Applying EWMA to first difference of a time series I am trying to fit an EWMA to the first difference but I am unsure how to properly fit the EWMA and how to assess if one model is better than another. I am trying to use the EWMA described in the forecast package: https://training-course-material.com/training/R_-_Forecasting
The first difference occurs always at integer values, most of which are zero, i.e. $$x_i=1,0,0,0,0,3,-2,1,0,0,0,...$$
Are there any standard ways to modelling first differences with EWMA? When I try to auto-fit it, I get an extremely tiny $\alpha$. Also, if I wanted to determine if another timer-series variable is correlated with this EWMA, how should I compute this?
 A: You would typically not apply Single Exponential Smoothing (also known as Exponentially Weighted Average) to differenced series. Instead, you'd directly apply Double Exponential Smoothing, i.e., smoothing with a level and a trend component. Here is the relevant section of Forecasting: Principles and Practice.
If you really want to smoothe first differences, just do so by feeding them into ets() or similar. If the first differences come out close to zero, then this looks like there may not be all that much trend or other nonstationarity in the mean to your original series. In such a case, the smoothing constant for the differenced series will quite reasonably be close to zero.
If your differenced series are always integer, your original series are probably also integer. You may want to look into count data models, or possible intermittent demand forecasting.
Finally, don't assess whether a different series is correlated with a smoothed series. Correlations with smoothed series are biased: Smoothing - when to use it and when not to?
