What is the right way to express the change in interest rate time series, if this time series contains both positive and negative rates?
The usual approaches used to deal with negative interest rates are:
a) the normal (Bachelier) model or Brownian motion, where $dr_t = \sigma dW_t$; this makes changes independent from the level of the interest rate,
b) shifted lognormal (displaced diffusion) model, where, instead of the ordinary Geometric Brownian Motion $dr_t = r_t \sigma dW_t$, we have $dr_t = (r_t + h) \sigma dW_t$ with for example $h=2\%$ or any similar value accepted by convention or made to fit the data; this keeps changes somewhat proportional to the rate level, while allowing negative values up to $-h$.
You can also look at this question, which has useful references.