An Empirical Extension of the Ridge Regression Theorem to Nonlinear Least Squares Biexponential Analysis, with Application to Myelin Mapping in the Human Brain*

Nonlinear least squares (NLLS) is a highly effective means of parameter estimation but can be extremely sensitive to noise in situations where the estimation problem is poorly-posed. To decrease the mean square error (MSE) of parameter estimation in such circumstances, we apply Tikhonov regularization, even though this is highly unconventional for low-dimensional NLLS problems with parameter estimates of qualitatively different types of parameters. In this sense, we are extending the use of the ridge regression theorem (RRT) for linear least squares, which states that a regularization parameter, λ, exists that reduces MSE in parameter estimation as compared with nonregularized linear least squares. We estimated parameter values with conventional NLLS and compared these with values obtained from regularized NLLS, with λ defined by generalized cross validation. We only regularized signals identified as biexponential by the Bayesian information criterion. Under conditions of modest SNR and relatively closely spaced exponential time constants, regularization substantially reduces variance and MSE across noise realizations. We applied this method to the challenging case of myelin mapping in the brain and found a 20% improvement in MSE. Finally, we note that our method is generalizable to many other NLLS analyses.