FARIMA processes with application to biophysical data

In this paper we show fractional autoregressive integrated moving average (FARIMA) time series with a negative memory parameter and stable non-Gaussian noise model movements of mRNA molecules inside live E. coli cells recorded by means of a single particle tracing experiment. The phenomenon of negative memory is related to the so-called subdiffusion which is often observed in crowded media. We fit the FARIMA process by using a variant of Whittle's method introduced by Kokoszka and Taqqu (1996 Ann. Statist. 24 1880) for the FARIMA stable case with a positive memory parameter, which we extend to the negative memory case. In order to show the goodness of fit we analyze residuals of the model. We check that they follow a non-Gaussian stable law and justify their independence. Finally, with the help of Monte Carlo simulations, we illustrate that the fitted FARIMA model reproduces statistical properties of the analyzed biophysical data.