I've tried searching for a general method for calculating a weighted average with multiple weight variables, but no luck so far (almost all search results are related to Excel implementation and I don't know Excel formulae).
Let's say I have a vector $X = (x_1, x_2, ..., x_n)$ and two weight vectors $U = (u_1, u_2, ..., u_n)$ and $V = (v_1, v_2, ..., v_n)$. The weighted average of $x_1, x_2, ..., x_n$ using just $U$ as weight would be $\frac{u_1 x_1 + u_2 x_2 + ... + u_n x_n}{u_1 + u_2 + ... + u_n}$.
But what's the weighted average of $x_1, x_2, ..., x_n$ using both $U$ and $V$ as weights? Would it just be $\frac{u_1 v_1 x_1 + u_2 v_2 x_2 + ... + u_n v_n x_n}{u_1 v_1 + u_2 v_2 + ... + u_n v_n}$?
(EDIT) To give some context, here's a crude example: let's say I have a bunch of students whose average marks I want to estimate. I could do so using past data on students' performance.
Assume that students have 2 "characteristics" - subject (can be either subject A or subject B) and school (school X or school Y). Now I know that the test group of students being analyzed is divided as follows: 60% of them study subject A and 40% subject B. Also 70% of them study in school X and 30% in school Y.
I have past data from which I calculate the average marks for:
- subject A students = 80,
- subject B students = 85,
- school X students = 70,
- school Y students = 90.
I want to estimate the average marks for the test set of students which reflects the proportions of the characteristics I described above. So let's say there was only 1 characteristic, i.e. the subject, then my estimate would simply be $80 \times .6 + 85 \times .4$, but what should my estimate be if I want to incorporate both the characteristics?
