MATLAB: Nonlinear curve fitting – propagation of data uncertainties to nlpredci confidence intervals

lsqcurvefit nlparci propagationMATLAB

Hi! I'm trying to fit a nonlinear function that has several parameters, some partially correlated. For simplicity, let's say the function is something like y = x * alpha^x, and the parameter to be fitted is alpha. The data are vectors X and Y. The data have measurement uncertainties that are Gaussian distributed, not correlated, and can have different standard deviations for each x_i or y_i.

I used LSQCURVEFIT to estimate 'alpha' for a given dataset, without applying weights (which ideally I'd like to do, but in R2010a the similar function NLINFIT does not support yet the 'Weights' parameter).

In any case, LSQCURVEFIT finds the best-fit 'alpha' and returns the residuals and a Jacobian. NLPARCI then can compute a 68% confidence interval for 'alpha'.

However, this 68% CI for 'alpha' is just for the unweighted fit itself.

How to get original uncertainties in both X and Y propagated into the 68% CI estimate of 'alpha'?

Best Answer

How to get original uncertainties in both X and Y propagated into the 68% CI estimate of 'alpha'?

Since alpha, the residuals, and the Jacobian are derived from X and Y, whatever uncertainty was originally in X and Y should already have had its impact on them.

Related Solutions

MATLAB: Calculate Uncertainty for fitted parameter from least squares fit

If your curve fit is unconstrained and your residual has uniform variance s2, then a common approximation to the covariance matrix of the parameters is

 Cov=inv(J'*J)*s2

where J is the Jacobian of the residual at the solution. Both LSQCURVEFIT and LSQNONLIN return the Jacobian as an optional output argument.

MATLAB: How to get the confidence intervals of regression coefficients in nlinfit

I looked into this somewhat recently and found two main strategies for computing confidence intervals of the coefficients.

using nlparci()
computing them directly by calculating the covariance matrix

Both methods use the jacobian and residual outputs and for my data there are very tiny differences in the results between the two methods.

Here are some sources I stored within the footnotes of my code and I recommend browsing through them to determine the best method for your problem.

Approximation of covar matrix of the params

Both covar matrix and nlparci methods

https://www.mathworks.com/matlabcentral/answers/155477-standard-deviation-of-parameters-in-lsqcurvefit

How covar matrix is used to get CI

https://stats.stackexchange.com/questions/241449/matrix-and-regression-model?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa

One method to approximate CIs using linear example, explains how to do with nonlinearly.

https://www.mathworks.com/matlabcentral/answers/94393-what-is-the-algorithm-used-in-nlparci-in-the-statistics-toolbox

nlparci can be used with lsqnonlin outputs (my project uses lsqnonlin but the outputs are similar for your project)

If you go with the nlparci() method, it's as easy as pluggin in your outputs...

   CI = nlparci(betaEstimates,residuals,'jacobian',jacobian);

If you go with the covar matrix method, here a snippet from my code that should be fairly self-explanatory.

    df = length(residuals) - numel(betaEstimates);   %degrees of freedom
    alpha = 0.05;                                    %for 95% conf int
    crit = tinv(1-alpha/2,df);                       %critical value
    covm = inv(jacobian'*jacobian) * var(residuals); %covariance matrix
    covmIdx = sub2ind(size(covm),1:size(covm,1),1:size(covm,2));  %indices of the diag of covm
    CI = nan(numel(betaEstimates),2);

I'm refraining for making a recommendation partially due to what @SK mentioned; there isn't a standard solution to getting CIs for the beta estimates, and also because I'm exploring these methods as a novice (and welcome feedback).

Best Answer

Related Solutions

MATLAB: Calculate Uncertainty for fitted parameter from least squares fit

MATLAB: How to get the confidence intervals of regression coefficients in nlinfit

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