Finding the function that produces i from f(x,y) - function analysis

I believe this is the correct forum for this query. Thank you in advance for your help!

I am trying to figure out what the function is that would generate var *i* .

It comes with a "stage" (var* x*) and "density" (var *y*)

Together these determine a level of mitigation, the result *i*.

Sample data:

Stage x, Density y, Mitigation i

2 9 8.26%

2 18 15.25%

3 19 11.24%

3 26 14.77%

4 41 17.01%

4 34 14.53%

4 27 11.89%

4 44 18.03%

5 45 15.25%

6 51 14.53%

6 103 25.56%

7 127 26.62%

7 144 29.15%

7 146 29.44%

The solution would be cool, but I am also interested in how to come to the solution.

So far i have:

i = f(x,y)

But that's all I've got. I think it uses an exponential curve. It definitely has a curve that makes density less effect per point as it scales upwards, and density points are less effective at higher stages, requiring more to provide the same level of mitigation.

If I need more information, let me know and I can collect a larger dataset.

\Thanks!

PS, it's been several years since I took math and to be honest I haven't retained too much of it, so if you have better language for anything I request please contribute. I did quite well, so hopefully I will get it without trouble.

Re: Finding the function that produces i from f(x,y) - function analysis

Hey themrrobert.

There are two extreme approaches to function fitting: the first one is that you assume absolutely nothing about the underlying model and the other extreme is that you assume some kind of model for the data and fit the data to that model.

The first method is what is known as interpolation and if you want to get a model that you can actually decipher to get patterns, then this method is pretty much useless, but you can do it if you want to.

The second method (or extreme) is based on projections where you start with a known form of the model (exponential function, Normal distribution, Mixed Normal, quadratic, whatever) and you construct an orthonormal basis and project your signal data to that to get the best approximation of your signal to that model.

If you are trying to fit probability data, then you would use what is called the EM algorithm (or similar ones) where you start off with a distribution that you assume is true and then use your data to estimate the best set of parameters that this distribution could take given the data.

So basically you have to decide where you want to be in this extreme: You can either assume quite a lot about the underlying model and fit the the whole thing to that model, or you can assume absolutely nothing and get something that is super complex that while fitting all the data to the interpolated function, will at the same time tell you absolutely nothing about the function itself.

There are middle grounds in this and it depends on how general your model and how much variation your model has: for example if you have a simple quadratic f(x) = ax^2 + bx + c then this has a lot less variation than say a quartic with f(x) = ax^4 + bx^3 + cx^2 + dx + e. The interpolation is one extreme that will fit a polynomial with the same degree as the number of points and if you have 1000 points, then you probably begin to see how useless it is.

Re: Finding the function that produces i from f(x,y) - function analysis

Use:

i = 12.268 + 0.169*density -0.958*stage

Re: Finding the function that produces i from f(x,y) - function analysis

Thank you both for your help and response :)

@chiro - yes, interpolation is NOT what I want, I remember that as a graph with many wavelengths, skewing up then down each time to match a point. useless in determining an unknown i from x,y. Any other take on this problem would be much appreciated :)

Things you can assume:

1) As each stage goes up, density is "worth less".

2) Regardless of stage, as you get more density, each density point is worthless. It sounds like it should fit on a quadratic-like function, but remember my math is horribly, welll... fallen victim to years of non-appliance.

@ MaxJasper - this seems close, but it doesn't seem to fit the dataset, at least not when i did it.

I used: =12.268 +(0.169*B44) - (0.958 * A44)

Where row b = denisty and row a = stage.

How did you get to that result?

@ Max:

Table when using your function for the 4th column (keep in mind that the mitigation is a percentage, i don't see how that could possibly matter other than shifting the coeffecients by /100 but i'm not the math expert :(

stage | density | Mitigation % | | MaxJasper's F(x,y) |

2 | 9 | 8.26 | % | 11.873 |

2 | 18 | 15.25 | % | 13.394 |

3 | 19 | 11.24 | % | 12.605 |

3 | 26 | 14.77 | % | 13.788 |

4 | 41 | 17.01 | % | 15.365 |

4 | 34 | 14.53 | % | 14.182 |

4 | 27 | 11.89 | % | 12.999 |

4 | 44 | 18.03 | % | 15.872 |

5 | 45 | 15.25 | % | 15.083 |

6 | 51 | 14.53 | % | 15.139 |

6 | 103 | 25.56 | % | 23.927 |

7 | 127 | 26.62 | % | 27.025 |

7 | 144 | 29.15 | % | 29.898 |

7 | 146 | 29.44 | % | 30.236 |

8 | 147 | 26.87 | % | 29.447 |

8 | 146 | 26.74 | % | 29.278 |

8 | 168 | 29.58 | % | 32.996 |

8 | 169 | 29.7 | % | 33.165 |

8 | 192 | 32.43 | % | 37.052 |

8 | 201 | 33.44 | % | 38.573 |

8 | 199 | 33.22 | % | 38.235 |

8 | 146 | 26.74 | % | 29.278 |

9 | 135 | 23.08 | % | 26.461 |

10 | 136 | 21.38 | % | 25.672 |

10 | 148 | 22.84 | % | 27.7 |

11 | 149 | 21.32 | % | 26.911 |

11 | 153 | 21.76 | % | 27.587 |

12 | 154 | 20.42 | % | 26.798 |

I'm thinking about getting 5 different Density (y) : Mitigation (i) measurements per Stage. In this case I would be able to draw (predict) the curve that the function is taking (per stage [x]) and hopefully find a pattern between the stages to adjust the function based on (x) stage.

Of course, I wouldn't remember/know how to find the functions in this case either, but at least (i think) it would be plenty and exactly the information you need to help me solve this. (But I'm also hoping you can do it with the dataset provided :D )

I have 2 more problems like this to solve, hopefully if I get the tools mastered here then I can do them on my own, otherwise I may come here looking for more help :)

1 of them is a bit simpler, with a quadratic with just 1 variable, and the other i believe takes a form similar to: chance = base + f(x,y), but i don't have the datasets quite ready on those yet.