From 7 inputs and 1 output, approximate a possible function

data analysisfunctionsinterpolation

I'm trying to approximate a car insurance quote algorithm/function:

It takes 7 input variables that I can change (Vehicle Cost, Post Code, Gender, Persons Age, Licence type, Licence age, and excess) and outputs a single numerical solution (Cost/week). I've already created a web scraper to collect data for me for one provider.

I'm looking for some direction on the possibilities of this; where to start; topics are relevant. Its a really interesting scenario that I can learn alot from

Eg n(0):
Vehicle cost = $10,000
Post Code = 1025
Gender = M
Age [Person] = 30
Licence Type = Full
Licence age = 10

Output = $45 / week

Function approximation = ???

Thank you

Best Answer

You need a bunch of data, not just one point. Several of the inputs are discrete, not continuous-sex, license type, post code. The others are continuous. The simplest assumption is that the cost is a linear function of each of the continuous variables and the functions are either added or multiplied. You can change each of the variables to find the linear function it creates and see if it is linear. Then you can change two of them and try to see how they are combined. The sex and license type don't have many options. The simplest is a fixed difference depending on the response. The post code may well impact the constants in the effects of the continuous variables.

It is research. Collect data, make a theory that explains it, and collect some more data to test the theory.

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You are solving a set of nonlinear simultaneous equations. As you have $2n$ values to evaluate (all the $a_n$'s and all the $b_n$'s) you need $2n$ pieces of data. In general there will be many solutions because of the $b_n$ exponents-do you really mean that? If all the $b_n$ are $1$ the problem is linear and much easier. For the linear problem the solutions are not hard and Excel's matrix tools will find the solution. For the nonlinear problem you will probably find too much of a mess without using a multidimensional function minimizer, found in many numerical analysis texts (and Excel's Goal Seek) but that gives a numeric answer, not an algebraic one.

[Math] How to determine a function from outputs

The first step is to look at the simple $XY$ plot of the data:

plot1

Whoa. What if we take the logarithm of $Y$?

logplot

This looks like a power of $X$ with a lot of noise. Averaging the ratio of $\ln\ln Y/\ln X$ gives $0.42$. Indeed, $\ln Y\approx X^{0.42}$ is a reasonable "smoothed" approximation.

power

But I don't think that curve-fitting is the way to go here. The nature of the function, and the fact that the argument runs from $1$ to $256$ suggest that some binary operations might be involved. And look, we have spikes at

113 = 0b01110001
159 = 0b10011111
242 = 0b11110010

Actually, I suspect we should subtract $1$ from $X$ to make it run from $0$ to $255$. Then these three spikes are at

112 = 0b01110000
158 = 0b10011110
241 = 0b11110001

Must be something about these chains of 0s and 1s.

Best Answer

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