# Fitting data to complicated model

1) For standard non-linear least squares, the standard approaches are to either
a) to use the jacobian to linearize, and proceed with linear regression, or to
b) linearize the raw data

2) When data needs to be "fit" to a complicated model (e.g. some non-linear differential equation) that has no closed form analytic solution, people typically run a simulation varying the model parameters until the least squares residuals is minimized. Fit parameter distributions are then estimated using monte carlo boostrap, or a polynomial approximation in the vicinity of the residual least squares minimum.

Has anyone here seen any literature about linearizing the data for case 2 and fitting with a difference equation approximation for the non-linear dif. e.q. ? For example let's say our data is described by y = A*exp(-t/tau)+c. The y data can be linearized by with a time difference approximation

y(t+1)-y(t)/dt = -y(t)/tau.

and the equation above can be use to fit it. This technique would also apply to more complicated non-linear diffy q's so long they can represented as difference equations.

A straightforward method to fit the function y = A*exp(-t/tau)+c is presented papes 16-17 in the paper :

https://fr.scribd.com/doc/14674814/Regressions-et-equations-intégrales

In this paper it is shown that fitting functions from differential equation is not robust in case of scattered data. It is suggested to use integral equation instead of differential (so, to first transform it into integral equation).

yumyumyum
Thanks for the reply jjac. Unfortunately i don't speak french :[

A better way of phrasing my question is. Can one transform the raw data into a phase space representation, and then fit the transformed data using equations describing phase space for the diffy q. This would be useful since extracting phase space equations from a non-linear differential equation is easier then solving the deffy q.

@ yumyumyum : Sorry, I have no answer on this way.

Svein
For example let's say our data is described by y = A*exp(-t/tau)+c. The y data can be linearized by with a time difference approximation

y(t+1)-y(t)/dt = -y(t)/tau.
As long as you have an idea of the mathematical relationship, you can use several techniques to determine the parameters. The problem is that your "idea of a mathematical relationship" will "pollute" the data. Usually, a set of wildly different mathematical relationships can be used to describe a data set with whatever degree of accuracy you want.

For a more stringent explanation, check out Bayesian statistics (https://en.wikipedia.org/wiki/Bayesian_statistics).

HallsofIvy