How to find uncertainties of estimated parameters in Levenberg- Marquardt algorithm
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Torsten il 4 Nov 2015
If you have the statistics toolbox, use nlinfit together with nlparci and nlpredci.
Star Strider il 5 Nov 2015
Finding the covariance matrix of a nonlinear regression requires that you calculate the Jacobian matrix at the convergence (the last parameter estimates calculated). In the context of the Wikipedia notation, the partial derivatives are with respect to each parameter at each data point, so the partial derivative of f(1) with respect to x(1) refers to the value of the partial derivative of the function at the first value of the independent variable with respect to the first parameter. It continues row-wise down the values of the independent variable, and column-wise across the parameters.
In a nonlinear regression, some or all of the partial derivatives of the function with respect to each parameter are by definition functions of themselves or of other parameters, so expect that in your result. (This is known as the model being ‘nonlinear in the parameters’.) This is easier if you have the Symbolic Math Toolbox, since it can calculate the symbolic Jacobian for you.
After that, you can plug your evaluated Jacobian matrix into the design matrix for a linear problem to calculate the covariance matrix, as explained in Least squares, regression analysis, and statistics. You can calculate the parameter confidence intervals from the diagonals of the covariance matrix and the t-distribution.