I have the following plot and have huge difficulties finding the corresponding function
$X=1,2,3,6,12,40,100$
$Y=-2,-1,0,1,2,3$
$R=$ the results
| $\mathbf{-2}$ | $\mathbf{-1}$ | $\mathbf{0}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$ | |
|---|---|---|---|---|---|---|
| $1$ | $1$ | $1.39$ | $1.58$ | $1.46$ | ||
| $2$ | $0.76$ | $1$ | $1.2$ | $1.26$ | $1.11$ | |
| $3$ | $0.52$ | $0.81$ | $1$ | $1.11$ | $1.12$ | $0.96$ |
| $6$ | $0.85$ | $0.94$ | $1$ | $1$ | $0.95$ | $0.77$ |
| $12$ | $1.02$ | $1.02$ | $1$ | $0.95$ | $0.85$ | $0.66$ |
| $20$ | $1.09$ | $1.06$ | $1$ | $0.92$ | $0.8$ | $0.61$ |
| $40$ | $1.15$ | $1.09$ | $1$ | $0.9$ | $0.76$ | $0.56$ |
| $100$ | $1.2$ | $1.11$ | $1$ | $0.89$ | $0.74$ | $0.54$ |
I think it needs to be a combination of a polynomial function and a power function as if I look at the graph for $Y$ vs $R$ by each $X$, these are nice polynomial graphs. If I look at $X$ vs $R$ by each $Y$ they sort of follow a power series graph, but I can not figure out how to "combine them".
So think I found a function and here is how I did it :) First I plottet the two graphs I mentioned above. I then got the following graphs
x vs r by y
y vs r by x
From the first, I see from trendlines that a powerseries on the form of A*x^B should sort of follow the plots.
For the second I find that a polynomial function in the form of A+By+Cy^2 very nicely follows the plots!
A was rather constant at 1.01, so kept it like that. I then looked at B and C pr X. Again I plottet the results ant looked at trendlines.
B and C by x
From here I find that B=-0.2+0.595x^-0.477 and C=-0.0897+0.0213ln(x)
The resulting function is then r=1.01+(-0.2+0.595*x^-0.477)y+(-0.0897+0.0213ln(x))*y^2
So, conclusion: Plot each variable with one series for each other variable. Choose the plots where you find functions that match the plotlines the best. Calculate the constants of these series depending on the second variable. Again, look at the plots for each and find the best matching function. Combine and there you have it :D