Statistical inference of one-dimensional persistent nonlinear time series and application to predictions

We introduce a method for reconstructing macroscopic models of one-dimensional stochastic processes with long-range correlations from sparsely sampled time series by combining fractional calculus and discrete-time Langevin equations. The is illustrated the ARFIMA(1,d,0) process nonlinear autoregressive toy model multiplicative noise. reconstruct daily mean temperature data recorded at Potsdam, Germany use it to predict first-frost date computing first passage reconstructed $0{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$ line, illustrating potential long-memory predictions in subseasonal-to-seasonal range.