Dear all,
first I would like to thank Leo for his help and advice. My sincere apologies for the delayed response; regrettably, I overlooked the email. In order to characterize the problem a bit better I want to give additional information and clarify the point where I have problems. First of all the equation system is defined as: h0 = h + hd + hg, d0 = d + hd, ga0 = ga + hg, kga = hg/(h*ga), kd = hd/(h*d) The aim is to fit a non-linear equation in the form: signal ~ I0 + IHD * hd + ID * d. The parameters which should be identified/optimized are: I0, ID, IHD and kd. However, the only information known is kga (33600.0), h0 (0.0000208), d0 (0.000079) and the guest concentration (= ga0). In Mathematica the equation system is repeatedly solved for hd and d. First the system 0 = h + hd + hga, 33600. = d + hd, ga0 = ga + hga, 0.000079 = hga/(ga*h), 0.0000208 = hd/(d*h) is passed to the Eliminate function which eliminates four variables. In the case for hd the variables h, d, hga and ga are eliminated. The resulting system is: 3.71035392*^12 + 2.635566*^8*hd - 3783.*hd^2 == ga0*(-1.72536*^8 + 5135.*hd). Afterwards this system is used by a numerical solver to get the result of hd. My problem is how to replicate the Eliminate function in R. Or is it possible to solely solve the system numerically? Thanks a lot in advance. All the best, Konrad On 04.12.23 23:10, Leo Mada wrote: > Fit NLE - was: [R] computer algebra in R > Original post: > https://stat.ethz.ch/pipermail/r-help/2023-November/478619.html > > Dear Kornad, > > I think I have started to understand what you try to achieve. The > problem is to fit a NLE and compute the parameters of the NL-Eq. I > have included the R Help-list back in the loop, as I am not an expert > in optimization. > > Goal: > y ~ I0 + IHD * hd + ID * d; > where: > y = given vector of measurements; > x = given vector of values; > I0, IHD, ID, kd = parameters to optimize; > hd = satisfies a polynomial of order 3; > > As d = d0 - hd, the previous formula can be written: > y ~ I0 + ID * d0 + (IHD - ID) * hd; > > f(x, hd, kd) = 0, > where f = a polynomial of order 3 in hd and order 2 in kd; > d0 (and other components of the polynomial) = given constants; > > 1) First Approach > I would back-substitute hd into the polynomial: > hd = (y - I0 - ID*d0) / (IHD - ID); > > f(x, hd, kd) becomes then f(x, y, kd, I0, ID, IHD) = 0; > - f is order 3 in y; > > You could fit: > (y^3) ~ f(x, y, kd, I0, ID, IHD) - y^3, > where you subtract the y^3 term from the function f, and add the (y^3) > values as a new columng to the data.frame: > data.frame(y3 = y^3, y=y, x=x) > > If the values of y are versy small (abs(y) << 1), then it may be wiser > to fit: > y ~ f(x, y^2, y^3, kd, I0, ID, IHD) - (y-term); > > But I am not an expert in these problems. Other R-users may be more > helpful. > > 2.) Approach 2: Math > I feel that the problem can be solved quasy-exactly as well. It is > much harder with 4 parameters to optimize: > - one needs to compute the 4 partial derivatives; > - solve the resulting system of 4 polynomial equations; > > The system is polynomial; although it looks ugly and I am not inclined > to do such calculations myself. > > I hope that you can get more useful answers from the R help-list. > > Sincerely, > > Leonard [[alternative HTML version deleted]] ______________________________________________ R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
