1 20th December 07:48 kavka External User   Posts: 1 Robust Parameter Estimation I am trying to implement a robust estimation algorithm instead of nonlinear least squares minimization: My nonlinear least-squares cost if of the form: J(e) = \sum_{i=1}^{N} [y_i - f(u_i, x)]^2, (Eq. 1) where f= f(u,x) is the model output for an input "u" and x is the vector (in \R^p) of model parameters. The variable y_i is the measured output for an input u_i at time i. I have an algorithm to solve the above LS problem given a nonlinear function f(u,x) and the set { (u_i,y_i), i = 1,...,N }. I want to use the same algorithm with minimal modifications to solve the following robust minimization problem (which I've seen in a book): \sum_{i=1}^{N} \psi(y_i - f(u_i,x)) grad_{x_k}(f) = 0, (similar to normal equations) (Eq. 2) k = 1, ..., p, where \psi(x) is the Huber function: \psi(x) = max(-k, min(x,k)), and grad_{x_k)(f) is the gradient vector of partial derivaties: df/dx(k), k = 1,...,p. My problem is that I don't know how to pose the robust problem, Eq. 2, (solution of the above "normal equations") in a similar form to the least squares minimizations problem, Eq. 1. I suspect it is NOT simply J_r(e) = \sum_{i=1}^{N} \psi(y_i - f(u_i, x))^2.

 2 20th December 07:48 arnold neumaier External User   Posts: 1 Robust Parameter Estimation It is J_r(e) = \sum_{i=1}^{N} \phi(y_i - f(u_i, x)), where phi is the integral of psi. Setting the gradient to zero gives the above conditions. Arnold Neumaier
 3 21st December 07:30 alienseducer External User   Posts: 1 Robust Parameter Estimation Hi I would like to know what Algorithm you use to solve this problem by nonlinear least-squares. Algorithm, you only provide (u_i,y_i) and the nonlinear function. Thanks. Alien+
 4 21st December 22:25 paul victor birke External User   Posts: 1 Robust Parameter Estimation of some interest is the smoothed huber function found in http://isi-eh.usc.es/trabajos/80_30_fullpaper.pdf
 5 22nd December 10:40 kavka External User   Posts: 1 Robust Parameter Estimation I use "lsqnonlin" in Matlab's Optimization Toolbox. I have an m-file, say fun.m, in which I implement my nonlinear function "f(u;x)" given the set of model parameters,x, and the set of inputs, {u_i}. The output of the model is \hat{y}_i, which is compared with experimental data, y_i. The differences are the residuals, which I try to minimize using the cost function J(e) given below.