Let's say I have a time series $ \{t_i, x_i\} $ where the $x_i$ are strictly increasing and the $t_i$ are not regularly spaced. I can define a differential operator as
$$ \Delta_i = \frac{x_i-x_{i-1}}{t_i-t_{i-1}} $$
but that is very noisy.
Müller et. alia define a different operator with characteristic time $\tau$ in terms of iterated EMAs $\epsilon^n_T$ of levels $n=1,2,4$ as
$$ \Delta_\tau = \gamma(\epsilon^1_{\alpha\tau} + \epsilon^2_{\alpha\tau} - 2\epsilon^4_{\alpha \beta \tau}) $$
where $ \gamma=1.22208,\beta=0.65, \alpha^{-1}=\gamma(8\beta-3)$.
This operator has a good-looking kernel with a nice small tail, but for monotonically increasing series $x$ it is not strictly positive. What differential operator works well but lacks this sign problem? Should I just be smoothing the straight differentials?
Example
Let's say I am tracking how frequently a stock has been mentioned in the news in the last 5 years. Some days there are no mentions, while on other days they are bountiful. The rate of mention is of course the derivative of the number of total times the stock has been mentioned.
I could just smooth the mention counts within a window and be done, but it seems elegant to apply a differential operator to the mention count. Unfortunately, as I noticed, I get some negative value which are clearly silly.
