What is first order difference in trend analysis? I am following this paper: Measuring, Predicting and Visualizing Short-Term Change in Word Representation and Usage in VKontakte Social Network where in Differencing Statistics section they describe that they performed first-order differences of their trends.
More specifically, for word frequecy time series and tf-idf time series they have taken subtraction, and for word embeddings they have used cosine similarity. 
My first question is why they have considered first-order difference without doing remaining tragectory clustering starightly using the original time-series data?
My second question is how first order difference is performed using substraction and cosine simialrity as it is not mentioned in the paper?
I am happy to provide more details if needed.
 A: Regarding the first question:
They want to cluster the time series into the categories: increasing, decreasing and constant. Those categories refer to change... the value of the time-series itself is not important, only the first derivative. That's why they calculate the first derivative.
Regarding the second question:
For the scalars $f$ and  $ \mathcal{X}$ subtraction is straight forward:
$$\Delta \tau_f(w) = \Big(\Delta f_{t_0,t_1}(w),\Delta f_{t_1,t_2}(w)..., \  \Delta f_{T-1,T}(w)\Big)=\Big(f_{t_1}(w)-f_{t_0}(w),  
 \ f_{t_2}(w)-f_{t_1}(w) ..., \ f_{T}(w)-f_{T-1}(w)\Big) $$
For the vector $e$ they use cosine similarity instead of the subtraction:
$$\Delta \tau_e(w) = \Big(\Delta e_{t_0,t_1}(w),\Delta e_{t_1,t_2}(w)..., \  \Delta e_{T-1,T}(w)\Big)=\Big(\text{cosim}\left(e_{t_1}(w), e_{t_0}(w)\right),  
 \ \text{cosim}\left(e_{t_2}(w), e_{t_1}(w)\right) ..., $$
$$\ \text{cosim}\left(e_{T}(w), e_{T-1}(w)\right)\Big) $$
How to actually calculate the cosine similarity you can find everywhere.
