Let's say we have two variables X and Y with n observations for each variable. We don't know the values of each observation, but we are given the difference between consecutive observation of each variable. For example:
X=[a,b,c], Y=[d,e,f] (a,b,c,d,e,f are unknown)
[b-a,c-b] and [e-d,f-e] are given and we know the values of b-a, c-b, e-d, f-e.
Is there any way to compute Pearson's correlation coefficient among X and Y by using this information? (one simple way could assign a constant value to a and d and construct observations by adding the given differences to each value and computing the Pearson correlation, but I'm looking for another approach which only uses the differences)
The Pearson correlation is invariant for translations, so the correlation of X and Y as defined in your example is the same as the correlation of [0,b-a,c-a] and [0,e-d,f-d]. And note that these last two can be written as [0,b-a,(c-b)+(b-a)] and [0,e-d,(f-e)+(e-d)]. So you can compute everything in terms of the differences.