Development of multivariable function from experimental data

[margaritaklag]

Hello! I am facing a difficulty into developing a multivariable function of a dependent variable "x".

Let's assume that "x" is a function of 6 independent variables a,b,c,d,e,f,g.

From experimental data i have developed 6 functions, each representing how "x" changes by each of the paremeters when the other 5 have a specific value.

So, what i have right now is for example: 1. x=f1(a,1,2,3,4,5)
2.x=f2(6,b,7,8,9,10)
3. x=f3(11,12,c,13,14,15)
......
6.x=f6(16,17,18,19,20,g)
What I need is to develop one single function of x where non of the variables will have a constant value.
x=f(a,b,c,d,e,f,g).
Any ideas or recommendations for where to look?
Thank you!

 

[tkhunny]

1) If x is 100% explained by these independent variables, then it can be done. This would be an unusual situation, 100% explained.
2) If these variables are 100% independent, then it can be done. This would be an unusual situation, 100% independent.
3) Is f a linear function of the independent variables?

 

[margaritaklag]

Dear tkhunny thank you for your answer!

1) You see x represents a physical quantity, so it is not 100% explained by 6 variables. However, in the model that i am trying to develop, i will make the assumption that it is.

2) Let's also assume that the variables are 100% independent.

3) I don't believe that f could be a linear function of the independent variables.

 

[tkhunny]

Dear tkhunny thank you for your answer!

1) You see x represents a physical quantity, so it is not 100% explained by 6 variables. However, in the model that i am trying to develop, i will make the assumption that it is.

2) Let's also assume that the variables are 100% independent.

3) I don't believe that f could be a linear function of the independent variables.

Those are VERY difficult answers.
1) This one is fine. We'll be able to measure how far from reality it is in the limited scope of available data.
2) We'll have to prove that - or at least that it's close enough to be a reasonable assumption. Otherwise, the model is mostly useless.
3) One would have to formulate what sorts of relationships exist. There are infinitely many possibilities.
-- 3a) How does the outcome relate directly to each variable? Linear Terms, a, b, c, etc...
-- 3b) How does the outcome relate directly to each squared variable? Quadratic Terms, a^2, b^2, etc...
-- 3c) How does the outcome relate to each combination of terms? Quadratic, but a*b, a*c, b*d, etc...
-- 3d) Will we need any higher-order terms? Doubtful, but possible: ab^2, c^3, c*d*f, etc...
-- 3e) Which pieces might be negatively correlated? Reciprocals, for example, 1/a, 1/b, etc...
-- 3f) Can any relationships be mapped to be something that acts more linear? square root or logarithm transformation, for example.
-- 3g) Is there a piece of the data that can be modeled easily and well and we should focus our energies on what is left?
-- 3h) Ever hear of Principal Component Analysis? Various software packages can help you with this.
-- 3i) Is the world of variables WAY TOO BIG? Six (6) can be a lot. If there are some dependencies, or one variable contributes very little, maybe we can throw something out.

I'm not quite trying to discourage you, but if we're going to build a good model, it takes quite a lot of theoretical work before we ever crunch a number.

 

[JeffM]

If you have the software to do it and if you have sufficient data, you can use repeated regressions to determine which variables have greatest explanatory power and what type of relationship seems best to fit the data. This calls for a great many numeric "experiments" with the data. (Do you know how to do regressions against non-linear functions?)

I might first try doing regressions of x against several simple types of function of a single independent variable. You may find that one or two variables seem to dominate. If so. try to build a model using just the most influential variables and the indicated functions.

If x is significantly influenced by several of the variables, then none of the single-variable regressions that you ran before is likely to give a good fit. So pick a few pairs that show the best fit and combine each with functions of a single additional variable.

At all times, keep an eye on theory and on the statistical significance of the coefficients.