Fitting data to complicated model

In summary, the conversation discusses the different methods for fitting data to a complicated model, such as a non-linear differential equation. The standard approaches involve using a linearized version of the model or running a simulation to minimize least squares residuals. However, there is also a technique of transforming the data into a phase space representation and fitting it using equations describing phase space. Additionally, the conversation mentions the potential limitations of fitting functions from differential equations and suggests using integral equations instead. The final point made is that determining the parameters for a model can be influenced by the chosen mathematical relationship and Bayesian statistics can be used to address this issue.
  • #1
yumyumyum
4
0
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.
 
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  • #2
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).
 
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  • #3
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.
 
  • #4
@ yumyumyum : Sorry, I have no answer on this way.
 
  • #5
yumyumyum said:
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).
 
  • #6
I've never heard of "fitting data to a model". I thought the objective was always the other way around- to fit the model to the data.
 

FAQ: Fitting data to complicated model

1. Can you explain the process of fitting data to a complicated model?

Fitting data to a complicated model involves finding the best possible parameters for the model in order to accurately represent the data. This is typically done using statistical methods such as least squares regression or maximum likelihood estimation.

2. How do you determine the accuracy of a fitted model?

The accuracy of a fitted model can be determined by comparing the predicted values from the model to the actual data points. This can be done using various metrics such as root mean square error or R-squared value.

3. What are some common challenges when fitting data to a complicated model?

Some common challenges when fitting data to a complicated model include overfitting, where the model fits the training data too closely and does not generalize well to new data, and underfitting, where the model is too simple and cannot accurately represent the data.

4. How do you choose the appropriate model for a given set of data?

Choosing the appropriate model for a given set of data involves considering the complexity of the data and the underlying relationships between the variables. It may also involve trying out different models and evaluating their performance on the data.

5. Can you provide an example of fitting data to a complicated model in real-world research?

One example of fitting data to a complicated model in real-world research is in epidemiology, where researchers use mathematical models to predict the spread of diseases and inform public health policies. These models often involve multiple variables and complex relationships between them, and require accurate fitting to accurately predict the spread of diseases.

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