I have two variables where each observation represents counts at some point in a discrete 1D space (along an RNA sequence). The space is finite, and the counts are highly zero-inflated compared to a Poisson model, and are probably also over-dispersed.
I assume that each variable is a noisy estimator of some underlying function on the space (the affinity of the some protein factor for that part of the RNA sequence). I assume that the value of these functions at adjacent positions in space are not independent (that is the affinity of the proteins are for regions greater than a single base).
I want to know if one underlying function is predictive of the other, but I only have the very noisy estimates of them.
For example: Lets say I have the following data:
var1: 0 1 0 5 1 5 0 1 0 1
var2: 1 1 0 1 5 0 5 0 0 1
position: 0 1 2 3 4 5 6 7 8 9
Both datasets clearly have higher values in the center of the space than at either end, but and correlation between them would be very low because of the sparseness of the data.
Anyone have any ideas how to deal with this?
EDIT:
At the risk of making things more complex, I'm going to try an formalise the question a bit to make things clearer.
An RNA sequence is a string of $N$ bases. Each base $n$ has an affinity $\lambda_{X,n}$ for protein A, and affinity $\lambda_{Y,n}$ for protein B. I wish to know if $\lambda_{X,n}$ predicts $\lambda_{Y,n}$.
$\lambda_{X,n}$ and $\lambda_{X,n+1}$ are positively dependent.i.e. $\lambda_{X,n}$ is high, I expect $\lambda_{X,n+1}$ to be high. I expect that the affinity for most bases is 0.
I have measurements of the affinities in the forms of counts of the number of times each protein has interacted with each base. These counts are random variables $X$ and $Y$ such that
$$ X_n \sim ZIP(\lambda_{X,n}, \pi_X) $$$$ Y_n \sim ZIP(\lambda_{Y,n}, \pi_Y) $$
where $\pi_X$,$\pi_Y$ represent the level of zero-inflation, which is high, such that a standard correlation, such as spearman, would probably be low or 0.