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An ARMA time series approach for analyzing long memory dynamics in measurements

A time series approach is often applied to obtain insight in the system dynamics or the signal filtering. Dynamical systems may exhibit a diffusion or dispersion components which reveals a long memory or long time dependency. This class of systems is called fractional order systems. It is typically difficult to detect that one is dealing with a fractional order system. In this paper, we show that one can start with a classical time series analysis based on Autoregressive-Moving average modeling. In the presence of a diffusion or dispersion component the fractional behavior forces the estimated poles and zeros of the ARMA model to form an alternating chain. This chain can be detected and consecutively be compressed to finally result in a fractional order version of the ARMA model which is called the Autoregressive Fractionally Integrated Moving Average or ARFIMA model. Note that the presented approach is fully measurement driven and requires no interaction from the user.