Auto-regressive modular processes and modeling methodology

ARM (Auto-Regressive Modular) processes constitute a broad class of nonlinear autoregressive schemes with modulo-1 reduction and additional transformations. Unlike their TES (Transform–Expand–Sample) precursors, which only admit independent, identically distributed innovation sequences, ARM processes admit dependent innovation sequences as well, so long as they are independent of the initial ARM variate. As such, the class of ARM processes constitutes a considerable generalization of the TES class, endowed with enhanced modeling flexibility. For example, a Markovian innovation sequence can model burstiness in traffic processes far better by making use of the Markovian state to capture the structure of bursts. This paper introduces ARM processes and derives their fundamental properties in terms of marginal distributions and autocorrelation functions. It defines several useful subclasses that illustrate the modeling flexibility of ARM classes. Finally, it outlines a modeling methodology of empirical time series that can simultaneously fit the empirical marginal distribution (histogram) and empirical autocorrelation function.