Higher Order System Function Analysis: Optimal Fitting and Statistical Estimates

The fitting of data to nonlinear models is discussed in the frequency domain with a least-squares cost function, or more general cost functions. Formulas are derived for the first order partial derivatives of higher order transfer functions obtained as transforms of Volterra or Wiener kernels, and for the second partial derivatives required in the estimation of dispersion and in optimization methods utilizing the Hessian matrix. Emphasis is placed on obtaining forms with the most rapid computer evaluations. A geometrical descent method is introduced, which is particularly effective for exhaustive searches of parameter spaces bounded by complicated constraints. The geometrical descent method is applied to the off-diagonal fitting of a second order Volterra transfer function to white noise admittance data of the squid giant axon membrane of Loligo pealii. The method is compared with the modified Levenberg-Marquardt and the Gauss-Sidel algorithms.