I've got this piece of work to do where we assign weightings to different variable to achieve a score:
var A = 60%
var B = 40%
var C = 20%
var D = 5%
The score was calculated as:
$(A^{0.6} * B^{0.4})=AB$
$AB^{(1-0.2)} * C^{0.2} =ABC$
$ABC^{(1-0.05)} * D^{0.05}$ = final score
This worked as the first two variables being multiplied had weights = 100% so for further multiplying of variable (c) we would raise the variable to their weight (0.2) and multiply it to the previous result (AB) with the weight of 1-the weight of the variable (1-0.2) to maintain an overall weight of 100% and so on and so on.
My issue is I have to design a new feature with 5 new variables, each with a weighting of 10%.
How would I incorporate it into the model, as I can't use the logic for previous variables, as the first 2 variable weights don't add up to 100% – and so it wouldn't make sense to do (1-weights) for adding in more variables to the model?
Best Answer
As I understand, you want to set this up such that with 5 variables, each will have an exponential weight of 0.10. Let the variables be $A, B, C, D, E$. Then we have
$$ A^{1-a_1} + B^{a_1} = AB, \\ AB^ {1-a_2} + C^{a_2} = ABC, \\ ABC^{1-a_3} + D^{a_3} = ABCD\\ ABCD^{1-a_4} + E^{a_4} = ABCDE $$
Clearly then we must have $ a_4 = 0.1$, $a_3(1-a_4) = 0.1$, $a_2(1-a_3)(1-a_4) = 0.1$, and so on. The constraint that these be normalized at every stage, (i.e. the weights add up to 1 at each stage) will be satisfied for any choice of $0 \leq a_i \leq 1$.
Please let us know if this is what you were looking for. This does seem a bit odd, not sure why you would want to do it like this instead of just weighing everything by 0.1.