An example of propagation of error associated with an iterated least squares solution

摘要

This report derives the covariance matrix associated with a nonlinear least squares problem f(x) = b where b is a vector of measurements and x is a vector of desired parameters. The linearized form of the equation is A(x)[x − x(0)] = b − b(0), where A = ∂f∂x and x(0) and b(0) = f(x(0)) are initial estimates. The linearized form is iterated until |x(r) − x(r − 1)| < ε, for some predetermined ε. The usual covariance matrix associated with the linearized form is σx = A†σb(A†)T, where σx is the covariance matrix of the parameters, σb is the covariance matrix of the measurements, and A† is the generalized inverse of A. In this report, it is shown that the covariance matrix is σ′x = σx + C, where C is a matrix resulting from the dependence of the coefficient matrix A on the measurements b. This dependence has not been considered previously in any literature encountered. A simple example is constructed and numerical calculations performed.