I'm having a bit of trouble with my thesis. I'm doing symbolic regression of implicit mathematical functions. The approach I went with uses estimation of derivatives provided by experimentally obtained data, which help in finding the underlying function of the data.
So far I have used fitting scattered data to conic sections for data table of any dimension size. For example, if I have 2D points, the least squares fit would look like: Ax^2 + By^2 + Cxy + Dx + Ey + 1 = 0. Out of the resulted cone, I would compare its derivatives to my wanted function which gives me a sanity check that the derivatives match and that I can undergo the symbolic regression with some other algorithms.
This method has served me well for spherical functions, for example Gauss function, circle, sphere, ellipse or any kind of other spherical object.
However for example, if I apply the same fitting algorithm to estimate the derivative of Z = X * Y, the derivatives are way off for each point.
I'm coming for a more technical and computer-science approach than the mathematical one so maybe I'm missing something. Can you please suggest some derivation estimation methods for multiple dimensional data, or give any advice to it?
See: https://books.google.com/books/about/Elements_of_the_Differential_and_Integra.html?id=0jcAAAAAYAAJ
and
https://www.bing.com/search?q=estimating+derivatives+from+a+graph&qs=NM&pq=estimation+of+derivates&sk=NM2&sc=6-23&cvid=F60BAA72A8A14E47AC5366B8510612ED&FORM=QBRE&sp=3&ghc=1
and
https://www.bing.com/search?q=Numerical+Calculus&qs=n&form=QBRE&sp=-1&pq=numerical+calculus&sc=8-18&sk=&cvid=E222ABD09B1147B8915BB1D3B984E9F1
Good luck, D. Booth
I'm sorry but that didn't quite answer my question. I'm not looking for studying the whole numerical, integral or differential calculus as I already know the basics, as well as estimating derivatives from the graph for 2D points (for which your query on bing provides results).
My question was specifical for multidimensional data, as I don't know what kind of problems lay in fitting scattered data to specific hyperdimensional planes, and I was looking for some proven methods that maybe fit points to some polynomial or other kind of hyperplane from which I can estimate the derivative experimentaly.
Thanks.
Pardon me. Then fit a multivariable loess smooth to your points and then take partial derivatives of your resulting curve. You may have to do do this numerically depending on your data set. This an application of the Generalized Additive model developed by Hastie and Tibshirani as well as Cleveland. https://www.bing.com/search?q=generalized+Additive+Models++multivariable+loess&qs=n&form=QBRE&sp=-1&pq=generalized+additive+models+multivariable+loess&sc=0-47&sk=&cvid=FB7CC8B411C544A7BC0F54063C08A59C
https://www.bing.com/search?FORM=U528DF&PC=U528&q=multivariable+loess
https://www.bing.com/search?q=multivariable+loess+W+S+Cleveland&qs=n&form=QBRE&sp=-1&pq=multivariable+loess+w+s+cleveland&sc=1-33&sk=&cvid=DB31D2A46B7C429FA63479D9412EC2EC
https://www.bing.com/search?q=numerical+partial+differentiation&qs=AS&pq=numerical+partial+different&sc=4-27&cvid=E65D5D2379C94E60A01AF9CA47C1ACE7&FORM=QBRE&sp=1
Mea Culpa again for my ignorance. D. Booth
PS here is a pedestrian application:
https://www.bing.com/search?q=Michaelis+Menten+model+fitted+by+GAMS%28Generalized+Additive+models%29&qs=n&form=QBRE&sp=-1&pq=michaelis+menten+model+fitted+by+gams%28generalized+additive+mod&sc=1-62&sk=&cvid=11CF249B64FB419BB7CF150A201CBE0B
and
https://www.bing.com/search?q=Michaelis+Menten+model+fitted+by+GAMS%28&qs=n&form=QBRE&sp=-1&pq=michaelis+menten+model+fitted+by+gams%28&sc=0-38&sk=&cvid=7D36B1A59AD44F88A075FD85C44D008A
and
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