  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 method is illustrated for the ARFIMA(1,d,0) process and a
nonlinear auto-regressive toy model with multiplicative noise. We reconstruct a
model for daily mean temperature data recorded at Potsdam (Germany) and use it
to predict the first frost date by computing the mean first passage time of the
reconstructed process and the zero degree Celsius temperature line,
illustrating the potential of long-memory models for predictions in the
subseasonal-to-seasonal range.

Data Analysis, Statistics and Probability