Unscented Transformation based estimation of parameters of nonlinear models using heteroscedastic data

• Derivative free approach of solving gradient weighted least squares problems.
• Improved approximation of the geometric distance for curve fitting.
• Improved performance of proposed technique demonstrated on three benchmarks.
• Monte Carlo used to benchmark the reduction in bias in estimated parameters.
• Relevant to model fitting problems with heteroscedastic data.

This paper addresses the issue of estimating the parameters of nonlinear models using heteroscedastic data. A weighted least squares problem formulation where the sum of the Mahalanobis distance for all measurements is minimized, forms the framework of this paper. Determining the Mahalanobis distance requires the gradient of the cost function with respect to the noisy measurements which can be computationally expensive and infeasible for model which are discontinuous. A derivative free approach to determine the Mahalanobis distance as an error metric is proposed using the Unscented Transformation. The advantages of using the proposed approach include: a black box approach to evaluate the gradient weighted objective function precluding the need for analytical gradients and an improved estimation of the covariance. Numerical results for various applications such as triangulation using radar measurements, ellipse, and super ellipse fitting demonstrate the benefits of the proposed approach. Heteroscedastic data resulting from real X-ray images are also used to illustrate the potential of the proposed approach.