I am looking for study material (books, articles, codes or even YouTube videos) on parameter estimation of differential equation models (using any method but preferably least squares). I would like to calibrate some mathematical models in the form of ODEs and PDEs; I have a time series data set for the dependent variables. I could simply Google but I would like material that has demonstrations for easy learning.
@Joseph, ODEs are not statistical models but least squares is. Now if you mean curve fitting and not regression, please be be more clear in your problem description. Perhaps a Google search of your question may be helpful but I don't know how. Best wishes, David Booth
@David, I mean curve fitting (model fitting to data). I've got a system of differential equations with several parameters and I have some data. I'd like to fit the model to data and as well estimate plausible parameter values. Yes, Google search returns lots of stuff but mostly articles that do not have the details.
On Google search I found I was wrong. A starting paper might be the JASA PAPER that I attached below. However your problem still needs work on explaining what you want. Best wishes David Booth
PS if I can fight through the Google search I bet you can too. DB
Use particle swarm optimization (PSO) or other metaheuristic algorithms for finding the optimal parameters of a system of ordinary differential equations (SODE) by minimizing the error (e.g. mean absolute percentage error or MAPE) between data and the model's solution. In Matlab, you can use ode45 package to solve the SODE. The step is:
0. Suppose you want to estimate the parameters in region interval I.
1. Generate m random vectors X in I, these vectors act as the initial guess of parameters.
2. Solve the SODE using ode45 for each vectors value. Then calculate the MAPE between SODE's solution with the data.
3. Find the minimum of the obtained MAPE. The vector vapue X* which gives the minimum of MAPE will act as the initial optimal parameters.
4. Next, you use the updating formula in the PSO to update the previous vectors X (you can find this in many books or article).
5. Redo the process 2-4 using the new vectors X.
6. At the desire stopping criteria, for example maximum iteration, you will get the estimated parameters value of your model.
Feel free to contact me for more detail explanation. I can help you.