Curve fitting for 1D vectors or ND matrix with any given model equations using a random search algorithm.
Codes for Levenberg–Marquardt algorithm have been provided as a comparison.
Renamed from jcfit to fitnguess 11/2021, stands for Fit with Natrual Guessing.
Rename back to (Jump-Chain Fitting) jcfit 11/09/2023 and modified searching precision setting from fixed precision to fixed significant figures.
Also update the convergence test procedure.


Developed and tested on software versions
MATLAB 2014b
Windows 10 

1. in folder jcfit_L1
    run jcfit_L1_test.m

2. in folder jcfit_L2
    run jcfit_L2_test.m
    
Both are the same code using different penalty terms:
1. L1 minimizes the absolute sum of residual, i.e. least absolute sum of variation/deviation: minimize(sum(abs(y-yguess))) 
2. L2 minimizes sum of square of residual, i.e. least square regression: minimize(sum(y-yguess)^2)

They both have different strengths for different problems. 
Least square regression jcfit_L2 is >10 times slower than the Levenberg-Marquardt (LM) (https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm) method for simple models such as exponential decays but is extremely easy to understand and to extend for complicated systems such as global fittings, fitting of multidimensional data, functions that are not continuous or differentiable, or systems with combined models. It is also very easy to be modified for parallel computing.

Basic idea: 
For a given raw data and a math model, use the inital guess of the parameters in the model to generate the guessed data and get the residual between the two sets of data.
The sum of the absolute values of all residuals (L1) or the sum of the square of the residuals (L2) is a function to the parameters for which the LM use an algorithm modified from Gauss-Newton algorithm (https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm) in finding the minimum. While jcfit will guess a series of possible values within the boundaries of each parameter and take the minimum as the initial guess for the next iteration. The searching space is exponentially distributed along the initial guess to the boundaries e.g. -10 to 10 with initial guess 0 and searching precision ~0.1: [-10, -5, -2, -1, -0.5, -0.2, -0.1, [0], 0.1, 0.2, 0.5, 1, 2, 5, 10]. It does not follow the assumption in LM that these is only one minimum. It is 5-20 times slower than LM in this example, but has a chance to find the "global" minimum in this parameter dimension, which unfortunitly may be different from the global minimum of all parameters. The searching goes through all parameters sequentially in each iteration making it a P searching algorithm (https://en.wikipedia.org/wiki/P_versus_NP_problem). 
This method is modified from the Random Search method especially the Friedman-Savage procedure (https://en.wikipedia.org/wiki/Random_search). (Friedman, M.; Savage, L.J. (1947). Planning experiments seeking maxima, chapter 13 of Techniques of Statistical Analysis, edited by Eisenhart, Hastay, and Wallis. McGraw-Hill Book Co., New York.)

Change your fitting model in the function 'mdl' in 'jcfitf.m' (moved to older versions 11/2021) if you use custom functions for a complicated model or global fitting.

cite:
A preprint manuscript link: PyJCFit: A Non-Linear Regression Random Search Algorithm for Chemistry Data Fitting, https://doi.org/10.26434/chemrxiv-2023-k9rvm

https://pubs.acs.org/doi/abs/10.1021/acs.jpcb.6b05697
Jixin Chen, Joseph R Pyle, Kurt Waldo Sy Piecco, Anatoly B Kolomeisky, Christy F Landes, A Two-Step Method for smFRET Data Analysis, J. Phys. Chem. B, 2016, 120 (29), pp 7128–7132

A semi-exhaustive and brutal searching algorithm using the least-square method to find the best fit of a curve. The least-square method can be changed easily to other residual treatment methods.

The fitting time of each iteration is scaled with the number of parameters of the fitting, the search speed is not very sensitive to the accuracy because of the exponential searching spacing design (not set as an option now and has to be changed manually inside the function). Each parameter is searched one by one within its boundaries. Thus a P^N question is reduced to PN with a cost of losing space coverage.

The code can be changed to fit multiple curves globally by introducing different models for different curves using the same parameters.

An example can be found at:
Juvinch R. Vicente, Ali Rafiei Miandashti, Kurt Sy Piecco, Joseph R. Pyle, Martin E. Kordesch, Jixin Chen*, Single-Particle Organolead Halide Perovskite Photoluminescence as a Probe for Surface Reaction Kinetics. ACS Applied Matierals & Interfaces, 2019, 11(19), 18034-18043.


3. in folder Fit_LM
    run fit_LM_test.m   or
        fit_LM_num_test.m
This is a reproduced Levenberg–Marquardt algorithm with literature citation given in the code. 
where fit_LM_test need provide analytical solution for the Jacobian matrix and fit_LM_num does not.
Fit_LM_num will automatically calculate the gradient numerically for any given function.


by Jixin Chen @ Ohio University 2021