Let $C(t)$ be a discrete, finite set of Customer IDs with $|C(t)| =n(t) >0$ denoting the number of customers in the set at time $t \in \mathbb{N}$. Consider the following time-dependent weighted average function:
$$W(t, C(t)) = \frac{ \sum_{c \in C(t)} w_c(t)x_c(t)}{\sum_{c \in C(t)}w_c(t)} $$
where $w_c(t), x_c(t) \geq 0 $ are positive real numbers.
This quantity arose from a data analysis problem in business intelligence. We are particularly interested in analyzing the time lagged difference $$\Delta W(t) := W(t, C(t)) - W(t-1, C(t-1))$$
In general, the goal is to understand which customers which are impacting this change the most. It is a bit tricky since the weights, values being averaged and set of customers all can change between time periods. New customers can appear and customers previously in $C(t)$ can leave in the next period.
More mathematically, we would like a ranking on the set of customers $C(t) \cup C(t-1)$ where each customers ranked according to there contribution to the change $\Delta W(t)$.
Unfortunately its a bit vague on what is meant by contribution. Of course, if the set of customers was static between periods, then for each $c \in C(t)$ the largest change to $\Delta W(t)$ will be from the customer with the largest $$w_c(t)x_c(t) - w_c(t-1)x_c(t-1)$$ however since new customers are added and $$w_c(t-1)x_c(t-1) = 0$$ for those customers. It its possible that customers with large $w_c(t)x_c(t)$ but similar to the overall average would appear high in the ranking which we do not want.
Can anyone see a good way of analyzing the drivers of $W(t, C(t))$?