Method
Nonlinear Partial Differential Equation Approach
In order to process the data we use the time series as the spatial forcing of a nonlinear extended system and look for the steady state.
The nonlinear system used is a Nonlinear Diffusion equation, or Ginzburg-Landau model:
where a is the diffusion coefficient (Smoothing), h is the input data scaled between 0 and 1, b (Data-strenght) is the forcing coefficient and h0 is called the reference level.
This equation has a bistable steady state; the idea is to choose the reference level h0 in such a way that the spatial points corresponding to the time series before the jump are attracted by the G-L dynamics to one steady state while the points after the jump are attracted to the other. At long times the solution of the partial differential equation approaches an steady state which is the processed time series.