How to figure out the contribution of each factor?

Question

Suppose we have value $S$ and then we multiply it by $xyz$. So,

$$T = Sxyz$$

How can we figure out how much each of $x,y,z$ contributes to $T$?

One way is to do:

$x$ contribution: $Sx - S$.
$y$ contribution: $Sxy - Sx$
$z$ contribution: $Sxyz - Sxy$

So,

$$T = S + (Sx-S) + (Sxy -Sx) + (Sxyz - Sxy) = Sxyz$$

The problem with this approach is that if we change order of $xyz$, for example say $T = Szyx$, then the contribution of $x,y,z$ changes using the method above.

How can this be done so that contribution of $x,y,z$ does not change if the order of $x,y,z$ changes? Am looking for general example that can expand to more than $3$ variables.

Just to let people know the application of this. It has one application towards insurance premiums. Say the base premium for home insurance for a customer is $1000, but they get discount of 10% for alarm system, and increase of 20% for having pool in backyard. So customer wants to know how much money they are paying or saving for each factor.

So customer with pool and alarm system pays 1000*0.9*1.2 = $1080.

They want to know how much of $1080 due to pool, and much saved due to alarm system.

Why isn't the contribution of $x$ equal to $S xyz - Syz$ instead? — Rodrigo de Azevedo, Nov 02 '19 at 13:35
Yes it could be if you do it that way. That is one way to do it. Maybe that is method I'm looking for. — steviekm3, Nov 02 '19 at 13:38

score 1 · Answer 1 · answered Nov 02 '19 at 13:39

1

A general approach that does not depend on order would be to make the contributions proportional: $Tx/(x+y+z)$, $Ty/(x+y+z)$, and $Tz/(x+y+z)$.

answered Nov 02 '19 at 13:39

RobPratt

45,619

Seems to work let me try it out. – steviekm3 Nov 02 '19 at 13:44

score 1 · Answer 2 · 2019-11-02T14:09:22.027

1

This question has no answer in terms of an amount added or deducted.

Without a discount, $\$1000$ become $\$1200$ when you add a pool, hence an increase of $\$200$ due to the pool.

With a discount, $\$900$ become $\$1080$, increase of $\$180$ due to the pool.

Unfortunately, not everyone understands the combination of percentages (for instance $-10\%+20\%\ne+10\%$). Because the true and best answer remains: $+20\%$ due to the pool.

edited Nov 02 '19 at 14:09

answered Nov 02 '19 at 14:03

Someone gave me a "solution" to this years ago by solving optimization problem. But it is not intuitive and I'm not sure it is right. I will try to write up solution to see if it makes sense to you. – steviekm3 Nov 02 '19 at 14:14
@steviekm3: the contribution of the factors is multiplicative, not additive. You don't want to play with logarithms, do you ? – Nov 02 '19 at 14:16
Yves - I actually tried this before using logarithms and did get answer. I posted answer someone else gave me. – steviekm3 Nov 03 '19 at 19:04
@steviekm3: this is surrealistic. Are you really going to answer your customers in terms of logarithmic Dollars ? – Nov 04 '19 at 07:56

score 0 · Answer 3 · answered Nov 03 '19 at 19:03

Someone gave me an answer before which is solution to optimization problem:

To explain by way of example mentioned in post we have

alarm system savings = 0.9*1000 - 1000 = -100

pool cost = 1.2*1000 - 1000 = 200

But -100 + 200 <> 80

So one solution is as follows

Find factors a, b so that

a(-100) + b(200) = 80.

Clearly there are many choices of a,b that work. But we want a, b to be close to 1.0 as possible, so costs don't change. Then this becomes optimization problem of solving

Minimize (1.0-a)^2 + (1.0-b)^2

subject to a(-100) + b(200) = 80

This is quadratic optimization problem. It can be solved by Lagrange multipliers using this method is

alarm savings = -110 pool cost = 190

How to figure out the contribution of each factor?

3 Answers3