RegressionTool©: for Linear and Nonlinear Regression Analysis
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The Levenberg–Marquardt algorithm is one of the most popular algorithms in nonlinear regression to fit a nonlinear function or model to a data set. This algorithm is very efficient in finding the solution of nonlinear least squares problems set-up for the regression analysis. Under suitable assumptions, the algorithm achieves global convergence, a competitive worst-case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems.[1] It is to be noted that the initial parameter guesses are important for finding well optimized parameter values as the Levenberg–Marquardt algorithm or in short LM is a local optimizer. LM does not find the global minima of the sum of squared errors. Instead, it finds a local minima guided by the initial guesses. The LM algorithm is also used in commercial data analysis software such as Origin, Sigmaplot, etc.
The users may use this webapp as per their own understanding of a regression work-flow to reach optimized values of the parameters. Regressiontool.com recommens the following steps towards finding optimized parameter values for a function fitted to a given data set:
Step 1. Copy and paste the (x,y) data to see vizualize the data on the plotting window.
Step 2. If there is no predetermined model or function, the user has to guess a suitable function or combination of functions that can form a shape or trend similar to the data.
Step 3. The user should use initial parameter values that yields (x,y) values that are not far away from the data set in terms of both absoulute values and the trend.
Step 4. After having vizualization of the input data and the function plot with initial parameter values, the run regression button will optimize the parameters as per the Levenberg–Marquardt algorithm.
References
[1] Bergou, E.H., Diouane, Y. & Kungurtsev, V. Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems. J Optim Theory Appl 185, 927–944 (2020). https://doi.org/10.1007/s10957-020-01666-1