weightings using exponents

Question

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?

Answer 1

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.

Answer 2

It isn't very clear what you intend. But starting from first principles, if you really want to use exponents as the weighting system (which is questionable in the first place, to me), then you could just do something like the following.

Suppose you have $n$ scores $s_1,s_2,\ldots,s_n$ and you want to use associated weights $w_1,w_2,\ldots,w_n$ (not necessarily summing to $1$).

Write $w=w_1+w_2+\cdots+w_n$, and define

$$s_{\textrm{avg}} = \left(s_1^{w_1} s_2^{w_2} \cdots s_n^{w_n}\right)^{1/w} = s_1^{w_1/w} s_2^{w_2/w}\cdots s_n^{w_n/w}$$

If the $w_i$ already sum to $1$, then this would simplify to $s_1^{w_1} s_2^{w_2}\cdots s_n^{w_n}$.

Is this the sort of thing you are after?