Statistical Analysis of Input Parameters

[hexa]

Hello,

I've been running a model with different combinations of imput parameters. Let's just assume they look like this:

1,2
1,3
1,4
3,4
1,2,3
2,3,4
1,2,3,4

As a result I receive a certain numerical value. Jus by looking at that value I can see if the result is good or not. But how do I decently analyse the influence of the input parameters on that result? As I know about nothing about statistics I can only cont how often every parameter appears with a good or bad result. But what about combinations of parameters? how do I analyse the meaning of a good result from a parameter which usually results in good and another that results in a bad result? Furthermore, some results are wonderful, some are not so good, some a not so bad, and some terribly bad.

I think it might be easier if I had a huge list of parameters and always only combinations of 2, but in fact I have only 5 parameters to play with, which results in 26 possible combinations.

Any ideas?

 

[Chris Hillman]

Fisher lives?

But how do I decently analyse the influence of the input parameters on that result? As I know about nothing about statistics

At first I thought you must be asking a garbled question about the theory of designs. But maybe not. In which case the obvious question is: what do you know about the theory of designs?

 

[EnumaElish]

How is it that the number of input parameters is changing from one trial to the next?

 

[hexa]

How is it that the number of input parameters is changing from one trial to the next?

Because I'm still trying to find out which combination works best, and if there's a minimum number of input parameters for a good result, and if that number is depended on the types of input parameters. I just don't know how to compare the results correctly now :(

 

[hexa]

At first I thought you must be asking a garbled question about the theory of designs. But maybe not. In which case the obvious question is: what do you know about the theory of designs?

Nothing. I don't know what you're talking about. If I knew how to formulate my question clearly then I think I would already be a step closer to solving my problem simply as I would have at elast some basic knowledge on statistics.

hexa

 

[EnumaElish]

Because I'm still trying to find out which combination works best, and if there's a minimum number of input parameters for a good result, and if that number is depended on the types of input parameters. I just don't know how to compare the results correctly now :(

I see. the easiest method would be to estimate a multiple regression of the type y = b0 + b1 x1 + ... + bk xk for each combination of k parameters (the x's) that you have used (and y is the observed output). You can then compare model fit across different k's and different parameter combinations by looking at the ADJUSTED R-SQUARED statistic of each equation.

 

[hexa]

I see. the easiest method would be to estimate a multiple regression of the type y = b0 + b1 x1 + ... + bk xk for each combination of k parameters (the x's) that you have used (and y is the observed output). You can then compare model fit across different k's and different parameter combinations by looking at the ADJUSTED R-SQUARED statistic of each equation.

Hello,

thanks a lot, that's something already. I'm just not quiet sure what to do with this.

lets assume I have
par1-par2-par3 = 80
par2-par3-par4 = 95

Please can you give me a few more hints? I understand just as much that I have to solve this in a matrix somehow, but what to solve for is a bit of a mystery. Yes, I'm rubbish with these things :(

Hexa

 

[EnumaElish]

Multiple regression will work only if either:
1. you take repeated measurements with each parameter combination and identical parameter values and each measurement is at least a little different from the others; or:
2. you assign different values to each parameter in a given combination of parameters and (as a result) record different output values.

If that is not the case, you'll be better off, say, taking the following "exact" measurements:

par1 par2 par3 = 80
par1 par2 par4 = 95
par1 par3 par4 = 70
par2 par3 par4 = 90

which is 4 equations in 4 unknowns and can be solved by:
[1 1 1 0] [a1] _ [80]
[1 1 0 1] [a2] = [95]
[1 0 1 1] [a3] _ [70]
[0 1 1 1] [a4] _ [90]

or in matrix notation M a = y, where each a is the contribution of the corresponding parameter to the output (the y's); and the solution is a = M^-1 y.

In case of multiple regression, you'd be changing parameter levels as well as the combination, so you'll end up with, say:

[10 10 10 0] _____ [80]
[20 10 10 0] _____ [85]
[25 10 10 0] _____ [90]
[10 15 0 10] [b1] _ [95]
[10 17 0 10] [b2] _ [85]
[10 19 0 10] [b3] = [75]
[10 0 10 10] [b4] _ [71]
[10 0 10 11] _____ [77]
[10 0 10 12] _____ [67]
[0 10 10 25] _____ [99]
[0 10 10 35] _____ [100]
[0 10 10 45] _____ [110]

or X b = y - u, where u is "random error" (which may include measurement error), and b is "estimated" as \hat {\bold b} = (X'X)^-1X'y.