of complex rational functions. We give a natural measure of the "distance" between two ARMA processes. The paper concentrates on the mathematics behind the problem and shows that the various algebraic structures endow the choice of metric with some interesting and remarkable properties, which we discuss. We suggest that the metric can be used in at least two circumstances: (i) in which we have signals arising from various models that are unknown (so we construct the distance matrix and perform cluster analysis) and (ii) where there are several possible models M/sub i/, all of which are known, and we wish to find which of these is closest to an observed data sequence modeled as M.
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机译：自回归移动平均（ARMA）模型试图将离散采样过程的系统函数表示为z域中的有理函数。将ARMA模型视为复杂的有理函数，我们讨论了在一组复杂的有理函数上定义的度量。我们给出了两个ARMA进程之间“距离”的自然度量。本文集中讨论了问题背后的数学，并显示了各种代数结构赋予度量选择具有一些有趣且引人注目的特性，我们将对此进行讨论。我们建议该度量标准至少可以在两种情况下使用：（i）我们有来自各种未知模型的信号（因此我们构造距离矩阵并执行聚类分析），并且（ii）有几种可能模型M / sub i /，所有这些都是已知的，我们希望找到其中哪个最接近建模为M的观测数据序列。
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