Estimating parameters of “unknown” non-linear function
I've got data of dice rolls. Every "experiment" consists of $n$ (independent) rolls with a three-sided dice (I'll call the results $A, B, C$ from now on). The chance to roll an $A$ in a single roll is what I am trying to get a better grasp of.
There are three explanatory variables ($x_1, x_2, x_3$). So $f_A(x_1,x_2,x_3)$ is the chance to roll an $A$ with a single roll. The chances for getting a $B$ or $C$ are equal, so $f_B := (1 - f_A) / 2$ for example.
My dataset consists of about 1 million samples of experiments with $n = 10$ (so there are 1 to 10 rolls per experiment). I always know the values of $x_1,x_2,x_3$ and how many $A$'s got rolled (I'll call that $outcome$) in that experiment (for example 2/7 or 0/1).
Trying to estimate what $f_A$ could look like, I regressed $outcome$ ~ $ g(x_1,x_2,x_3)$. Since I don't know what $g$ looks like (but I can make some pretty solid educated guesses), I tried stuff like:
$1/x_1$, $1/(log(x_2) + x_1)$, $(x_1 + x_2)/(x_1 + c*x_2)$
These come up with some "nice" results - some of them seem to be statistically significant and they explain the outcome (reasonably) well. My only problem are the estimated coefficients....
Since someone initially came up with that kind of experiment (it's part of an old program), I doubt he/she used $0.82 / log(x_1) + 7.83 * x_2^{4.35}$ for an easy probability. It's much more likely that it is something like $20/(x_1+15) + 1/x_2$.
"Simple" regression is the only thing I really know of, to do stuff like that. What other options are there to find better ways of estimating $f_A$?
statistics regression parameter-estimation
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
add a comment |
I've got data of dice rolls. Every "experiment" consists of $n$ (independent) rolls with a three-sided dice (I'll call the results $A, B, C$ from now on). The chance to roll an $A$ in a single roll is what I am trying to get a better grasp of.
There are three explanatory variables ($x_1, x_2, x_3$). So $f_A(x_1,x_2,x_3)$ is the chance to roll an $A$ with a single roll. The chances for getting a $B$ or $C$ are equal, so $f_B := (1 - f_A) / 2$ for example.
My dataset consists of about 1 million samples of experiments with $n = 10$ (so there are 1 to 10 rolls per experiment). I always know the values of $x_1,x_2,x_3$ and how many $A$'s got rolled (I'll call that $outcome$) in that experiment (for example 2/7 or 0/1).
Trying to estimate what $f_A$ could look like, I regressed $outcome$ ~ $ g(x_1,x_2,x_3)$. Since I don't know what $g$ looks like (but I can make some pretty solid educated guesses), I tried stuff like:
$1/x_1$, $1/(log(x_2) + x_1)$, $(x_1 + x_2)/(x_1 + c*x_2)$
These come up with some "nice" results - some of them seem to be statistically significant and they explain the outcome (reasonably) well. My only problem are the estimated coefficients....
Since someone initially came up with that kind of experiment (it's part of an old program), I doubt he/she used $0.82 / log(x_1) + 7.83 * x_2^{4.35}$ for an easy probability. It's much more likely that it is something like $20/(x_1+15) + 1/x_2$.
"Simple" regression is the only thing I really know of, to do stuff like that. What other options are there to find better ways of estimating $f_A$?
statistics regression parameter-estimation
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
add a comment |
I've got data of dice rolls. Every "experiment" consists of $n$ (independent) rolls with a three-sided dice (I'll call the results $A, B, C$ from now on). The chance to roll an $A$ in a single roll is what I am trying to get a better grasp of.
There are three explanatory variables ($x_1, x_2, x_3$). So $f_A(x_1,x_2,x_3)$ is the chance to roll an $A$ with a single roll. The chances for getting a $B$ or $C$ are equal, so $f_B := (1 - f_A) / 2$ for example.
My dataset consists of about 1 million samples of experiments with $n = 10$ (so there are 1 to 10 rolls per experiment). I always know the values of $x_1,x_2,x_3$ and how many $A$'s got rolled (I'll call that $outcome$) in that experiment (for example 2/7 or 0/1).
Trying to estimate what $f_A$ could look like, I regressed $outcome$ ~ $ g(x_1,x_2,x_3)$. Since I don't know what $g$ looks like (but I can make some pretty solid educated guesses), I tried stuff like:
$1/x_1$, $1/(log(x_2) + x_1)$, $(x_1 + x_2)/(x_1 + c*x_2)$
These come up with some "nice" results - some of them seem to be statistically significant and they explain the outcome (reasonably) well. My only problem are the estimated coefficients....
Since someone initially came up with that kind of experiment (it's part of an old program), I doubt he/she used $0.82 / log(x_1) + 7.83 * x_2^{4.35}$ for an easy probability. It's much more likely that it is something like $20/(x_1+15) + 1/x_2$.
"Simple" regression is the only thing I really know of, to do stuff like that. What other options are there to find better ways of estimating $f_A$?
statistics regression parameter-estimation
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
I've got data of dice rolls. Every "experiment" consists of $n$ (independent) rolls with a three-sided dice (I'll call the results $A, B, C$ from now on). The chance to roll an $A$ in a single roll is what I am trying to get a better grasp of.
There are three explanatory variables ($x_1, x_2, x_3$). So $f_A(x_1,x_2,x_3)$ is the chance to roll an $A$ with a single roll. The chances for getting a $B$ or $C$ are equal, so $f_B := (1 - f_A) / 2$ for example.
My dataset consists of about 1 million samples of experiments with $n = 10$ (so there are 1 to 10 rolls per experiment). I always know the values of $x_1,x_2,x_3$ and how many $A$'s got rolled (I'll call that $outcome$) in that experiment (for example 2/7 or 0/1).
Trying to estimate what $f_A$ could look like, I regressed $outcome$ ~ $ g(x_1,x_2,x_3)$. Since I don't know what $g$ looks like (but I can make some pretty solid educated guesses), I tried stuff like:
$1/x_1$, $1/(log(x_2) + x_1)$, $(x_1 + x_2)/(x_1 + c*x_2)$
These come up with some "nice" results - some of them seem to be statistically significant and they explain the outcome (reasonably) well. My only problem are the estimated coefficients....
Since someone initially came up with that kind of experiment (it's part of an old program), I doubt he/she used $0.82 / log(x_1) + 7.83 * x_2^{4.35}$ for an easy probability. It's much more likely that it is something like $20/(x_1+15) + 1/x_2$.
"Simple" regression is the only thing I really know of, to do stuff like that. What other options are there to find better ways of estimating $f_A$?
statistics regression parameter-estimation
statistics regression parameter-estimation
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
asked Jan 5 at 21:56
KchnkrmlKchnkrml
1
1
add a comment |
add a comment |
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