Material on how to go from data to differential equation

[marellasunny]

Is there any material or book that explains how one could go from data to differential equation comprehensively?
More like functional data analysis+differential equations

 

[Stephen Tashi]

That's an interesting question. My first instinct is to say "No, there is no single technique, you have to combine a variety of methods, beginning with creating a mathematical model for the data." However, I find this presentation on the web http://www.google.com/url?sa=t&rct=...8cDmYmAcXqdLHtg&bvm=bv.44158598,d.aWM&cad=rja which says there are "new" methods for fitting differential equations to data. Perhaps another forum member will comment on that.

 

[JJacquelin]

Hi !

unfortunately, I think that there are few material dealing with this subject.
Not excatly in the scope of what you are asking for, the pdf paper "Regressions and equations integrales" :
http://www.scribd.com/JJacquelin/documents
It is written in French. Up to now, only the abstract is translated :
<< The main aim of this paper is to draw attention to a method rarely used to solve some regression problems.
In many cases, a differential and/or integral equation allows to turn a difficult problem of non-linear regression into a simple linear regression, which is the key part of the presentation.
The computation process is fundamentally different from the usual ones, since it isn't recursive. So, it doesn't requires an iterative loop.
In order to give a more concrete view, some exemple of non linear regressions are treated with detailed numerical examples : functions power, exponential, logarithm and some functions currently used in statistics : Gaussian Function, Weibull distribution >>

Some exemples of various forms of differential equations and integral equationsare provided (attachment)

 

[marellasunny]

Hi !


http://www.scribd.com/JJacquelin/documents
<<
In many cases, a differential and/or integral equation allows to turn a difficult problem of non-linear regression into a simple linear regression, which is the key part of the presentation.
>>

I really don't have the need to go into linear integral equations. So,if you could please explain in a gist what you are trying to do with integral equations and how it could help with the regression,that would be helpful!

 

[JJacquelin]

I really don't have the need to go into linear integral equations. So,if you could please explain in a gist what you are trying to do with integral equations and how it could help with the regression,that would be helpful!

The referenced document deals with differential equations and/or integral equations.
In your case, only differential equations are considered. So, you don't need to consider the integrals appearing in the paper, but only the differentials.
As explained, from a given data, it is possible to compute the coefficients of a differential equation in order to obtain an optimized fitting between the solution of the differential equation and the given data.
As it is written in my first answer, I am aware that this paper is not excatly in the scope of what you are asking for. Nevertheless, I hoppe it will suggest you a new way of search.

 

[marellasunny]

To JJacquelin and others,
I'm trying to learn the terms in your paper,so could you please answer the following questions.

In your paper, you say one is given another function g(x) in addition to the fitting function y(x). What is g(x), and how does one arrive at it?(page 3)

How do you arrive at S_k in page 3. What is this mathematical procedure termed as?
I guess T_k is also similar to S_k and used in the integral equation. What are they termed as in mathematics?

I understand that the aim of this paper is the to eliminate the need of recursive iteration process in nonlinear regression,which intern means this method eliminates the need to choose an initial condition as close to the real solution as possible. Am I right?Could you send me a program code and I could have a more visual understanding.

 

[JJacquelin]

In your paper, you say one is given another function g(x) in addition to the fitting function y(x). What is g(x), and how does one arrive at it?(page 3)

In the differential equation which we have to fit to given data ( see the first equation on top of page 4), some functions are likely to be present. g(x) is one of them. Possibly, g(x)=1 or g(x)=0. So g(x) is a known function.

How do you arrive at S_k in page 3. What is this mathematical procedure termed as?
I guess T_k is also similar to S_k and used in the integral equation. What are they termed as in mathematics?

Since your equation is purely a differential equation, there is no integral term in it. As a consequence S_k doesn’t exist in your case and you don’t need to compute it.
Same answer about T_k which is another notation of SS_k

I understand that the aim of this paper is the to eliminate the need of recursive iteration process in nonlinear regression,which intern means this method eliminates the need to choose an initial condition as close to the real solution as possible. Am I right?

Quite right. In fact, the aim of the paper is to show how one can transform a non-linear regression problem into a linear regression problem (not always, but in some cases). Since a linear regression doesn’t need recursive process, the consequence is that the transformed non-linear regression non longer needs a recursive process. That is an advantage of the method, but there are some drawbacks.

Could you send me a program code and I could have a more visual understanding.

Several examples of very simple algorithms are shown page 7, 17, 19. One can easily write them in program code. These examples correspond to the cases of integral equations. On can understand what are the similar algorithms in cases of differential equations ( i.e. : computation of D_k instead of S_k , page 3).
I have no ready made code in case of differential equations because, in practice, I mainly treated some cases of integral equations. As it is explained in the paper (with an example page 27), the method is more reliable for the fitting of the integral equations than for the fitting of the differential equations.